ISSN (Print) 0473-453X

Discussion Paper No. 1090

ISSN (Online) 2435-0982

DYNAMIC LEGISLATIVE BARGAINING

Hülya Eraslan
Kirill S. Evdokimov
Jan Zápal

June 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanDynamic Legislative Bargaining *Hülya Eraslan, Kirill S. Evdokimov and Jan Zápal
June 10, 2020

Abstract
This article surveys the theoretical literature on legislative bargaining with endogenous status-quo. These are the legislative bargaining situations in which ineach period a new policy is decided and the policy implemented in the event of noagreement is endogenously determined by the outcome of bargaining in the previous
period. After describing a general framework, we discus bargaining over redistributive policies, bargaining over spatial policies, existence issues, efficiency issues andopen questions.
JEL codes: A3, C7, D7
Keywords: legislative bargaining, multilateral bargaining, spatial bargaining, redistributive bargaining, endogenous status-quo, dynamic political economy

1

Introduction

In this paper, we survey the literature on legislative bargaining with endogenous statusquo. Since the term “endogenous status-quo” is used to express different things bydifferent authors, we start by proposing a typology to distinguish “endogenous statusquo” from “evolving status-quo.”1 By bargaining with endogenous status-quo, we mean
dynamic bargaining situations where in each period a new policy is decided and thepolicy implemented in the event of no agreement is endogenously determined by theoutcome of bargaining in the previous period. By bargaining with evolving status-quo, wemean dynamic bargaining situations in which a single policy is decided during the entire
bargaining time horizon, the status-quo policy in each round of bargaining depends on
the history up to that bargaining round, and at the final bargaining round, in the eventof no agreement, the status-quo at that round is implemented.
As in Eraslan and Evdokimov (2019), by “legislative bargaining” we mean multilateral bargaining situations in which agreement requires less than unanimous consent* We thank Kyle Hyndman and Emin Karagözoğlu for the invitation to write this survey and an anony-mous referee for an insightful feedback on an earlier version. Eraslan thanks the International Joint Research Promotion Program (Osaka University). Eraslan: Department of Economics, Rice University andInstitute of Social and Economic Research, Osaka University, eraslan@rice.edu. Evdokimov: Departmentof Economics, Rice University, kirill.evdokimov@rice.edu. Zapal: CERGE-EI, j.zapal@cerge-ei.cz.
1
Anesi and Seidmann (2014, Section 6.1) discusses some of the literature using these two concepts
without introducing new terminology to distinguish between them.

1and agreement on a proposal binds all parties. They survey the literature on legislative
bargaining with exogenous status-quo in which players decide on a policy through voting
between an exogenously given status-quo and a proposal offered by a proposer. This
protocol is repeated in each period until an agreement is reached. Here we survey theliterature on legislative bargaining with endogenous status-quo. In these models the protocol just described is the stage game in a dynamic model in which, in each period, a
new policy is decided and the status-quo is endogenously determined by the outcome of
the bargaining in the previous period.
Baron (1996) is typically credited as the first paper to start the literature on legislative
bargaining with endogenous status-quo. Two important early precursors are Ingberman(1985) and Epple and Riordan (1987) who study bargaining over spatial and redistributive
policy respectively. The key difference between these two papers and the subsequent workis their focus on deterministic proposer recognitions, and in the case of the latter paper,on equilibria supported by history-dependent punishments.
We start by introducing a general framework that incorporates two main strands of theliterature we discuss in detail: bargaining over redistributive policy and bargaining overspatial policy. After discussing the equilibrium existence issues, we use this framework
to study the many interesting questions the literature on legislative bargaining withendogenous status-quo has raised: How does the endogeneity of the status-quo affect the
incentives of players? What is the equilibrium dynamics of policies? Do the policies
converge? If so, to which policy? If not, what is their long-run behavior? What is theappropriate efficiency concept? Are equilibrium policies (Pareto) efficient? We end oursurvey by briefly discussing some avenues for future research.
The general framework we study incorporates related political economy models withbilateral bargaining, bargaining with evolving status-quo, random proposal models, costly
policy change, models in which either a dictator decides policy or median chooses policy.2
We do not review these papers due to space limitations. We also ignore other related
literature on elections,3 dynamic linkages via macro variables4 or power structures,5 andcoalitional bargaining.6
2

With apologies for inevitable omissions, see, for example, Bernheim, Rangel, and Rayo (2006); Anesi
and Seidmann (2014) on bargaining with evolving status-quo, Roberts (2007); Penn (2009); HortalaVallve (2011); Acharya and Ortner (2017) on random proposal models, Gersbach and Tejada (2018);
Dziuda and Loeper (2019); Eraslan and Piazza (2019); Gersbach, Muller, and Tejada (2019); Gersbach,Jackson, and Tejada (2020) on costly policy change, Tabellini and Alesina (1990); Gieczewski (2017) on
models in which median chooses policy.
3See, for example, Alesina and Tabellini (1990); Baron, Diermeier, and Fong (2012); Cho (2014);
Forand (2014); Callander and Raiha (2017); Nunnari and Zápal (2017); Baron (2018); Duggan and
Forand (2018a,b); Gersbach, Jackson, Muller, and Tejada (2018).
4
See, for example, Persson and Svensson (1989); Krusell and Rı́os-Rull (1996, 1999); Leblanc, Snyder,and Tripathi (2000); Hassler, Rodrı́guez Mora, Storesletten, and Zilibotti (2003a); Hassler, Storesletten,
and Zilibotti (2003b); Battaglini and Coate (2007, 2008, 2016); Klein, Krusell, and Rı́os-Rull (2008);
Azzimonti (2011); Battaglini, Nunnari, and Palfrey (2012, 2014, forthcoming); Azzimonti, Battaglini,and Coate (2016).
5
See, for example, Jack and Lagunoff (2006); Lagunoff (2009); Acemoğlu, Egorov, and Sonin (2010,2012, 2018); Bai and Lagunoff (2011); Diermeier, Prato, and Vlaicu (2015, 2016); Gibilisco (2015); Jeon(2015); Karakas (2016, 2017).
6
See, for example, Konishi and Ray (2003); Gomes (2005); Gomes and Jehiel (2005); Hyndman and
Ray (2007); Acemoğlu et al. (2010, 2012). We refer the reader to Ray and Vohra (2015) for an excellent
survey of this literature.

22

General framework

In a typical model of dynamic legislative bargaining, legislators in set N = {1, . . . , n}must choose a policy xt in each period t = 0, 1, . . . , T , where T can be finite or infinite.
The policy space is X. When X ⊆ R, bargaining is over a one-dimensional spatial policy.
When X is the (n − 1)-dimensional simplex, bargaining is over redistributive policy.
The game in each period t proceeds as follows. First, period t starts with a publicly
observed state denoted by st . The state st is a vector whose components include xt−1 but
it can include other (potentially stochastic) variables as well. One of the players κ(st ) is
the proposer and she makes a proposal x ∈ X(st ) where X(st ) ⊆ X is the set of feasible
policies in state st . All players vote to either accept or reject x. If the set of players who
accept x is in W(st ), which is the collection of winning coalitions in state st , then thepolicy xt implemented at time t is x, i.e., xt = x. Otherwise, the policy xt implemented
at time t is the status-quo. We let ζ(st ) to denote the status-quo in period t when thestate is st , where ζ is some known function.
Stage utility of player i from policy x in any period when the state is s is ui (x, s).
Player i discounts the future at a rate δi , thus the utility of player i from a sequence ofpolicies (xt )Tt=0 and states (st )Tt=0 is given by(1 − δi )TXδit ui (xt , st ).

t=0

The papers we survey fit the framework outlined. For example, Baron (1996) and Kalandrakis (2004) study, respectively, spatial and distributive settings with simple majority
and random proposer recognitions. Under our notation, they assume that st includes thepolicy xt−1 from the previous period and identity of the proposer in the current period,}, and ζ(st ) = xt−1 .
and, for any state st , X(st ) = X, W(st ) = {C ⊆ N : |C| ≥ n+1
2
The framework naturally incorporates more complex models with state-dependent stage
utilities (as in Riboni and Ruge-Murcia, 2008; Dziuda and Loeper, 2016; Bowen, Chen,
Eraslan, and Zápal, 2017), models in which the status-quo in period t equals the (t − 1)period policy with some noise, i.e., ζ(st ) = xt−1 + ε (as in Duggan and Kalandrakis,2012), models in which players redistribute pie with size that stochastically changes overtime (as in the stochastic bargaining models in the spirit of Merlo and Wilson, 1995),
or models with state-dependent winning coalitions and policy spaces (as in Chen andEraslan, 2017).
Histories, strategies and subgame-perfect equilibrium for the class of games above canbe defined in the standard way. The papers we survey restrict attention to (stationary)Markov strategies and equilibria.7 A Markov (behavioral) proposal strategy of player i
specifies a distribution of proposals πi (s, t) over X(s) for each period t and each state sin which i is the proposer. A Markov (behavioral) voting strategy of player i specifies
the probability αi (x, s, t) that player i accepts proposal x in period t when the state is s.
Infinite horizon models further restrict attention to stationary strategies that do not allow
dependence on t. Moreover, the strategies are usually allowed to depend only on certain(payoff relevant) components of the state, e.g., the previous-period policy or preference
7

There is also a strand of the literature that analyzes subgame perfect equilibria in dynamic political
economy models. See, for example, Dixit, Grossman, and Gül (2000); Acemoğlu, Golosov, and Tsyvinski(2008, 2011); Aguiar and Amador (2011); Halac and Yared (2014).

3

shocks, but not allowed to depend on others, e.g., the identity of the proposer.8
A Markov strategy of player i is denoted σi = (πi , αi ) and a profile of Markov strategies
is denoted σ = (σi )i∈N . A profile of strategies σ induces a dynamic utility V σ (x, s, t) frompolicy x accepted in period t when the state is s, which satisfies the following recursive
relationZσ(1)
Vi (x, s, t) = (1 − δi )ui (x, s) + δi Viσ (x0 , s0 , t + 1)dP σ (x0 , s0 |x, s, t)
where P σ (x0 , s0 |x, s, t) is the probability measure over (t + 1)-period policies and states
induced by profile σ and policy x in period t when the state is s. Denote by α(x, s, t)
the probability that a proposal x is accepted in period t when the state is s. Formally, itsatisfies
X YYα(x, s, t) =αi (x, s, t)(1 − αi (x, s, t)).
(2)
C∈W(s) i∈Ci∈N \CA profile of strategies σ constitutes a Markov Perfect equilibrium (MPE) if
i) for each i, s in which i is the proposer, t and p in the support of πi (s, t),p ∈ arg max α(x, s, t)Viσ (x, s, t) + (1 − α(x, s, t))Viσ (ζ(s), s, t),x∈X(s)

ii) for each i, x, s and t, if Viσ (x, s, t) > Viσ (ζ(s), s, t), then αi (x, s, t) = 1, and if
Viσ (x, s, t) < Viσ (ζ(s), s, t), then αi (x, s, t) = 0.
A profile of strategies σ is a stationary MPE (SMPE) if it is an MPE and πi and αi are
independent of t for each player i.
The first condition in the definition of MPE requires that the proposal strategies are
optimal while the second condition requires that the voting strategies are optimal. As isstandard in the literature, the definition of MPE uses the one-stage deviation principle,
and hence implicitly assumes that it holds.9 The voting strategies in the definition assume
that players vote as if pivotal. This rules out implausible equilibria that support arbitrary
outcomes because no voter is pivotal. Some work further focuses on voting strategies
with indifferent players voting for the proposal. This implies that any proposal is either
accepted or rejected with probability one, and hence allows one to focus on proposal
strategies that generate proposals that are always accepted.10

3

Existence of equilibria

Existence of SMPE in dynamic legislative bargaining models is an open issue, outside
special cases represented by finite-horizon models with finite policy spaces, for which MPE
existence follows by standard backward induction arguments. The literature studying
discounted stochastic games, of which the dynamic legislative bargaining models are a8

See Maskin and Tirole (2001) for a rigorous definition of Markov equilibria and payoff relevant
histories.
9
Discounting and bounded stage utilities suffice for the one-stage deviation principle to apply (seeFudenberg and Tirole, 1991, Theorem 4.2).
10
If the voting strategies are pure and ζ(s) ∈ X(s) for each state s, it is without loss of generality
to assume that proposers choose policies only from the set of policies that would be accepted. This isbecause proposing a policy that would be rejected is identical to proposing the default outcome ζ(s).

4special case, includes numerous conditions for SMPE existence (see He and Sun, 2017, fora recent contribution) but also examples of SMPE non-existence (see Levy, 2013; Levy
and McLennan, 2015). Moreover, none of the known conditions for SMPE existence
applies to legislative bargaining models (see Duggan and Kalandrakis, 2007, for further
discussion of the issues involved).
One possible approach to proving SMPE existence in an infinite-horizon stationary
dynamic legislative bargaining model is to work with a one-shot auxiliary game in which
the payoff of player i from policy xt in state st combines her stage payoff (1 − δi )ui (xt , st )with the expected dynamic utility δi Ui (xt , st ). The expected dynamic utility of a player
represents her payoff from playing the dynamic legislative bargaining game starting fromperiod t + 1 before the state st+1 is drawn, knowing that the distribution of st+1 depends
on the state st and policy xt at t. That is, it corresponds to the second term on the righthand side of (1).11 Fixing a profile (Ui )i∈N in the one-shot auxiliary game, one derives
the Nash equilibrium payoffs of this game and integrates these payoffs, as in (1), into a
new profile (Ũi )i∈N . Any fixed point of the implied mapping Γ from (Ui )i∈N to (Ũi )i∈N
corresponds to an SMPE of the original game.
The map Γ is well-defined once one assumes that (Ui )i∈N are continuous in both of
its arguments, which ensures that the proposer i’s problem of choosing xt to maximize(1−δi )ui (xt , st )+δi Ui (xt , st ) has a solution. However, continuity of Ũi in its first argument
is not guaranteed even when (Ui )i∈N are continuous, which implies that no fixed-point
theorem can be used to conclude that Γ has a fixed point.12 Ũi (xt−1 , st−1 ) might not be
continuous in xt−1 because a small change in xt−1 in general leads to a small change inthe state st that parametrizes the proposer’s problem in the auxiliary game, and evena small change in this state might induce a large change in the policy that solves the
proposer’s problem and hence in the value of being a responder.
The difficulties described typically arise in models where ζ(st ) = xt−1 for all states
st induced by the policy xt−1 , that is, in models where the status-quo in period t equals
the policy in period t − 1. One approach that overcomes these difficulties is then to workwith models where xt−1 induces a distribution over ζ(st ) that changes smoothly whenxt−1 changes. This is how Duggan and Kalandrakis (2012) prove a very general SMPE
existence result for dynamic legislative bargaining models; they require shocks to statusquo transitions and players’ preferences. Similarly, Duggan (2017a) assumes smoothtransitions from a continuous policy space to a countable state space. An alternative
strategy is to assume finite policy space (Anesi, 2010; Diermeier and Fong, 2012), which
automatically guarantees existence of a solution to the proposer’s problem.
Because SMPE is not guaranteed to exist in a general dynamic legislative bargaining
model, most papers in the literature construct an equilibrium and study its properties. In
the next two sections we display such constructions in two canonical settings, distributive
and spatial.

4

Distributive policy

In this section, we review the results for the case of distributive policies. This type ofmodel with infinite policy sequences raises many interesting questions regarding equilib11

Expected dynamic utilities are sometimes called continuation values.
This is because all fixed-point theorems require functions or correspondences between identicalspaces. Constructing the fixed point in the strategy space, not in the expected dynamic utility space,would face similar problems.
125rium strategies and outcomes in addition to those questions raised in the introduction.
Specifically, what is the structure of winning coalitions? For example, does the size principle of Riker (1962)13 hold? Is the entire surplus shared in every period in equilibrium oris there waste? More generally, is there inefficiency in equilibrium? Most existing papers
use constructive arguments to find SMPE to answer these questions.
As we will see, even for a given set of primitives this model has a great variety ofequilibria, some of which differ drastically in their answer to the above questions. To
emphasize this point, we will focus on two equilibrium constructions that have been
influential in the literature: Kalandrakis (2004) which is an early paper that features
an intuitive construction, and Anesi and Seidmann (2015) which is a very thoroughinvestigation of a more general model.14
Kalandrakis (2004) is the first to describe an SMPE in a dynamic legislative bargaining
game in which three players bargain over redistribution of a surplusPof size 1. The policy
space is the two-dimensional unit simplex, that is, X = {x ∈ R3+ : 3i=1 xi = 1}. Under
our earlier notation, st includes the policy xt−1 from the previous period, and, for any
state st , X(st ) = X, W(st ) = {C ⊆ N : |C| ≥ 2}, and ζ(st ) = xt−1 . Player i’s utility
from policy x is ui (x) = xi and players discount the future at a common rate δ.
We describe the equilibrium constructed by Kalandrakis (2004) with the help of Figure
1. The figure depicts an equilateral triangle with each vertex designated for one of theplayers. Each edge has length √23 , and therefore the distances between an arbitrary
point inside the triangle and the edges sum to unity.15 For example, x represents theequal allocation ( 31 , 31 , 13 ), x0 and x00 represent allocations in which player 3 and player 2
respectively receive zero share, and x000 represent the unequal allocation (1, 0, 0) favorable
to player 1.
Figure 1: Equilibrium dynamics in the distributive model

1

x0

x000

x

2

x00

3

The figure represents the equilibrium dynamics. For four different status-quo policies
x, x0 , x00 and x000 , the arrows point to the policy proposed by player 1 (solid line), player
13

Riker’s size principle states that in n-person zero-sum games with perfect information, when side
payments are permitted, only minimum winning coalitions can occur.
14
The earliest paper we know of is Epple and Riordan (1987) which characterizes subgame perfectequilibria in a three-player game to show that a wide range of outcomes can be sustained in equilibriumusing punishment strategies.
15
This is known as Viviani’s theorem.

62 (dashed line) and player 3 (dotted line).16 The germane feature of the equilibrium thefigure illustrates is the dynamics that moves policies from strictly inside the triangle toits edges and from the edges to its vertices. That is, the equilibrium dynamics moves overallocations that differ in the number of players who receive zero share: starting with no,then with one and then with two players receiving zero share. Eventually, the equilibrium
reaches a state in which the entire resource is claimed by the randomly chosen proposer.
The literature has dubbed this a rotating dictator equilibrium.
The equilibrium dynamics arises due to the the nature of winning coalitions. We
explain the intuition with the help of Figure 2. All four figures show indifference curvesof the equilibrium dynamic utility. The left column is for player 1 and the right column
is for player 3 (higher utility levels closer to the player’s vertex). The top row is for δ = 0
6
and the bottom row is for δ = 10.
Start with the simple case when δ = 0, so that the model we study is an infinitelyrepeated one-shot dictator game with three players and simple majority. Consider player
1’s proposer problem assuming she seeks the approval of player 3, given some status-quo
utility level of player 3. Using top row of Figure 2, the optimal proposal of player 1 will bea policy at the north-eastern edge of the policy space that gives player 3 the same utility as
the status-quo, allocates zero share to player 2 and allocates the entire remaining resources
to player 1. When, in addition, the status-quo is located at the north-western edge of thepolicy space, the status-quo utility of player 3 is zero and player 1 will propose a policy
in which she obtains the entire pie. That is, the equilibrium dynamics moves policies
from within the policy space to its edges and from the edges to the vertices. Inspection
6, althoughof the bottom row of Figure 2 shows that similar forces operate when δ = 10
now the equilibrium is considerably more complex and its construction significantly more
involved.17
The rotating dictator equilibrium constructed by Kalandrakis (2004) has several interesting properties. First, players prefer policies that maintain their share of the resource
but make the other players’ shares unequal. For example, in the left bottom part of Figure 2, starting on the dotted line and moving horizontally towards the edges of the policy
space typically increases player 1’s dynamic utility. This preference arises endogenously;
the more unequal players 2 and 3 are, the more accommodating will the disadvantagedof the two players be as a responder and hence the stronger the bargaining position ofplayer 1 if recognized as a proposer in the next period.
Second, the strategies supporting the rotating dictator equilibrium are continuous in
the status-quo. This implies that the equilibrium dynamic utilities are continuous in
the status-quo. However, as Kalandrakis (2004) shows, the dynamic utilities lack quasiconcavity and might be constant over some parts of the policy space (the shaded areasin the bottom row of Figure 2). This implies that the equilibrium acceptance sets, whichcontain the policies that would be accepted given some status-quo, in general are notconvex and that the acceptance correspondence, which maps status-quos to acceptancesets, is not lower hemicontinuous. These results highlight the difficulties in working with
16

The equilibrium shown is symmetric, which allows one to work out the entire equilibrium policy
dynamics starting from x. In particular, given a status-quo at the vertex, all players allocate the entire
pie to themselves. And given a status-quo at the edge close to one of the vertices, both players with
non-zero status-quo share allocate the entire pie to themselves and the player with zero status-quo shareallocates zero share to the player with the largest status-quo share.
17
The construction in Kalandrakis (2004) involves mixed proposal strategies for status-quos that allocate similar shares to two or more players. His discussion of the construction shows that this mixing isrequired for his construction (page 303).

7

Figure 2: Indifference curves of the dynamic utility in the rotating dictator equilibrium
1

2

1

3

2

1

2

3
1

3

2

3

the dynamic legislative bargaining games because the endogenous objects (e.g., dynamic
utilities and acceptance sets) are not guaranteed to be well-behaved.
Finally, Kalandrakis (2004) notes that the equilibrium he constructs is not unique,although all the additional equilibria he finds are payoff equivalent to the equilibrium hestudies in detail.18 The subsequent work finds other equilibria and culminates in Anesi
and Seidmann (2015) who study a significant generalization of the model from Kalandrakis (2004) and show that large number of outcomes can be supported by SMPE whenplayers become arbitrarilyP
patient.19 Their model features general number of players, poln
icy space X = {x ∈ R+ : i∈N xi ≤ 1} that allows waste, any quota voting rule between
simple majority and unanimity, and heterogeneous recognition probabilities, discountfactors and nonlinear utilities.
The argument in Anesi and Seidmann (2015) is constructive and we describe its, n). Thus,
main features for n odd and a threshold voting rule with quota q ∈ [ n+12W(st ) = {C ⊆ N : |C| ≥ q}. The key ingredient of their construction is a concept
called simple solutions. A simple solution consists of n policies. For each player i, eachof these policies allocates either a bad share bi or a good share gi > bi to i, with policy
xi being a good policy and at least one of the other policies being a bad policy for i.
In addition, each of these policies is a good policy for at least q players. Formally, a18

Kalandrakis (2010) extends the rotating dictator equilibrium of Kalandrakis (2004) to an odd numberof at least five players with heterogeneous recognition probabilities and mildly concave utilities.
19
Bowen and Zahran (2012) construct a compromise equilibrium in which more than a minimum
winning coalition is allocated a positive share in each period. Their construction relies on risk-sharing
incentives provided by concave utilities. By additionally allowing waste, Richter (2014) constructs anequal division equilibrium. Baron and Bowen (2015) independently uses a construction similar to Anesi
and Seidmann (2015). Our discussion focuses on the latter since its construction is also applicable tospatial bargaining (Anesi and Duggan, 2018). Baron (2019) simplifies the construction in Baron and
Bowen (2015) and studies equilibrium coalitions and proposal power.

8

simple solution S = (xj )j∈N consist of n policies such that (i) xj ∈ X for all j, (ii) for
any i, xji ∈ {bi , gi } for all j, where bi < gi , with xii = gi and xji = bi for some j, and (iii)|{i ∈ N : xji = gi }| ≥ q for all j. Given a simple solution S, let Bi = {x ∈ S : xi = bi }and Gi = {x ∈ S : xi = gi } be the set of bad and good policies in S for i.
Given a simple solution S, let σ be the following strategy profile. Player i recognized
to propose given status-quo xt−1 proposes xi if xt−1 ∈/ S and proposes xt−1 when xt−1 ∈ S.
If player i is a responder given status-quo xt−1 and proposal x 6= xt−1 , he votes accept
if and only if either (i) xt−1 ∈ Bi , or (ii) xt−1 ∈/ S and x ∈ Gi , or (iii) xt−1 , x ∈/ S andui (x) ≥ ui (xt−1 ). Notice the policy dynamics induced by σ: starting from any status-quo
xt−1 ∈/ S, xt is a random variable that takes the value xj with probability equal to theprobability of recognition of player j, while it takes the value xt−1 starting from a statusquo xt−1 ∈ S. That is, starting from any status-quo, the equilibrium policy reaches S
in at most one period and stays constant thereafter. Anesi and Seidmann (2015) showthat the profile σ constitutes an SMPE for sufficiently patient players. Their argument issurprisingly straightforward due to the simplicity of S and σ and because patient players
care mostly about the limit of policy dynamics induced by σ.
Figure 3: A simple solution in the distributive model
1

x1

x3
x2

2

3

Studying simple solutions thus provides information about the equilibrium outcomesin the dynamic distributive legislative bargaining
P3 model. Figure 3 draws part of the3three-player policy space such that {x ∈ R+ : i=1 xi = c}, where c ∈ (0, 1], along witha simple solution S = (x1 , x2 , x3 ) that corresponds to vertices of an inverted equilateraltriangle inscribed into the policy space. With patient players, there exists an SMPE withequilibrium dynamics converging on S in at most one period from any initial status-quo.
However, a continuum of simple solutions exists and hence large variety of outcomesare compatible with SMPE. For example, vertices of any inverted equilateral triangle
inscribed into the policy space in Figure 3 correspond to a simple solution, irrespective of
the size of the inscribed triangle and irrespectiveof c used to draw the policy space. ThisP
implies that any policy in {x ∈ R3+ : 3i=1 xi ≤ 1} can be supported as an equilibriumoutcome including policies with strictly positive share allocated to all three players orwith waste. The only policies that cannot be supported in equilibrium are the policies
that assign zero share to two or three players.20
20

Outside the three-player example, the only policies ruled out by equilibrium are those in whichn − q + 1 or more players receive zero share.

9

5

Spatial policy

The spatial dynamic legislative bargaining literature starts with Baron (1996) who analyzes a model in which players choose the scale of collective goods, and each player i has astrictly concave utility with bliss-point θi . The policy space is the positive real line, that
is, X = R+ . The state st includes the policy xt−1 from the previous period, and, for any}, and ζ(st ) = xt−1 . Players discountstate st , X(st ) = X, W(st ) = {C ⊆ N : |C| ≥ n+1
2
the future at a common rate δ. Kalandrakis (2016) and Zápal (2016) analyze models
similar to Baron (1996) focusing on the case of three players. This is the version of themodel we study here: three players with quadratic stage utilities, bliss-points θL = −1,
θM = 0 and θR = 1 and equal recognition probabilities choose policies from R.21
As shown by Zápal (2016), this model admits an SMPE. An equilibrium consists ofpure proposal strategies, mapping status-quo into the proposed policy, and pure voting
strategies, specifying voting behavior given any status-quo proposed-policy pair. We first
discuss the asymmetric equilibrium shown in Figure 4 and later turn to other (symmetricand asymmetric) equilibria the game admits.
Figure 4: Asymmetric pure strategy equilibrium in the spatial modelxt
πR (xt−1 )

1 θR

θL

θR

−1

1
− 12 θ̂L

xt−1

πL (xt−1 )−1 θL

In Figure 4, for relevant subset of the policy space, the white area is the acceptancecorrespondence induced by the equilibrium voting strategies while the πL and πR functions are the equilibrium proposal strategies of players L and R. We do not show theequilibrium proposal strategy of the median player M ; for any status-quo, she proposesher bliss-point θM = 0. (For example, given status-quo xt−1 = −1, a proposal is accepted
if and only if it falls into [−1, 1], player L proposes − 12 , player M proposes 0 and player
R proposes 1.)
The shape of the acceptance correspondence is identical to the one that would arise
with a myopic median player: the median player accepts any proposal that is weakly
closer to her bliss-point θM = 0 than the status-quo. To see this, first note that the21

Baron (1996) assumes existence of a ‘dynamic median’ voter, which is guaranteed to exist when stage
utilities are quadratic (Banks and Duggan, 2006; Duggan, 2014).

10

shape of the acceptance correspondence is driven by the dynamic preferences of player
M . This follows because under quadratic utilities the decisiveness of player M in voting
over deterministic alternatives extends into the decisiveness of player M in voting overstochastic policy paths (Banks and Duggan, 2006; Duggan, 2014). Second, the medianplayer accepts any proposal that is closer to her bliss-point θM since both her stage
utility uM and her expected dynamic utility are both single-peaked at θM . The latter isdetermined by the entire profile of strategies and cannot be assumed to be single-peaked.
Nevertheless, the intuition behind its single-peakedness is not hard to grasp: the proposal
strategies of all the players are such that (i) given any status-quo, any proposal made is
accepted, and (ii) as the status-quo moves away from θM , the equilibrium proposals moveaway from θM , and hence the expected dynamic utility of the median player decreases.
The shape of the proposal strategies of players L and R are also related to those thatwould arise with myopic players as in Romer and Rosenthal (1979): for any status-quo,player R proposes a policy that is as close as possible to his bliss-point θR = 1 out of
those the median player accepts, that is, out of those policies that are weakly closer
to θM than the status-quo. In this asymmetric equilibrium, player L behaves similarly,
except that she is not using her bliss-point θL = −1 and uses her ‘strategic’ bliss-pointθ̂L = − 12 instead. Because θ̂L = − 12 is closer to θM = 0, that is, it is a more moderate
than θL = −1, we call this behavior moderation.
Why does player L moderate? Her incentive to do so is strategic and is driven by
the endogenous status-quo. Consider status-quo xt−1 = −1. We claim player L proposesθ̂L = − 12 in equilibrium. To see this, consider a one-stage deviation to propose θL = −1,
which is the policy that maximizes L’s static utility. Both of these policies are accepted
given the status-quo. Player L proposes θ̂L = − 21 instead of θL = −1 in order to constrain
the proposed policy of player R if this player is recognized in the next period: player R’s
policy will be 1 if player L proposes θL = −1 now, while player R’s policy will be 12 if
player L proposes θ̂L = − 21 now (Figure 4 highlights these policies by dots in its left half).
At the same time, player L is not constraining herself if she is recognized next period:
player L’s policy will be − 12 irrespective of whether she proposes θL or θ̂L now. That is,player L moderates in equilibrium using the acceptance set of the median player M andthe endogenous status-quo to constrain the future policies of her opponent player R.
Why does not player R moderate? Player R has the same incentive to constrain thepolicies of player L. Consider status-quo xt−1 = 1. We claim player R proposes θR = 1
in equilibrium. To see this, consider a one-stage deviation to propose xt = 21 . Both ofthese policies are accepted given the status-quo. However, the only effect of moderating
and proposing xt = 12 rather than θR = 1 is for player R to constrain herself if she isrecognized next period: player R’s policy will be 21 if she proposes xt = 12 now and willbe 1 if she proposes θR = 1 (Figure 4 highlights these policies by circles in its right half).
In order to constrain the policy of player L, player R would have to moderate to some
proposal below 12 , which is too costly for her in terms of the foregone static utility. Thatis, the incentive to moderate is a strategic substitute: because player L moderates, she
is effectively constraining herself and player R has no incentive to moderate. Because
player R does not moderate, her strategic bliss-point coincides with her bliss-point.
Equilibrium moderation and its strategic substitute nature are two key insights ofthe model. But the model allows us to make several additional observations. First, inorder to simplify the exposition we have chosen δ = 34 , identical recognition probabilities
and particular values for the bliss-points such that the strategic bliss-point of player L isθ̂L = − 21 . For general values of these parameters, Zápal (2016) shows that the strategic
11

bliss-point of a non-median player i is θi (1−2δr−i ), where r−i is the recognition probabilityof player {L, R} \ {i}. That is, the strategic bliss-point is a point where two forces offset
each other. The first force is standard: policies are pushed towards players’ bliss-points.
The second force is strategic: players moderate in order to constrain their opponents. The
second force intensifies and the strategic bliss-point of a player becomes more moderate
when the player becomes more patient and when the probability of recognition of her
opponent increases. In fact, when 1 − 2δr−i < 0, player i’s incentive to moderate isstrong enough for this player to propose θM = 0 for any status-quo in equilibrium, thatis, to propose in the identical way as the median player M does.
Second, because the game underlying Figure 4 is symmetric, it is intuitive that a mirror equilibrium exists in which player L does not moderate and uses a proposal strategy
with her bliss-point θL = 1, while player R moderates and uses a proposal strategy withstrategic bliss-point θ̂R = 21 . This multiplicity is driven by the strategic substitute natureof moderation: when player L moderates player R has no incentive to do so, while whenplayer L does not moderate player R has incentive to do so. However, results in Zápal(2020) suggest that this multiplicity is non-generic and restricted to symmetric games.
Namely, he shows that, generically, at most one equilibrium exists in a class of equilibria in which the proposal strategy of each player is pinned down by a single strategic
bliss-point (as in Figure 4).
Figure 5: Symmetric mixed strategy equilibrium in the spatial model
xt

0
4

0
4

1
4

1
4
2
4
3
4
4
4

4
4

3
4

2
4

1
4

2
4

1 θR

0
4

1
2

θ̂R

3
4
4
4
0
4

1
4

θL
4
4

3
4

2
4

− 12
1
4

0
4
4
4

3
4
2
4

3
4

4
4

θR

−1

− 23

2
4

1

1
2

− 12 θ̂L

0
4

1
4

2
4

3
2
3
4

4
4

4
4

πR (xt−1 )xt−1
πL (xt−1 )

3
4

−1 θL

2
4

1
4

1
4

0
4

0
4

Third, Kalandrakis (2016) shows that the game underlying Figure 4 admits a symmetric mixed strategy
 3SMPE.
 Figure 1 3  5 shows key features of this equilibrium. For status-quos
1
outside Xm = − 2 , − 2 ∪ 2 , 2 , both players L and R use strategies similar to thosefrom Figure 4, except that both players use strategies with moderate strategic bliss-pointsθ̂L = − 21 and θ̂R = 21 respectively. For any status-quo xt−1 ∈ Xm , player R mixes overtwo proposals |xt−1 | and θ̂R = 21 putting probability |xt−1 | − 21 on the latter. Player L
mixes in the similar way over −|xt−1 | and θ̂L = − 21 . (Figure 5 shows the mixing probabilities for xt−1 ∈ Xm .) This equilibrium displays two interesting features: first, it is12

symmetric and does not inherit the strategic substitute nature of moderation of the purestrategy equilibrium from above, and, second, policies outside the set of statically Pareto
efficient policies [−1, 1] are proposed and accepted with strictly positive probability for a
non-negligible set of status-quos.
The game underlying Figures 4 and 5 features three symmetric players and admits
multiple SMPE. Going beyond this setup opens new questions, some of which has beentackled by existing work. First, Zápal (2016) studies a game with three players, without
restricting their bliss-points and recognition probabilities. He proves an SMPE existence
result and shows that the equilibria he describes are generically unique, but only in a
restrictive class of equilibria with proposal strategies pinned down by strategic blisspoints. The equilibria he describes all involve moderation that is a strategic substitute,that is, in equilibrium only one of the non-median players moderates. Moreover, when
the patience and the recognition probabilities of the players are the same, it is the player
with less extreme bliss-point who moderates.
Second, do all equilibria involve moderation driven by strategic incentives? In fact,
Dziuda and Loeper (2016, 2018) highlight that bargaining with an endogenous statusquo leads to strategic polarization in a bilateral bargaining under unanimity. The logic
behind their polarizing effect and the moderating effect just discussed is the same: with an
endogenous status-quo a policy determines both the current and future policies. Consider
player L and two policies x < x0 both in [θL , θM ]. In the equilibrium described in Figure
4, both x and x0 are revised when player R proposes, but x is revised to a policy thatis worse for L than the policy x0 is revised to. Although player L prefers x to x0 , policy
x brings about worse policies than x0 , and hence L’s preference for x becomes moderateor even reverses as a result of an endogenous status-quo. Dziuda and Loeper (2016)
study equilibria in which policies are unlikely to be revised, because acceptance of apolicy requires unanimity. In these equilibria, player L prefers x to x0 and x brings about
better policies than x0 , and hence L’s preference for x becomes stronger as a result of an
endogenous status-quo. An endogenous status-quo thus induces moderation over policies
that are revised when they become status-quo and polarization over policies that remainunchanged. The contrast between Dziuda and Loeper (2016, 2018) and Zápal (2016)highlights the importance of voting rule on polarization.
Third, Kalandrakis (2016) and Zápal (2016) restrict attention to three players. Whether
equilibria exist when there are more than three players, and whether the equilibria, if
they exists, involve moderation that is a strategic substitute, are open questions. Zápal(2020) provides affirmative but partial answer to these questions. The key to his analysis
is an algorithm that derives strategic bliss-points for games with more than three players.
The algorithm is based on the insight that the incentive of players to moderate is driven
by their opponents, players with bliss-points on the other side of the median’s bliss-point,
who moderate to a smaller extent. The reason Zápal (2020) does not provide a complete
answer is because his equilibrium construction fails for some games.22
Fourth, Baron (1996) and subsequently Kalandrakis (2016) and Zápal (2016) analyze
simple majority voting. The existence and structure of equilibria in spatial model withsupermajority, unanimity or more general voting rules remain an open question.
Fifth, little is known about existence and properties of equilibria with general utility
functions or multi-dimensional policy spaces. The key complication associated with gen22

His construction fails in games with players who have bliss-points far from the median and strong
incentives to moderate. The resulting large interval between the bliss-point and the strategic bliss-point
is challenging for the equilibrium construction because it might give rise to profitable deviations.

13eral utility functions is that the median player might cease to be decisive in voting overlotteries over policy paths. Partial progress has been made with multi-dimensional policy
spaces; Zápal (2014) uses similar construction as in Zápal (2016, 2020) to construct equilibria with moderation that is strategic substitute, while Anesi and Duggan (2018) show
generic indeterminacy of Markovian equilibria in games with patient players and policy
spaces with sufficient number of dimensions. Moreover, Baron (2018), using similar techniques as Anesi and Seidmann (2015) and Baron and Bowen (2015), constructs equilibria
in a two-dimensional model with three parties with bliss-points located at vertices of an
equilateral triangle. Baron and Herron (2003) study a similar model using numericalmethods.
Finally, our understanding of dynamic bargaining models with changing preferences
is very limited, despite the obvious appeal of such extension. Several papers advanced inthis direction but only partially (Riboni and Ruge-Murcia, 2008 assume fixed proposer;
Duggan and Kalandrakis, 2011 compute equilibria numerically; Dziuda and Loeper, 2016,2018 have only two policies in the policy space; Austen-Smith, Dziuda, Harstad, andLoeper, 2019 have three policies; Bowen et al., 2017; Buisseret and Bernhardt, 2017,2018 focus on two-period models). Similarly limited is our understanding of models witha stochastic drift in policies (Callander and Krehbiel, 2014; Callander and Martin, 2017),
or models with endogenously changing policy spaces (Chen and Eraslan, 2017).

6

Efficiency

Are the equilibrium outcomes of the dynamic legislative bargaining models efficient or
not? Answering this question requires a framework for thinking about (in)efficiency of political outcomes. One possibility is to follow Besley and Coate (1998) who draw a parallelwith general equilibrium theory: one defines feasible allocations and uses players’ preferences to decide, without reference to the process that determines which allocations arise,
whether an allocation is Pareto efficient or not, and, only then, looks at the allocations
that arise as a result of an economic or political process.
Under our notation, assuming momentarily that, in any period t, the state in period t is xt−1 and that the stage utilities and policy spaces are state independent, an
allocation x = (xt )Tt=0 is a sequence of policies and is feasible if x ∈ X T . A feasible allocationPx is dynamically ParetoPefficient if there is no feasible allocation y suchthat (1 − δi ) Tt=0 δit ui (yt ) ≥ (1 − δi ) Tt=0 δit ui (xt ) for all i ∈ N , with at least one inequality strict. When states include other variables, let ωt be those other variables int, i.e., st = (xt−1 , ωt ), and let ω = (ωt )Tt=0 ∈ Ω. An allocation now is a contingentTωωsequence of policies (xω )ω∈Ω = ((xωt )t=0 )ω∈Ω and is feasible if xt ∈ X(xt−1 , ωt ) for23ω
all t and ω. A feasible allocation (x )ω∈Ω is ex-ante dynamicallyefficient if
PT t Paretoωωω
there is no feasible allocation (y )ω∈Ω such that (1 − δi )Eω [ t=0 δi ui (yt , (yt−1 , ωt ))] ≥Pω(1 − δi )Eω [ Tt=0 δit ui (xωt , (xt−1 , ωt ))] for all i ∈ N , with at least one inequality strict. Forωgiven ω, feasible x isPex-post dynamically Pareto efficientif there is no feasible
PT in ω
T
ω
t
ω
ω
t
ω
y such that (1 − δi ) t=0 δi ui (yt , (yt−1 , ωt )) ≥ (1 − δi ) t=0 δi ui (xωt , (xt−1 , ωt )) for alli ∈ N , with at least one inequality strict. Looking at a single period t, xt ∈ X(st ) is23

The variable ω is what Mas-Colell, Whinston, and Green (1995, chapter 19) call a state and should beunderstood as a ‘complete description of a possible outcome of uncertainty’ (page 688). Our contingent
policy sequences are analogous to (state) contingent commodities, one of the central concepts in generalequilibrium under uncertainty.

14

statically Pareto efficient in st if there is no yt ∈ X(st ) such that ui (yt , st ) ≥ ui (xt , st ) for
all i ∈ N , with at least one inequality strict.
P
In the distributive setting with policy space {x ∈ Rn+ : i∈N xi ≤ 1}, linear preferences and homogeneous discount factors, i.e., ui (x) =δi = δ for all i, (xω )ω∈Ω
Pxi andis ex-ante dynamically Pareto efficient if and only if i∈N xω
t,i = 1 for all t and ω, that
ωis, if (x )ω∈Ω consists of policies without waste. This implies that the rotating dictator equilibrium of Kalandrakis (2004) is ex-ante dynamically Pareto efficient. This also
implies that the equilibria constructed by Anesi and Seidmann (2015), in case of linear
preferences and homogeneous discount factors, are ex-ante dynamically Pareto efficient
if and only if based on simple solutions that consist of policies without waste.
General characterization of Pareto efficient policy sequences, outside special cases
like the one just discussed, is an open question.24 Nevertheless, we make the following
general observations. First, in the distributive setting, heterogenous discount factors
require efficient policy sequences to allocate larger shares to more patient players in later
periods. Failure of equilibria to generate this patter is one of the sources of inefficiency
in the distributive setting of Anesi and Seidmann (2015).
Second, with risk averse players, when states include variables that do not affectplayers’ preferences and the policy space, the efficient policy sequences do not vary withthese variables. That is, if (xω )ω∈Ω is ex-ante dynamically Pareto efficient and if ωand ω 0 differ only in the preference and policy space irrelevant variables, then xω =0xω . Standard example of variables included in states that are preference and policy
space irrelevant are proposer identities. Dynamic ex-ante Pareto efficiency then requiresthat equilibrium policies do not depend on proposer identities, which they often do.
The resulting inefficiency stemming from sensitivity of equilibrium policies to proposer
identities is one of the sources of inefficiency in Anesi and Seidmann (2015) and is what
Bowen et al. (2017) call political risk. The converse arises when states include preferenceor policy space relevant variables. Ex-ante dynamic Pareto efficiency then requires thatequilibrium policies track the preference or policy space relevant variables. However, inmost spatial models the equilibrium policy in the current period is often the status-quo
because change to any other policy is unacceptable to a sufficient number of players. The
resulting inefficiency stemming from insensitivity of equilibrium policies to preferenceor policy space relevant variables is what Bowen et al. (2017) call gridlock.25 However,gridlock can also improve efficiency as Piguillem and Riboni (2013, 2015) show. They
study environments where government would benefit from ability to commit to futurelevels of taxation. In the absence of commitment, endogenous status-quo and the resulting
gridlock represents a form of (endogenous) commitment that improves efficiency.
Third, it is not hard to see that if (xω )ω∈Ω is dynamically ex-ante Pareto efficient,then, for each ω, the policy sequence xω is dynamically ex-post Pareto efficient in ω,ω26
which in turn implies that xω
This
t is statically Pareto efficient in (xt−1 , ωt ) for each t.
means that the static Pareto inefficiencies noticed by Riboni and Ruge-Murcia (2008),
Dziuda and Loeper (2016, 2018) and Kalandrakis (2016) imply dynamic ex-post and ex24
See Anesi and Seidmann (2015) for the discussion of the issues involved. Standard approach tocharacterization of Pareto efficient allocations is via an optimization problem in which utility of one
player is maximized subject to a lower bound on the utilities of the other players. See Mas-Colell et al.
(1995, chapter 16) and Duggan (2017b) for details of this approach.
25
Bowen et al. (2017) link political risk and gridlock to dynamic ex-post Pareto inefficiency. Because
ex-post inefficiency implies ex-ante inefficiency, as we discuss in the next paragraph, political risk andgridlock are linked to ex-ante inefficiency as well.
26
See Mas-Colell et al. (1995, chapter 19.D) for sketch of the argument.

15

ante Pareto inefficiencies. The sources of inefficiency in Riboni and Ruge-Murcia (2008)
and Dziuda and Loeper (2016, 2018) are similar. These papers analyze models in which
players’ preferences are subject to stochastic shocks. In some periods the shocks are suchthat players agree, i.e., have identical stage utilities, while in other periods the shocks are
such that players disagree, i.e., have different stage utilities. An important insight of these
papers is that players might fail to implement policies that maximizes their common stage
utility even in the periods when they agree. This is because with endogenous status-quo,
the implemented policy acquires two roles: it constitutes a policy and hence enters stage
utility of the players, but it also becomes the status-quo for the next period and hence
determines players’ future bargaining position. And agreeing to implement the policythat maximizes players’ stage utility might compromise future bargaining position forsome of them.
In light of Riboni and Ruge-Murcia (2008) and Dziuda and Loeper (2016, 2018), one
might think that endogenous status-quo leads to inefficient policies. Bowen, Chen, andEraslan (2014) show that this is not true. In their model two parties bargain repeatedly
over the division of a fixed budget into public good and private transfers. Contrasting
equilibria with exogenous and endogenous status-quo level of public goods, they showthat having endogenous status-quo increases the equilibrium level of public goods andimproves efficiency. The effect arises because higher status-quo level of public goods provides players with higher status-quo utility and hence protects them from expropriation
when not recognized as proposers. These contrasting results show that the link between
endogenous status-quo and efficiency is not fully understood, despite the partial progress
made by Bowen et al. (2017) who suggest that efficiency might be achieved by combining
exogenous and endogenous status-quo. This can be attained, for example, by having
public goods with endogenous status-quo along with public goods with exogenous statusquo. Alternatively, a generalized endogenous status-quo in the form of sunset provisions
might lead to efficiency.
Ultimately, we are interested in efficiency of political institutions. The models wewrite capture key features of these institutions. Because any given model can give rise tomultiple outcomes, either due to multiplicity of equilibria or due to equilibrium mixing,
one has to take a stand on whether to require one or all possible outcomes to be Pareto
efficient.

7

Open questions

We end the survey with a list of what we deem to be open questions, beyond the ones
already mentioned in Sections 5 and 6. First, Anesi and Duggan (2018) show that thelarge multiplicity of SMPE observed by Anesi and Seidmann (2015) for the distributive
model extends to other environments. Their result requires infinite horizon, continuous
policy space, patient players, absence of veto players, and strictly positive recognition
probabilities and hence does not apply to a number of papers.27 Nevertheless, theirwork highlights that the literature we survey lacks an equilibrium refinements that would
restore the predictive power of the models it studies. Eraslan and Piazza (2019) propose
27

For example, Anesi (2010); Diermeier and Fong (2011); Dziuda and Loeper (2016, 2018); Anesi andDuggan (2017); Diermeier, Egorov, and Sonin (2017) study models with finite policy spaces, Buisseretand Bernhardt (2017, 2018); Chen and Eraslan (2017) focus on finite horizon, Duggan, Kalandrakis,
and Manjunath (2008); Diermeier et al. (2017); Nunnari (2018) include veto players, and Riboni and
Ruge-Murcia (2008); Riboni (2010); Diermeier and Fong (2012) feature fixed proposers.

16

refining MPE to ensure that the property chosen to refine MPE is satisfied in (i) a single
player version of the model, and (ii) a finite horizon version of the model. Specifically,
they restrict attention to the class of equilibria in which equilibrium strategies satisfy ashape restriction that is also satisfied by the optimal policy of a dictator. Remarkably,
the refinement allows them to fully characterize equilibria and establish its uniquenesswhen it is unique. Interestingly, the identical shape refinement allows Gersbach et al.
(2018) to establish a unique equilibrium as well. They also provide a micro-foundationof this refinement using a finite horizon version of their model.
Second, although the surveyed literature includes a number of applications, further,
especially economically relevant ones, should provide novel insights.28 In particular, thedynamic nature of the surveyed models seems to be well suited to the study of debt (seePiguillem and Riboni, 2018; Bouton, Lizzeri, and Persico, forthcoming, for headways),
dynamic issues of taxation (see Piguillem and Riboni, 2013; Ma, 2014, for existing work),
and should be embedded in realistic macro models (see Grechyna, 2017; Azzimonti, Karpuska, and Mihalache, 2020, for early examples).
Third, a study of incomplete information is missing, as in the literature with exogenous
status-quo. This is an obvious gap that should be addressed. Possible developments inthis spirit include dynamic bargaining over pie with size that stochastically changes overtime (as in Merlo and Wilson, 1995), or dynamic bargaining with privately-observed
type-dependent preferences (see Anesi, 2018, for an initial contribution).
Finally, very little effort has been devoted to testing the predictions of the surveyed
models. The limited existing evidence comes from laboratory experiments (Battagliniand Palfrey, 2012; Battaglini et al., 2012, forthcoming; Baron, Bowen, and Nunnari, 2017;
Nunnari, 2018; Agranov, Cotton, and Tergiman, 2020, see also the chapter on laboratory
experiments in this volume by Marina Agranov) and we are not aware of any evidence
using observational data.

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