Sequential Exchange with
Stochastic Transaction Costs∗
Yuichiro Kamada†

Neel Rao‡

August 22, 2018

Abstract
We analyze the bilateral trade of divisible goods in the presence of stochastic transaction costs. Mutual best-response conditions are applied to a modelof optimal investment under uncertainty. The first-best solution involves a single transaction, but such behavior is not incentive compatible without courtenforceable contracts. We solve for a second-best policy in which some gains
from trade can be realized through a gradual transfer. When the transaction cost follows a geometric Brownian motion, the optimal path of play in a
subgame-perfect equilibrium is unique, and a closed-form solution can be obtained. A number of comparative statics and welfare implications are presented
as well as real-world examples.

JEL Classification: C73, C78, D23, D86, L14
Keywords: dynamic games, continuous time, transaction costs, stochastic processes∗

We thank Eduardo Faingold, Drew Fudenberg, Oliver Hart, Michihiro Kandori, Akitada Kasahara, Phillip Strack, Takuo Sugaya, Satoru Takahashi, Juuso Toikka, Yasutora Watanabe, andseminar participants at Chinese University of Hong Kong, Harvard University/Massachusetts Institute of Technology, Hitotsubashi University, University of California at Davis, University of Tokyo,
and the Summer Workshop on Economic Theory 2018 at Otaru University of Commerce for helpfulcomments.
†
University of California, Berkeley, Haas School of Business, Berkeley, CA 94720. E-mail: y.
cam.24@gmail.com‡
University at Buffalo, SUNY, Department of Economics, Buffalo, NY 14260. E-mail: neelrao@
buffalo.edu

11Introduction

This paper is motivated by situations like the following. Two firms operating indifferent markets find it potentially profitable to exchange trade secrets, but there is acost for transferring knowledge from one firm to the other. This cost might represent
the expense of encrypting data to protect secrets from outsiders or the resources
spent on administrative paperwork. Due to the intangible nature of information, it
is infeasible for the two parties to write a court-enforceable contract specifying thegoods to be traded. Moreover, if one party immediately reveals all of its information
to the other party, then the latter would have no incentive to reveal its information
to the former, because such transactions are costly. In this setting, how can the twoparties arrange trade with each other?
We study the exchange of divisible goods in a continuous-time environment. Eachof two agents is endowed with an equal amount of a good that only the other party
values. In order to transfer some of its good to the other party, an agent must pay atransaction cost that evolves according to a stochastic process. The first-best solution
requires each agent to make a one-shot transfer of the entire stock of its good to theother agent. Nonetheless, such a policy cannot be supported in a subgame-perfectequilibrium. It is not incentive compatible because the agents are unwilling to incur
the transaction cost if positive future transfers are impossible.
Our ultimate objective is to solve for a second-best transaction scheme that enables
the agents to realize positive gains from trade. We establish conditions under which
the game has a nondegenerate equilibrium, and the general form of a solution isdetermined. Assuming that the cost process follows a geometric Brownian motion,a closed-form expression can be obtained for the unique path of play in a maximal
equilibrium.1 It involves a potentially infinite and gradually decreasing sequence oftransfers. We derive a number of comparative statics and welfare implications.
The basic idea behind the implementation of a self-enforcing agreement is simple.
In equilibrium, when the two agents make transfers, they withhold some amount of
the goods. The players use grim-trigger strategies, whereby they continue to transfer
the withheld goods when the transaction cost reaches a certain threshold if and only
if both parties have made the prescribed transfers in the past. Because the contin1

A symmetric equilibrium is said to be maximal if no symmetric equilibrium yields a strictly
higher expected payoff to each agent.

2

uation value is positive if and only if each party transfers its goods as stipulated,
the agents are willing to incur a positive transaction cost, enabling trade to occur inequilibrium. Thus, the fundamental mechanism for sustaining cooperation is akin tothat in repeated games, where future rewards and punishments are used to prevent
deviation.2
Nonetheless, our analysis differs from much existing work on stochastic games inthat the state transition in our model is non-irreducible.3 In particular, as agents
make transfers, the remaining stock of each good decreases irreversibly. A related
complication is that a single agent can obstruct a state transition. For example, aplayer can prevent the supply of its good from falling simply by refusing to make any
further transfers.
This sort of non-irreducibility generates a key tradeoff between the size of the nexttransfer and the waiting time until the transfer. In order for agents to anticipate ahigh payoff, they must exchange a large quantity of each good at the next transaction.
Hence, the amount withheld must be low at that transaction. Because the resulting
continuation value is low, the agents are willing to incur only a small cost when making
the transaction. However, the expected waiting time for a small cost to realize is high.
Since agents discount the future, a lengthy waiting time reduces the expected payoff.
We solve for a subgame-perfect equilibrium that optimizes the tradeoff between thetransfer size and waiting time.
Methodologically, we combine the theory of investment under uncertainty withthe theory of repeated games by imposing mutual best-response constraints on investment decisions.4 That is, the set of admissible investment policies is restrictedby the incentive-compatibility requirements of a subgame-perfect equilibrium. Comparing the first- and second-best outcomes of our model, we show that the problem
of incentive compatibility delays the realization of the gains from trade.
The game is specified in continuous time, which has a number of advantages.5
First, a closed-form solution for the maximal equilibrium is obtained with further
assumptions on the cost process. The resulting expression uniquely determines the2

For example, see Fudenberg and Maskin (1986).
See Dutta (1995), Fudenberg and Yamamoto (2011), and Hörner, Sugaya, Takahashi, and Vielle(2011) for folk theorems in stochastic games at varying levels of generality.
4
See footnote 9 for references to the literature on investment under uncertainty.
5
Continuous-time modeling also involves some complications as noted by Simon and Stinchcombe(1989). The methodology newly developed by Kamada and Rao (2018) is applied to suitably definestrategy spaces in the game here.
33

cost incurred and amount transferred on each transaction. This facilitates the analysis
of comparative statics. In addition, if the true model is one in which the transaction
cost evolves continuously over time, then welfare might be lower when players arelimited to moving at discrete times instead of responding instantly. Rational agents
may be unwilling to constrain their behavior in this way.
Besides the opening example, our framework fits a number of social situations in
the real world where exchanges are involved. For instance, consider the transfer ofprisoners or hostages between groups that are enemies or countries that are fighting.
The two agents in the model would be representatives of the parties who are negotiating for the release of detainees. Each party wishes to trade captives from the otherside for the return of prisoners from its own side. The stochastic transaction costmight be a measure of the political climate or degree of tensions between the twogroups. When tensions are high and the relationship is hostile, each group has a highcost of releasing prisoners of war. On the other hand, the cost is low when tensions
are low and the relationship is not hostile.
In many cases, exchanges of prisoners occur in a gradual manner, as our model
predicts. One example comes from the negotiations over the repatriation of prisoners
towards the end of the Korean War. A release from the Associated Press (1953)
describes a plan under which “the United Nations will return 500 disabled North
Koreans and Chinese daily in exchange for 100 United Nations sick and wounded,”
resulting in a total of “over 600 United Nations men in exchange for 5,800 Communists.” Likewise, the transfer of captives during the Civil War in El Salvador proceeded in several stages. According to a report by LeMoyne (1984), “The wounded
guerillas left in four groups, numbering 15 each. As each group was permitted toleave El Salvador the rebels freed two army officers.” Finally, a similar principle wasinstrumental in the resolution of the Lebanon hostage crisis. In an article by Haberman (1991), a senior official is quoted as stating that “this will be the first of a seriesof stages at the end of which the question of the hostages, of the prisoners of war, ofthe missing in action, will come to a conclusion.” The chief hostage negotiator also
remarks, “We believe that this cannot be done in one shot. This is a step-by-stepprocess. We must take it one step at a time.”
A further prediction of our model is that the size of transfers will decrease overtime. An example of this pattern in the real world is provided by Article Two of
the Second Treaty of Indian Springs (1825). The following extract describes how the4

United States intended to compensate the Creek Indians for lands ceded to the stateof Georgia:
The United States agree to pay to the nation emigrating from the lands hereinceded the sum of four hundred thousand dollars; of which amount there shall be
paid to said party of the second part, as soon as practicable after the ratificationof this treaty, the sum of two hundred thousand dollars. And as soon as thesaid party of the second part shall notify the Government of the United Statesof their readiness to commence their removal, there shall be paid the further
sum of one hundred thousand dollars. And the first year after said emigrating
party shall have settled in their new country, they shall receive, of the amountfirst above named, the further sum of twenty-five thousand dollars; and thesecond year, the sum of twenty-five thousand dollars; and annually thereafter,
the sum of five thousand dollars, until the whole is paid.

According to the agreement, the United States would make a gradually decreasing
series of monetary payments to the Creek Indians as they progressively surrendered
their existing lands and relocated to a different territory.
The remainder of this paper is organized as follows. Immediately after the introduction is a review of the related literature. Section 2 outlines our basic model.
Section 3 analyzes the model in several steps. We start by identifying general conditions on the cost process under which a nondegenerate equilibrium does and does notexist. Thereafter, we characterize the basic properties of a maximal equilibrium. In
the case where the cost process follows a geometric Brownian motion, a closed-form
expression can be obtained for the optimal path of play. Section 4 presents comparative statics for the unique solution. Section 5 investigates the welfare implications of
the model, and section 6 discusses some modeling extensions and robustness checks
in the online appendix. Section 7 concludes. The proofs of the results in the body ofthe paper are provided in the main appendix.
1.1

Related Literature

Our paper connects with a line of research on gradualism in contribution games andconcession bargaining.6 In this literature, parties arrive at an agreement in a step6
See Admati and Perry (1991); Compte and Jehiel (1995, 2003, 2004); Marx and Matthews(2000). See also Gale (1995, 2001) and Gueron (2015) for monotone games.

5

by-step fashion, and there is an efficiency loss due to delay in reaching an agreement.
Likewise, cooperation between the two players in our model is sustained through a
gradual sequence of transactions over time. There is a key modeling difference between
our paper and, for example, Admati and Perry (1991). In their setup, a benefit
from cooperation is realized only when a joint project is completed, whereas in ourframework, a benefit is received every time a transaction takes place. Additionally, ourmodel differs from much of this literature in that transactions continue indefinitely inequilibrium; so that, the game we study cannot be solved using an iterated-dominanceprocedure.
In our model, agents incur costs whenever they take actions. Thus, another pointof comparison is the literature on repeated interaction with switching costs. In particular, Lipman and Wang (2000, 2009) show that equilibrium dynamics are nontrivial
in repeated games with finite and infinite horizons where switching costs are large
compared to stage-game payoffs. Caruana and Einav (2008) derive similar results forthe case where agents can revise their actions over a finite horizon with increasing
switching costs. Nonetheless, the role of switching costs in those papers is different
from the effect of transaction costs in our model. In those papers, the fact thatswitching costs are large in the future is important for the results, as such costs canserve as a commitment device. By contrast, no cooperation occurs in the equilibriumof our model if the transaction cost is bounded away from zero with probability one.
Exchange mechanisms are extensively studied in many different contexts thatinvolve trading favors. The seminal paper on this topic is Mobius (2001). Later work
includes: Abdulkadiroglu and Bagwell (2012) and Hauser and Hopenhyan (2005)
for the “chips mechanism”; Johnson, Simchi-Levi, and Sun (2014) for the “scrip
system”; and Wolitzky (2015) for the exchange of tokens. The distinguishing featureof our mechanism is that devices such as chips or scrips or tokens are not needed tosustain cooperation. It is the remaining supply of each good that is used to reward
cooperators.
Pitchford and Snyder (2004) consider a holdup problem between a buyer and aseller in which no investment occurs in the equilibrium of the static game. In a
dynamic version, positive investment can be supported in an equilibrium where theseller’s investment and the buyer’s repayment take place alternately. Those authors
observe that such an equilibrium would not exist in their deterministic setting if agents
must incur a fixed cost for transacting that does not depend on the amount invested.
6

We resolve this issue by allowing for uncertainty in the transaction cost, in whichcase agents may be able to realize positive gains from trade in a subgame-perfectequilibrium.7 The stochastic nature of the cost may be more realistic in practice as
economic conditions or the political climate may vary over time.8 Furthermore, thewelfare implications of our model are somewhat more nuanced than in Pitchford and
Snyder (2004), where the equilibrium converges to the efficient outcome as discounting frictions disappear. In our setup, the fixed cost of making a transfer creates anadditional friction, and so such convergence is not guaranteed. We identify conditions on the cost process under which the efficient outcome can be approximated inequilibrium as agents become infinitely patient.
Several theoretical results are presented in the context of general stochastic processes in continuous time. To obtain an explicit solution for the optimal strategies,
we also consider the special case where the transaction cost follows a geometric Brownian motion. The assumption of geometric Brownian motion is standard in the finance literature on investment under uncertainty.9 Similarly, a number of papers on
the principal-agent problem in continuous time, including Holmstrom and Milgrom(1987) and Sannikov (2008), assume that the output process follows an arithmetic
Brownian motion.
Although a continuous-time setting is useful in obtaining closed-form solutions
and comparative statics, there are subtle technicalities in defining strategy spaces
due to the possibility of instantaneous responses. Simon and Stinchcombe (1989)
and Bergin and MacLeod (1993) introduced methods to overcome such a problem
in deterministic models, but they do not fit our stochastic environment. For thisreason, we use the framework recently introduced by Kamada and Rao (2018), which
also resolves technical issues related to the measurability of stochastic processes incontinuous time.
7

As noted in section 6, the online appendix also examines a setting in which the quantity of goods
available for trade can vary randomly over time. With uncertainty in the supplies of goods, positivetransfers may be sustainable in equilibrium even if the transaction cost is a positive constant.
8
We give further justification for stochastic transaction costs in section 6.
9
For example, geometric Brownian motion is used by McDonald and Siegel (1986) to model themarket value of an investment, by Dixit (1991) to model a demand parameter, and by Bertola andCaballero (1994) to model an index of business conditions. He (2009) studies optimal managerial
compensation under the assumption of a geometric Brownian motion in firm size. Empirical evidence
is presented in support of the model. Merton (1971) derives optimal portfolio rules under thebenchmark hypothesis of a geometric Brownian motion in asset prices. He also discusses the effectof alternative modeling assumptions.

7

2

Model

There are two agents, 1 and 2, who take actions in continuous time t ∈ [0, ∞). The
discount rate is ρ > 0. There are two divisible goods, 1 and 2. The allocation of the
1,2
2,1 2,2i,jgoods at time t is represented by [(s1,1t , st ), (st , st )], where st denotes the amountof good j that agent i possesses at time t.10 The total supply q > 0 of each good is2,j11
constant over time; so that, s1,jThe
t + st = q > 0 for j ∈ {1, 2} and t ∈ [0, ∞).
initial endowment vector is s0 = [(q, 0), (0, q)].12 That is, agent 1 is endowed with allof good 1, and agent 2 is endowed with all of good 2.13
In addition, there is a positive transaction cost ct for transferring goods between
the two parties, which changes over time according to some stochastic process.14 For
ease of exposition, let c0 > q.15 We assume that the cost process evolves independently
from the actions of the agents.16
In every instant of time, each agent observes the current realization of the costand chooses an amount of her good to transfer to the other agent. For i ∈ {1, 2}
and t ∈ [0, ∞), let fti ∈ [0, q] represent the amount of good i that agent i transfers
at time t. A history of the game is represented as h = {ct , (ft1 , ft2 )}t∈[0,∞) . That is,a history consists of the realization of the cost process along with the transfers madeby the agents at each time. Given a history h, a history hu up to time u is defined asis the amount of good j that agent i possesses
If a transaction occurs at time t, then si,jt
immediately after this transaction.
11
As described in section 6, the online appendix presents a model in which the supply of eachgood can vary over time.
12
The equilibria of the model do not necessarily depend on the assumption that the agents havesymmetric endowments of goods. If π is an SPE when each agent i is endowed with the quantity qof good i, then a model in which each agent i is endowed with any quantity qi ≥ q of good i also has
the SPE π as well as an SPE that is the same as π except that each agent i transfers the additionalamount qi − q at the first transaction on the equilibrium path.
2,213This assumption is without loss of generality provided that s1,1
0 = s0 .
14
The equilibria of the model are robust to some relaxations of the assumption that both agents
face the same transaction cost. Let {cit }t∈[0,∞) for each i ∈ {1, 2} be a cost process such thatcit (ω) ≤ ct (ω) for all t ∈ [0, ∞) and every ω ∈ Ω. Any SPE in grim-trigger strategies when thetransaction cost is {ct }t∈[0,∞) for each party is also an SPE of a model in which the cost is {c1t }t∈[0,∞)for agent 1 and {c2t }t∈[0,∞) for agent 2.
15
This condition will help ensure that the initial value of the transaction cost is sufficiently highthat it is a strictly dominated strategy for an agent to make a transaction right at the beginning of
the game.
16
To be clear, {ct }t∈[0,∞) is a stochastic process on the probability space (Ω, F, P ), where eachrandom variable ct for t ∈ [0, ∞) takes values in the state space S ⊆ R++ . The probability space andstate space do not depend in any way on the transactions of the agents. As described in section 6, theonline appendix presents a model in which the transaction cost depends on the amount transferred.
10

8


{ct }t∈[0,u] , {(ft1 , ft2 )}t∈[0,u) . Note that hu includes information about the cost at time
u but does not contain information about the transfers at time u. By convention,
h0 is used to denote the null history (c0 , {}) at the start of the game. Letting Ht
S
be the set of all histories up to time t, define H = t∈[0,∞) Ht . A strategy for agenti ∈ {1, 2} is a map πi : H → R+ that assigns an amount to transfer to each history
up to a given time.17 Let Πi with generic element πi represent the set of all possible
strategies for agent i.18
Throughout the analysis, we restrict attention to feasible strategies that satisfy
traceability, frictionality, and calculability as defined by Kamada and Rao (2018). The
set of agent i’s strategies satisfying these properties is denoted by Π̄C
i . Restricting
attention to Π̄Ci avoids technical difficulties arising from instantaneous responses andnonmeasurable behavior in continuous time.19 With this restriction, each strategy
Cprofile in Π̄C
1 × Π̄2 induces a unique path of play that can be represented by aprogressively measurable stochastic process, and the expected payoffs to each agent
are well defined (Kamada and Rao, 2018).20 A formal definition of the histories andstrategy spaces for which we define calculable strategies is provided in Appendix A.1.
It is helpful to introduce some notation to describe the path of play. Let π =Let(π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} be a profile of calculable strategies.
ku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) be any history up to an arbitrary time u. Given
that {ct }t∈[0,u] = {gt }t∈[0,u] , let {φit (ku , {cτ }τ ∈(u,∞) , π)}t∈[0,∞) be a path of transfers foreach agent i ∈ {1, 2} such that φit (ku , {cτ }τ ∈(u,∞) , π) = gt for all t ∈ [0, u) and each i ∈
{1, 2} and such that πi cv v∈[0,t] , [φ1v (ku , {cτ }τ ∈(u,∞) , π), φ2v (ku , {cτ }τ ∈(u,∞) , π)] v∈[0,t)= φit (ku , {cτ }τ ∈(u,∞) , π) for all t ∈ [u, ∞) and each i ∈ {1, 2}. With conditional prob17

Simon and Stinchcombe (1989) allow for sequential moves at a single moment of time. The
definition of the strategy space here rules out such behavior. This restriction is innocuous in thepresent setting because of the maximality assumption that we will impose: given a symmetricequilibrium in which sequential transactions by either or both agents occur at some history up toa given instant in time, one can find a symmetric equilibrium generating a weakly higher expected
payoff to each agent and requiring only a one-shot simultaneous transaction at any such history.
18
The definition of strategies thus far does not eliminate the possibility of a flow transfer, whereby
the amount transferred at each instant of time is zero, but the total amount transferred over aninterval of time may be positive. However, current behavior cannot be conditioned on such flows
given the set of admissible histories. The restriction to frictional strategies precludes flow exchange.
19
Since the cost process can be specified such that it changes only at a certain countable set oftimes and the agents can be restricted to moving only at these times, our continuous-time setup
encompasses discrete-time models as a special case.
20
Another possible definition of strategy spaces might involve the notion of inertia from Berginand MacLeod (1993), but the maximal equilibrium for which we solve violates inertia, as explained
in section 3.3.

9

ability one, the traceability and frictionality assumptions guarantee the existence anduniqueness of such a path of transfers, and the set {t ∈ [u, v] : φit (ku , {cτ }τ ∈(u,∞) , π) >0} is finite for all v ≥ u and each i ∈ {1, 2}. In addition, denote by h(ku , {ct }t∈(u,∞) , π)
= ct , [φ1t (ku , {cτ }τ ∈(u,∞) , π), φ2t (ku , {cτ }τ ∈(u,∞) , π)] t∈[0,∞) the history that results when
ku is the history up to time u, {ct }t∈(u,∞) is the cost after time u, and strategy profile
π is played from time u onwards.
To define expected payoffs and equilibrium concepts, we begin by describing theevolution of the stocks of the goods. Choose any history h = {ct , (ft1 , ft2 )}t∈[0,∞) suchthat the set {t ∈ [0, ∞) : fti > 0} is finite for each i ∈ {1, 2}. Recall that si,jt denotes
the amount of good j that agent i possesses immediately after time t. The relationshipPiof the stocks of the goods to the transfer paths is specified as si,jt = q−{τ ∈[0,t]:fτi >0} fτPjif i = j and as si,jt ={τ ∈[0,t]:fτj >0} fτ if i 6= j.
We define payoffs as follows. Letting h = {ct , (ft1 , ft2 )}t∈[0,∞) be a history for which
the set {t ∈ [0, ∞) : fti > 0} is finite for each i ∈ {1, 2}, the realized payoff to agent i
at time t is given by:
Vti (h) =X

e−ρ·(τ −t) fτ−i −{τ ∈[t,∞):fτ−i >0}X

e−ρ·(τ −t) cτ .

(1){τ ∈[t,∞):fτi >0}

Note that each agent i values not its own good i, but the good −i possessed by theother agent −i, so that there exist gains to trade, provided that the transaction costis sufficiently small.
The first term in the preceding expression represents the discrete benefits generated by the good received from the other agent. The second term captures the fixed
costs incurred by an agent when making transactions. The restriction of each agent
i’s strategy to Π̄Ci implies that, with probability one, the number of transactions madeby each agent is countable (Kamada and Rao, 2018).
Under this specification, each agent seemingly consumes the good from the otheragent immediately upon receipt. In section 6, however, we argue that the game canbe straightforwardly reformulated so that the goods being exchanged are regarded as
durable and trade yields a stream of flow benefits.21 We also interpret these different
formulations in some detail, explaining how they capture different features of thetransaction process.
21
The online appendix formally establishes the equivalence between the original model with discrete benefits and the modified game with flow benefits.

10

Next choose any strategy profile π = (π1 , π2 ) such that πi ∈ Π̄Ci for i ∈ {1, 2}.

1 2
Let ku = {gt }t∈[0,u] , {(bt , bt )}t∈[0,u) be any history up to time u for which the set{t ∈ [0, u) : bit > 0} is finite for each i ∈ {1, 2}. The expected payoff to agent i at ku
can be expressed as:
Vi (ku , π) = E{ct }t∈(u,∞) {Vui [h(ku , {ct }t∈(u,∞) , π)]|{ct }t∈[0,u] = {gt }t∈[0,u] }.

(2)

The left-hand side of the equation above is the expected payoff at ku . On the righthand side, the conditional expectation is taken with respect to {ct }t∈(u,∞) given that{ct }t∈[0,u] = {gt }t∈[0,u] . The expected payoff at ku is well defined because the restriction of each agent i’s strategy to Π̄Ci ensures that the realized payoff can be uniquely
computed with conditional probability one and that the conditional expectation exists.
Since there is no uncertainty regarding past play and past events, we use subgameperfect equilibrium (SPE) as our equilibrium concept. Formally, a strategy profile(π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} is a subgame-perfect equilibrium if for any
history ku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) up to time u such that the set {t ∈ [0, u) :bit > 0} is finite for each i ∈ {1, 2}, the expected payoff to agent i ∈ {1, 2} at ku
satisfies Vi [ku , (πi , π−i )] ≥ Vi [ku , (πi0 , π−i )] for any πi0 ∈ Π̄Ci .
In some parts of the analysis, we restrict attention to symmetric strategies. A
strategy profile (π1 , π2 ) with πi ∈ Πi for i ∈ {1, 2} is symmetric if π1 (ku ) = π2 (ku )for any history ku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) up to time u such that b1t = b2t fort ∈ [0, u). The symmetry assumption requires that the amount of good 1 transferredby agent 1 at time u is equal to the amount of good 2 transferred by agent 2 at time u
given that players 1 and 2 have made symmetric transfers up to time u.22 Given any
symmetric strategy profile (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} as well as any historyku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) up to time u such that the set {t ∈ [0, u) : bit > 0} is
finite for each i ∈ {1, 2}, we define V (ku , π) = V1 (ku , π) = V2 (ku , π).
Let Π∗ denote the nonempty set of all symmetric SPE.23 We next define optimal
behavior. A symmetric SPE π ∈ Π∗ is maximal in the class of symmetric SPE if
there is no π 0 ∈ Π∗ such that V (h0 , π 0 ) > V (h0 , π).
22Note that this definition does not impose any restriction on behavior following a unilateraldeviation from a symmetric path of play.
23
The strategy profile in which each agent never makes a transfer conditional on any history up
to an arbitrary time is an element of Π∗ ; therefore, the set Π∗ is nonempty.

11

Most of our analysis focuses on maximal equilibrium because we consider a situation in which the two parties have made an informal agreement with each other.
Hence, it is reasonable to assume that agents can coordinate their play so as to induce their preferred outcome as long as the incentive constraints of each party are
not violated. Restricting attention to such an equilibrium makes it possible to obtaina unique solution, which enables us to derive meaningful comparative statics.
3

Analysis of Model

The solution of the model proceeds as follows. Section 3.1 presents conditions underwhich the model does and does not have a nondegenerate equilibrium. Section 3.2
characterizes the general form of a maximal equilibrium. Section 3.3 explicitly derives
the optimal policy subject to the incentive constraints under the assumption that thecost process follows a geometric Brownian motion.
3.1

Nondegenerate Equilibrium

The following theorem is an impossibility result. If the transaction cost is bounded
below by a positive number and the quantity of each good available for trade is fixed,
then there is no equilibrium in which an agent receives a positive expected payoff.
Theorem 1. Assume that {ct }t∈[0,∞) is an arbitrary cost process and that each randomvariable ct for t ≥ 0 takes values in the state space S ⊆ R with inf(S) > 0. If thestrategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} is an SPE, then there isiprobability one that φt (h0 , {cτ }τ ∈(0,∞) , π) = 0 for i ∈ {1, 2} and all t ∈ [0, ∞).
Note that this result does not depend on assuming symmetric or maximal equilibrium strategies. Furthermore, the proof only relies on an induction argument. Thatis, for any given S in the statement of the theorem, only a finite hierarchy of knowledge about rationality is needed to establish the theorem. In what follows, we relax
the assumption that the transaction cost is bounded below by a positive number.24
The next result identifies a general condition under which the game has a nondegenerate equilibrium.25
24

As explained in section 6, the online appendix outlines an alternative approach. We examine amodel with uncertainty in the supply of each good, demonstrating that the impossibility result may
not hold in such a setting, even with a positive lower bound on the cost.
25
In the following statement, a stochastic process is said to be right continuous if its sample pathis almost surely continuous from the right everywhere.

12

Theorem 2. Assume that {ct }t∈[0,∞) is an arbitrary right-continuous cost process
and that each random variable ct for t ≥ 0 takes values in the state space S ⊆ R++ .
Suppose that one can find p > 0, r < 1, and v > 0 for which given any realization of
the cost process {ct }t∈[0,u] up to an arbitrary time u, there is conditional probability
no less than p that the cost process {ct }t∈(u,∞) after time u is such that cτ ≤ rcu forsome τ ∈ (u, u + v). Then there exists a symmetric SPE π such that, with positive
probability, φit (h0 , {cτ }τ ∈(0,∞) , π) > 0 for i ∈ {1, 2} and some t ∈ [0, ∞).
An example of a cost process with the specified property is geometric Brownian
motion. Note that it may be permissible for there to be zero probability of the cost
approximating zero. That is, the condition admits some cases in which the probabilityof a cost less than c ever being reached approaches 0 in the limit as c goes to 0.26
The proof constructs a grim-trigger equilibrium strategy profile in which no further transfers occur following a deviation. On the path of play, each agent makes
a transaction at the first time that the cost process reaches a sufficiently low value.
Thereafter, each agent makes a transaction whenever the cost process reaches a value
no greater than a fraction r of the cost incurred on the preceding transaction. Moreover, the amount transferred by each agent on a given transaction is a fraction r of
the previous transfer.
3.2

Optimal Solution

This section describes the basic structure of a maximal symmetric SPE of the game.
First, we formally define the concept of a grim-trigger strategy profile, and we justify
restricting the analysis to this class of strategies. Next, we define stationary strategy
profiles and identify some important properties of stationary maximal symmetric SPE
when the cost process is continuous and has the Markov property.
Given a strategy profile π = (π1 , π2 ) with πi ∈ Πi for i ∈ {1, 2}, the strategy πi
is a grim trigger if πi (ht ) = 0 for every history h and any time t such that h is notconsistent with π1 or π2 at some time u < t. The next result provides justification
for restricting attention to SPE in grim-trigger strategies. Given any SPE, there26

In addition, it is straightforward to generalize the preceding theorem to allow the state space forthe cost process to include zero. However, this result does not necessarily extend to the case where
the state space for the cost process includes negative numbers. If the transaction cost is negative,then it may be possible for an agent to secure an arbitrarily high expected payoff by making asufficiently large number of transfers. An SPE may not exist in this situation.

13

exists an SPE in grim-trigger strategies that achieves the same path of play.27 Thus,
when characterizing the equilibrium strategies on the path of play, there is no loss ofgenerality from limiting the analysis to grim-trigger strategies.
Proposition 1. Given an arbitrary SPE π, there exists an SPE π 0 in grim-trigger
strategies such that, with probability one, φit (h0 , {cτ }τ ∈(0,∞) , π) = φit (h0 , {cτ }τ ∈(0,∞) , π 0 )for all t ∈ [0, ∞) and i ∈ {1, 2}.
Note that the proposition above does not require symmetric or maximal equilibrium strategies to be played. The basic idea behind the proof is that each agent canalways obtain a continuation value of zero by transferring nothing. If an opponent
uses a grim-trigger strategy, then zero is the maximum payoff that can be achieved
after any deviation. Thus, the grim-trigger strategy is the severest punishment available.
Hereafter, we restrict attention to symmetric SPE in grim-trigger strategies. In
order to characterize the basic properties of a maximal symmetric SPE, it is helpful to focus on stationary strategy profiles. However, we ultimately show that nonon-stationary maximal symmetric SPE exists given that the cost process follows
a geometric Brownian motion. Hence, there is no loss of generality from requiring
stationarity given the assumption about the form of the cost process.
Stationary strategy profiles are formally defined as follows. For any c ∈ (0, ∞)and s1 , s2 ∈ [0, q], let H̃τ (c, s1 , s2 ) denote the set consisting of every history kτ ={gυ }υ∈[0,τ ] , {(b1υ , b2υ )}υ∈[0,τ ) up to time τ such that the set {υ ∈ [0, τ ) : biυ > 0} isPP
finite for each i ∈ {1, 2}, q − {υ∈[0,τ ):b1υ >0} b1τ = s1 , {υ∈[0,τ ):b2υ >0} b2τ = s2 , and gτ = c.
S
Define H̃(c, s1 , s2 ) = τ ∈[0,∞) H̃τ (c, s1 , s2 ).
The strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} is stationary if††††
there exists a strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} satisfying the000
following two conditions. For any c ∈ (0, ∞) and s1 , s2 ∈ [0, q], π1† (ku0 ) = π1† (ku00 )
0
00
0
00
and π2† (ku0 ) = π2† (ku00 ) for all ku0 , ku00 ∈ H̃(c, s1 , s2 ). There is probability one thatφit (h0 , {cτ }τ ∈(0,∞) , π) = φit (h0 , {cτ }τ ∈(0,∞) , π † ) for all t ∈ [0, ∞) and i ∈ {1, 2}.
In words, a state is a triple (c, s1 , s2 ), where c is the current value of the cost
process, and s1 and s2 are the amounts of the goods remaining untransferred. Ac27

By path of play, we mean the following. Consider any strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci
for i ∈ {1, 2}. The path of play induced by π is the function that maps each realization of the cost process {cτ }τ ∈[0,∞) to transfer paths {φ1t (h0 , {cτ }τ ∈(0,∞) , π)}t∈[0,∞) and {φ2t (h0 , {cτ }τ ∈(0,∞) , π)}t∈[0,∞)for agents 1 and 2.

14

cordingly, two histories up to given times are regarded as being in the same state if
the current value of the transaction cost process and the total amount of each goodpreviously transferred are the same at these two histories. The set H̃(c, s1 , s2 ) consists of every history belonging to the state (c, s1 , s2 ). The set of stationary strategy
profiles includes any strategy profile that requires the agents to transfer the sameamount at any two histories belonging to the same state. This set also includes any
strategy profile that almost surely induces the same path of play as some strategy
profile that satisfies the condition in the previous sentence.28
For example, the maximal symmetric SPE in theorem 4 is a stationary strategy
profile. Noting that the agents play grim-trigger strategies, this SPE requires theagents to transfer different amounts at some two histories belonging to the same state.
However, this SPE almost surely induces the same path of play as some strategy profilethat requires the agents to transfer the same amount at any two histories belonging
to the same state. Moreover, the preceding definition is such that zero probability
events do not affect the stationarity of a strategy profile.29
The next theorem establishes some basic properties of an optimal solution. The
result requires the cost to follow a continuous Markov process.30 It does not rely on theassumption of geometric Brownian motion, which is introduced later. The theorem
shows that in any stationary maximal symmetric equilibrium, there is probability one
that the incentive constraints are binding and the costs incurred are decreasing.
In order to explain what is meant by binding incentive constraints, we must first
define the continuation value after each history. Let π = (π1 , π2 ) be any strategy

1 2
profile such that πi ∈ Π̄Ci for i ∈ {1, 2}. Let ku = {gt }t∈[0,u] , {(bt , bt )}t∈[0,u) be any
history up to time u for which the set {t ∈ [0, u) : bit > 0} is finite for each i ∈ {1, 2}.
Define Yi (ku , π) = Vi (ku , π) − [π−i (ku ) − gu ]. That is, Yi (ku , π) represents the expected
payoff to agent i immediately after any transaction at ku has been made. If thestrategy profile π is also symmetric, then we denote Y (ku , π) = Y1 (ku , π) = Y2 (ku , π),28

Two strategy profiles π a and π b are said to almost surely induce the same path of play if there iszero probability that the realization of the cost process {cτ }τ ∈[0,∞) is such that there exist t ∈ [0, ∞)and i ∈ {1, 2} for which φit (h0 , {cτ }τ ∈(0,∞) , π a ) 6= φit (h0 , {cτ }τ ∈(0,∞) , π b ).
29
The insensitivity of stationarity to zero probability events is relevant when showing that a
non-stationary maximal symmetric SPE does not exist when the cost follows a geometric Brownian
motion. Without this insensitivity property, it may be trivial to construct a non-stationary maximal
symmetric SPE given a stationary maximal symmetric SPE.
30
A stochastic process is said to be Markov if the conditional probability distribution of future
values of the process depends only on its current value. A stochastic process is said to be continuous
if its sample path is almost surely continuous everywhere.

15

provided that ku is such that b1t = b2t for t ∈ [0, u).
In the case where π is symmetric, let p be the probability that the realization of the cost process {cτ }τ ∈[0,∞) is such that there exists a time t for whichφit (h0 , {cτ }τ ∈(0,∞) , π) > 0 and Y [ht ({cτ }τ ∈[0,∞) , π), π] > ct . If p > 0, then π is said tohave a positive probability of the incentive constraint being slack at some transaction. If p = 0, then π is said to have probability one of the incentive constraint being
binding at every transaction. In essence, an incentive constraint is binding if thecontinuation value after a transaction is equal to the cost paid on the transaction.
The meaning of decreasing costs incurred is straightforward. Let π = (π1 , π2 )be any symmetric strategy profile such that πi ∈ Π̄Ci for i ∈ {1, 2}. Suppose that
there is zero probability of the realization of the cost process {cτ }τ ∈[0,∞) being suchthat there exist times t1 and t2 with t1 < t2 for which φit1 (h0 , {cτ }τ ∈(0,∞) , π) > 0,
φit2 (h0 , {cτ }τ ∈(0,∞) , π) > 0, and ct2 > ct1 . Then we say that there is probability one of
the cost incurred being decreasing between any two consecutive transactions.
Theorem 3. Assume that {ct }t∈[0,∞) is a continuous Markov cost process and thateach random variable ct for t ≥ 0 takes values in the state space S ⊆ R++ . Then
any stationary maximal symmetric SPE in grim-trigger strategies is such that there
is probability one of the incentive constraint being binding at every transaction and of
the cost incurred being decreasing between any two consecutive transactions.
The theorem is a consequence of three lemmata, which are stated and proved in themain appendix.31 The first of these lemmata exploits the Markov property of the cost
process to show that every non-stationary symmetric SPE is weakly Pareto dominatedby a stationary symmetric SPE. To understand the intuition for the proof, supposethat the agents are playing a symmetric SPE in which the next transaction may occur
at multiple cost levels, each of which is less than the current value of the cost. Since
the cost is Markov, another symmetric SPE with a weakly higher continuation payoff
can be constructed so that the next transaction can occur at only one such cost level.
This construction can be performed iteratively so as to define a stationary symmetric
SPE with an expected payoff no lower than the original symmetric SPE. The actual
proof is more complicated because the next transaction in a symmetric SPE may
occur at a cost level greater than or equal to the current value of the cost.
31

For each lemma, the online appendix provides an example that illustrates the intuition behindthe proof.

16

The second lemma in the proof states that any stationary symmetric SPE withpositive probability of the incentive constraint being slack at some transaction isstrictly Pareto dominated by a stationary symmetric SPE in which every incentive
constraint is binding. Intuitively, if the incentive constraint were slack at some transaction in a stationary symmetric SPE, then a symmetric SPE with a higher expected
payoff could be constructed by raising the amount transferred on that transaction
and lowering the amount transferred on successive transactions.
Given binding incentive constraints, the third lemma shows that any stationary
symmetric SPE with positive probability of the cost incurred being nondecreasing between successive transactions is strictly Pareto dominated by a stationary symmetric
SPE in which the cost incurred is decreasing. The intuition is that if, for example, thecosts incurred were increasing between some two consecutive transactions at which
the incentive constraints were binding, then a symmetric SPE with a higher expected
payoff could be generated by combining the two transfers into a single transaction at
the greater of the two costs. Thereby, some of the goods would be transferred earlier,and the probability of such a transfer occurring may rise as well.
0
Let Π denote the set consisting of every stationary symmetric SPE π ∈ Π∗ ingrim-trigger strategies for which there is probability one of the incentive constraint
being binding at every transaction and there is probability one of the cost incurred
being decreasing between any two consecutive transactions. According to theorem
3, any stationary maximal symmetric SPE in grim-trigger strategies belongs to the00
set Π . Given any strategy profile π ∈ Π for which there is positive probability ofa transaction occurring, let {c̃k (π), f˜k (π)}∞
k=1 be the unique sequence of cost cutoffs
and amounts transferred such that c̃k (π) ∈ (0, c0 ) and f˜k (π) ∈ (0, q) for all k, and thefollowing holds. The path of play induced by π is such that, with probability one,the k th transaction is made when the cost reaches c̃k (π) for the first time, and theamount f˜k (π) is transferred by each agent at this transaction.
Some further properties of a maximal symmetric equilibrium can be established
as corollaries of theorem 3. The result below shows that the sequences of potential
amounts transferred and costs incurred converge to zero in a stationary maximal
symmetric equilibrium in which a transaction occurs with positive probability.
Corollary 1. Assume that {ct }t∈[0,∞) is a continuous Markov cost process and thateach random variable ct for t ≥ 0 takes values in the state space S ⊆ R++ . Let π ∈ Π∗
be a stationary maximal symmetric SPE in grim-trigger strategies for which there is17

positive probability of the cost realization {ct }t∈[0,∞) being such that φit (h0 , {cτ }τ ∈(0,∞) , π)
> 0 for i ∈ {1, 2} and some t ∈ [0, ∞). Then limk→∞ c̃k (π) = 0 and limk→∞ f˜k (π) =0.
Intuitively, if the amount transferred did not converge to zero, then the stock ofeach good would be completely exhausted with positive probability. Consequently, if
the cost incurred did not converge to zero while the amount transferred did, then there
would exist some transaction at which the cost incurred is greater than the amountremaining to be potentially transferred. However, the incentive constraint for thattransaction would be violated because the continuation value from the relationshipwould necessarily be smaller than the transaction cost.
The next result shows that in a stationary maximal symmetric equilibrium inwhich a transaction occurs with positive probability, the quantities of each goodpotentially transferred sum up to the total quantity q of each good available.
Corollary 2. Assume that {ct }t∈[0,∞) is a continuous Markov cost process and thateach random variable ct for t ≥ 0 takes values in the state space S ⊆ R++ . Let π ∈ Π∗
be a stationary maximal symmetric SPE in grim-trigger strategies for which there ispositive probability of the cost realization {ct }t∈[0,∞) being such that φit (h0 , {cτ }τ ∈(0,∞) , π)P
˜
> 0 for i ∈ {1, 2} and some t ∈ [0, ∞). Then ∞k=1 fk (π) = q.
This property is a consequence of the maximality condition. For example, consider
an equilibrium in which a transaction occurs with positive probability but the sum of
the amounts of each good potentially transferred is at most q̃ < q. A Pareto superior
equilibrium can be constructed by requiring the agents to transfer the additionalamount q − q̃ at the first transaction.
The preceding corollary has a further implication. Let π be a stationary maximal
symmetric SPE. The set consisting of each transaction time when the agents play
π and the realized cost is {cτ }τ ∈[0,∞) is denoted by J({cτ }τ ∈[0,∞) , π) = {υ ∈ [0, ∞) :φiυ (h0 , {cτ }τ ∈(0,∞) , π) > 0}. Some continuous Markov cost processes have the property
that the probability of the cost ever reaching the value c is one for all c ∈ (0, c0 ).32
P
This condition is necessary and sufficient for υ∈J({cτ }τ ∈[0,∞) ,π) φiυ (h0 , {cτ }τ ∈(0,∞) , π)
to be equal to q almost surely. That is, the entire stock of each good is almost surely
32

A geometric Brownian motion with drift µ and volatility σ satisfies this condition for µ ≤ σ 2 /2
but not for µ > σ 2 /2.

18

transferred in the limit as time goes to infinity if and only if there is probability oneof the transaction cost ever approximating zero.
3.3

Explicit FormulaThis section derives the optimal strategies given the form of the cost process. Specifically, we assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian motion
dct = µct dt + σct dzt with arbitrary drift µ and positive volatility σ.33 It is useful todefine the parameter
q

µ
µ
1 2
1
−
+ 2 σρ2 .
β = 2 − σ2 −2
σ2
We now provide a closed-form solution. The subsequent sections analyze thesolution in greater detail by calculating comparative statics and characterizing welfare
properties.
Theorem 4. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian
motion with arbitrary drift µ and positive volatility σ. Then there exists a stationary
maximal symmetric SPE in grim-trigger strategies. Moreover, any maximal symmetric SPE in grim-trigger strategies is characterized by a sequence {c∗k , fk∗ }∞k=1 satisfying
c∗k =



q
1−β



β
β−1

k−β
and

fk∗ =



q
1−β



β
β−1

k−1

such that, with probability one, the k th transaction is made when the cost reaches c∗kfor the first time, and the amount fk∗ is transferred by each agent at this transaction.
The preceding result is derived by solving a constrained optimization problem.
The three lemmata in the proof of theorem 3 are used in order to express the objective function and constraints in a tractable form. Specifically, the first of theselemmata allows attention to be restricted to stationary symmetric SPE without lowering the expected payoffs that can be attained. In addition, the second and third
lemmata imply that only stationary symmetric SPE with binding incentive constraints
and decreasing costs incurred need to be considered when solving the maximization
33
By specifying that the value of the cost process evolves according to a geometric Brownian
motion, we are assuming that the growth in the transaction cost follows a Brownian motion withdrift. This is an important baseline case because a Brownian motion with drift is the only continuousLévy process. As noted by Sannikov and Skrzypacz (2010), a Lévy process has independent andidentically distributed increments. It is the continuous-time analog of a random walk.

19

problem. The incentive compatibility conditions provide one set of constraints, which
require that the cost incurred on each transaction equals the continuation value afterthat transaction. Another constraint is feasibility, whereby the sum of the amounts
transferred cannot exceed the total amount of each good available. Because the incentive constraints are binding, the objective function is simply the expected payoff
to each agent from the transfer at the first transaction.
The solution to this maximization problem delivers an expression for the uniquely
optimal path of play in any stationary maximal symmetric SPE. Furthermore, weshow that there is no non-stationary maximal symmetric SPE given that the cost
process follows a geometric Brownian motion. Therefore, the path of play specified
in the statement of the theorem applies to any maximal symmetric SPE.
The intuition behind the non-existence of a non-stationary maximal symmetric
SPE is illustrated by an example in the online appendix. If there were to exist a
non-stationary maximal symmetric SPE, then such a strategy profile could be used
to find two stationary maximal symmetric SPE that do not almost surely induce thesame path of play. However, this would contradict the result that the path of play ina stationary maximal symmetric SPE is unique up to probability zero events.
As justified above, attention is restricted to grim-trigger strategies. Although
there exist maximal symmetric SPE not in grim-trigger strategies, any such SPE
induces the path of play described in the preceding theorem.34
The optimal solution in theorem 4 has a proportional structure, in which the cost
incurred decreases by a factor of (1 − β)/β from the current transaction to the next.
The amount transferred is likewise becoming smaller.
∗
if n > m.
Corollary 3. fn∗ < fm

Intuitively, the incentive to deviate is falling over successive transactions because the prescribed cost payment is decreasing, making cooperation sustainable withsmaller future transfers. The decline in the transaction size is consistent with the example of land transferral in the introduction.
Note that the strategy profile in theorem 4 violates the inertia condition of Berginand MacLeod (1993), which would require in our context that for any history up to
an arbitrary time t, there exists  > 0 such that no transactions occur in the time34

To understand how to construct a maximal symmetric SPE not in grim-trigger strategies, see
the online appendix for an example.

20

interval (t, t + ). To see this, fix a history up to time t such that neither player has
deviated in the past, each player has transacted k times, and the cost is currently
greater than c∗k+1 . Then for any  > 0, there is positive probability that the costreaches c∗k+1 during the time interval (t, t + ), in which case a transaction takes place
under the maximal equilibrium.35
4Comparative Statics

Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian motion with
arbitrary drift µ and positive volatility σ. The path of play in a maximal symmetric
SPE is unique. Using the closed-form expression in theorem 4, a number of comparative statics can be obtained. In particular, we describe how the drift and volatility of
the cost process affect the sequence of costs incurred and amounts transferred. The
role of the discount rate is also investigated.
The corollary below describes how the size of each transfer changes with thediscount rate and the parameters of the cost process.
Corollary 4. If k < 1 + |β|, then fk∗ is decreasing in µ and ρ but increasing in σ. If
k > 1 + |β|, then fk∗ is increasing in µ and ρ but decreasing in σ.
This is an intuitive result.36 If the drift µ of the cost process decreases, then thecost is more likely to fall enough in the near future for the agents to make anothertransaction. The greater proximity of a future transaction raises the continuation
value of the relationship and relaxes the incentive constraints for the problem. Likewise, a lower discount rate ρ increases the continuation value, thereby weakening
the incentive constraints. Thus, agents can make larger transfers at early stages asstated in the first part of the corollary. However, if more of each good is transferred
at earlier stages, then less of each good is remaining at later stages. Therefore, thetransfers at later stages must be smaller as stated in the second part of the corollary.
If the volatility σ of the cost process increases, then both extremely high and low
35

Bergin and MacLeod (1993) also propose a weaker condition involving the completion of the setof inertia strategies. However, this approach does not apply here either. As shown in the online
appendix, the maximal equilibrium of our model cannot be represented as the limit of a Cauchy
sequence of strategy profiles satisfying inertia.
36
Note that the ratio of the amount transferred by an agent on each transaction to the stock ofeach good remaining before the transaction is (1 − β)−1 . Because β < 0 is decreasing in µ and ρ aswell as increasing in σ, the fraction of the remaining stock transferred on each transaction decreases
with µ and ρ but increases with σ.

21

realizations of the cost process become more likely. Because the solution has a cutoff
form, the favorable impact of low cost realizations dominates the adverse impact ofhigh cost realizations. This option-value argument suggests that a high volatility σ
has a similar effect on the solution as a low drift µ.
The next corollary characterizes the impact of the cost parameters and discountrate on the sequence of transaction costs paid.
Corollary 5. There exists an increasing function n(|β|) with lim|β|→0 n(|β|) = 0 and
lim|β|→∞ n(|β|) = ∞ such that the following holds. If k < n(|β|), then c∗k is decreasingin µ and ρ but increasing in σ. If k > n(|β|), then c∗k is increasing in µ and ρ but
decreasing in σ.
This result exemplifies a tradeoff between two effects. Suppose that the drift µ
decreases. On the one hand, low values of the transaction cost become more likely,
which encourages the agents to wait longer before transacting. Thereby, the cost
payment can be reduced. This suggests that the cost cutoff c∗k should decrease. On theother hand, the continuation value of the relationship rises, which enables the agentsto transact at a higher cost without violating the incentive constraints. Thereby, thewaiting time can be reduced. This suggests that the cost cutoff c∗k should increase. As
has already been noted, a high volatility σ has an impact similar to a low drift µ. A
smaller discount rate ρ also exerts two effects. More patient agents are willing to wait
longer for a low realization of the transaction cost, but greater patience increases thecontinuation value of the relationship, enabling higher cost payments to be sustainedin equilibrium.
The dominant effect depends on the parameters of the model. Intuitively, larger
absolute values of β reflect greater frictions. If |β| is low, then the first effect isimportant, in which case the cost thresholds are increasing in µ and ρ but decreasing in
σ. If |β| is high, then the second effect is relatively strong. Hence, the cost thresholds
early in the relationship are decreasing in µ and ρ but increasing in σ. However,a decrease in µ, an increase in σ, or a decrease in ρ implies that the supplies ofuntransferred goods are depleted earlier in the relationship. Given a small remaining
stock and a low continuation value, the cost cutoffs tend to decline at later stages.
An implication of the foregoing result is that each cost threshold is nonmonotonic
in the drift parameter and possibly also the volatility and discount rate. As the drift µ
increases from −∞ to ∞, the absolute value of β increases from 0 to ∞. Hence, each22

cost cutoff c∗k first rises with µ and then falls with µ. As the volatility σ decreases from∞ to 0, the absolute value of β increases from 0 to ∞ when µ ≥ 0 and from 0 to −ρ/µwhen µ < 0. Therefore, the threshold c∗k exhibits an inverted U-shaped relationshipwith σ if µ is nonnegative. Otherwise, c∗k can be decreasing or nonmonotonic in σ,depending on specific parameter values. An increase in the discount rate ρ from 0 to
∞ causes β to rise in absolute value from 0 to ∞ for µ ≤ σ 2 /2 and from 2µ/σ 2 − 1
to ∞ for µ > σ 2 /2. It follows that the relationship between c∗k and ρ displays aninverted U-shape if µ ≤ σ 2 /2. Otherwise, c∗k decreases with ρ for some parametervalues, but the effect is nonmonotonic in other cases.
This nonmonotonicity is on display in figure 1. The first cost threshold is plottedagainst the drift parameter µ when the volatility is fixed at a given level σ and theinitial stock of each good is 1. Different values of the discount rate ρ are considered
in comparison to the limiting case as ρ approaches 0. As explained above, the cutoffc∗1 is first rising and then falling in µ. In addition, the respective curves depicting
c∗1 for ρ = 0.01 and ρ = 0.1 intersect. Accordingly, c∗1 is increasing in ρ when µ is
low and so |β| is small, whereas the opposite holds for high values of µ, in whichcase |β| is large. This property generates the potentially nonmonotonic relationshipbetween the cost threshold and the discount rate mentioned above. Furthermore,c∗1 essentially becomes insensitive to ρ when µ is high relative to σ.37 The intuitionis that as µ increases, the upward drift of the cost becomes a bigger friction in theexchange process than the discounting of future transactions. Hence, the discountrate plays less of a role in determining the optimal trading policy.
The following corollary examines how the transfers made and costs incurred behave in the limits as the discount rate goes to zero and to infinity.
Corollary 6. If µ ≤ σ 2 /2, then limρ→0 f1∗ = q, limρ→0 fk∗ = 0 for k ≥ 2, andlimρ→0 c∗k = 0 for all k. If µ > σ 2 /2, then limρ→0 f1∗ < q, limρ→0 fk∗ > 0 for k ≥ 2,
and limρ→0 c∗k > 0 for all k. In addition, limρ→∞ fk∗ = limρ→∞ c∗k = 0 for all k.
These results follow immediately from theorem 4. First, consider the case whereµ ≤ σ 2 /2, meaning that the drift of the cost process is low relative to the volatility.
Given any positive integer k, let c∗k (ρ, µ, σ) denote the cost incurred on the k th transaction in
the maximal symmetric SPE when the discount rate is ρ and the cost process follows a geometric
Brownian motion with drift µ and volatility σ. It is straightforward to show that for any ρ0 , ρ00 > 0along with any σ > 0, the ratio c∗k (ρ0 , µ, σ)/c∗k (ρ00 , µ, σ) converges to one in the limit as µ approaches
∞.
37

23

Figure 1: c∗1 vs. µ for fixed σ > 0 and q = 1
As the agents become infinitely patient, it is optimal to wait for an extremely low
cost realization before making a transfer. Furthermore, a large initial transfer can besupported in equilibrium, because the incentive constraints are weak for low valuesof the transaction cost and so a high continuation value is not needed to prevent
deviation. Next, suppose that µ > σ 2 /2, in which case the drift is high comparedwith the volatility. The transaction cost is trending upward, and a fortuitously low
realization of the cost is improbable. Because a favorable trading opportunity is
unlikely, it is not optimal for even infinitely patient agents to wait an extremely
long time before transacting. Moreover, since transaction costs are still significant
in the limit, a substantial continuation value is needed to ensure that cooperation is
incentive compatible, which restricts the size of the initial transfer.
Finally, the corollary states that as the agents become infinitely impatient, it
becomes impossible to induce them to incur any strictly positive transaction cost. In
this situation, the continuation value from the relationship is low and so the incentive
to deviate is high. Therefore, the size of each transfer must also be small, in order toensure that the continuation value is sufficiently high to sustain cooperation.
Several of the comparative statistics in this section can be interpreted in terms of
the applications described in the introduction. For instance, consider the example ofprisoner exchange between two hostile parties, where the transaction cost represents

24the level of tensions between the opponents. The solution to the model suggests a
prisoner exchange protocol in which the prevailing level of tensions and the quantityof prisoners traded are smaller at the next transfer than at the current transfer. If thelevel of tensions decreases more quickly or becomes more volatile, then larger transfersof prisoners would take place at earlier stages of the relationship, and smaller transfers
would happen at later stages. These changes may cause either a rise or a fall in thelevel of tensions at which earlier transfers occur, but tensions at later transfers would
decrease.38 An interesting insight to arise from the analysis is that greater volatility
in the level of tensions often has the same effect on the transaction scheme as a more
rapid decrease in the level of tensions.
5

Welfare Properties

Section 5.1 describes the efficient trading policy in the absence of incentive constraints.
Section 5.2 examines whether the efficient outcome can be approximated as agents
become infinitely patient. A number of comparative statics regarding welfare levels
are presented.
5.1

Efficient Outcome

This section characterizes the efficient outcome of the model. It thereby establishesa benchmark for the welfare analysis. When the cost process follows a geometric
Brownian motion, a closed-form expression can be obtained.
We begin with a formal definition of efficiency. A symmetric strategy profile
π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} is efficient if there is no symmetric strategy
0
0
0
0
0
profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} such that V (h0 , π ) > V (h0 , π).
Efficiency is defined similarly to maximality, but non-equilibrium strategy profiles
must also be considered when identifying an optimum.
The result below shows that efficiency requires there to be probability one thatat most one transaction occurs. This is in contrast to the path of play in any nondegenerate equilibrium, in which there is no bound on the number of transactions thatcan occur with positive probability.
38

Note that the cutoff n(|β|) in corollary 5 satisfies lim|β|→0 n(|β|) = 0 and lim|β|→∞ n(|β|) = ∞,where the parameter β is such that limµ→−∞ |β| = 0 and limµ→∞ |β| = ∞.

25

Proposition 2. Assume that {ct }t∈[0,∞) is an arbitrary cost process and that eachrandom variable ct for t ≥ 0 takes values in the state space S ⊆ R++ . Let π be any
efficient symmetric strategy profile. Then there is probability one of the realizationof the cost process {ct }t∈[0,∞) being such that either φit (h0 , {cτ }τ ∈(0,∞) , π) = 0 fori ∈ {1, 2} and all t ∈ [0, ∞) or φit (h0 , {cτ }τ ∈(0,∞) , π) = q for i ∈ {1, 2} and somet ∈ [0, ∞).
The explanation is straightforward. Suppose that the agents reach a history atwhich a transaction occurs but some of each good is remaining after the transaction.
Then the agents can increase the expected payoffs at that history by instead transferring all of the goods. Hence, efficiency requires there to be zero probability that
goods remain untransferred after a transaction.
Assuming that the transaction cost evolves according to a geometric Brownian
motion, the result below solves for the path of play induced by any efficient symmetricstrategy profile.
Proposition 3. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian motion with arbitrary drift µ and positive volatility σ. Then there exists anefficient symmetric strategy profile. Moreover, any efficient symmetric strategy profile is characterized by a cost cutoff c̄ given by

c̄ =

β
β−1


q

such that, with probability one, a transaction is made when the cost reaches c̄ for the
first time, and the amount q is transferred by each agent at this transaction.
Note that there is a critical value of the cost such that the agents almost surelytransfer all of their goods at the first time the cost reaches that value. The online
appendix documents the following comparative statics. First, the efficient cost cutoffc̄ is increasing in the drift parameter and discount rate but decreasing in the volatilityof the cost process. Second, this threshold approaches the initial endowment of eachgood in the limit as the discount rate goes to infinity. Finally, it may or may notconverge to zero as agents become infinitely patient, depending on the parameters of
the cost process.
Comparing proposition 3 with theorem 4, the efficient cost threshold is greaterthan the first cost cutoff in the maximal symmetric SPE.
26

Corollary 7. c̄ > c∗1 .
Intuitively, an efficient path of play requires the agents to transfer all of the goods
at the first time cost reaches the threshold c̄. However, such a policy is not incentive
compatible because the transaction cost incurred, which is positive, is less than thequantity remaining untransferred, which is zero. Since it is profitable for each agent
to deviate by transferring nothing, this is not an equilibrium. In order to satisfy the
incentive constraint in a maximal equilibrium, the amount remaining after the firsttransaction is increased, and the cost at which the first transaction occurs is decreased.
The online appendix presents a theorem that generalizes this remark to the case of
a continuous Markov cost process. We relax the specific assumption of geometric
Brownian motion, requiring instead that the cost obeys a scaling rule according towhich the incremental change in the cost is proportional to the current value of the
cost.
5.2

Approximating Efficiency

This section evaluates the welfare implications of the model. We begin by presenting
formulae for the first- and second-best expected payoffs, which are then used to derivecomparative statics. Although the analysis in this section is confined to the case inwhich the transaction cost follows a geometric Brownian motion, the online appendix
relaxes the restriction on the functional form.
Consider the closed-form solutions for the maximal symmetric equilibrium in theorem 4 and for the efficient symmetric strategy profile in proposition 3. We use theterm second-best to refer to the expected payoff of each agent when the path of playspecified in theorem 4 is followed. In this case, the behavior of the players is limited
by incentive constraints. We use the term first-best to refer to the expected payoff ofeach agent when the path of play specified in proposition 3 is followed. The proposition below provides closed-form expressions for the expected payoffs to the agents inthe first- and second-best cases.
Proposition 4. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian
motion with arbitrary drift µ and positive volatility σ. Then the first-best expected
payoff M f b and the second-best expected payoff M sb are given by
M f b = θq 1−β cβ0 and M sb = θ1−β q 1−β cβ0 ,27whereθ = (−β)−β (1 − β)−(1−β) .
The aforementioned formulae are used to derive a number of corollaries. The
result below provides comparative statics for the effects of the parameters on the
expected payoffs in the first- and second-best cases.
Corollary 8. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian
motion with arbitrary drift µ and positive volatility σ. Then M f b and M sb are bothdecreasing in µ and ρ but increasing in σ.
The intuition is straightforward. If the drift µ decreases or the volatility σ increases, then the probability of a low realization of the transaction cost rises. Hence,
an opportunity for profitable trade becomes more likely, and so the expected payoff
to each agent increases in both the first- and second-best cases. In addition, morepatient agents receive higher expected payoffs because they assign greater value tofuture benefits.
The next corollary confirms that the second-best expected payoff converges to thefirst-best expected payoff in the limit as the discount rate approaches zero, provided
that the drift of the cost process is not excessively high relative to the volatility.
Corollary 9. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian
motion with arbitrary drift µ and positive volatility σ. If µ ≤ σ 2 /2, then M f b andM sb both converge to q in the limit as ρ goes to zero. If µ > σ 2 /2, then M f b and M sb
converge neither to q nor to each other in the limit as ρ goes to zero.
On the one hand, if trading frictions are low in the sense that the drift of thecost process is sufficiently small relative to the volatility, then the first-best outcome
can be approximated as the the agents become infinitely patient. On the other hand,
if the drift is high and the volatility is low, then the trading environment tends todeteriorate over time, and there is uncertainty about whether a future transaction willtake place. In this case, the second-best solution fails to converge to the first-best
outcome.
The condition µ ≤ σ 2 /2 is critical to the welfare analysis because it determines
whether there is probability one of the transaction cost ever approximating zero. The
probability of the cost c ever being reached is one for all c ∈ (0, c0 ) if and only if µ ≤σ 2 /2. For µ ≤ σ 2 /2, the cost incurred on the first transaction converges to zero in both
28

the first- and second-best solutions as agents become infinitely patient. In the limiting
case, the amount of each good transferred on the first transaction in the second-bestsolution approximates the entire amount, just as in the first-best solution.39 Forµ > σ 2 /2, the respective costs incurred on the first transaction converge to different
positive real numbers in the first- and second-best cases. In addition, the incentive
constraint requires the continuation value after the first transaction to be no lessthan the cost incurred on the first transaction. Hence, the amount transferred byeach agent on the first transaction does not converge to the total stock of each goodin the second-best solution.
The following corollary investigates the situation in which the first- and secondbest outcomes differ when agents are infinitely patient. It examines how the drift andvolatility of the cost process affect the relative values of the first- and second-best
expected payoffs.
Corollary 10. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian
motion with arbitrary drift µ and positive volatility σ. If µ > σ 2 /2, then in the limit
as ρ approaches zero, M sb /M f b is decreasing in µ and increasing in σ.
Intuitively, a lower drift or a higher volatility results in a more favorable trading
environment in which future transactions are more likely to occur. Consequently,
the continuation value from the relationship is higher, and larger transfers can besustained in equilibrium. The result above shows that the second-best expected payoff
increases as a fraction of the first-best expected payoff.
To illustrate the welfare properties of the model, figure 2 displays how the expected
payoffs vary with µ for a given value of σ. The graph fixes the initial value of the costat 3/2 and the original amount of each good at 1. In the limit as the discount rate
approaches zero, M f b and M sb coincide with each other if µ ≤ σ 2 /2. If µ > σ 2 /2,then M sb becomes an increasingly small fraction of M f b as µ increases relative to
σ. With strictly positive discounting, M f b is greater than M sb . Finally, the diagramsuggests that the discount rate has little effect on welfare when µ is high comparedto σ.40 Intuitively, when the transaction cost is rapidly trending upward with little
39

An implication of the preceding corollary for µ ≤ σ 2 /2 is that given any profile of expected
payoffs (V1 , V2 ) ∈ (0, M f b )2 , there exists ρ̄ > 0 such that for all ρ < ρ̄, (V1 , V2 ) can be attainedin an SPE. This follows because the incentive constraints on the first transaction are binding withprobability one in the maximal equilibrium, and so any equilibrium payoff profile (V1 , V2 ) ∈ (0, M sb )2
can be achieved by appropriately lowering the amounts transferred on the first transaction.
40
More precisely, let M f b (ρ, µ, σ) and M sb (ρ, µ, σ) respectively denote the first- and second-best29

Figure 2: M f b and M sb vs. µ for fixed σ > 0 and (c0 , q) = (3/2, 1)
chance of dropping, then a transaction, if any, is likely to occur early in the game.
This tends to lessen the impact of discounting on the expected payoffs.
The online appendix proves that the following general condition on the cost process suffices for all the potential gains from trade to be realized in the limit as thediscount rate approaches zero. Consider a right-continuous cost process. Supposethat there exists a constant r < 1 for which the conditional probability of the cost
becoming a fraction r of its current value during a time interval of length v approaches
one in the limit as v goes to infinity. Then one can find a discount rate and a symmetric equilibrium such that the expected payoff to each agent is arbitrarily close to
the total stock of each good. Note that a geometric Brownian motion satisfies theaforementioned condition if and only if its drift and volatility parameters are suchthat µ ≤ σ 2 /2. Hence, the first part of corollary 9 is simply an application of therelevant theorem in the online appendix.
6

Discussions and Extensions

Alternative specification of payoffs
expected payoffs when the discount rate is ρ and the cost process follows a geometric Brownian
motion with drift µ and volatility σ. It is straightforward to show that for any ρ0 , ρ00 > 0 along withany σ > 0, the ratios M f b (ρ0 , µ, σ)/M f b (ρ00 , µ, σ) and M sb (ρ0 , µ, σ)/M sb (ρ00 , µ, σ) both converge to
one in the limit as µ approaches ∞.

30

As mentioned in section 2, the payoff function in equation (1) has an alternative
formulation. The online appendix considers a modified model in which the transferredgood provides the recipient with a stream of flow payoffs. In particular, if an agent
transfers the amount fti of her good at time t, then she incurs the fixed cost Ct = ct /ρ
at time t, and her opponent enjoys a flow benefit of fti starting from time t. We showthat any SPE of the original game is also an SPE of the model with flow payoffs,
and vice versa. That is, the equilibrium set remains unchanged, and a maximal
equilibrium in the original game is an optimal solution to the modified model. Thisidea is formalized in the online appendix.
These different specifications of the game can be related to some of the examples
from the introduction, such as prisoner exchange between countries. In the modified
framework, goods are treated as durable, and a country enjoys a flow payoff fromhaving possession of returned prisoners. This flow payoff might represent a streamof emotional benefits to individuals in each country from being reunited with family
members who were being held captive by the other country. Alternatively, eachcountry might receive a stream of economic benefits as former captives reenter thelabor force. The original setup effectively requires the consumption of goods instantly
upon delivery. In this case, a country obtains a single discrete payoff when prisoners
from its side are released. For example, the return of prisoners might increase thepopularity of political leaders, or individuals might experience a sudden feeling of
happiness upon meeting their relatives again.
Variations of basic model
The online appendix describes two variations of the basic model. First, we allow
for growth or volatility in the supply of each good available for trade. We show thateven if the transaction cost is bounded away from zero, positive transfers may besupported in equilibrium, provided that there is some variability in the supply ofeach good. In particular, grim-trigger strategies are used to generate an equilibrium
path in which the agents make a transaction whenever the remaining stock of eachgood reaches a given threshold. Second, we study a setting in which the cost incurred
at a given transaction depends on the amount of each good transferred. Even in sucha situation, positive transfers may be supported in equilibrium. On the path of playthat we specify, the number of transactions is potentially infinite, and the transfer
size and transaction cost are gradually decreasing.

31Additional explanation of transaction cost
Transactions may involve various costs related to transportation, communication,and monitoring. In the example of prisoner exchange between countries, the transaction cost may be interpreted in political terms as a loss of popular support for thegovernment. This political cost may be increasing in the number of prisoners released
and may fluctuate depending on the level of public approval for the administration.
Under a democratic system, if public approval is closer to the threshold for electoral
victory, then the political cost of releasing prisoners may be higher as the election
outcome is more likely to be adversely affected. In the case where the cost follows
a geometric Brownian motion, the drift and volatility parameters may reflect thepattern of public approval ratings for the government over time. These parameters
may also be related to changes in economic factors that influence public opinion, suchas the unemployment rate, wage growth, stock market returns, and inflation. Suchfactors could be correlated across countries to some extent, and the common transaction cost, which might be linked to indices of global economic conditions, serves asa simple way to capture this correlation.
7

Conclusion

This paper studied the exchange of divisible goods between two agents facing astochastic transaction cost. We developed a model of trade that applies techniquesfrom the literature on investment under uncertainty to the theory of repeated games.
In our setup, the first-best policy requires a single transaction, whereas the secondbest solution involves a decreasing sequence of transfers of each good. Several comparative statics were presented that illustrate how the properties of the cost process
affect optimal behavior. We also examined the convergence of equilibria to the efficient outcome as discounting disappears. Our framework can be applied to real-world
situations involving the gradual exchange of trade secrets, captured prisoners, or landclaims.
Continuous-time modeling made the analysis of a stochastic game tractable. In
particular, a closed-form solution can be obtained given that the cost process evolves
according to a geometric Brownian motion, and we used the resulting formula to
study comparative statics. Interestingly, we found that the waiting time until thefirst transaction is nonmonotonic in the drift and maybe also the volatility of the32stochastic process. Moreover, the waiting time is longer than in the first-best case,where the entire stock of each good is transferred at once.
The analysis suggests a number of possibilities for further research. For example,
the framework could be extended to incorporate asymmetries between players in the
key elements of the model, such as the payoff functions, initial endowments, cost
processes, and equilibrium strategies. Allowing for mixed strategies might increase
the gains from trade in equilibrium. It would also be interesting to study incentives
to cooperate when players have imperfect information about past actions. Althoughthese questions may be beyond the scope of the current paper, we hope our analysis
serves as a basis for tackling them.
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35

A
A.1

Appendix
Formal Definitions of Histories and Strategy Spaces

This section provides formal definitions of histories and strategy spaces. In doing so,
we follow the notation and terminology in Kamada and Rao (2018).
The action z stands for “no move.” In the present paper’s model, we replace theaction of transferring the amount 0 with action z, so that the transaction amount fti
is defined over {z} ∪ (0, q]. Choose any time t ∈ [0, ∞) and cost process {cτ }τ ∈[0,t]
up to that time. A history up to time t is represented by {cτ }τ ∈[0,t] , {fτ1 , fτ2 }τ ∈[0,t) ,where {fτi }τ ∈[0,t) denotes the action path of agent i ∈ {1, 2} up to time t.
The set of all histories up to an arbitrary time is denoted by H. For each i ∈ {1, 2}
0
and any q 0 ∈ R++ , let H i,q ⊆ H be the set consisting of every history up to any timeP
t of the form {cτ }τ ∈[0,t] , {fτ1 , fτ2 }τ ∈[0,t) where q 0 = q − {τ ∈[0,t):fτi 6=z} fτi .
Let Āi (ht ) denote the set of feasible actions for agent i ∈ {1, 2} at history ht ∈ H.
0
If ht ∈ H i,q for some q 0 ∈ R++ , then let Āi (ht ) = {z} ∪ (0, q 0 ]. Otherwise, letĀi (ht ) = {z}.
The set of feasible strategies is for each i = 1, 2:
Π̄i = {πi : H → {z} ∪ (0, q]|πi (ht ) ∈ Āi (ht ) for all ht ∈ H}.
The set of traceable, frictional, calculable, and feasible strategies can then be definedand is denoted by Π̄Ci for each i = 1, 2.
The shock process st is formally defined as a pair comprising the cost ct andcalendar time t. The instantaneous utility function vi used in the definition of the
expected payoff is specified as follows for each i = 1, 2:



0




−c
τ
vi [(fτ1 , fτ2 ), sτ ] =
−i


fτ




f −i − cττA.2

if fτi = fτ−i = z
if fτi 6= z and fτ−i = z
if fτi = z and fτ−i 6= z.

if fτi 6= z and fτ−i 6= z

Proof of Theorem 1

Proof. Denote inf(S) = c > 0. Let H̃τ denote the set consisting of every historykτ = {gυ }υ∈[0,τ ] , {(b1υ , b2υ )}υ∈[0,τ ) up to time τ such that the set {υ ∈ [0, τ ) : biυ > 0}36

S
is finite for each i ∈ {1, 2}. Define H̃ = τ ∈[0,∞) H̃τ . We prove by induction that nostrategies in which some agent i makes a positive transfer fti at some ht ∈ H̃ can beplayed in any SPE.
i,i
i,i
i
Consider any ht ∈ H̃. Define ŝi,it = st + ft . That is, ŝt ∈ [0, q] represents
the amount of good i that agent i possesses immediately before time t. If ŝt1,1 < c,then no strategy where agent 2 makes a positive transfer at ht can be played in any
SPE, because such a strategy would give agent 2 an expected payoff no greater thanŝt1,1 − c < 0, whereas agent 2 could obtain an expected payoff of at least zero bymaking no transfers after reaching ht . Because no strategy where agent 2 makes apositive transfer at some ht with ŝt1,1 < c can be played in any SPE, agent 1 would
obtain an expected payoff no greater than −c < 0 by making a positive transfer at an
ht with ŝ1,1
t < c but would obtain an expected payoff of zero by making no transfers
after such an ht . Thus, no strategy where agent 1 makes a positive transfer at someht ∈ H̃ in which ŝ1,1< c can be played in any SPE. A symmetric argument showstthat no strategy where agent 1 or 2 makes a positive transfer at some ht ∈ H̃ in whichŝt2,2 < c can be played in any SPE.
Suppose now that for some integer n ≥ 1, no strategies where agent 1 or 2 makes< nc can be played
a positive transfer at some ht ∈ H̃ in which ŝ1,1< nc or ŝ2,2tt
in any SPE. Given this assumption, we show that no strategies where agent 1 or 2
makes a positive transfer at some ht ∈ H̃ in which nc ≤ ŝt1,1 < (n + 1)c or nc ≤ŝt2,2 < (n + 1)c can be played in any SPE. Consider in particular any ht ∈ H̃ in whichnc ≤ ŝ1,1t < (n + 1)c. If agent 1 is using a strategy that can be played in an SPE, thenthe greatest amount of the good that agent 1 can transfer at ht is ŝt1,1 − nc, because
if agent 1 made a transfer greater than ŝt1,1 − nc at ht , then with probability one, theremaining amount ŝ1,1u of agent 1’s good for u > t would be such that agent 2 makes
no further transfers, implying that agent 1 could obtain a higher expected payoff by
instead making no transfers after reaching ht . Thus, if agent 1 is using a strategy
that can be played in an SPE and ht is the history up to time t, then with probability
one, the history hu up to any time u > t must be such that ŝ1,1u ≥ nc, in which casethe total amount transferred by agent 1 after reaching ht is at most ŝt1,1 − nc < c.
It follows that no strategy in which agent 2 makes a positive transfer at ht can beplayed in any SPE, because making a positive transfer would give agent 2 an expected
payoff no greater than ŝ1,1t − nc − c < 0, whereas agent 2 could obtain an expected
payoff of at least zero by making no transfers after reaching ht . Thus, no strategy
37where agent 1 makes a transfer at ht can be played in any SPE, because agent 1
would obtain an expected payoff no greater than −c < 0 by making a transfer at ht
but would obtain an expected payoff of zero by making no transfers after reaching
ht . Thus, no strategies where agent 1 or 2 makes a transfer at some ht ∈ H̃ in whichnc ≤ ŝ1,1t < (n + 1)c can be played in any SPE. A symmetric argument holds for any
ht ∈ H̃ in which nc ≤ ŝ2,2t < (n + 1)c. This completes the induction.
A.3

Proof of Theorem 2

Proof. Consider the symmetric grim-trigger strategy profile ψ defined as follows. Recall that q denotes the amount of good i that agent i initially possesses. Letting
δ = e−ρv , choose any c∗ > 0 no greater than δpqr(1 − r). The first transaction occurs
at the first time that the current value of the cost is less than or equal to c∗ . If theprevious transaction occurred at cost ĉ, then the next transaction occurs at the firsttime that the cost is less than or equal to rĉ. For every positive integer k, each agent
transfers the amount rk−1 q(1 − r) on the k th transaction. If an agent deviates fromthe path of play described above, then neither agent makes any transactions following
the deviation.
We argue that strategy profile ψ is an SPE. Note that the strategies in ψ areP
k−1
q(1 − r) = q. We next show that the incentive compatifeasible because ∞k=1 rbility constraint is satisfied at each transaction when playing ψ. Choose any positive
integer l. Suppose that the agents have followed strategy profile ψ up to the current time, l − 1 transactions have happened in the past, and ψ specifies transaction
l will occur at the first time the cost reaches ĉ. If the agents follow strategy profile
ψ, then the cost incurred by each agent on transaction l is no greater than rl−1 c∗ ,and the expected payoff to each agent immediately after transaction l is no less thanP∞m m l+m−1
q(1 − r) − rl+m−1 c∗ ]. Hence, the incentive compatibility constraint
m=1 δ p [ris satisfied for transaction l if the following holds:
r

l−1 ∗

c ≤

∞
X

δ m pm [rl+m−1 q(1 − r) − rl+m−1 c∗ ],m=1which reduces to:δpr[q(1 − r) − c∗ ]⇔ c∗ ≤ δpqr(1 − r).
c ≤1 − δpr∗38

The last inequality is true by assumption, confirming that the incentive compatibility
constraint is satisfied.
A.4

Proof of Proposition 1

Proof. Let H̃τ denote the set comprising every history kτ = {gυ }υ∈[0,τ ] , {(b1υ , b2υ )}υ∈[0,τ )up to time τ such that the set {υ ∈ [0, τ ) : biυ > 0} is finite for each i ∈ {1, 2}. De
Sfine H̃ = τ ∈[0,∞) H̃τ . Given any history ht = {cτ }τ ∈[0,t] , {(fτ1 , fτ2 )}τ ∈[0,t) up to anarbitrary time t, let h = {gτ , (b1τ , b2τ )}τ ∈[0,∞) be any history such that {gτ }τ ∈[0,t] ={cτ }τ ∈[0,t] and {(b1τ , b2τ )}τ ∈[0,t) = {(fτ1 , fτ2 )}τ ∈[0,t) . Then ht is said to be on the equilibrium path induced by π if h is consistent with π1 and π2 at each time τ ∈ [0, t).
Fix any SPE π = (π1 , π2 ). We construct an SPE that induces the same equilibrium
path of play using grim-trigger strategies. Given any ht ∈ H̃, recall that Vi (ht , π) isthe expected payoff to agent i when both agents follow the strategy profile π fromtime t onwards, and let Xi (ht , π) denote the supremum of the expected payoffs toagent i from any set of deviations when π−i is fixed. Because π is an SPE, it must bethat Vi (ht , π) ≥ Xi (ht , π).
Now consider the grim-trigger strategy profile π 0 = (π10 , π20 ) such that [π10 (ht ), π20 (ht )]= [π1 (ht ), π2 (ht )] for any ht ∈ H on the equilibrium path induced by π and such that[π10 (ht ), π20 (ht )] = (0, 0) for any history ht ∈ H not on the equilibrium path inducedby π. We show that π 0 is an SPE strategy profile.41
Suppose that ht ∈ H̃ is on the equilibrium path induced by π. If πi0 (ht ) = 0,then the expected payoff to agent i from a one-shot deviation at ht must be negative.
However, the expected payoff to agent i when both agents follow the strategy profile
π 0 from ht onwards must be nonnegative, because π 0 (ku ) = π(ku ) for any ku ∈ H on
the equilibrium path induced by π, where π is an SPE. Hence, if πi0 (ht ) = 0, thenagent i does not have an incentive to make a one-shot deviation at ht .
0
Assume instead that πi0 (ht ) > 0. By definition, if agent −i is using strategy π−i,then the expected payoff to agent i from following πi0 from ht onwards is Vi (ht , π);0so that, Vi (ht , π 0 ) = Vi (ht , π). In addition, if agent −i is using strategy π−i, then a0
one-shot deviation at ht gives agent i an expected payoff of at most π−i(ht ), because
agent −i makes no transfers at any history not on the equilibrium path induced by
0π. Furthermore, agent i can ensure that she receives an expected payoff of π−i(ht ) byWhen confirming that the grim-trigger strategy profile π 0 is an SPE, it is sufficient to consider
only one-shot deviations from π 0 .
41

39

0
transferring nothing from time t onward. Hence, X̃i (ht , π 0 ) = π−i(ht ), where X̃i (ht , π 0 )is the supremum of the expected payoffs to agent i at ht from any one-shot deviation0
is fixed.
when π−i0(ht ) because if agent −i is using strategy
Now note that Xi (ht , π) ≥ π−i (ht ) = π−i
π−i , then agent i can always obtain at history ht an expected payoff of at least π−i (ht )by transferring nothing from time t onward. Thus, we have Vi (ht , π 0 ) = Vi (ht , π) ≥0
Xi (ht , π) ≥ π−i(ht ) = X̃i (ht , π 0 ) for any ht ∈ H̃ on the equilibrium path induced by πsuch that πi0 (ht ) > 0. Moreover, for any ht ∈ H̃ not on the equilibrium path inducedby π, we have Vi (ht , π 0 ) ≥ Xi (ht , π 0 ) because Vi (ht , π 0 ) = 0 = Xi (ht , π 0 ). Hence, thestrategy profile π 0 constitutes an SPE. Since π 0 is defined so as to agree with π on theequilibrium path, this completes the proof.

A.5

Lemmata in Proof of Theorem 3

It is helpful to introduce some additional terminology and notation. Consider any
Cb
strategy profiles π a = (π1a , π2a ) and π b = (π1b , π2b ) with πia ∈ Π̄Ci and πi ∈ Π̄i for i ∈{1, 2}. The strategy profiles π a and π b are said to induce different paths of play withprobability p if p is the probability that the realization of the cost process {cτ }τ ∈[0,∞)is such that there exist t ∈ [0, ∞) and i ∈ {1, 2} for which φit (h0 , {cτ }τ ∈(0,∞) , π a ) 6=
φit (h0 , {cτ }τ ∈(0,∞) , π b ). If there is probability zero that the strategy profiles π a and π b
induce different paths of play, then π a and π b are said to induce the same path ofplay with probability one.
Next consider any strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2}. We
denote:ht ({cτ }τ ∈[0,∞) , π) = {cτ }τ ∈[0,t] , {[φ1τ (h0 , {cυ }υ∈(0,∞) , π), φ2τ (h0 , {cυ }υ∈(0,∞) , π)]}τ ∈[0,t) .
That is, ht ({cτ }τ ∈[0,∞) , π) represents the history up to time t when the strategy profile
π is played and the realization of the cost is {cτ }τ ∈[0,∞) . We also define:iq − P{υ∈[0,t]:φiυ (h0 ,{cτ }τ ∈(0,∞) ,π)>0} φυ (h0 , {cτ }τ ∈(0,∞) , π), if i = ji,jσt ({cτ }τ ∈[0,∞) , π) = P
.
j

j
φ
(h
,
{c
}
,
π),
if
i
=
6
j
0
τ τ ∈(0,∞){υ∈[0,t]:φυ (h0 ,{cτ }τ ∈(0,∞) ,π)>0} υWe also denote σ̂ti,j ({cτ }τ ∈[0,∞) , π) = limυ→t− συi,j ({cτ }τ ∈[0,∞) , π), letting σ̂0i,j ({cτ }τ ∈[0,∞) ,π) = σ0i,j ({cτ }τ ∈[0,∞) , π) = q if i = j and σ̂0i,j ({cτ }τ ∈[0,∞) , π) = σ0i,j ({cτ }τ ∈[0,∞) , π) = 040

if i 6= j. That is, σ̂ti,j ({cτ }τ ∈[0,∞) , π) and σti,j ({cτ }τ ∈[0,∞) , π) represent the amount ofgood j that agent i possesses immediately before and after time t, respectively.
We now define the concept of a dual cutoff form. Some further notation is useful.
Given a symmetric strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2}, lettk ({cτ }τ ∈[0,∞) , π) signify the k th smallest time t such that φit (h0 , {cτ }τ ∈(0,∞) , π) > 0,
where k is a positive integer. If no such time exists, then let tk ({cτ }τ ∈[0,∞) , π) = ∞.
That is, tk ({cτ }τ ∈[0,∞) , π) is the time of the k th transaction when the strategy profile
π is played and the realization of the cost is {cτ }τ ∈[0,∞) . For notational purposes, let
t0 ({cτ }τ ∈[0,∞) , π) = 0.
In addition, we abbreviate:σ̂ i,i ({cτ }τ ∈[0,∞) , π, k) = σ̂ti,ik ({cτ }τ ∈[0,∞) ,π) ({cτ }τ ∈[0,∞) , π)
σ i,i ({cτ }τ ∈[0,∞) , π, k) = σti,ik ({cτ }τ ∈[0,∞) ,π) ({cτ }τ ∈[0,∞) , π)φi ({cτ }τ ∈[0,∞) , π, k) = φitk ({cτ }τ ∈[0,∞) ,π) ({cτ }τ ∈[0,∞) , π)c({cτ }τ ∈[0,∞) , π, k) = ctk ({cτ }τ ∈[0,∞) ,π) ,assuming that tk ({cτ }τ ∈[0,∞) , π) < ∞. That is, σ̂ i,i ({cτ }τ ∈[0,∞) , π, k), σ i,i ({cτ }τ ∈[0,∞) ,π, k), φi ({cτ }τ ∈[0,∞) , π, k), and c({cτ }τ ∈[0,∞) , π, k) respectively denote the amount ofgood i possessed by agent i immediately before the k th transaction, the amount ofgood i possessed by agent i immediately after the k th transaction, the amount of goodi transferred by agent i on the k th transaction, and the value of the cost process at
the k th transaction when the strategy profile π is played and the realization of thecost is {cτ }τ ∈[0,∞) .
Definition 1. Choose any symmetric strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci
for i ∈ {1, 2}. Let k be a positive integer. Transaction k has a dual cutoff
form when playing π if there is probability one that tk−1 ({cτ }τ ∈[0,∞) , π) = ∞ or
both tk−1 ({cτ }τ ∈[0,∞) , π) < ∞ and the following holds. Given the realizations ofc({cτ }τ ∈[0,∞) , π, n) and σ̂ i,i ({cτ }τ ∈[0,∞) , π, n) for n ∈ {0, 1, . . . , k − 1}, there exist:
1. ca with 0 ≤ ca ≤ c({cτ }τ ∈[0,∞) , π, k − 1),
2. φa with 0 < φa ≤ σ i,i ({cτ }τ ∈[0,∞) , π, k − 1),
3. cb with c({cτ }τ ∈[0,∞) , π, k − 1) ≤ cb ≤ ∞,4. φb with 0 < φb ≤ σ i,i ({cτ }τ ∈[0,∞) , π, k − 1).
41for which there is conditional probability one that the cost process {cτ }τ ∈[0,∞) is suchthat one of the three sets of requirements below is satisfied:
1. tk ({cτ }τ ∈[0,∞) , π) < ∞, c({cτ }τ ∈[0,∞) , π, k) = ca , φi ({cτ }τ ∈[0,∞) , π, k) = φa , there
does not exist u with tk−1 ({cτ }τ ∈[0,∞) , π) < u < tk ({cτ }τ ∈[0,∞) , π) such thatcu = ca or cu = cb .
2. tk ({cτ }τ ∈[0,∞) , π) < ∞, c({cτ }τ ∈[0,∞) , π, k) = cb , φi ({cτ }τ ∈[0,∞) , π, k) = φb , there
does not exist u with tk−1 ({cτ }τ ∈[0,∞) , π) < u < tk ({cτ }τ ∈[0,∞) , π) such thatcu = ca or cu = cb .
3. tk ({cτ }τ ∈[0,∞) , π) = ∞, there does not exist u > tk−1 ({cτ }τ ∈[0,∞) , π) such thatcu = ca or cu = cb .
The following is a verbal description of this concept. The first transaction has adual cutoff form when playing a symmetric strategy profile π if there exists ca ∈ [0, c0 ],i,ibcb ∈ [c0 , ∞] and φa ∈ (0, si,i0 ], φ ∈ (0, s0 ] such that, with probability one, the agents
transfer the amount φa at the first positive time the cost reaches ca if and only if cb
has not been reached earlier, the agents transfer the amount φb at the first positive
time the cost reaches cb if and only if ca has not been reached earlier, and the agents
do not make any transfers until a cost of ca or cb has been reached.42 For any positive
integer k, the (k + 1)th transaction has a dual cutoff form when playing a symmetricstrategy profile π if there is probability one that a k th transaction does not occur orboth a k th transaction occurs and the first transaction in the subgame immediately
following the k th transaction has a dual cutoff form when playing π.
The concept of a dual cutoff form is closely related to the concept of stationarity
for strategy profiles. The stationarity property implies a dual cutoff form, but notvice versa. The result below shows that each transaction has a dual cutoff form whenplaying a stationary symmetric strategy profile.
Claim 1. If π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} is a stationary symmetricstrategy profile, then transaction k has a dual cutoff form when playing π, where k is
an arbitrary positive integer.
Proof. Let π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} be a stationary symmetric strategy
profile. By definition, there exists a symmetric strategy profile π † = (π1† , π2† ) with42

If ca = 0, then there is zero probability of the first transaction occurring at a cost less than c0 .
If c = ∞, then there is zero probability of the first transaction occurring at a cost higher than c0 .
b42πi† ∈ Π̄Ci for i ∈ {1, 2} satisfying the following two conditions. For any c ∈ (0, ∞) and
0
00
0
00
0
00
s1 , s2 ∈ [0, q], π1† (ku0 ) = π1† (ku00 ) and π2† (ku0 ) = π2† (ku00 ) for all ku0 , ku00 ∈ H̃(c, s1 , s2 ).
There is probability one that φit (h0 , {cτ }τ ∈(0,∞) , π) = φit (h0 , {cτ }τ ∈(0,∞) , π † ) for allt ∈ [0, ∞) and i ∈ {1, 2}. Choose any positive integer k. If k > 1, then assume
that for every positive integer n < k, transaction n has a dual cutoff form whenplaying π † . We argue that transaction k has a dual cutoff form when playing π † .
Because π † induces the same path of play as π with probability one, it then follows
that transaction k has a dual cutoff form when playing π.
If tk−1 ({cτ }τ ∈[0,∞) , π † ) = ∞ with probability one, then transaction k has a dual
cutoff form when playing π † . Therefore, assume that tk−1 ({cτ }τ ∈[0,∞) , π † ) < ∞ withpositive probability. Choose any c† and s† for which there is positive probability of therealization of the cost process {cτ }τ ∈[0,∞) being such that both c({cτ }τ ∈[0,∞) , π † , k −1) = c† and σ̂ i,i ({cτ }τ ∈[0,∞) , π † , k−1) = s† . Let {c∗τ }τ ∈[0,∞) be any cost realization suchthat c({c∗τ }τ ∈[0,∞) , π † , k − 1) = c† and σ̂ i,i ({c∗τ }τ ∈[0,∞) , π † , k − 1) = s† . Let T be the set
consisting of every c for which there exists {c̃τ }τ ∈[0,∞) such that tk ({c̃τ }τ ∈[0,∞) , π † ) <∞, c({c̃τ }τ ∈[0,∞) , π † , k) = c, and c̃t = c∗t for t ≤ tk−1 ({c∗τ }τ ∈[0,∞) , π † ).
Let ca denote the supremum of the set consisting of every c ≤ c† such that c ∈ T .
Let cb denote the infimum of the set consisting of every c ≥ c† such that c ∈ T . If
there is no c ∈ T with c ≤ c† , then let ca = 0. If there is no c ∈ T with c ≥ c† ,then let cb = ∞. Note that there is no c ∈ T with c < ca and there is no c ∈ Twith c > cb . Otherwise, there would exist some c ∈ (0, ∞) and s ∈ [0, q] such that
0
00
0
00
0
00
π1† (ku0 ) 6= π1† (ku00 ) and π2† (ku0 ) 6= π2† (ku00 ) for some ku0 , ku00 ∈ H̃(c, s, s). This would
contradict the properties of the strategy profile π † .
For x ∈ {a, b}, let {cxτ }τ ∈[0,∞) be any cost realization such that tk ({cxτ }τ ∈[0,∞) , π † ) <∞, c({cxτ }τ ∈[0,∞) , π † , k) = cx , and cxt = c∗t for t ≤ tk−1 ({cxτ }τ ∈[0,∞) , π † ). For x ∈ {a, b},define φx = φi ({cxτ }τ ∈[0,∞) , π † , k), provided that the cost realization {cxτ }τ ∈[0,∞) in theprevious sentence exists. Suppose that the realization of the cost process {cτ }τ ∈[0,∞)satisfies c({cτ }τ ∈[0,∞) , π † , k − 1) = c† and σ̂ i,i ({cτ }τ ∈[0,∞) , π † , k − 1) = s† . Then{cτ }τ ∈[0,∞) is either such that tk ({cτ }τ ∈[0,∞) , π † ) = ∞ or such that tk ({cτ }τ ∈[0,∞) , π † ) <∞, c({cτ }τ ∈[0,∞) , π † , k) = cx , and φi ({cτ }τ ∈[0,∞) , π † , k) = φx for some x ∈ {a, b}.
Moreover, if tk ({cτ }τ ∈[0,∞) , π † ) < ∞, then there exists no u with tk−1 ({cτ }τ ∈[0,∞) , π † ) <u < tk ({cτ }τ ∈[0,∞) , π † ) such that cu = cx for some x ∈ {a, b}. Hence, transaction k
has a dual cutoff form when playing π † .
The next result provides an example to show that not every symmetric strategy
43

profile in which each transaction has a dual cutoff form is stationary.
Claim 2. There exists a non-stationary symmetric strategy profile π = (π1 , π2 ) with
πi ∈ Π̄Ci for i ∈ {1, 2} such that for every positive integer k, transaction k has a dual
cutoff form when playing π.
Proof. Consider the non-stationary symmetric grim-trigger strategy profile π defined
as follows. First, choose any c1 ∈ (0, c0 ) and φ1 ∈ (0, q). The agents each transfer theamount φ1 at the first time that the cost reaches c1 , and the agents do not make anytransfers until a cost of c1 has been reached.
Second, choose any c2,a ∈ (0, c1 ), c2,b ∈ (c1 , ∞), and φ2 ∈ (0, q − φ1 ). Eachagent transfers the amount φ2 at the first time the cost reaches c2,a after the firsttransaction by each agent if and only if c2,b has not been reached earlier following thefirst transaction, each agent transfers the amount φ2 at the first time the cost reaches
c2,b after the first transaction by each agent if and only if c2,a has not been reachedearlier following the first transaction, and the agents do not make any transfers after
the first transaction until a cost of c2,a or c2,b has been reached following the firsttransaction.
Third, choose any c3 ∈ R++ with c3 6= c2,a and c3 6= c2,b , and choose any φ3,a ∈(0, q − φ1 − φ2 ) and φ3,b ∈ (0, q − φ1 − φ2 ) with φ3,a 6= φ3,b . If the cost incurred on
the second transaction is c2,a , then the agents each transfer the amount φ3,a at thefirst time after the second transaction by each agent that the cost reaches c3 , andthe agents do not make any transfers after the second transaction until a cost of c3
has been reached following the second transaction. If the cost incurred on the second
transaction is c2,b , then the agents each transfer the amount φ3,b at the first time after
the second transaction by each agent that the cost reaches c3 , and the agents do notmake any transfers after the second transaction until a cost of c3 has been reachedfollowing the second transaction.
Each agent makes at most three transactions. If an agent deviates from the pathof play described above, then neither agent makes any transactions following thedeviation. The strategy profile π is such that for every positive integer k, transaction
k has a dual cutoff form when playing π.
The following are the statements and proofs of the three lemmata that imply
theorem 3. The result below shows that given any non-stationary symmetric SPE

440π, one can find a stationary symmetric SPE π that generates at least as high an
expected payoff to each agent.
Here is an outline of the proof. Let π be an arbitrary symmetric SPE. We begin
by using the Markov property of the cost process to construct a symmetric SPE in
which the payoff to each agent is at least as high as under π and the first transaction
has a dual cutoff form. The next step is to find a symmetric SPE that satisfies theconditions in the preceding sentence as well as the additional requirement that the
incentive constraint is binding on the first transaction. It is then straightforward tospecify a symmetric SPE in which the first transaction has a dual cutoff form, theincentive constraint is binding on the first transaction, and the payoff to each agent
is the same as under π. An iterative procedure is next used along with the Markov
property of the cost to define a symmetric SPE in which every transaction has a dual
cutoff form, the incentive constraint is binding on every transaction, and the payoff to0each agent is the same as under π. Finally, a stationary symmetric SPE π is derived
in which the payoff to each agent is at least as high as under π. In addition, the cost0
incurred between any two consecutive transactions is decreasing when playing π .
Lemma 1. Given any symmetric SPE π ∈ Π∗ in grim-trigger strategies, there exists
00a stationary symmetric SPE π ∈ Π∗ in grim-trigger strategies such that V (h0 , π ) ≥V (h0 , π). Moreover, π 0 can be chosen such that the cost incurred between any twoconsecutive transactions is decreasing.
Proof. Let the strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} be an arbitrary
symmetric SPE in grim-trigger strategies. Define the symmetric SPE π 0 = (π10 , π20 )in grim-trigger strategies as follows. For any history ku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u)up to an arbitrary time u, let πi0 (ku ) = πi (ku ) if πi (ku ) ≥ gu , and let πi0 (ku ) = 0 if
πi (ku ) < gu , where i ∈ {1, 2}. Note that V (h0 , π 0 ) ≥ V (h0 , π).
We begin by finding a symmetric SPE π ∗ ∈ Π∗ in grim-trigger strategies with
V (h0 , π ∗ ) ≥ V (h0 , π 0 ) such that the first transaction has a dual cutoff form whenplaying π ∗ . There are three cases to consider. First, π 0 is a stationary symmetric
SPE in grim-trigger strategies. Second, π 0 is a non-stationary maximal symmetric
SPE in grim-trigger strategies. Third, π 0 is a non-stationary strategy profile thatis not a maximal symmetric SPE in grim-trigger strategies. The strategy profile π ∗is constructed as follows in each case. In the first case, where π 0 is a stationary
symmetric SPE in grim-trigger strategies, simply let π ∗ = π 0 .
45Next, consider the second case, where π 0 is a non-stationary maximal symmetric
SPE in grim-trigger strategies. Either there is zero probability that t1 ({cτ }τ ∈[0,∞) , π 0 ) <∞ and c({cτ }τ ∈[0,∞) , π 0 , 1) < c0 , or there exists a realization {caτ }τ ∈[0,∞) of the cost process as well as a time ta such that t1 ({caτ }τ ∈[0,∞) , π 0 ) = ta , cata < c0 , and V [ht ({caτ }τ ∈[0,∞) ,π 0 ), π 0 ] ≥ V [ht ({caτ }τ ∈[0,∞) , π 0 ), π̂] for every symmetric SPE π̂ in grim-trigger strategies along with any time t ≤ ta . If such a cost realization {caτ }τ ∈[0,∞) exists, then let
00ca = cata . Otherwise, let ca = 0. Either there is zero probability that t1 ({cτ }τ ∈[0,∞) , π 0 )< ∞ and c({cτ }τ ∈[0,∞) , π 0 , 1) ≥ c0 , or there exists a realization {cbτ }τ ∈[0,∞) of thecost process as well as a time tb such that t1 ({cbτ }τ ∈[0,∞) , π 0 ) = tb , cbtb ≥ c0 , andV [ht ({cbτ }τ ∈[0,∞) , π 0 ), π 0 ] ≥ V [ht ({cbτ }τ ∈[0,∞) , π 0 ), π̂] for every symmetric SPE π̂ ingrim-trigger strategies along with any time t ≤ tb . If such a cost realization {cbτ }τ ∈[0,∞)00exists, then let cb = cbtb . Otherwise, let cb = ∞. Define the strategy profile π ∗ ∈ Π∗
in grim-trigger strategies as follows. The agents do not make any transactions until00
the first time t∗ that the current value of the cost process is ca or cb . If no suchtime t∗ exists, then the agents never make a transaction. Consider the case where
00
there exists such a time t∗ . If ca ∈ (0, c0 ) and ct∗ = ca , then each agent transfers
the amount φita (h0 , {caτ }τ ∈(0,∞) , π 0 ) at time t∗ . After this transaction, the agents playaccording to strategy profile π 0 , behaving as if the history up to the time of thistransaction were hta ({caτ }τ ∈[0,∞) , π 0 ) and strategy profile π 0 had always been followed
00from the beginning of the game. If cb ∈ [c0 , ∞) and ct∗ = cb , then each agent transfers the amount φitb (h0 , {cbτ }τ ∈(0,∞) , π 0 ) at time t∗ . After this transaction, the agentsplay according to strategy profile π 0 , behaving as if the history up to the time of thistransaction were htb ({cbτ }τ ∈[0,∞) , π 0 ) and strategy profile π 0 had always been followedfrom the beginning of the game.
Now, consider the third case, where π 0 is a non-stationary strategy profile that is
not a maximal symmetric SPE in grim-trigger strategies. In this case, one can find asymmetric SPE π̃ = (π̃1 , π̃2 ) in grim-trigger strategies such that V (h0 , π̃) > V (h0 , π 0 ).
Define the symmetric SPE π̃ 0 = (π̃10 , π̃20 ) in grim-trigger strategies as follows. For any
history ku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) up to an arbitrary time u, let π̃i0 (ku ) = π̃i (ku )if π̃i (ku ) ≥ gu , and let π̃i0 (ku ) = 0 if π̃i (ku ) < gu , where i ∈ {1, 2}. Note thatV (h0 , π̃ 0 ) ≥ V (h0 , π̃). Let δ = V (h0 , π̃ 0 ) − V (h0 , π 0 ).
Consider the strategy profile π̃ 0 . Let T be the set consisting of each cost level c
for which there exists a realization {c̃τ }τ ∈[0,∞) of the cost process as well as a time ũ
i0such that c̃ũ = c, σ̂ũi,i ({c̃τ }τ ∈[0,∞) , π̃ 0 ) = si,i
0 , and φũ (h0 , {c̃τ }τ ∈(0,∞) , π̃ ) > 0. If the set
46

T is empty, then the expected payoff to each agent from playing strategy profile π̃ 0
is zero, because no transfers are made on the equilibrium path. However, this would
contradict the fact that π 0 is an SPE, because π̃ 0 generates a strictly higher expected
payoff to each agent than π 0 , and no SPE can generate an expected payoff to eachagent lower than zero. Therefore, it can be assumed that the set T is non-empty.
For any cost level c in T , let K(c) be the set consisting of every history ku =
{gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) up to an arbitrary time u for which gu = c, π̃i0 (ku ) > 0,
biv = π̃i0 {gt }t∈[0,v] , {(b1t , b2t )}t∈[0,v) = 0 for all v < u. Define Q(c) = supku ∈K(c) V (ku ,π̃ 0 ). A state is a pair (c, s), where c is the current value of the cost process, and s
is the amount of each good remaining untransferred. For any cost levels ca and cb
ain T , let R(ca , cb ) denote the value of an asset at state (c0 , si,i0 ) that pays Q(c ) at
the first time the cost reaches ca if and only if cb has not been reached earlier andthat pays Q(cb ) at the first time the cost reaches cb if and only if ca has not beenreached earlier. Letting X = supca ,cb ∈T R(ca , cb ), it follows from the properties of thecost process that X ≥ V (h0 , π̃ 0 ).
To understand the last claim, consider the following argument. If the agents play
strategy profile π̃ 0 , then Q(c) is an upper bound on the expected payoff to each agent
when a state (c, si,i0 ) with c in T is reached and the agents choose to make a transfer
at this time. Hence, V (h0 , π̃ 0 ) ≤ B, where B denotes the supremum of the set of
possible expected payoffs to each agent in the following problem. Assume that thei,iinitial state is (c0 , si,i0 ). At each time a state (c, s0 ) with c ∈ T is reached, the agentscan choose either to accept or to decline, provided that the agents have not chosen
to accept in the past. If the agents accept, then each agent receives an immediate
payoff of Q(c), and the problem ends. If the agents decline, then each agent receives
an immediate payoff of zero, but the problem continues. Given any state (c, si,i0 ) withc ∈ T , let H(c) be the supremum of the set of possible expected payoffs to each agent
upon reaching state (c, si,i0 ) if the agents follow policies in which they choose to decline
at this time.
Let Z be the set consisting of each cost level c ∈ T such that Q(c) ≥ H(c). Let z ldenote the supremum of the set consisting of every cost level c ∈ Z such that c ≤ c0 .
Let z h denote the infimum of the set consisting of every cost level c ∈ Z such thatc ≥ c0 . For any  > 0, let Z l () denote the set consisting of every cost level c ∈ Z
for which c ∈ [z l − , z l ], and let Z h () denote the set consisting of every cost levelc ∈ Z for which c ∈ [z h , z h + ]. For any  > 0, define Ql () = supc∈Z l () Q(c) and47

Qh () = supc∈Z h () Q(c). Let D() denote the value of an asset at state (c0 , si,i0 ) thathhlpays Q () at the first time the cost reaches z +  if and only if z −  has not beenreached earlier and that pays Ql () at the first time the cost reaches z l −  if and only
if z h +  has not been reached earlier. Note that lim→0 D() = B. For any  > 0 and0000
η > 0, there exist cost levels ca and cb for which the expected payoff to each agent
is greater than D() − η when the agents follow a policy of accepting at the first time00
00
the cost reaches ca if and only if cb has not been reached earlier and accepting at
00
00
the first time the cost reaches cb if and only if ca has not been reached earlier.
Define the strategy profile π ∗ as follows. Choose any realizations {cpτ }τ ∈[0,∞) and{cqτ }τ ∈[0,∞) of the cost process and times tp and tq such that σ̂ti,ip ({cpτ }τ ∈[0,∞) , π̃ 0 ) = si,i
0 ,
i,i
i,i
σ̂tq ({cqτ }τ ∈[0,∞) , π̃ 0 ) = s0 , φitp (h0 , {cpτ }τ ∈(0,∞) , π̃ 0 ) > 0, φitq (h0 , {cqτ }τ ∈(0,∞) , π̃ 0 ) > 0,and X − P ({cpτ }τ ∈[0,∞) , {cqτ }τ ∈[0,∞) ) < δ/2, where P ({cpτ }τ ∈[0,∞) , {cqτ }τ ∈[0,∞) ) is the00pvalue of an asset at state (c0 , si,i0 ) that pays V [htp ({cτ }τ ∈[0,∞) , π̃ ), π̃ ] at the firsttime the cost reaches cptp if and only if cqtq has not been reached earlier and that pays
V [htq ({cqτ }τ ∈[0,∞) , π̃ 0 ), π̃ 0 ] at the first time the cost reaches cqtq if and only if cptp has notbeen reached earlier. The strategy profile π ∗ requires the agents to make no transfers
until the cost level cptp or cqtq is reached. If the cost level cptp is reached and the cost levelcqtq has not been reached earlier, then π ∗ requires the agents to play according to thestrategy profile π̃ 0 from the first time that the cost level cptp is reached, behaving as
if the history up to this time were htp ({cpτ }τ ∈[0,∞) , π̃ 0 ). If the cost level cqtq is reachedand the cost level cptp has not been reached earlier, then π ∗ requires the agents toplay according to the strategy profile π̃ 0 from the first time that the cost level cqtq
is reached, behaving as if the history up to this time were htq ({cqτ }τ ∈[0,∞) , π̃ 0 ). The
strategy profile π ∗ requires the agents to make no transfers following a deviation fromthe path of play described above.
Consider the three cases above. In the first case, the stationarity of π 0 implies that the first transaction has a dual cutoff from when playing π ∗ . Recall thatc0 > q. In the second case, the feasibility of πi implies that φit (h0 , {c̃τ }τ ∈(0,∞) , π) < c0
for any cost realization {c̃τ }τ ∈[0,∞) and time t such that c̃t = c0 . It follows thatφit (h0 , {c̃τ }τ ∈(0,∞) , π 0 ) = 0 for any such {c̃τ }τ ∈[0,∞) and t. In the third case, the feasibility of π̃i implies that φit (h0 , {c̃τ }τ ∈(0,∞) , π̃) < c0 for any cost realization {c̃τ }τ ∈[0,∞)and time t such that c̃t = c0 . It follows that φit (h0 , {c̃τ }τ ∈(0,∞) , π̃ 0 ) = 0 for any such{c̃τ }τ ∈[0,∞) and t. Hence, it must be that πi∗ (h0 ) = 0 in the second and third cases. It
follows that the first transaction has a dual cutoff form when playing π ∗ . Moreover,48

π ∗ ∈ Π∗ is a symmetric SPE in grim-trigger strategies with V (h0 , π ∗ ) ≥ V (h0 , π 0 ).
It is helpful to introduce some terminology. Given a symmetric SPE π̂ ∈ Π∗ aswell as any positive integer k, let p̂ be the probability that tk−1 ({cτ }τ ∈[0,∞) , π̂) < ∞and there exists a time t with tk−1 ({cτ }τ ∈[0,∞) , π̂) < t < tk+1 ({cτ }τ ∈[0,∞) , π̂) for whichφit (h0 , {cτ }τ ∈(0,∞) , π̂) > 0 and Y [ht ({cτ }τ ∈[0,∞) , π̂), π̂] > ct . If p̂ > 0, then π̂ is said tohave a positive probability of the incentive constraint being slack at transaction k. Ifp̂ = 0, then π̂ is said to have probability one of the incentive constraint being bindingat transaction k.
We next construct a symmetric SPE π ∗∗ ∈ Π∗ in grim-trigger strategies with
V (h0 , π ∗∗ ) ≥ V (h0 , π ∗ ) such that the first transaction has a dual cutoff form whenplaying π ∗∗ and such that π ∗∗ has probability one of the incentive constraint being
binding at the first transaction. If π ∗ has probability one of the incentive constraint
being binding at the first transaction, then simply let π ∗∗ = π ∗ . If π ∗∗ has a positive
probability of the incentive constraint being slack at the first transaction, then thestrategy profile π ∗∗ is constructed as described below.
It is helpful to introduce some notation. Let t̂1 and t̂2 be any two nonnegative real
numbers such that t̂1 ≤ t̂2 . Let π̂ ∈ Π∗ be a symmetric SPE in grim-trigger strategies.
Given a realization {c̃τ }τ ∈[0,∞) of the cost process, let Σ[ht̂1 ({c̃τ }τ ∈[0,∞) , π̂); {c̃τ }τ ∈[t̂1 ,t̂2 ] ;t̂1 , t̂2 ; π̂] be the sum of the transfers that an agent would make between times t̂1 andt̂2 inclusive if ht̂1 ({c̃τ }τ ∈[0,∞) , π̂) is the history up to time t̂1 , the cost path {c̃τ }τ ∈[t̂1 ,t̂2 ]is realized between times t̂1 and t̂2 , and the strategy profile π̂ is played by the agents.
The following is how π ∗∗ is defined in this case. The agents play according tostrategy profile π ∗ until the first time t for which the realization {c̃τ }τ ∈[0,∞) of thecost process is such that φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) > 0 and Y [ht ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗ ] > c̃t .
If no such time t exists, then the agents simply follow strategy profile π ∗ . Otherwise,
let t̃1 denote the first time t that this condition holds. At time t̃1 , each agent transfers
the amount φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) + ζ(c̃t̃1 ), where ζ(c̃t̃1 ) is a positive real number noi∗greater than si,i0 − φt̃1 (h0 , {c̃τ }τ ∈(0,∞) , π ) that can depend on c̃t̃1 . After time t̃1 , theagents do not make any transfers at any time t > t̃1 such that the realization of thecost process {c̃τ }τ ∈[t̃1 ,t] between times t̃1 and t satisfies φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) = 0 or
Σ[ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ); {c̃τ }τ ∈[t̃1 ,t] ; t̃1 , t; π ∗ ] − φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) − ζ(c̃t̃1 ) < c̃t .
Let t̃2 denote the first time t such that φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) > 0 and Σ[ht̃1 ({c̃τ}τ ∈[0,∞) , π ∗ ); {c̃τ }τ ∈[t̃1 ,t] ; t̃1 , t; π ∗ ] − φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) − ζ(c̃t̃1 ) ≥ c̃t . If such a timet does not exist, then the agents do not make any further transactions. Other49

wise, proceed as follows. At time t̃2 , each agent transfers an amount equal to
the lesser of φit̃2 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) and Σ[ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ); {c̃τ }τ ∈[t̃1 ,t̃2 ] ; t̃1 , t̃2 ; π ∗ ] −φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) − ζ(c̃t̃1 ). Thereafter, the agents play according to strategy profile π ∗ , behaving as if the history up to time t̃1 were ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ), the realizationof the cost process between times t̃1 and t̃2 were {c̃τ }τ ∈[t̃1 ,t̃2 ] , and strategy profile π ∗were followed up to and including time t̃2 . The agents do not make any transfers
following a deviation from the path of play specified above.
If each ζ(c̃t̃1 ) is chosen appropriately, then π ∗∗ is a symmetric SPE in grim-trigger
strategies with V (h0 , π ∗∗ ) > V (h0 , π ∗ ), and π ∗∗ has probability one of the incentive
constraint being binding at the first transaction. Such a choice of ζ(c̃t̃1 ) is possible
because V [ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗∗ ] is increasing in ζ(c̃t̃1 ), Y [ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗∗ ]approaches Y [ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗ ] as ζ(c̃t̃1 ) goes to 0, Y [ht̃1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗∗ ]i∗∗∗∗equals 0 for ζ(c̃t̃1 ) = si,i0 − φt̃1 (h0 , {c̃τ }τ ∈(0,∞) , π ), and Y [ht̃1 ({c̃τ }τ ∈[0,∞) , π ), π ] is acontinuous function of ζ(c̃t̃1 ).
We now observe that it is possible to construct a symmetric SPE π ∗∗∗ in grimtrigger strategies with V (h0 , π ∗∗∗ ) = V (h0 , π) such that the first transaction has adual cutoff form when playing π ∗∗∗ and such that π ∗∗∗ has probability one of the
incentive constraint being binding at the first transaction. If there is zero probabilityof a transaction occurring when playing π ∗∗ , then simply let π ∗∗∗ = π ∗∗ . If there isa positive probability of a transaction occurring when playing π ∗∗ , then a symmetric
SPE π ∗∗∗ satisfying the prescribed conditions can be constructed from π ∗∗ by appropriately lowering or leaving unchanged the amount of each good transferred at thefirst transaction.
Let π 1,∗∗∗ = π ∗∗∗ . Choose any positive integer k. Assume that there exists a
symmetric SPE π k,∗∗∗ in grim-trigger strategies with V (h0 , π k,∗∗∗ ) = V (h0 , π) suchthat the nth transaction has a dual cutoff form when playing π k,∗∗∗ and such thatπ k,∗∗∗ has probability one of the incentive constraint being binding at the nth transaction, where n can be any positive integer no greater than k. Furthermore, assume that π k,∗∗∗ is such that φit (h0 , {c̃τ }τ ∈(0,∞) , π k,∗∗∗ ) ≥ c̃t for any cost realization{c̃τ }τ ∈[0,∞) and time t > tk ({c̃τ }τ ∈[0,∞) , π k,∗∗∗ ) at which φit (h0 , {c̃τ }τ ∈(0,∞) , π k,∗∗∗ ) >0. Note that this condition holds for k = 1. It will be shown below that there
exists a symmetric SPE π k+1,∗∗∗ in grim-trigger strategies with V (h0 , π k+1,∗∗∗ ) =V (h0 , π) such that the nth transaction has a dual cutoff form when playing π k+1,∗∗∗and such that π k+1,∗∗∗ has probability one of the incentive constraint being bind50

ing at the nth transaction, where n can be any positive integer no greater thank + 1. Moreover, π k+1,∗∗∗ can be constructed such that φit (h0 , {cτ }τ ∈(0,∞) , π k+1,∗∗∗ ) =φit (h0 , {cτ }τ ∈(0,∞) , π k,∗∗∗ ) for all t ≤ tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ). Additionally, π k+1,∗∗∗ will
be such that φit (h0 , {c̃τ }τ ∈(0,∞) , π k+1,∗∗∗ ) ≥ c̃t for any cost realization {c̃τ }τ ∈[0,∞) andtime t > tk+1 ({c̃τ }τ ∈[0,∞) , π k+1,∗∗∗ ) with φit (h0 , {c̃τ }τ ∈(0,∞) , π k+1,∗∗∗ ) > 0. It will00
then follow that there exists a symmetric SPE π in grim-trigger strategies with00
V (h0 , π ) = V (h0 , π) such that the nth transaction has a dual cutoff form when play00
00
ing π and such that π has probability one of the incentive constraint being bindingat the nth transaction, where n can be any positive integer.
Given the strategy profile π k,∗∗∗ described in the preceding paragraph, a strategy
profile π k+1,∗∗∗ with the aforementioned properties can be constructed by applying theprocedure above to each subgame on the equilibrium path immediately following thefirst transaction when playing π k,∗∗∗ . Let π k,∗∗∗ = π k,∗∗∗ . In particular, the strategy
profile π k+1,∗∗∗ is derived as follows. Either tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ) = ∞ with probability
one, or tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ) < ∞ with positive probability. If tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ) =∞ with probability one, then simply let π k+1,∗∗∗ = π k,∗∗∗ . Otherwise, proceed as below. Let {c†n }kn=1 be any sequence of costs incurred on the first k transactions for which{c({cτ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {c†n }kn=1 with positive probability. Let {fn† }kn=1 be any
sequence of amounts transferred on the first k transactions for which there is positive
probability of {cτ }τ ∈[0,∞) being such that both {c({cτ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {c†n }kn=1
and {φi ({cτ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {fn† }kn=1 . Let {g̃τ }τ ∈[0,∞) be any realization of thecost process such that {c({g̃τ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {c†n }kn=1 , {φi ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {fn† }kn=1 , and Yu† [hu† ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ), π k,∗∗∗ )] = c†k , where u† = tk ({g̃τ }τ ∈[0,∞) ,π k,∗∗∗ ).
Consider a modified game in which the initial value of the cost process is c†k insteadPof c0 and the original amount of each good is q − kn=1 fn† instead of q. Let h00 denotethe null history in this modified game. Define the symmetric SPE ψ in grim-trigger
strategies as follows. At time 0, the agents do not make any transactions. After time
0, the agents play according to strategy profile π k,∗∗∗ ∈ Π∗ , behaving as if the history
at time 0 were hu† ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ). The agents do not make any transfers following
a deviation from the path of play described above. Note that ψ is such that ψi (ht ) ≥ gt
for any history ht = {gτ }τ ∈[0,t] , {(b1τ , b2τ )}τ ∈[0,t) up to an arbitrary time t for which
ψi (ht ) > 0. Suppose that a procedure similar to that described in the first nine
paragraphs in the proof of the lemma is applied to the strategy profile ψ. By doing51

so, it is possible to construct a symmetric strategy profile ψ ∗ in grim-trigger strategies
such that the following statement holds. Either ψ ∗ is a symmetric SPE in grim-trigger
strategies with V (h00 , ψ ∗ ) ≥ V (h00 , ψ) such that the first transaction has a dual cutoff
form when playing ψ ∗ , or each agent makes a transfer at time 0 with probability one.

Note that ψ ∗ is such that ψi∗ (ht ) ≥ gt for any history ht = {gτ }τ ∈[0,t] , {(b1τ , b2τ )}τ ∈[0,t)up to an arbitrary time t for which ψi∗ (ht ) > 0.
In the former case, a procedure similar to that described in the eleventh throughsixteenth paragraphs can be used to construct a symmetric SPE ψ ∗∗ in grim-trigger
strategies with V (h00 , ψ ∗∗ ) ≥ V (h00 , ψ) such that the first transaction has a dual cutoff
form when playing ψ ∗∗ and such that ψ ∗∗ has probability one of the incentive constraint being binding at the first transaction. As in the seventeenth paragraph, thestrategy profile ψ ∗∗ can then be adjusted so as to generate a symmetric SPE ψ ∗∗∗ ingrim-trigger strategies with V (h00 , ψ ∗∗∗ ) = V (h00 , ψ) such that the first transaction has
a dual cutoff form when playing ψ ∗∗∗ and such that ψ ∗∗∗ has probability one of the
incentive constraint being binding at the first transaction. If this case holds, then
the next three paragraphs should be skipped after constructing the strategy profileψ ∗∗∗ . Note that ψ ∗∗∗ is such that φit (h00 , {c̃τ }τ ∈(0,∞) , ψ ∗∗∗ ) ≥ c̃t for any cost realization{c̃τ }τ ∈[0,∞) and time t > t1 ({c̃τ }τ ∈[0,∞) , ψ ∗∗∗ ) at which φit (h00 , {c̃τ }τ ∈(0,∞) , ψ ∗∗∗ ) > 0.
In the latter case, proceed as follows. Consider another modified game in which
the initial value of the cost process is equal to c0 and the original amount of each goodP0
is q − kn=1 fn† . Define the symmetric SPE ψ ∗ in grim-trigger strategies as follows.
0
The agents do not make any transactions until the first time t∗ that the current
0value of the cost process is c†k . At time t∗ , each agent transfers the amount ψi∗ (h00 ).
After this transaction, the agents play according to strategy profile ψ ∗ , behaving as
if the history at the time of this transaction were h00 and strategy profile ψ ∗ hadalways been followed. The agents do not make any transfers following a deviationfrom the path of play described above. A procedure similar to that described in
the eleventh through sixteeth paragraphs can be used to construct a symmetric SPE0ψ ∗∗ in grim-trigger strategies such that the first transaction has a dual cutoff form
00when playing ψ ∗∗ , such that ψ ∗∗ has probability one of the incentive constraint being
binding at the first transaction, and for which there is probability one of {cτ }τ ∈[0,∞)00being such that either t1 ({cτ }τ ∈[0,∞) , ψ ∗∗ ) = ∞ or both t1 ({cτ }τ ∈[0,∞) , ψ ∗∗ ) < ∞000
and V [ht ({cτ }τ ∈[0,∞) , ψ ∗∗ ), ψ ∗∗ ] ≥ V (h00 , ψ) for t = t1 ({cτ }τ ∈[0,∞) , ψ ∗∗ ). Note that00
ψ ∗ is such that ψi∗ (ht ) ≥ gt for any history ht = {gτ }τ ∈[0,t] , {(b1τ , b2τ )}τ ∈[0,t) up to
52

0

0

0

an arbitrary time t for which ψi∗ (ht ) > 0 and that ψ ∗∗ is such that ψi∗∗ (ht ) ≥ gt
for any history ht = {gτ }τ ∈[0,t] , {(b1τ , b2τ )}τ ∈[0,t) up to an arbitrary time t for which0ψi∗∗ (ht ) > 0.
0
Let {p̃τ }τ ∈[0,∞) be any realization of the cost process such that t1 ({p̃τ }τ ∈[0,∞) , ψ ∗∗ ) <00
00∞ and both Y [ht ({p̃τ }τ ∈[0,∞) , ψ ∗∗ ), ψ ∗∗ ] = p̃t and V [ht ({p̃τ }τ ∈[0,∞) , ψ ∗∗ ), ψ ∗∗ ] ≥0
V (h00 , ψ) for t = t1 ({p̃τ }τ ∈[0,∞) , ψ ∗∗ ). Now consider the original game in which theinitial value of the cost process is c0 and the original amount of each good is q. Define0
the strategy profile π k,∗∗∗ as follows. The agents play according to strategy profile
π k,∗∗∗ until the history up to the current time is such that each agent has made k − 1transactions in the past, the respective transaction costs incurred on these transactions were {c†n }k−1
n=1 , the respective amounts transferred on these transactions were0
† k−1
{fn }n=1 , and the current value of the cost process is c†k . Let tk,∗∗∗ denote the first00such time if such a time exists. Let fk† = fk† + φi ({p̃τ }τ ∈[0,∞) , ψ k,∗∗ , 1). Note that000φi ({p̃τ }τ ∈[0,∞) , ψ k,∗∗ , 1) ≥ c†k . At time tk,∗∗∗ , each agent transfers the amount fk† .
00After time tk,∗∗∗ , the agents play according to strategy profile ψ ∗∗ , behaving as if00
the history at this transaction were ht ({p̃τ }τ ∈[0,∞) , ψ ∗∗ ) for t = t1 ({p̃τ }τ ∈[0,∞) , ψ ∗∗ )0
and as if strategy profile ψ ∗∗ had always been followed. The agents do not make anytransfers following a deviation from the path of play described above.
Return to the nineteenth paragraph in the proof of the lemma. Redefine the0strategy profile π k,∗∗∗ as π k,∗∗∗ . However, do not redefine π k,∗∗∗ . Choose {c†n }kn=1 and††0{fn† }k−1n=1 to be the same as before, but redefine fk as fk . Given these redefinitions, let{g̃τ }τ ∈[0,∞) be any realization of the cost process such that {c({g̃τ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1= {c†n }kn=1 , {φi ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {fn† }kn=1 , and Yu† [hu† ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ),π k,∗∗∗ )] = c†k , where u† = tk ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ). Follow the instructions from thetwentieth paragraph of the proof.
0
Recall that φi ({p̃τ }τ ∈[0,∞) , ψ k,∗∗ , 1) ≥ c†k in the twenty-third paragraph from theP
proof of the lemma. Hence, it would be the case that q − kn=1 fn† < c†k after only
finitely many repetitions of the procedure in the nineteenth to twenty-fourth paragraphs. In this case, the strategy profile ψ ∗ described in the twentieth paragraphcould not be such that each agent makes a transaction at time zero with probability one. It follows that the procedure in the nineteenth to twenty-fourth paragraphs
needs to be repeated only finitely many times in order to generate the strategy profileψ ∗∗∗ described in the twentieth paragraph. In addition, this procedure can be applied to any sequence {c†n }kn=1 of costs incurred on the first k transactions for which53{c({cτ }τ ∈[0,∞) , π k,∗∗∗ , n)}kn=1 = {c†n }kn=1 with positive probability. Given such a se† kquence {c†n }kn=1 , let ψ ∗∗∗,{cn }n=1 denote the strategy profile ψ ∗∗∗ derived by applying
this procedure to {c†n }kn=1 .
Now define the strategy profile π ∗∗∗,k+1 as follows. The agents play according
to strategy profile π ∗∗∗,k until k transactions have occurred. Let t∗∗∗,k denote thetime of the k th transaction if such a time exists, and proceed as follows in this case.
Let {c̃†n }kn=1 denote the sequence of costs incurred on the first k transactions. After† ktime t∗∗∗,k , the agents play according to strategy profile ψ ∗∗∗,{c̃n }n=1 , behaving as if† k
the history at time t∗∗∗,k were h00 and strategy profile ψ ∗∗∗,{cn }n=1 had always been
followed. The agents do not make any transfers following a deviation from the pathof play described above. As explained in the eighteenth paragraph in the proof of00
the lemma, it follows that there exists a symmetric SPE π in grim-trigger strategies
00with V (h0 , π ) = V (h0 , π) such that the nth transaction has a dual cutoff form when
00
00
playing π and such that π has probability one of the incentive constraint being
binding at the nth transaction, where n can be any positive integer.
In the sixteenth paragraph in the proof, note that the strategy profile π ∗∗ is suchthat πi∗∗ (ht ) ≥ gt for any history ht = {gτ }τ ∈[0,t] , {(b1τ , b2τ )}τ ∈[0,t) up to an arbitrary
time t for which πi∗∗ (ht ) > 0. Similarly, the strategy profile ψ ∗∗ in the twenty-firstparagraph satisfies ψi∗∗ (ht ) ≥ gt for any history ht = {gτ }τ ∈[0,t] , {(b1τ , b2τ )}τ ∈[0,t) up to00
an arbitrary time t for which ψi∗∗ (ht ) > 0. Given the procedure used to construct π , it
Pk
P
00
00
i
follows that ∞n=1 c({c̃τ }τ ∈[0,∞) , π , n) for any cost
n=k+1 φ ({c̃τ }τ ∈[0,∞) , π , n) ≤ q −00
realization {c̃τ }τ ∈[0,∞) and any positive integer k, where we define φi ({c̃τ }τ ∈[0,∞) , π , n)0000
= 0 and c({c̃τ }τ ∈[0,∞) , π , n) = 0 if tn ({c̃τ }τ ∈[0,∞) , π ) = ∞.
0
We finally explain how to construct a stationary symmetric SPE π ∈ Π∗ in grim0
trigger strategies with V (h0 , π ) ≥ V (h0 , π) for which the cost incurred between any
00
two consecutive transactions is decreasing. Recall the strategy profile π constructed000
above. To simplify the exposition, it is helpful to define the symmetric SPE π000
in grim-trigger strategies as follows. With probability one, φit (h0 , {cτ }τ ∈(0,∞) , π ) =00000
φit (h0 , {cτ }τ ∈(0,∞) , π ) for all t ≥ 0. Moreover, the strategy profile π satisfies φit1 (h0 ,000000{c̃1τ }τ ∈(0,∞) , π ) = φit2 (h0 , {c̃2τ }τ ∈(0,∞) , π ) for any pair of cost realizations {c̃1τ }τ ∈[0,∞)and {c̃2τ }τ ∈[0,∞) along with times t1 ≥ 0 and t2 ≥ 0 such that c̃1t1 = c̃2t2 and σ̂ti,i1 ({c̃1τ }τ ∈[0,∞) ,
000
000
000
000
π ) = σ̂ti,i2 ({c̃2τ }τ ∈[0,∞) , π ) and such that c({c̃1τ }τ ∈[0,∞) , π , k) = c({c̃2τ }τ ∈[0,∞) , π , k)000000
and σ i,i ({c̃1τ }τ ∈[0,∞) , π , k) = σ i,i ({c̃2τ }τ ∈[0,∞) , π , k) for any integer k ≥ 0 satisfying
000000
both tk ({c̃1τ }τ ∈[0,∞) , π ) < t1 and tk ({c̃2τ }τ ∈[0,∞) , π ) < t2 .
54

Consider any cost realization {c̃τ }τ ∈[0,t] up to an arbitrary time t. Let R({c̃τ }τ ∈[0,t] )denote the set consisting of every cost level c such that the following holds. Given that{c̃τ }τ ∈[0,t] is the realization of the cost process up to time t, there is positive conditional
probability that the cost realization {c̃τ }τ ∈(t,∞) after time t is such that there exists
0000u > t with φiu (h0 , {c̃τ }τ ∈(0,∞) , π ) > 0 and c̃u = c. The strategy profile π is defined000
as follows. If π is such that there is zero probability of a transaction occurring,
0
000
000
then simply let π = π . In the case where π is such that a transaction occurs with0
positive probability, the strategy profile π is constructed using the procedure below.
0
The first transaction when playing strategy profile π is specified as follows. Let r1
denote the supremum of the set R(c0 ). If r1 ∈ R(c0 ), then let {c̃1τ }τ ∈[0,∞) be any realization of the cost process for which there exists a time t̃1 such that φit (h0 , {c̃1τ }τ ∈(0,∞) ,000π ) > 0 and c̃1t = r1 for t = t̃1 and such that every element of the set R({c̃1τ }τ ∈[0,t̃1 ] )0is less than r1 . For the case where r1 ∈ R(c0 ), the strategy profile π requires eachagent to make a transaction at the first time that the cost reaches r1 , and each agent
000
should transfer the amount q − σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π ) on this transaction.
If r1 ∈/ R(c0 ), then let g̃1 denote the unique cost level g for which there is positive
probability that the cost realization {c̃τ }τ ∈[0,∞) is such that there exists a time t000000
satisfying φit (h0 , {c̃τ }τ ∈(0,∞) , π ) > 0, c̃t = g, and φiu (h0 , {c̃τ }τ ∈(0,∞) , π ) = 0 for allu ≤ t. Choose any cost level r̃1 ∈ [g̃1 , r1 ) for which there exists a cost realization000{c̃1τ }τ ∈[0,∞) along with a time t̃1 such that φit (h0 , {c̃1τ }τ ∈(0,∞) , π ) > 0 and c̃1t = r̃1 fort = t̃1 and such that every element of the set R({c̃1τ }τ ∈[0,t̃1 ] ) is less than r̃1 . For the case0
where r1 ∈/ R(c0 ), the strategy profile π requires each agent to make a transaction
at the first time that the cost reaches r̃1 , and each agent should transfer the amount000
q − σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π ) on this transaction.
Pi
In each case, such a choice of {c̃1τ }τ ∈[0,∞) and t̃1 is possible because ∞n=k+1 φ ({c̃τ
P
000
000
}τ ∈[0,∞) , π , n) ≤ q− kn=1 c({c̃τ }τ ∈[0,∞) , π , n) for any cost realization {c̃τ }τ ∈[0,∞) and000
000
any positive integer k, where φi ({c̃τ }τ ∈[0,∞) , π , n) = 0 and c({c̃τ }τ ∈[0,∞) , π , n) = 0 if
000
tn ({c̃τ }τ ∈[0,∞) , π ) = ∞. Otherwise, if no such {c̃1τ }τ ∈[0,∞) and t̃1 existed, then there
P000
iwould exist k such that ∞n=k+1 φ ({c̃τ }τ ∈[0,∞) , π , n) < 0 for some {c̃τ }τ ∈[0,∞) , which
is impossible.
0
The second transaction when playing strategy profile π is specified as follows.
Let r2 denote the supremum of the set R({c̃1τ }τ ∈[0,t̃1 ] ). If r2 ∈ R({c̃1τ }τ ∈[0,t̃1 ] ), thenlet {c̃2τ }τ ∈[0,t̃2 ] be any realization of the cost process with {c̃2τ }τ ∈[0,t̃1 ] = {c̃1τ }τ ∈[0,t̃1 ] for000
which there exists a time t̃2 > t̃1 such that φit (h0 , {c̃2τ }τ ∈(0,∞) , π ) > 0 and c̃2t = r2 for55

t = t̃2 and such that every element of the set R({c̃2τ }τ ∈[0,t̃2 ] ) is less than r2 . For the0case where r2 ∈ R({c̃1τ }τ ∈[0,t̃1 ] ), the strategy profile π requires each agent to make atransaction at the first time that the cost reaches r2 , and each agent should transfer
000000
the amount σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π ) − σt̃i,i2 ({c̃2τ }τ ∈[0,∞) , π ) on this transaction.
If r2 ∈
/ R({c̃1τ }τ ∈[0,t̃1 ] ), then let g̃2 denote the unique cost level g such that thefollowing holds. Given that the realization of the cost process {c̃τ }τ ∈[0,t̃1 ) up to timet̃1 is such that {c̃τ }τ ∈[0,t̃1 ) = {c̃1τ }τ ∈[0,t̃1 ) , there is positive conditional probability thatthe cost realization {c̃τ }τ ∈(t̃1 ,∞) after time t̃1 is such that there exists a time t sat000000
isfying φit (h0 , {c̃τ }τ ∈(0,∞) , π ) > 0, c̃t = g, and φiu (h0 , {c̃τ }τ ∈(0,∞) , π ) = 0 for allu ∈ (t̃1 , t). Choose any cost level r̃2 ∈ [g̃2 , r2 ) for which there exists a cost realization {c̃2τ }τ ∈[0,∞) along with a time t̃2 > t̃1 such that {c̃2τ }τ ∈[0,t̃1 ] = {c̃1τ }τ ∈[0,t̃1 ] ,000such that φit (h0 , {c̃2τ }τ ∈(0,∞) , π ) > 0 and c̃2t = r̃2 for t = t̃2 , and such that every element of the set R({c̃2τ }τ ∈[0,t̃2 ] ) is less than r̃2 . For the case where r2 ∈/01
R({c̃τ }τ ∈[0,t̃1 ] ), the strategy profile π requires each agent to make a transaction at
the first time that the cost reaches r̃2 , and each agent should transfer the amount
000
000
σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π ) − σt̃i,i2 ({c̃2τ }τ ∈[0,∞) , π ) on this transaction. In each case, such a
choice of {c̃2τ }τ ∈[0,∞) and t̃2 is possible because of the property described above.
0
Proceeding in this way, the k th transaction when playing strategy profile π is}τ ∈[0,t̃k−1 ] ). If
specified as follows. Let rk denote the supremum of the set R({c̃k−1
τkk−1
rk ∈ R({c̃τ }τ ∈[0,t̃k−1 ] ), then let {c̃τ }τ ∈[0,t̃k ] be any realization of the cost processwith {c̃kτ }τ ∈[0,t̃k−1 ] = {c̃k−1}τ ∈[0,t̃k−1 ] for which there exists a time t̃k > t̃k−1 such that
τ
000
i
k
φt (h0 , {c̃τ }τ ∈(0,∞) , π ) > 0 and c̃kt = rk for t = t̃k and such that every element of}τ ∈[0,t̃k−1 ] ), thethe set R({c̃kτ }τ ∈[0,t̃k ] ) is less than rk . For the case where rk ∈ R({c̃k−1
τ0strategy profile π requires each agent to make a transaction at the first time that the000
cost reaches rk , and each agent should transfer the amount σt̃i,i ({c̃k−1}τ ∈[0,∞) , π ) −
τ
k−1
000
σt̃i,i ({c̃kτ }τ ∈[0,∞) , π ) on this transaction.
k
If rk ∈
/ R({c̃k−1}τ ∈[0,t̃k−1 ] ), then let g̃k denote the unique cost level g such that theτfollowing holds. Given that the realization of the cost process {c̃τ }τ ∈[0,t̃k−1 ) up to timet̃k−1 is such that {c̃τ }τ ∈[0,t̃k−1 ) = {c̃k−1}τ ∈[0,t̃k−1 ) , there is positive conditional probaτbility that the cost realization {c̃τ }τ ∈(t̃k−1 ,∞) after time t̃k−1 is such that there exists
000000
a time t satisfying φit (h0 , {c̃τ }τ ∈(0,∞) , π ) > 0, c̃t = g, and φiu (h0 , {c̃τ }τ ∈(0,∞) , π ) = 0
for all u ∈ (t̃k−1 , t). Choose any cost level r̃k ∈ [g̃k , rk ) for which there exists a
cost realization {c̃kτ }τ ∈[0,∞) along with a time t̃k > t̃k−1 such that {c̃kτ }τ ∈[0,t̃k−1 ] =000{c̃k−1}τ ∈[0,t̃k−1 ] , such that φit (h0 , {c̃kτ }τ ∈(0,∞) , π ) > 0 and c̃kt = r̃k for t = t̃k , andτ
56such that every element of the set R({c̃kτ }τ ∈[0,t̃k ] ) is less than r̃k . For the case where0rk ∈/ R({c̃τk−1 }τ ∈[0,t̃k−1 ] ), the strategy profile π requires each agent to make a transaction at the first time that the cost reaches r̃k , and each agent should transfer the000000}τ ∈[0,∞) , π ) − σt̃i,i ({c̃kτ }τ ∈[0,∞) , π ) on this transaction. In eachamount σt̃i,i ({c̃k−1
τkk−1
case, such a choice of {c̃kτ }τ ∈[0,∞) and t̃k is possible because of the property describedabove.
The agents do not make any transfers following a deviation from the path of play0
described above. It is straightforward to confirm that π is a stationary symmetric0
SPE in grim-trigger strategies with V (h0 , π ) ≥ V (h0 , π) for which the cost incurred
between any two consecutive transactions is decreasing.
The next result shows that given a stationary symmetric SPE π for which there
is positive probability of the incentive constraint being slack at some transaction,0
one can find a stationary symmetric SPE π in which each agent receives a higher
expected payoff and there is probability one of the incentive constraint being bindingat every transaction.
Lemma 2. Given a stationary symmetric SPE π ∈ Π∗ in grim-trigger strategies forwhich there is a positive probability of the incentive constraint being slack at some0transaction, there exists a stationary symmetric SPE π ∈ Π∗ in grim-trigger strate0
gies with V (h0 , π ) > V (h0 , π) for which there is probability one of the incentive
constraint being binding at every transaction.
Proof. Consider any stationary symmetric SPE π ∈ Π∗ in grim-trigger strategies forwhich there is a positive probability of the incentive constraint being slack at sometransaction. We first explain how to construct a symmetric SPE π̃ ∈ Π∗ in grimtrigger strategies such that V (h0 , π̃) > V (h0 , π).
It is helpful to introduce some notation. Let t̂1 and t̂2 be any two nonnegative real
numbers such that t̂1 ≤ t̂2 . Let π̂ ∈ Π∗ be a symmetric SPE in grim-trigger strategies.
Given a realization {c̃τ }τ ∈[0,∞) of the cost process, let Σ[ht̂1 ({c̃τ }τ ∈[0,∞) , π̂); {c̃τ }τ ∈[t̂1 ,t̂2 ] ;t̂1 , t̂2 ; π̂] be the sum of the transfers that an agent would make between times t̂1 andt̂2 inclusive if ht̂1 ({c̃τ }τ ∈[0,∞) , π̂) is the history up to time t̂1 , the cost path {c̃τ }τ ∈[t̂1 ,t̂2 ]is realized between times t̂1 and t̂2 , and the strategy profile π̂ is played by the agents.
The following is how π̃ is defined in this case. The agents play according to strategy profile π until the first time t for which the realization {c̃τ }τ ∈[0,∞) of the cost
process is such that φit (h0 , {c̃τ }τ ∈(0,∞) , π) > 0 and Y [ht ({c̃τ }τ ∈[0,∞) , π), π] > c̃t . If no57

such time t exists, then the agents simply follow strategy profile π. Otherwise, lett̃1 denote the first time t that this condition holds. At time t̃1 , each agent transfers
the amount φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π) + ζ[ht ({c̃τ }τ ∈[0,∞) , π)], where ζ[ht ({c̃τ }τ ∈[0,∞) , π)]
is a positive real number no greater than σt̃i,i1 ({c̃τ }τ ∈[0,∞) , π) that can depend onht ({c̃τ }τ ∈[0,∞) , π). After time t̃1 , the agents do not make any transfers at any timet > t̃1 such that the realization of the cost process {c̃τ }τ ∈[t̃1 ,t] between times t̃1
and t satisfies φit (h0 , {c̃τ }τ ∈(0,∞) , π) = 0 or Σ[ht̃1 ({c̃τ }τ ∈[0,∞) , π); {c̃τ }τ ∈[t̃1 ,t] ; t̃1 , t; π] −φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π) − ζ[ht ({c̃τ }τ ∈[0,∞) , π)] < c̃t .
Let t̃2 denote the first time t such that φit (h0 , {c̃τ }τ ∈(0,∞) , π) > 0 and Σ[ht̃1 ({c̃τ }τ ∈[0,∞) ,π); {c̃τ }τ ∈[t̃1 ,t] ; t̃1 , t; π] − φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π) − ζ[ht̃1 ({c̃τ }τ ∈[0,∞) , π)] ≥ c̃t . If sucha time t does not exist, then the agents do not make any further transactions.
Otherwise, proceed as follows. At time t̃2 , each agent transfers an amount equal
to the lesser of φit̃2 (h0 , {c̃τ }τ ∈(0,∞) , π) and Σ[ht̃1 ({c̃τ }τ ∈[0,∞) , π); {c̃τ }τ ∈[t̃1 ,t̃2 ] ; t̃1 , t̃2 ; π] −φit̃1 (h0 , {c̃τ }τ ∈(0,∞) , π) − ζ[ht̃1 ({c̃τ }τ ∈[0,∞) , π)]. Thereafter, the agents play according tostrategy profile π, behaving as if the history up to time t̃1 were ht̃1 ({c̃τ }τ ∈[0,∞) , π), therealization of the cost process between times t̃1 and t̃2 were {c̃τ }τ ∈[t̃1 ,t̃2 ] , and strategy
profile π were followed up to and including time t̃2 . The agents do not make anytransfers following a deviation from the path of play specified above.
If each ζ[ht̃1 ({c̃τ }τ ∈[0,∞) , π)] > 0 is chosen to be sufficiently small, then π̃ is a symmetric SPE in grim-trigger strategies such that V (h0 , π̃) > V (h0 , π). Such a choiceof ζ[ht̃1 ({c̃τ }τ ∈[0,∞) , π)] is possible because V [ht̃1 ({c̃τ }τ ∈[0,∞) , π), π̃] is increasing inζ[ht̃1 ({c̃τ }τ ∈[0,∞) , π)], and Y [ht̃1 ({c̃τ }τ ∈[0,∞) , π), π̃] approaches Y [ht̃1 ({c̃τ }τ ∈[0,∞) , π), π]as ζ[ht̃1 ({c̃τ }τ ∈[0,∞) , π)] goes to 0.
Having constructed a symmetric SPE π̃ in grim-trigger strategies such that V (h0 , π̃)
˜ in grim> V (h0 , π), lemma 1 shows that there exists a stationary symmetric SPE π̃˜ ≥ V (h0 , π̃) for which the cost incurred between any
trigger strategies with V (h0 , π̃)
two consecutive transactions is decreasing. To simplify the exposition, it is helpful todefine the stationary symmetric SPE π ∗ in grim-trigger strategies as follows. With˜ for all t ≥ 0. Moreover,probability one, φit (h0 , {cτ }τ ∈(0,∞) , π ∗ ) = φit (h0 , {cτ }τ ∈(0,∞) , π̃)
the strategy profile π ∗ satisfies φit1 (h0 , {c̃1τ }τ ∈(0,∞) , π ∗ ) = φit2 (h0 , {c̃2τ }τ ∈(0,∞) , π ∗ ) for
any pair of cost realizations {c̃1τ }τ ∈[0,∞) and {c̃2τ }τ ∈[0,∞) along with times t1 ≥ 0 andt2 ≥ 0 such that c̃1t1 = c̃2t2 and σ̂ti,i1 ({c̃1τ }τ ∈[0,∞) , π ∗ ) = σ̂ti,i2 ({c̃2τ }τ ∈[0,∞) , π ∗ ). We next
0
explain how to construct a stationary symmetric SPE π ∈ Π∗ in grim-trigger strate0
gies with V (h0 , π ) = V (h0 , π ∗ ) for which there is probability one of the incentive
58

constraint being binding on every transaction and the cost incurred between any twoconsecutive transactions is decreasing.
It is helpful to introduce some terminology. Given a symmetric SPE π̂ ∈ Π∗ aswell as any positive integer k, let p̂ be the probability that tk−1 ({cτ }τ ∈[0,∞) , π̂) < ∞and there exists a time t with tk−1 ({cτ }τ ∈[0,∞) , π̂) < t < tk+1 ({cτ }τ ∈[0,∞) , π̂) for whichφit (h0 , {cτ }τ ∈(0,∞) , π̂) > 0 and Y [ht ({cτ }τ ∈[0,∞) , π̂), π̂] > ct . If p̂ > 0, then π̂ is said tohave a positive probability of the incentive constraint being slack at transaction k. Ifp̂ = 0, then π̂ is said to have probability one of the incentive constraint being bindingat transaction k.
Let the symmetric strategy profile π ∗∗ be defined as follows. If π ∗ is such that there
is probability one of the incentive constraint being binding at the first transaction,then simply define π ∗∗ = π ∗ . If π ∗ has a positive probability of the incentive constraint
being slack at the first transaction, then the strategy profile π ∗∗ is constructed asfollows. The agents play according to strategy profile π ∗ until the first time t for which
the realization {c̃τ }τ ∈[0,∞) of the cost process is such that φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) > 0
and Y [ht ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗ ] > c̃t . If no such time t exists, then the agents simply
follow strategy profile π ∗ . Otherwise, let t∗1 denote the first time t that this condition
holds. At time t∗1 , each agent transfers the amount φit∗1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) + ζ, wherei∗∗ζ is a positive real number no greater than si,i0 − φt∗1 (h0 , {c̃τ }τ ∈(0,∞) , π ). After time t1 ,the agents do not make any transfers at any time t > t∗1 such that the realization of
the cost process {c̃τ }τ ∈[t∗1 ,t] between times t∗1 and t satisfies φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) = 0
or Σ[ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ); {c̃τ }τ ∈[t∗1 ,t] ; t∗1 , t; π ∗ ] − φit∗1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) − ζ < c̃t .
Let t∗2 denote the first time t such that φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) > 0 and Σ[ht∗1 ({c̃τ }τ ∈[0,∞) ,π ∗ ); {c̃τ }τ ∈[t∗1 ,t] ; t∗1 , t; π ∗ ] − φit∗1 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) − ζ ≥ c̃t . If such a time t does
not exist, then the agents do not make any further transactions. Otherwise, proceed as follows. At time t∗2 , each agent transfers an amount equal to the lesser ofφit∗2 (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) and Σ[ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ); {c̃τ }τ ∈[t∗1 ,t∗2 ] ; t∗1 , t∗2 ; π ∗ ] − φit∗1 (h0 , {c̃τ}τ ∈(0,∞) , π ∗ ) − ζ. Thereafter, the agents play according to strategy profile π ∗ , behaving as if the history up to time t∗1 were ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ), the realization of thecost process between times t∗1 and t∗2 were {c̃τ }τ ∈[t∗1 ,t∗2 ] , and strategy profile π ∗ were
followed up to and including time t∗2 . The agents do not make any transfers following
a deviation from the path of play specified above.
If each ζ is chosen appropriately, then π ∗∗ is a symmetric SPE in grim-trigger
strategies with V (h0 , π ∗∗ ) > V (h0 , π ∗ ), and π ∗∗ has probability one of the incentive
59

constraint being binding at the first transaction. Such a choice of ζ is possible because
V [ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗∗ ] is increasing in ζ, Y [ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗∗ ] approaches
Y [ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗ ] as ζ goes to 0, Y [ht∗1 ({c̃τ }τ ∈[0,∞) , π ∗ ), π ∗∗ ] equals 0 for ζ =
i
∗
∗
∗∗
si,i
0 − φt∗1 (h0 , {c̃τ }τ ∈(0,∞) , π ), and Y [ht∗1 ({c̃τ }τ ∈[0,∞) , π ), π ] is a continuous functionof ζ.
We now observe that it is possible to construct a stationary symmetric SPE π ∗∗∗in grim-trigger strategies with V (h0 , π ∗∗∗ ) = V (h0 , π ∗ ) such that π ∗∗∗ has probability
one of the incentive constraint being binding at the first transaction and the cost
incurred between any two consecutive transactions is decreasing. If there is zeroprobability of a transaction occurring when playing π ∗∗ , then simply let π ∗∗∗ = π ∗∗ .
If there is a positive probability of a transaction occurring when playing π ∗∗ , then asymmetric SPE π ∗∗∗ satisfying the prescribed conditions can be constructed from π ∗∗by appropriately lowering or leaving unchanged the amount of each good transferredat the first transaction.
Let π 1,∗∗∗ = π ∗∗∗ . Choose any positive integer k. Assume that there exists a stationary symmetric SPE π k,∗∗∗ in grim-trigger strategies with V (h0 , π k,∗∗∗ ) = V (h0 , π ∗ )such that the cost incurred between any two consecutive transactions is decreasing andsuch that π k,∗∗∗ has probability one of the incentive constraint being binding at thenth transaction, where n can be any positive integer no greater than k. Note that thisproperty holds for k = 1. It will be shown below that there exists a stationary symmetric SPE π k+1,∗∗∗ in grim-trigger strategies with V (h0 , π k+1,∗∗∗ ) = V (h0 , π ∗ ) such thatthe cost incurred between any two consecutive transactions is decreasing and such that
π k+1,∗∗∗ has probability one of the incentive constraint being binding at the nth transaction, where n can be any positive integer no greater than k + 1. Moreover, π k+1,∗∗∗can be constructed such that φit (h0 , {cτ }τ ∈(0,∞) , π k+1,∗∗∗ ) = φit (h0 , {cτ }τ ∈(0,∞) , π k,∗∗∗ )for all t ≤ tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ). It will then follow that there exists a stationary
00symmetric SPE π in grim-trigger strategies with V (h0 , π ) > V (h0 , π) such that the0
cost incurred between any two consecutive transactions is decreasing and such that π
has probability one of the incentive constraint being binding at the nth transaction,
where n can be any positive integer.
Given the strategy profile π k,∗∗∗ described in the preceding paragraph, a strategy
profile π k+1,∗∗∗ with the aforementioned properties can be constructed by applying
the procedure above to each subgame on the equilibrium path immediately following
the first transaction when playing π k,∗∗∗ . In particular, the strategy profile π k+1,∗∗∗60

is derived as follows. Either tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ) = ∞ with probability one, ortk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ) < ∞ with positive probability. If tk ({cτ }τ ∈[0,∞) , π k,∗∗∗ ) = ∞with probability one, then simply let π k+1,∗∗∗ = π k,∗∗∗ . Otherwise, proceed as below.
Let {g̃τ }τ ∈[0,∞) be any realization of the cost process for which there exists a timet̃ < ∞ such that tk ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ) = t̃.
Consider a modified game in which the initial value of the cost process is g̃t̃ insteadof c0 and the original amount of each good is σt̃i,i ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ) instead of q. Let
h00 denote the null history in this modified game. Define the symmetric SPE ψ ∗ ingrim-trigger strategies as follows. At time 0, the agents do not make any transactions.
After time 0, the agents play according to strategy profile π k,∗∗∗ ∈ Π∗ , behaving as if
the history at time 0 were ht̃ ({g̃τ }τ ∈[0,∞) , π k,∗∗∗ ). The agents do not make any transfers
following a deviation from the path of play described above. A procedure similar
to that described above can be used to construct a stationary symmetric strategy
profile ψ ∗∗∗ in grim-trigger strategies with V (h00 , ψ ∗∗∗ ) = V (h00 , ψ ∗ ) such that ψ ∗∗∗ hasprobability one of the incentive constraint being binding at the first transaction andthe cost incurred between any two consecutive transactions is decreasing.
Now define the strategy profile π ∗∗∗,k+1 as follows. The agents play according tostrategy profile π ∗∗∗,k until k transactions have occurred. Let t∗∗∗,k denote the time of
the k th transaction if such a time exists, and proceed as follows in this case. After timet∗∗∗,k , the agents play according to strategy profile ψ ∗∗∗ , behaving as if the history attime t∗∗∗,k were h00 and strategy profile ψ ∗∗∗ had always been followed. The agents do
not make any transfers following a deviation from the path of play described above.
0
As previously explained, it follows that there exists a stationary symmetric SPE π in0grim-trigger strategies with V (h0 , π ) = V (h0 , π ∗ ) such that the cost incurred between
0
any two consecutive transactions is decreasing and such that π has probability oneof the incentive constraint being binding at the nth transaction, where n can be any
positive integer.
The next result shows that given a stationary symmetric SPE π for which there
is probability one of the incentive constraint being binding at every transaction andthere is a positive probability of the cost incurred being nondecreasing between some0
two consecutive transactions, one can find a stationary symmetric SPE π in which
each agent receives a higher expected payoff, there is probability one of the incentive
constraint being binding at every transaction, and there is probability one of the cost
incurred being decreasing between any two consecutive transactions.
61

Lemma 3. Given a stationary symmetric SPE π ∈ Π∗ in grim-trigger strategies
for which there is probability one of the incentive constraint being binding at everytransaction and there is a positive probability of the cost incurred being nondecreasing
between some two consecutive transactions, there exists a stationary symmetric SPE00π ∈ Π∗ in grim-trigger strategies with V (h0 , π ) > V (h0 , π) for which there is probability one of the incentive constraint being binding at every transaction and there
is probability one of the cost incurred being decreasing between any two consecutive
transactions.
Proof. Let π ∈ Π∗ be any stationary symmetric SPE in grim-trigger strategies forwhich there is probability one of the incentive constraint being binding at everytransaction and there is a positive probability of the cost incurred being nondecreasing
between some two consecutive transactions. We start by explaining how to constructa stationary symmetric SPE π ∗ ∈ Π∗ in grim-trigger strategies with V (h0 , π ∗ ) >V (h0 , π) for which the cost incurred between any two consecutive transactions isdecreasing.
Suppose first that there is positive probability that the cost realization {c̃t }t∈[0,∞)is such that there exists a time t for which 0 < φit (h0 , {c̃τ }τ ∈(0,∞) , π) < c̃t . In thiscase, define the symmetric SPE π 0 in grim-trigger strategies as follows. For any
history ku = {gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) up to an arbitrary time u, let πi0 (ku ) = πi (ku )if πi (ku ) ≥ gu , and let πi0 (ku ) = 0 if πi (ku ) < gu , where i ∈ {1, 2}. Note thatV (h0 , π 0 ) > V (h0 , π). In addition, lemma 1 implies that there exists a stationary
symmetric SPE π ∗ in grim-trigger strategies with V (h0 , π ∗ ) ≥ V (h0 , π 0 ) for which thecost incurred between any two consecutive transactions is decreasing.
Suppose next that there is zero probability that the cost realization {c̃t }t∈[0,∞) issuch that there exists a time t for which 0 < φit (h0 , {c̃τ }τ ∈(0,∞) , π) < c̃t . To simplify
the exposition, it is helpful to define the stationary symmetric SPE π̃ in grim-trigger
strategies as follows. With probability one, φit (h0 , {cτ }τ ∈(0,∞) , π̃) = φit (h0 , {cτ }τ ∈(0,∞) , π)
for all t ≥ 0. Moreover, the strategy profile π̃ satisfies φit1 (h0 , {c̃1τ }τ ∈(0,∞) , π̃) =φit2 (h0 , {c̃2τ }τ ∈(0,∞) , π̃) for any pair of cost realizations {c̃1τ }τ ∈[0,∞) and {c̃2τ }τ ∈[0,∞) alongwith times t1 ≥ 0 and t2 ≥ 0 such that c̃1t1 = c̃2t2 and σ̂ti,i1 ({c̃1τ }τ ∈[0,∞) , π̃) = σ̂ti,i2 ({c̃2τ }τ ∈[0,∞) ,P
Pkiπ̃). Note that π̃ satisfies the condition ∞n=k+1 φ ({c̃τ }τ ∈[0,∞) , π̃, n) ≤ q −n=1 c({c̃τ}τ ∈[0,∞) , π̃, n) for any cost realization {c̃τ }τ ∈[0,∞) and any positive integer k, where we
define φi ({c̃τ }τ ∈[0,∞) , π̃, n) = 0 and c({c̃τ }τ ∈[0,∞) , π̃, n) = 0 if tn ({c̃τ }τ ∈[0,∞) , π̃) = ∞.
In this case, the strategy profile π ∗ is constructed using the procedure below.
62

Consider any cost realization {c̃τ }τ ∈[0,t] up to an arbitrary time t. Let R({c̃τ }τ ∈[0,t] )denote the set consisting of every cost level c such that the following holds. Given
that {c̃τ }τ ∈[0,t] is the realization of the cost process up to time t, there is positive
conditional probability that the cost realization {c̃τ }τ ∈(t,∞) after time t is such that
there exists u > t with φiu (h0 , {c̃τ }τ ∈(0,∞) , π̃) > 0 and c̃u = c.
The first transaction when playing strategy profile π ∗ is specified as follows.
Let r1 denote the supremum of the set R(c0 ). If r1 ∈ R(c0 ), then let {c̃1τ }τ ∈[0,∞)be any realization of the cost process for which there exists a time t̃1 such thatφit (h0 , {c̃1τ }τ ∈(0,∞) , π̃) > 0 and c̃1t = r1 for t = t̃1 and such that every element of the set
R({c̃1τ }τ ∈[0,t̃1 ] ) is less than r1 . For the case where r1 ∈ R(c0 ), the strategy profile π ∗requires each agent to make a transaction at the first time that the cost reaches r1 ,and each agent should transfer the amount q − σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π̃) on this transaction.
If r1 ∈/ R(c0 ), then let g̃1 denote the unique cost level g for which there is positive
probability that the cost realization {c̃τ }τ ∈[0,∞) is such that there exists a time tsatisfying φit (h0 , {c̃τ }τ ∈(0,∞) , π̃) > 0, c̃t = g, and φiu (h0 , {c̃τ }τ ∈(0,∞) , π̃) = 0 for allu ≤ t. Choose any cost level r̃1 ∈ [g̃1 , r1 ) for which there exists a cost realization{c̃1τ }τ ∈[0,∞) along with a time t̃1 such that φit (h0 , {c̃1τ }τ ∈(0,∞) , π̃) > 0 and c̃1t = r̃1 fort = t̃1 and such that every element of the set R({c̃1τ }τ ∈[0,t̃1 ] ) is less than r̃1 . For the casewhere r1 ∈/ R(c0 ), the strategy profile π ∗ requires each agent to make a transaction
at the first time that the cost reaches r̃1 , and each agent should transfer the amountq − σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π̃) on this transaction.
Pi
In each case, such a choice of {c̃1τ }τ ∈[0,∞) and t̃1 is possible because ∞n=k+1 φ ({c̃τPk}τ ∈[0,∞) , π̃, n) ≤ q − n=1 c({c̃τ }τ ∈[0,∞) , π̃, n) for any cost realization {c̃τ }τ ∈[0,∞) andany positive integer k, where φi ({c̃τ }τ ∈[0,∞) , π̃, n) = 0 and c({c̃τ }τ ∈[0,∞) , π̃, n) = 0 if
tn ({c̃τ }τ ∈[0,∞) , π̃) = ∞. Otherwise, if no such {c̃1τ }τ ∈[0,∞) and t̃1 existed, then therePiwould exist k such that ∞n=k+1 φ ({c̃τ }τ ∈[0,∞) , π̃, n) < 0 for some {c̃τ }τ ∈[0,∞) , which
is impossible.
The second transaction when playing strategy profile π ∗ is specified as follows.
Let r2 denote the supremum of the set R({c̃1τ }τ ∈[0,t̃1 ] ). If r2 ∈ R({c̃1τ }τ ∈[0,t̃1 ] ), thenlet {c̃2τ }τ ∈[0,t̃2 ] be any realization of the cost process with {c̃2τ }τ ∈[0,t̃1 ] = {c̃1τ }τ ∈[0,t̃1 ] forwhich there exists a time t̃2 > t̃1 such that φit (h0 , {c̃2τ }τ ∈(0,∞) , π̃) > 0 and c̃2t = r2 fort = t̃2 and such that every element of the set R({c̃2τ }τ ∈[0,t̃2 ] ) is less than r2 . For thecase where r2 ∈ R({c̃1τ }τ ∈[0,t̃1 ] ), the strategy profile π ∗ requires each agent to make atransaction at the first time that the cost reaches r2 , and each agent should transfer
63the amount σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π̃) − σt̃i,i2 ({c̃2τ }τ ∈[0,∞) , π̃) on this transaction.
If r2 ∈
/ R({c̃1τ }τ ∈[0,t̃1 ] ), then let g̃2 denote the unique cost level g such that the following holds. Given that the realization of the cost process {c̃τ }τ ∈[0,t̃1 ) up to timet̃1 is such that {c̃τ }τ ∈[0,t̃1 ) = {c̃1τ }τ ∈[0,t̃1 ) , there is positive conditional probability
that the cost realization {c̃τ }τ ∈(t̃1 ,∞) after time t̃1 is such that there exists a time
t satisfying φit (h0 , {c̃τ }τ ∈(0,∞) , π̃) > 0, c̃t = g, and φiu (h0 , {c̃τ }τ ∈(0,∞) , π̃) = 0 for allu ∈ (t̃1 , t). Choose any cost level r̃2 ∈ [g̃2 , r2 ) for which there exists a cost realization {c̃2τ }τ ∈[0,∞) along with a time t̃2 > t̃1 such that {c̃2τ }τ ∈[0,t̃1 ] = {c̃1τ }τ ∈[0,t̃1 ] , such thatφit (h0 , {c̃2τ }τ ∈(0,∞) , π̃) > 0 and c̃2t = r̃2 for t = t̃2 , and such that every element of the set
R({c̃2τ }τ ∈[0,t̃2 ] ) is less than r̃2 . For the case where r2 ∈/ R({c̃1τ }τ ∈[0,t̃1 ] ), the strategy profile π ∗ requires each agent to make a transaction at the first time that the cost reachesr̃2 , and each agent should transfer the amount σt̃i,i1 ({c̃1τ }τ ∈[0,∞) , π̃) − σt̃i,i2 ({c̃2τ }τ ∈[0,∞) , π̃)on this transaction. In each case, such a choice of {c̃2τ }τ ∈[0,∞) and t̃2 is possible becauseof the property described above.
Proceeding in this way, the k th transaction when playing strategy profile π ∗ is
specified as follows. Let rk denote the supremum of the set R({c̃k−1}τ ∈[0,t̃k−1 ] ). Ifτ}τ ∈[0,t̃k−1 ] ), then let {c̃kτ }τ ∈[0,t̃k ] be any realization of the cost processrk ∈ R({c̃k−1τ}τ ∈[0,t̃k−1 ] for which there exists a time t̃k > t̃k−1 such thatwith {c̃kτ }τ ∈[0,t̃k−1 ] = {c̃k−1τφit (h0 , {c̃kτ }τ ∈(0,∞) , π̃) > 0 and c̃kt = rk for t = t̃k and such that every element of theset R({c̃kτ }τ ∈[0,t̃k ] ) is less than rk . For the case where rk ∈ R({c̃k−1}τ ∈[0,t̃k−1 ] ), theτ0strategy profile π requires each agent to make a transaction at the first time that the}τ ∈[0,∞) , π̃) −cost reaches rk , and each agent should transfer the amount σt̃i,i ({c̃k−1
τ
k−1
i,i
σt̃ ({c̃kτ }τ ∈[0,∞) , π̃) on this transaction.
k
If rk ∈
/ R({c̃k−1}τ ∈[0,t̃k−1 ] ), then let g̃k denote the unique cost level g such thatτ
the following holds. Given that the realization of the cost process {c̃τ }τ ∈[0,t̃k−1 ) up
to time t̃k−1 is such that {c̃τ }τ ∈[0,t̃k−1 ) = {c̃τk−1 }τ ∈[0,t̃k−1 ) , there is positive conditional
probability that the cost realization {c̃τ }τ ∈(t̃k−1 ,∞) after time t̃k−1 is such that there
exists a time t satisfying φit (h0 , {c̃τ }τ ∈(0,∞) , π̃) > 0, c̃t = g, and φiu (h0 , {c̃τ }τ ∈(0,∞) , π̃) =0 for all u ∈ (t̃k−1 , t). Choose any cost level r̃k ∈ [g̃k , rk ) for which there exists
a cost realization {c̃kτ }τ ∈[0,∞) along with a time t̃k > t̃k−1 such that {c̃kτ }τ ∈[0,t̃k−1 ] ={c̃k−1}τ ∈[0,t̃k−1 ] , such that φit (h0 , {c̃kτ }τ ∈(0,∞) , π̃) > 0 and c̃kt = r̃k for t = t̃k , and suchτthat every element of the set R({c̃kτ }τ ∈[0,t̃k ] ) is less than r̃k . For the case where rk ∈/k−1∗
R({c̃τ }τ ∈[0,t̃k−1 ] ), the strategy profile π requires each agent to make a transaction
at the first time that the cost reaches r̃k , and each agent should transfer the amount64}τ ∈[0,∞) , π̃) − σt̃i,i ({c̃kτ }τ ∈[0,∞) , π̃) on this transaction. In each case, such aσt̃i,i ({c̃k−1
τkk−1
choice of {c̃kτ }τ ∈[0,∞) and t̃k is possible because of the property described above.
The agents do not make any transfers following a deviation from the path ofplay described above. Note that π ∗ is a stationary symmetric SPE in grim-trigger
strategies with V (h0 , π ∗ ) > V (h0 , π̃) for which the cost incurred between any twoconsecutive transactions is decreasing.
To understand why V (h0 , π ∗ ) > V (h0 , π̃), consider the following argument. Because there is a positive probability when playing π̃ that the cost incurred is nondecreasing between some two consecutive transactions, there exists a positive integer
m for which one can find a positive integer n greater than m such that there is apositive probability of the cost incurred at the nth transaction being at least as large
as the cost incurred at the mth transaction. Let m∗ denote the least such integer m.
If m∗ = 1, then the first transaction occurs at least as early when playing π ∗ as whenplaying π̃, and the amount transferred at the first transaction is larger when playing
π ∗ than when playing π̃. Noting that the incentive constraint at the first transaction
is binding when playing π̃, it follows that V (h0 , π ∗ ) > V (h0 , π̃) if m∗ = 1.
If m∗ > 1, then both π̃ and π ∗ induce the same path of play up to the (m∗ −1)th transaction. That is, tm−1 ({c̃τ }τ ∈[0,∞) , π̃) = tm−1 ({c̃τ }τ ∈[0,∞) , π ∗ ) for any cost
realization {c̃τ }τ ∈[0,∞) , and φit (h0 , {c̃τ }τ ∈(0,∞) , π̃) = φit (h0 , {c̃τ }τ ∈(0,∞) , π ∗ ) for any cost
realization {c̃τ }τ ∈[0,∞) along with any time t such that t ≤ tm−1 ({c̃τ }τ ∈[0,∞) , π̃) =tm−1 ({c̃τ }τ ∈[0,∞) , π ∗ ). Moreover, the m∗th transaction occurs at least as early whenplaying π ∗ as when playing π̃, and the amount transferred at the m∗th transaction islarger when playing π ∗ than when playing π̃. Noting that the incentive constraint at
the m∗th transaction is binding when playing π̃, it follows that V (h0 , π ∗ ) > V (h0 , π̃)
if m∗ > 1.
Finally, there may be a positive probability of the incentive constraint being slack
at some transaction when playing π ∗ . However, the procedure described in the lastnine paragraphs from the proof of lemma 2 can be applied to π ∗ to generate a station00
ary symmetric SPE π in grim-trigger strategies with V (h0 , π ) = V (h0 , π ∗ ) for which
there is probability one of the incentive constraint being binding at every transaction
and there is probability one of the cost incurred being decreasing between any twotransactions.

65

A.6
A.6.1

Proof of Corollaries to Theorem 3
Proof of Corollary 1

Proof. Suppose that the sequence {f˜k (π)}∞
k=1 does not converge to zero. Then there
Pn ˜
exists n such that k=1 fk (π) > q. However, this violates the requirement for the
strategies in π to be feasible. It follows that limk→∞ f˜k (π) = 0.
Now suppose that limk→∞ f˜k (π) = 0 but that the sequence {c̃k (π)}∞
does notP∞ ˜ k=1
converge to zero. For any  > 0, there exists n such that k=m fk (π) <  for allm ≥ n. Moreover, there exists η > 0 such that for any n, one can find m ≥ n forP
˜
which c̃m (π) > η. Hence, there exists l such that ∞k=l fk (π) < c̃l (π).
It follows that, with positive probability, the cost c̃l (π) is reached for the first time,
and the expected payoff to each agent is negative at that time. However, an agent
can secure an expected payoff of at least zero by transferring nothing at such a time.
This contradicts the fact that π is an SPE. It follows that limk→∞ c̃k (π) = 0.
A.6.2

Proof of Corollary 2P00
˜
Proof. Suppose that ∞k=1 fk (π) < q. Then there exists a symmetric SPE π ∈ Π000such that f˜k (π ) = f˜k (π) for k ≥ 2, c̃k (π ) = c̃k (π) for k ≥ 1, and f˜1 (π ) = f˜1 (π) +P0˜[q − ∞
k=1 fk (π)]. Note that V (h0 , π ) > V (h0 , π). This contradicts the fact that π isP˜a maximal symmetric SPE. It follows that ∞k=1 fk (π) = q.
A.7

Proof of Theorem 4

We first restrict attention to stationary maximal symmetric SPE in grim-trigger
0
strategies. In particular, we show that there exists a strategy profile π ∗ ∈ Π such0that V (h0 , π ∗ ) > V (h0 , π) for every π ∈ Π that does not almost surely induce thesame path of play as π ∗ . Moreover, we characterize the path of play induced by π ∗ .
We then show that there does not exist a non-stationary maximal symmetric SPE in
grim-trigger strategies.
0
Choose any strategy profile π ∈ Π for which there is positive probability of00
a transaction occurring. Let π ∈ Π denote the strategy profile in which the k thtransaction is made when the cost reaches c̃k (π) for the first time and the amountf˜k (π) is transferred by each agent at this transaction. Note that π almost surely
induces the same path of play as π 0 .

66Consider the expected payoff to each agent when both agents play strategy profile
π . Assume that each agent has previously made n ≥ 0 transactions and that thecurrent value of the cost is c ≥ c̃n+1 (π). Let Mn (c, π) be the expected payoff to each0agent if both agents play strategy profile π from the current time onwards. The valueMn (c, π) is defined as follows. Let {gt , (b1t , b2t )}t∈[0,∞) be any history that is consistent
with π10 and π20 at each t ∈ [0, ∞) and such that there exists u ≥ 0 for which gu = c
and the set {t ∈ [0, u) : b1t > 0 and b2t > 0} contains exactly n elements. Then
Mn (c, π) = V [({gt }t∈[0,u] , {(b1t , b2t )}t∈[0,u) ), π 0 ].
The sequence {c̃k (π), f˜k (π)}∞
k=1 characterizing π must have the following properties. Because there is probability one of the cost incurred being decreasing between
any two consecutive transactions, it must be that c̃k+1 (π) < c̃k (π) for all k ≥ 1. InP
˜order for π to be feasible for the agents to play, it must be that ∞k=1 fk (π) ≤ q.
Because there is probability one of the incentive constraint being binding at everytransaction, it must be that c̃k (π) = Mk [c̃k (π), π] for all k ≥ 1.
It follows that Mn (c, π) is simply the value of an asset that pays f˜n+1 (π) at the firsttime that the cost reaches c̃n+1 (π) when the current value of the cost is c ≥ c̃n+1 (π).
Calculating Mn (c, π) is a basic asset pricing problem.43 The Bellman equation forthis problem is:ρMn (c, π)dt = E(dMn ),(3)0with the boundary condition:
Mn [c̃n+1 (π), π] = f˜n+1 (π).

(4)

The result below describes the solution to this asset pricing problem under the assumption that the cost process follows a geometric Brownian motion.
Lemma 4. The unique solution to the Bellman equation (3) with the boundary condition (4) is:βc
Mn (c, π) = f˜n+1 (π),
c̃n+1 (π)
where
β = 12 − µ/σ 2 −q(µ/σ 2 − 21 )2 + 2ρ/σ 2 .

43

McDonald and Siegel (1986) perform a similar calculation under the assumption of geometric
Brownian motion when solving for the optimal timing of a single irreversible investment.

67

Proof. A straightforward application of Ito’s lemma to the right-hand side of equation(3) yields:∂Mn (c, π) 1 2 2 ∂ 2 Mn (c, π)ρMn (c, π) = µc
+ σ c
,
∂c
2
∂c2
which provides a second-order linear differential equation for Mn (c, π). The boundary
condition is Mn [c̃n+1 (π), π] = f˜n+1 (π). We seek a solution of the form g[c, c̃n+1 (π), f˜n+1(π)] = B[c̃n+1 (π), f˜n+1 (π)]cβ̃ . The following quadratic equation is obtained by substituting the functional form into the differential equation:
1 2
σ β̃(β̃ − 1) + µβ̃ − ρ = 0,2

whose solution is given by:β̃ = 12 − µ/σ 2 ±q(µ/σ 2 − 21 )2 + 2ρ/σ 2 .

Letting β̃ + and β̃ − respectively denote the positive and negative roots of the quadratic,+
the general solution to the differential equation is Mn (c, π) = B + [c̃n+1 (π), f˜n+1 (π)]cβ̃ +−B − [c̃n+1 (π), f˜n+1 (π)]cβ̃ . It must be the case that B + [c̃n+1 (π), f˜n+1 (π)] = 0, because Mn (c, π) would otherwise become unboundedly large in absolute value as cgoes to ∞. Moreover, the boundary condition Mn [c̃n+1 (π), π] = f˜n+1 (π) yields−
B − [c̃n+1 (π), f˜n+1 (π)] = f˜n+1 (π)/c̃n+1 (π)β̃ . Hence, the solution to the Bellman equation (3) is:
β̃ −

c
˜
.
Mn (c, π) = fn+1 (π)
c̃n+1 (π)
Denoting β = β̃ − gives the expression in the statement of the lemma. Uniquenessobtains because the solution to a second-order linear differential equation is uniquein general.
0

We search for a strategy profile π̂ ∈ Π such that V (h0 , π̂) ≥ V (h0 , π̃) for every0π̃ ∈ Π . The strategy profile π satisfies this condition if and only if the trading policy{c̃k (π), f˜k (π)}∞
k=1 solves the optimization problem:
β
∞
X
c0
max
, subject tof˜k (π) ≤ qc̃(π){c̃k (π),f˜k (π)}∞
1
k=1
k=1

β
c̃n (π)and c̃n (π) = f˜n+1 (π)
for all n ≥ 1,c̃n+1 (π)
f˜1 (π)



68

(5)

where c̃k (π) ∈ (0, c0 ) and f˜k (π) ∈ (0, q) for all k, and c̃k (π) is decreasing in k.
Our approach for solving the optimization problem above is as follows. We impose
the additional constraint that limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0. We then identify
the unique sequence that solves the maximization problem in expression (5) with thisadditional constraint. We finally argue that this sequence is the unique solution tothe original maximization problem without the extra constraint.
The result below describes how the maximization problem in (5) can be simplified
by applying an extra constraint.
Lemma 5. Given the additional constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0, thetrading policy {c̃k (π), f˜k (π)}∞
k=1 solves the maximization problem in expression (5) if
and only if it solves:
k−1
∞ 
∞
X
X
β
˜
f˜k (π) ≤ qmaxln[fk (π)], subject toβ−1{c̃k (π),f˜k (π)}∞
k=1 k=1
k=1


∞
j
X
β
and ln[c̃n (π)] = −(β − 1)−1ln[f˜n+j+1 (π)] for all n ≥ 1.
β
−
1
j=0

(6)

Proof. Note that each incentive compatibility constraint in expression (5) can be
rewritten as:
β ln[c̃n+1 (π)] − ln[f˜n+1 (π)]
ln[c̃n (π)] =.
β−1
Iterating the incentive compatibility constraint x times starting with transfer n, it
can be shown by induction that c̃n (π) must satisfy:ln[c̃n (π)] =

β
β−1

x

−1

ln[c̃x+n (π)] − (β − 1)

j
x−1 
X
β
ln[f˜n+j+1 (π)].
β
−
1
j=0

Assuming that limx→∞ [β/(β−1)]x ln[c̃x+n (π)] = 0, the term ln[c̃n (π)] can be expressed
as follows:
j
∞ 
X
β
−1
ln[c̃n (π)] = −(β − 1)ln[fn+j+1 (π)].
β−1j=0Substituting for ln[c̃1 (π)] after taking the logarithm of the maximand in expression(5), we obtain the optimization problem in expression (6), where the constant termβ ln(c0 ) is eliminated from the logarithm of the maximand.

69

The maximization problem in (6) is solved in three parts. The result below
provides the solution for the sequence of amounts transferred when the constraint
limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0 is not imposed.
Lemma 6. Dropping the additional constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0,
the unique sequence of amounts transferred that solves the maximization problem in
expression (6) is given by:
fk∗ =



q
1−β



β
β−1

k−1
for all k ≥ 1.

Proof. The sequence {f˜k (π)}∞
k=1 of amounts transferred solves the maximization problem in expression (6) if and only if it solves the simplified version of the problem below:
k−1
∞ 
∞
X
X
β
˜
max
f˜k (π) ≤ q.
ln[fk (π)], subject to
∞
˜
β
−
1
{fk (π)}k=1
k=1
k=1
The Lagrangian for the maximization problem in the preceding expression is given
by:
!
k−1
∞ 
∞
X
X
β
f˜k (π) ,L=
ln[f˜k (π)] + λ q −
β
−
1
k=1
k=1
which provides a first-order condition for each index k ≥ 1:1
f˜k (π) =
λ



β
β−1

k−1
.

Substitution of this result into the budget constraint yields:
∞

1X
q=
λ k=1



β
β−1

k−1
=

1−β
.
λ

Thus, we obtain λ = (1 − β)/q, leading to the following sequence of amounts transferred:

k−1
q
β
˜
fk (π) =for all k ≥ 1.
1−β β−1

Given the amount transferred at each transaction, the next result states the solu70

tion for the sequence of cost cutoffs when the constraint limx→∞ [β/(β−1)]x ln[c̃x+n (π)]= 0 is not imposed.
Lemma 7. Dropping the additional constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0,
the unique sequence of cost cutoffs that solves the maximization problem in expression(6) is given by:

k−β

β
q
∗
for all k ≥ 1.
ck =
1−β
β−1
Proof. Substituting the expression for f˜k (π) obtained in lemma 6 into the expression
for ln[c̃n (π)] stated in lemma 5, we have:j " 
j+n #∞ Xβqβln[c̃n (π)] = (1 − β)−1
ln
+ ln
β
−
1
1
−
β
β−1
j=0




q
β
+ (n − β) ln
,
= ln
1−β
β−1
where the standard formulas for geometric series and their derivatives provide:−1(1 − β)

j 



∞ 
X
q
q
β
ln
= ln
β
−
1
1
−
β
1−β
j=0

and
(1 − β)

−1


j 
j+n
X

j
∞
∞ 
X
β
β
β
β
−1
ln
= (1 − β) ln(j + n)
β
−
1
β
−
1
β
−
1
β−1
j=0
j=0


β
−1
= (1 − β) ln[n(1 − β) − β(1 − β)]
β−1


β
.
= (n − β) ln
β−1Thus, the sequence of cost cutoffs is given by:c̃k (π) =

q
1−β



β
β−1

k−β
for all k ≥ 1.

Using the solution for the sequence of cost cutoffs, we can confirm that the constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0 is satisfied, even though the constraint was71not imposed when solving the maximization problem.
Lemma 8. The solution {c∗k , fk∗ }∞k=1 in lemmata 6 and 7 satisfies the technical conx
dition limx→∞ [β/(β − 1)] ln[c̃x+n (π)] = 0.
Proof. Substituting the expression for {c∗k } from lemma 7 into the technical condition,
we have:xβlim
ln(c∗x+n )x→∞β−1
x  




β
q
β
= lim
ln
+ (x + n − β) ln
x→∞
β−1
1−β
β−1
x 


x



q
β
β
β
ln
+ lim
(x + n − β) ln
= lim
x→∞
x→∞
β−1
1−β
β−1
β−1

x


β
β
= lim
(x + n − β) ln
x→∞
β−1
β−1
= 0.

Hence, the sequence {c∗k , fk∗ }∞k=1 solves the maximization problem in expression(6) when the constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0 is imposed. The lemma
below implies that {c∗k , fk∗ }∞k=1 is the unique solution to the optimization problem in
expression (5).
Lemma 9. If {c̃k (π), f˜k (π)}∞
k=1 is the unique sequence that solves the maximization
problem in expression (6) with the constraint limx→∞ [β/(β −1)]x ln[c̃x+n (π)] = 0, then{c̃k (π), f˜k (π)}∞
k=1 is the unique solution to the maximization problem in expression (5).
Proof. In the proof of lemma 5, it was shown that any trading policy {c̃k (π), f˜k (π)}∞
k=1satisfying the constraints for the maximization problem in expression (5) must solve
the equation:ln[c̃n (π)] =

β
β−1

x

−1

ln[c̃x+n (π)] − (β − 1)

j
x−1 
X
β
ln[f˜n+j+1 (π)].
β
−
1
j=0

(7)

First, note that there must exist an index B > 1 such that f˜b (π) < 1 for all b ≥ B.
P
˜
Otherwise, if there were no such B, then the feasibility constraint ∞k=1 fk (π) ≤ q
72

would be violated. Second, it must be that c̃b (π) < 1 for all b ≥ B − 1. Noting that[c̃n (π)/c̃n+1 (π)]β < 1 for all n ≥ 1, this claim follows from the incentive compatibility
constraint c̃n (π) = f˜n+1 (π)[c̃n (π)/c̃n+1 (π)]β for all n ≥ 1 along with the fact thatf˜b (π) < 1 for all b ≥ B.
Because ln[f˜n+j+1 (π)] is negative for j sufficiently large, there exists X such thatthe sum
j
x−1 
X
β
−1
ln[f˜n+j+1 (π)]−(β − 1)β−1
j=0
is decreasing in x for x ≥ X. This fact along with equation (7) implies that [β/(β −1)]x ln[c̃x+n (π)] is increasing in x for x ≥ X. Because ln[c̃x+n (π)] is negative for x
sufficiently large, it follows from the monotone convergence theorem that the term[β/(β − 1)]x ln[c̃x+n (π)] converges to some number L ≤ 0 as x goes to infinity.
Now suppose that the additional constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0
is not imposed in the proof of lemma 5. We take the logarithm of the maximandof the optimization problem in expression (5), after which we substitute for ln[c̃1 (π)]
using equation (7) in the limit as x goes to infinity. Doing so and eliminating theconstant term β ln(c0 ) from the logarithm of the maximand, the optimization problem
becomes:
k−1
∞ 
X
β
ln[c̃x+n (π)] +ln[f˜k (π)],
max−β limx→∞β−1{c̃k (π),f˜k (π)}∞
k=1
k=1
β

∞
X
c̃
(π)
n
for all n ≥ 1.
subject tof˜k (π) ≤ q and c̃n = f˜n+1 (π)
c̃n+1 (π)
k=1


β
β−1

x

(8)

Choose any trading policy {c̃k (π), f˜k (π)}∞
k=1 that satisfies the constraints in the preceding expression. Consider the objective function in optimization problem (8). The
first term in this objective function cannot be greater than 0 when evaluated at{c̃k (π), f˜k (π)}∞
k=1 . Moreover, the second term in this objective function when evaluated at {c̃k (π), f˜k (π)}∞
k=1 cannot be greater than the maximum of the problem in
expression (6) when the additional constraint limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0 is
not imposed.
Recall that the solution {c∗k , fk∗ }∞k=1 in lemmata 6 and 7 is the unique trading
policy that achieves the maximum of problem (6) without the additional constraint
limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0. Moreover, the sequence {c∗k , fk∗ }∞k=1 satisfies the73technical condition limx→∞ [β/(β − 1)]x ln[c̃x+n (π)] = 0. Hence, {c∗k , fk∗ }∞k=1 achieves
the upper bounds for the first and second terms of the objective function in expression(8). In addition, {c∗k , fk∗ }∞k=1 satisfies all of the constraints in expression (8). It∗ ∞∗follows that {ck , fk }k=1 is the unique solution to maximization problem (8), which isequivalent to optimization problem (5).
It follows that a stationary maximal symmetric SPE in grim-trigger strategies
exists and that the sequence {c∗k , fk∗ }∞k=1 characterizes any such strategy profile.
We now show that there does not exist a non-stationary maximal symmetric SPE.
Hence, any maximal symmetric SPE induces the aforementioned path of play.
Lemma 10. Assume that the cost process {ct }t∈[0,∞) follows a geometric Brownian
motion with arbitrary drift µ and positive volatility σ. Then there does not exist a
non-stationary maximal symmetric SPE in grim-trigger strategies.
Proof. Let π ∗ be any stationary maximal symmetric SPE in grim-trigger strategies.
Recall that any stationary maximal symmetric SPE induces the path of play specified
in the statement of theorem 4. Suppose that π 0 is a non-stationary maximal symmetric
SPE in grim-trigger strategies. Let g be the greatest positive integer for which there
is probability one that the realization of the cost process {cτ }τ ∈[0,∞) is such thatφit (h0 , {cτ }τ ∈(0,∞) , π 0 ) = φit (h0 , {cτ }τ ∈(0,∞) , π ∗ ) for all t ≤ tg−1 ({cτ }τ ∈[0,∞) , π ∗ ). Suchan integer g must exist because π ∗ is a stationary SPE and π 0 is a non-stationary
SPE; so that, π ∗ and π 0 must induce different paths of play with positive probability.
Noting that the SPE π 0 is non-stationary, there must be positive probability that{cτ }τ ∈[0,∞) is such that tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞. Recall the definitions of c∗k and fk∗ inthe statement of theorem 4. Let c∗0 = c0 for definitional purposes. At least one of thefour claims in the following paragraph must be true.
First, there is positive probability that {cτ }τ ∈[0,∞) is such that tg ({cτ }τ ∈[0,∞) , π 0 ) <∞ and c({cτ }τ ∈[0,∞) , π 0 , g) > c∗g . Second, there is positive probability that {cτ }τ ∈[0,∞)is such that tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) < c∗g . Third, there
is probability one of {cτ }τ ∈[0,∞) being either such that tg ({cτ }τ ∈[0,∞) , π 0 ) = ∞ orsuch that tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) = c∗g , and there is positive probability of {cτ }τ ∈[0,∞) being such that there exists υ < tg ({cτ }τ ∈[0,∞) , π 0 ) for
which cυ = c∗g . Fourth, there is probability one of {cτ }τ ∈[0,∞) being either such thattg ({cτ }τ ∈[0,∞) , π 0 ) = ∞ or such that tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) =c∗g , there is zero probability of {cτ }τ ∈[0,∞) being such that there exists υ < tg ({cτ }τ ∈[0,∞) ,74

π 0 ) for which cυ = c∗g , and there is positive probability of {cτ }τ ∈[0,∞) being such thattg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and φi ({cτ }τ ∈[0,∞) , π 0 , g) 6= fg∗ . We show below that there is acontradiction if one or more of the preceding claims is true. It will then follow that
there cannot exist a non-stationary maximal symmetric SPE π 0 .
Consider the case where the first of the four claims holds. We argue in the paragraph below that there is zero probability of {cτ }τ ∈[0,∞) being such that tg ({cτ }τ ∈[0,∞) ,π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) ≥ c∗g−1 ; so that, there is probability one that eithertg ({cτ }τ ∈[0,∞) , π 0 ) = ∞ or both tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) < c∗g−1 .
Because π 0 is a maximal symmetric SPE for which tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ andc({cτ }τ ∈[0,∞) , π 0 , g) ∈ (c∗g , c∗g−1 ) with positive probability, there exists a realization
1∗
1∗
0
1∗
{c1∗
τ }τ ∈[0,∞) of the cost process as well as a time u such that tg ({cτ }τ ∈[0,∞) , π ) = u ,
∗
1∗
0
0
1∗
0
00
c1∗
u1∗ > cg , and V [ht ({cτ }τ ∈[0,∞) , π ), π ] ≥ V [ht ({cτ }τ ∈[0,∞) , π ), π ] for every symmetric SPE π 00 ∈ Π∗ in grim-trigger strategies along with any time t ≤ u1∗ . A stationary
maximal symmetric SPE π 1∗ ∈ Π∗ in grim-trigger strategies can be constructed asfollows. The agents play strategy profile π 0 up to and including the time t1∗ whentransaction g − 1 occurs. Thereafter, the agents do not make any transactions until1∗∗
the first time t1∗∗ that the current value of the cost process is c1∗,
u1∗ . At time t
01∗ieach agent transfers the amount φu1∗ (h0 , {cτ }τ ∈(0,∞) , π ). After this transaction, theagents follow the path of play specified in the statement of theorem 4, behaving as
if the game just started at the time of this transaction with the initial value of thei,i
1∗0
cost process being c1∗
u1∗ and the amount of each good being σu1∗ ({cτ }τ ∈[0,∞) , π ). The
agents do not make any transfers following a deviation from the path of play describedabove. The fact that π 1∗ and π ∗ induce different paths of play with positive probability contradicts the assumption that every stationary maximal symmetric SPE induces
the path of play specified in the statement of theorem 4.
We now argue as noted above that there is probability one that either tg ({cτ }τ ∈[0,∞) ,0π ) = ∞ or both tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) < c∗g−1 . Suppose to the contrary that tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) ≥ c∗g−1with positive probability. For any k ≥ 1, it follows from c0 > q that there is
probability one that either tk ({cτ }τ ∈[0,∞) , π 0 ) = ∞ or both tk ({cτ }τ ∈[0,∞) , π 0 ) < ∞and c({cτ }τ ∈[0,∞) , π 0 , k) < c0 . Otherwise, if there were to exist k ≥ 1 such thattk ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , k) > c0 with positive probability, thenthere would be positive probability of {cτ }τ ∈[0,∞) being such that tk ({cτ }τ ∈[0,∞) , π 0 ) <∞ and V [hu ({cτ }τ ∈[0,∞) , π 0 ), π 0 ] < 0 for u = tk ({cτ }τ ∈[0,∞) , π 0 ), which would con75

tradict the assumption that π 0 is a symmetric SPE. It is immediate that g 6= 1.
Assume therefore that g > 1. Given that π 0 is a maximal symmetric SPE forwhich tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) ≥ c∗g−1 with positive probability, there exists a realization {c̃1τ }τ ∈[0,∞) of the cost process as well as a timeũ1 such that tg ({c̃1τ }τ ∈[0,∞) , π 0 ) = ũ1 , c̃1ũ1 ≥ c∗g−1 , and V [hũ1 ({c̃1τ }τ ∈[0,∞) , π 0 ), π 0 ] ≥V [hũ1 ({c̃1τ }τ ∈[0,∞) , π 0 ), π 00 ] for every symmetric SPE π 00 ∈ Π∗ in grim-trigger strategies. Define the symmetric SPE π 00 ∈ Π∗ in grim-trigger strategies as follows. The
agents play strategy profile π 0 up to but not including the first time t̃1 that the current value of the cost process is c̃1ũ1 . At time t̃1 , the agents each transfer the amountŝi,i− σũi,i1 ({c̃1τ }τ ∈[0,∞) , π 0 ). After this transaction, the agents play according to thet̃1strategy profile π 0 , behaving as if the history up to the time of this transaction werehũ1 ({c̃1τ }τ ∈[0,∞) , π 0 ) and strategy profile π 0 had always been followed from the beginning of the game. The agents do not make any transfers following a deviation fromthe path of play described above. It can easily be seen that V (h0 , π 00 ) > V (h0 , π ∗ ),which contradicts the assumption that π ∗ is a maximal symmetric SPE.
Consider the case where the second of the four claims holds. Because π 0 is a maximal symmetric SPE for which tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) < c∗gwith positive probability, there exists a realization {c2∗τ }τ ∈[0,∞) of the cost process as
2∗
∗
0
2∗
2∗
well as a time u2∗ such that tg ({c2∗τ }τ ∈[0,∞) , π ) = u , cu2∗ < cg , and V [ht ({cτ }τ ∈[0,∞) ,
0
00
00
∗
π 0 ), π 0 ] ≥ V [ht ({c2∗τ }τ ∈[0,∞) , π ), π ] for every symmetric SPE π ∈ Π in grim-trigger
strategies along with any time t ≤ u2∗ . A stationary maximal symmetric SPE
π 2∗ ∈ Π∗ in grim-trigger strategies can be constructed as follows. The agents play
strategy profile π 0 up to and including the time t2∗ when transaction g − 1 occurs.
Thereafter, the agents do not make any transactions until the first time t2∗∗ that2∗∗
the current value of the cost process is c2∗, each agent transfers theu2∗ . At time ti
2∗0amount φu2∗ (h0 , {cτ }τ ∈(0,∞) , π ). After this transaction, the agents follow the path ofplay specified in the statement of theorem 4, behaving as if the game just started at
the time of this transaction with the initial value of the cost process being c2∗
u2∗ andi,i
2∗0
the amount of each good being σu2∗ ({cτ }τ ∈[0,∞) , π ). The agents do not make anytransfers following a deviation from the path of play described above. We argue in theparagraph below that there is probability one that either tg+1 ({cτ }τ ∈[0,∞) , π 2∗ ) = ∞or both tg+1 ({cτ }τ ∈[0,∞) , π 2∗ ) < ∞ and c({cτ }τ ∈[0,∞) , π 2∗ , g + 1) < c2∗
u2∗ . The fact that2∗∗
π and π induce different paths of play with positive probability contradicts theassumption that every stationary maximal symmetric SPE induces the path of play76

specified in the statement of the theorem 4.
We argue as mentioned in the previous paragraph that there is probability one of{cτ }τ ∈[0,∞) being such that either tg+1 ({cτ }τ ∈[0,∞) , π 2∗ ) = ∞ or both tg+1 ({cτ }τ ∈[0,∞) ,π 2∗ ) < ∞ and c({cτ }τ ∈[0,∞) , π 2∗ , g + 1) < c2∗
u2∗ . To do so, it suffices to show that02∗β−1
σ i,i ({c2∗would almost surely be theτ }τ ∈[0,∞) , π , g) ≤ s̃, where s̃ = −cu2∗ β[β/(β −1)]amount of each good remaining untransferred after the first transaction if the agents
follow the path of play specified in the statement of theorem 4 in a game where theinitial value of the cost process is c0 and the initial stock of each good is such that thefirst transaction almost surely occurs at a cost of c2∗
u2∗ . Suppose to the contrary that
i,i
2∗
0
20
i,i
2∗
σ ({cτ }τ ∈[0,∞) , π , g) > s̃. Let z = σ ({cτ }τ ∈[0,∞) , π 0 , g) − s̃. Define the symmetric0
SPE π 2 ∈ Π∗ in grim-trigger strategies as follows. The agents play strategy profile0
π 0 until the history up to the current time is hu2∗ ({c2∗τ }τ ∈[0,∞) , π ). Upon reaching
0
20
0
i
2∗
hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), the agents each transfer the amount φ ({cτ }τ ∈[0,∞) , π , g) + z .
After this transaction, the agents follow the path of play specified in the statementof theorem 4, behaving as if the game just started at the time of this transaction with the initial value of the cost process being c2∗
u2∗ and the amount of eachgood being s̃. The agents do not make any transfers following a deviation from0
the path of play described above. There exists a realization {c2τ }τ ∈[0,∞) of the cost
0
i,i
20
20
process along with a time v 2∗ such that c2v2∗ = c2∗u2∗ , σ̂v 2∗ ({cτ }τ ∈[0,∞) , π ) = s̃, and
0
0
0
0
0
V [hv2∗ ({c2τ }τ ∈[0,∞) , π 2 ), π 2 ] ≥ V [hv2∗ ({c2τ }τ ∈[0,∞) , π 2 ), π 00 ] for every symmetric SPE
0
20
20
20
20
π 00 ∈ Π∗ . Observe that V [hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), π ] = V [hv 2∗ ({cτ }τ ∈[0,∞) , π ), π ] +
0
2∗
0
2∗
[z 2 + φi ({c2∗τ }τ ∈[0,∞) , π , g) − cu2∗ ]. We argue in the paragraph below that V [hu2∗ ({cτ
0
0
0
0
0
2∗
}τ ∈[0,∞) , π 0 ), π 0 ] − V [hv2∗ ({c2τ }τ ∈[0,∞) , π 2 ), π 2 ] < [z 2 + φi ({c2∗τ }τ ∈[0,∞) , π , g) − cu2∗ ],
0
0
2
2∗
0
0
which implies that V [hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), π ] > V [hu2∗ ({cτ }τ ∈[0,∞) , π ), π ]. This
0
0
0
00
2∗
contradicts the fact that V [hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), π ] ≥ V [hu2∗ ({cτ }τ ∈[0,∞) , π ), π ] forevery symmetric SPE π 00 ∈ Π∗ in grim-trigger strategies. It follows that σ i,i ({c2∗τ }τ ∈[0,∞) ,0π , g) ≤ s̃.
0
We now argue as mentioned in the previous paragraph that V [hu2∗ ({c2∗τ }τ ∈[0,∞) , π ),
0
0
0
0
0
2∗
2∗
π 0 ] − V [hv2∗ ({c2τ }τ ∈[0,∞) , π 2 ), π 2 ] < [z 2 + φi ({c2∗τ }τ ∈[0,∞) , π , g) − cu2∗ ]. Let Σ[hu2∗ ({cτ}τ ∈[0,∞) , π 0 ); {cτ }τ ∈[v2∗ ,t] ; u2∗ , u2∗ +(t−v 2∗ ); π 0 ] be the sum of the transfers that an agent
0
would make between times u2∗ and u2∗ + (t − v 2∗ ) inclusive if hu2∗ ({c2∗τ }τ ∈[0,∞) , π ) is
the history up to time u2∗ , the cost process {cτ }τ ∈[v2∗ ,t] is realized between times
u2∗ and u2∗ + (t − v 2∗ ), and strategy profile π 0 is played by the agents. Define thesymmetric SPE π̃ 0 ∈ Π∗ as follows. The agents do not make any transactions until
77

0

0

0

0

the history up to the current time is hv2∗ ({c2τ }τ ∈[0,∞) , π 2 ). After hv2∗ ({c2τ }τ ∈[0,∞) , π 2 )has been reached, the agents do not make any transfers at any time t for which
the realization of the cost process {cτ }τ ∈[v2∗ ,t] between times v 2∗ and t is such that
0
2∗
2∗
2∗
0
20
i
2∗
0
Σ[hu2∗ ({c2∗τ }τ ∈[0,∞) , π ); {cτ }τ ∈[v 2∗ ,t] ; u , u +(t−v ); π ] ≤ [z +φ ({cτ }τ ∈[0,∞) , π , g)].
02∗
2∗Each agent transfers the amount Σ[hu2∗ ({c2∗+ (t −τ }τ ∈[0,∞) , π ); {cτ }τ ∈[v 2∗ ,t] ; u , u
2∗
0
20
i
2∗
0
v ); π ] − [z + φ ({cτ }τ ∈[0,∞) , π , g)] at the first time t for which the realization of
the cost process {cτ }τ ∈[v2∗ ,t] between times v 2∗ and t is such that Σ[hu2∗ ({c2∗τ }τ ∈[0,∞) ,
0
2∗
2∗
2∗
0
20
i
2∗
0
π ); {cτ }τ ∈[v2∗ ,t] ; u , u + (t − v ); π ] > [z + φ ({cτ }τ ∈[0,∞) , π , g)]. Let w2∗ denote the first time t that this condition holds. Thereafter, the agents play according to strategy profile π 0 , behaving as if this transaction happened at time0
u2∗ + (w2∗ − v 2∗ ), the history up to time u2∗ were hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), the cost
process {cτ }τ ∈[v2∗ ,w2∗ ] were realized between times u2∗ and u2∗ + (w2∗ − v 2∗ ), andstrategy profile π 0 were followed up to and including time u2∗ + (w2∗ − v 2∗ ). The
agents do not make any further transfers if either agent has deviated from the path of0
0
play described above after reaching hv2∗ ({c2τ }τ ∈[0,∞) , π 2 ). It can easily be seen that
0
0
0
20
0
20
i
2∗
20
V [hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), π ]−V [hv 2∗ ({cτ }τ ∈[0,∞) , π ), π̃ ] < [z +φ ({cτ }τ ∈[0,∞) , π , g)−
20
20
20
20
20
00
c2∗
u2∗ ]. Noting that V [hv 2∗ ({cτ }τ ∈[0,∞) , π ), π ] ≥ V [hv 2∗ ({cτ }τ ∈[0,∞) , π ), π ] for every
0020
symmetric SPE π 00 ∈ Π∗ , it follows that V [hu2∗ ({c2∗τ }τ ∈[0,∞) , π ), π ]−V [hv 2∗ ({cτ }τ ∈[0,∞) ,
0
0
0
0
2∗
π 2 ), π 2 ] < [z 2 + φi ({c2∗τ }τ ∈[0,∞) , π , g) − cu2∗ ].
Consider the case where the third of the four claims holds. Because π 0 is a maximal
symmetric SPE for which tg ({cτ }τ ∈[0,∞) , π 0 ) < ∞ and c({cτ }τ ∈[0,∞) , π 0 , g) = c∗g withpositive probability, there exists a realization {c3∗τ }τ ∈[0,∞) of the cost process as well as
0
3∗ 3∗
∗
3∗
0
0
a time u3∗ such that tg ({c3∗τ }τ ∈[0,∞) , π ) = u , cu3∗ = cg , and V [hu3∗ ({cτ }τ ∈[0,∞) , π ), π ]
0
00
00
∗
≥ V [hu3∗ ({c3∗τ }τ ∈[0,∞) , π ), π ] for every symmetric SPE π ∈ Π in grim-trigger strategies. Define the symmetric SPE π 3∗ ∈ Π∗ in grim-trigger strategies as follows. The
agents play strategy profile π 0 up to and including the time t3∗ when transaction g − 1
occurs. Thereafter, the agents do not make any transactions until the first time t3∗∗3∗∗that the current value of the cost process is c3∗, each agent transfers theu3∗ . At time ti
3∗0amount φu3∗ (h0 , {cτ }τ ∈(0,∞) , π ). After this transaction, the agents play according to
the strategy profile π 0 , behaving as if the history up to the time of this transaction were00
hu3∗ ({c3∗τ }τ ∈[0,∞) , π ) and strategy profile π had always been followed from the beginning of the game. The agents do not make any transfers following a deviation fromthe path of play described above. It can easily be seen that V (h0 , π 3∗ ) > V (h0 , π 0 ),which contradicts the assumption that π 0 is a maximal symmetric SPE.
78

Consider the case where the last of the four claims holds. A stationary maximal
symmetric SPE π 4∗ ∈ Π∗ in grim-trigger strategies can be constructed as follows.
The agents play strategy profile π 0 up to and including the time t4∗ when transaction
g occurs. Thereafter, the agents follow the path of play specified in the statementof theorem 4, behaving as if the game just started with the initial value of the costi,i
process being c4∗
t4∗ and the amount of each good being st4∗ . The agents do not makeany transfers following a deviation from the path of play described above. The factthat π 4∗ and π ∗ induce different paths of play with positive probability contradicts
the assumption that every stationary maximal symmetric SPE induces the path ofplay specified in the statement of theorem 4.
A.8
A.8.1

Proof of Corollaries to Theorem 4
Proof of Corollary 4

Proof. The sign of ∂fk∗ /∂β can be determined as follows:

sgn

∂fk∗
∂β



( "


k−1 #  )βq= sgn ∂∂β
1−β β−1= sgn −(k − 1)(β)k−2 (β − 1)k + k(β)k−1 (β − 1)k−1= sgn [−(k − 1)(β − 1) + kβ](β)k−2 (β − 1)k−1= sgn [(k − 1)(β − 1) − kβ] = sgn (1 − k − β) .

Thus, ∂fk∗ /∂β is positive if k < 1 + |β| and negative if k > 1 + |β|. In addition, we
have ∂β/∂µ < 0, ∂β/∂σ > 0, and ∂β/∂ρ < 0. Combining these facts, we obtain thedesired result.

79A.8.2

Proof of Corollary 5

Proof. The sign of ∂c∗k /∂β can be determined as follows:

sgn

∂c∗k
∂β




∂ ln (c∗k )
= sgn
∂β


∂{ln (q) − ln (1 − β) + (k − β) [ln (−β) − ln (1 − β)]}
= sgn
∂β





1
1
1
−β
= sgn
+ (k − β)
+
− ln
.
1−β
β 1−β
1−β


Let k ∗ be the value of k that solves the following equation:1+ (k − β)
1−β





1
1
+
β 1−β


− ln

−β
1−β


= 0,

which gives:∗k = −β(1 − β) ln

1−β
−β


> 0.

Note that ∂c∗k /∂β is positive if k < k ∗ and negative if k > k ∗ . In addition, we have∂β/∂µ < 0, ∂β/∂σ > 0, and ∂β/∂ρ < 0. Hence, c∗k is decreasing in µ and ρ as wellas increasing in σ for k < k ∗ , and c∗k is increasing in µ and ρ as well as decreasing in
σ for k > k ∗ .
We next examine how k ∗ varies with β. The partial derivative with respect to β
is given by:∂c∗k= 1 − (1 − 2β) ln
∂β



1−β
−β





−β
β
< 1 − (1 − 2β) 1 −
=
< 0,
1−β
1−β

where the general result ln(x) > 1 − 1/x for all x > 1 is used to establish the firstinequality. Hence, k ∗ is increasing in |β|.
The limit as β goes to −∞ can be calculated as:∗lim k = lim −β(1 − β) ln

β→−∞

β→−∞

1−β
−β





−β
≥ lim −β(1 − β) 1 −
β→−∞
1−β ,

= lim −β = ∞β→−∞where the general result ln(x) > 1−1/x for all x > 1 is used to establish the inequality.
Hence, lim|β|→∞ k ∗ = ∞.
80

The limit as β goes to 0 can be calculated as:






1−β
1−β
1
lim k = lim− −β(1 − β) ln= lim− ln
β→0−
β→0
β→0
−β
−β
−β(1 − β)
 
 


1−β
∂
1
∂
ln
,
= lim−
β→0 ∂β−β∂β −β(1 − β)



1
1
(−β)(1 − β)1
= lim−= lim−
=0
−
2
2
β→0
β→0
−β(1 − β)β(1 − β)1 − 2β∗where the third equality applies L’Hôpital’s rule. Hence, lim|β|→0 k ∗ = 0.
A.9

Proof of Proposition 2

Proof. Choose any symmetric strategy profile π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2}.
Suppose that there is positive probability of the cost realization {ct }t∈[0,∞) being suchthat φ1t (h0 , {cτ }τ ∈(0,∞) , π) = φ2t (h0 , {cτ }τ ∈(0,∞) , π) ∈ (0, q) for some t ∈ [0, ∞). Let0000π = (π1 , π2 ) with πi ∈ Π̄Ci for i ∈ {1, 2} be a symmetric strategy profile that requires
the agents to behave as follows. At any history where a transaction occurs for the
first time when playing π, each agent transfers the amount q. At any other history,0each agent transfers the amount 0. Note that V (h0 , π ) > V (h0 , π). Hence, π is notefficient.
A.10

Proof of Proposition 3

Proof. We identify the path of play induced by an efficient strategy profile. In theabsence of incentive constraints, the efficient policy requires the agents to almost
surely make at most one transfer. Attention can be restricted to a stationary thresholdpolicy. The efficient path of play can be obtained by choosing ĉ ∈ (0, q) so as tomaximize the value L(ĉ) of an asset that pays q − ĉ when the cost first reaches ĉstarting from c0 . From the earlier analysis, the value of such an asset is:L(ĉ) = (q − ĉ)(ĉ/c0 )−β .
Differentiating with respect to ĉ yields the following first-order condition for c̄:−β−1−β(q − c̄)c−1− (c̄/c0 )−β = 0 ⇒ c̄ = −β(1 − β)−1 q,
0 (c̄/c0 )81

where the derivative is positive for ĉ < c̄ and negative for ĉ > c̄.
A.11
A.11.1

Proofs of Proposition 4 and Its Corollaries
Proof of Proposition 4

Proof. We begin by computing the solution to the second-best problem. From theproof of theorem 4, the value function M sb of each agent for the second-best problem
is:
f1∗ β
M sb = ∗βc0 ,c1
where f1∗ and c∗1 are defined as follows:
f1∗ =

1
q
and c∗1 =
1−β
1−β



−β
1−β

1−β
,

and β < 0 is given by:
β = 12 − σµ2 −

q


µ
1 2
−
+ 2 σρ2 .
2σ2

Substituting for f1∗ and c∗1 in the expression for M sb results in:M sb = θ1−β q 1−β cβ0 ,where the constant θ ∈ (0, 1) is given by:θ = (−β)−β (1 − β)−(1−β) .
We next compute the solution to the first-best problem. From the proof of proposition 3, the value function M f b of each agent for the first-best problem is:
M fb =

q − c̄ βc ,c̄β 0where c̄ is defined as follows:c̄ = −β(1 − β)−1 q.

82

Substituting for c̄ in the expression for M f b results in:M f b = θq 1−β cβ0 .

A.11.2

Proof of Corollary 8

Proof. The logarithm of M f b is:ln(M f b ) = −β ln(−β) − (1 − β) ln(1 − β) + (1 − β) ln(q) + β ln(c0 ),and the logarithm of M sb is:ln(M sb ) = −β(1 − β) ln(−β) − (1 − β)2 ln(1 − β) + (1 − β) ln(q) + β ln(c0 ).
Differentiating ln(M f b ) with respect to β yields:∂ ln(M f b )= ln(1 − β) − ln(−β) + ln(c0 ) − ln(q),∂βand differentiating ln(M sb ) with respect to β yields:∂ ln(M sb )= (2 − 2β) ln(1 − β) − (1 − 2β) ln(−β) + ln(c0 ) − ln(q).
∂βNoting that 0 < q < c0 and β < 0, we have ∂ ln(M f b )/∂β > 0 and ∂ ln(M sb )/∂β > 0.
Hence, M f b and M sb are increasing in β. Since β is decreasing in µ and ρ as well as
increasing in σ, this completes the proof.
A.11.3

Proof of Corollary 9

Proof. We now show that M f b and M sb converge to q as ρ approaches zero from theright if and only if the condition µ ≤ σ 2 /2 is satisfied. The limiting value of β is asfollows:0,
if µ ≤ σ 2 /2
lim+ β =.
1 − 2µ/σ 2 , if µ > σ 2 /2
ρ→083

Note that limβ→0− θ = limβ→0− (−β)−β · limβ→0− (1 − β)−(1−β) , where limβ→0− (1 −β)−(1−β) is clearly equal to one, and limβ→0− (−β)−β is easily shown to be one bytaking the logarithm and applying L’Hôpital’s rule. Thus, the limiting value of θ isgiven by:
1,
if µ ≤ σ 2 /2.
lim+ θ =(2µ/σ 2 − 1)2µ/σ2 −1 (2µ/σ 2 )−2µ/σ2 , if µ > σ 2 /2
ρ→0
It follows that:
lim M

fb


q,

if µ ≤ σ 2 /2,
(lim + θ)q (1−limρ→0+ β) c(limρ→0+ β) , if µ > σ 2 /2
ρ→0
0

q,
if µ ≤ σ 2 /2sb
lim M =.
(lim + θ)(1−limρ→0+ β) q (1−limρ→0+ β) c(limρ→0+ β) , if µ > σ 2 /2
ρ→0+

ρ→0+

=

ρ→0

0

Note that c0 > q > 0 by assumption. In addition, −∞ < limρ→0+ β < 0 and
0 < limρ→0+ θ < 1 for µ > σ 2 /2. Hence, 0 < limρ→0+ M sb < limρ→0+ M f b < q if
µ > σ 2 /2.
A.11.4

Proof of Corollary 10

Proof. Assume that µ > σ 2 /2, and let γ = µ/σ 2 . Then the ratio of limρ→0+ M sb tolimρ→0+ M f b is given by:( lim+ M sb )/( lim+ M f b ) = ( lim+ θ)−(limρ→0+ β) = [(2γ − 1)2γ−1 /(2γ)2γ ]−(1−2γ) .
ρ→0

ρ→0

ρ→0

Taking the logarithm of limρ→0+ θ, we obtain:ln( lim+ θ) = (2γ − 1) ln(2γ − 1) − (2γ) ln(2γ).
ρ→0Differentiating with respect to γ, we have:∂ ln(limρ→0+ θ)= 2[ln(2γ − 1) − ln(2γ)] < 0.
∂γ

84Hence, limρ→0+ θ is decreasing in γ. Moreover, limρ→0+ β = 1 − 2γ is decreasing in γ.
Noting that 0 < limρ→0+ θ < 1 and −∞ < limρ→0+ β < 0, the ratio is decreasing in
γ. Since γ is increasing in µ and decreasing in σ, this completes the proof.

85