Global COE Hi-Stat Discussion Paper Series 063

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Enforcing International Trade Agreements with
Imperfect Private Monitoring:
Private Trigger Strategies and a Possible Role for the WTO

Hi-Stat Discussion Paper

Jee-Hyeong Park

April 2009

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Enforcing International Trade Agreements with Imperfect Private Monitoring:
Private Trigger Strategies and a Possible Role for the WTO
byJee-Hyeong Park*
First Draft: June 2004
Revised: February, 2009
Abstract
International trade disputes often involve the WTO as a third party that generates impartial opinions ofpotential violations when countries receive imperfect and private signals of violations. To identify therole that the WTO plays in enforcing trade agreements, this paper first characterizes what countries canachieve alone in a repeated bilateral trade relationship in which they can secretly raise their protectionlevels through concealed trade barriers. In particular, countries adopt “private trigger strategies (PTS)”
under which each country triggers a punishment phase by imposing an explicit tariff based on its
privately observed imperfect signals of such barriers. This paper identifies the condition under whichcountries can restrain the use of concealed barriers based on simple PTS, where each country imposes its
static optimal tariff in all periods under any punishment phase: The sensitivity of private signals rises in
response to an increase in concealed protection. Any equilibrium payoff under almost strongly
symmetric PTS will be identical to the one under simple PTS, as long as the initial punishment is
triggered by a static optimal tariff, justifying the paper’s focus on simple PTS. With countries
maximizing their expected payoffs under the optimal PTS, they will not push down the cooperative
protection level to its minimum attainable level, thus not setting it to the free trade level even when it isattainable. To analyze a possible role of the WTO, this paper considers “third-party trigger strategies(TTS)” under which the WTO allows each country to initiate a punishment phase based on the WTO’s
judgment (signals) about potential violations. The WTO changes the nature of punishment-triggering
signals from private into public, enabling countries to use punishment phases of any length under TTS,
which in turn facilitates a better cooperative equilibrium. The optimal TTS will involve an asymmetricand minimum punishment if the probability of a punishment phase being triggered is low enough, but it
will entail punishments involving a permanent Nash tariff war if the probability of a punishment being
triggered is high enough. A numerical comparison of the optimal TTS and optimal PTS indicates thatthe contribution of the WTO is likely to be significant when the signals of potential violations are
relatively accurate. The WTO enables countries to adopt a more efficient punishment, such as theasymmetric and minimum punishment, which in turn enables countries to be less tolerant of potential
violations and attain a higher level of cooperation as a result.
JEL Classification Code: F020; F130
*Current address: Department of Economics, Princeton University, Princeton, NJ 08544, USA; tel: +1609-258-6851; fax: +1-609-258-1374; email: Jeepark@princeton.edu; Home Institution: Department of
Economics, Seoul National University, Seoul, 151-746, Korea; email: j-hpark@snu.ac.kr. For helpfulcomments, I would like to thank Pol Antras, Kyle Bagwell, Eric Bond, Kevin Cotter, Carl Davidson, In
Ho Lee, Giovanni Maggi, Robert Staiger, and other participants of seminars at Ewha Woman’s
University, Princeton University, Seoul National University, Yale University, Yonsei University, andWayne State University, and at Hitotsubashi COE Conference, International Conference on Game
Theory at Stony Brook, International Economic Institutions Workshop in Seoul, Midwest International
Economics and Economic Theory Meetings, and 2004 Far Eastern Meeting of the Econometric Society.
I also would like to thank the editor and two anonymous referees for comments that led to substantial
improvements in the article.

1. Introduction
Enforcing international trade agreements often entails disputes in which countries present
different opinions about potential deviations from the agreements. Differences in opinion may
take various forms, such as disagreement over the existence of concealed trade barriers as indisputes between the U.S. and Japan during 1980s, or disagreement over the legitimacy ofantidumping duties, a frequent theme in the dispute settlement procedure of the World Trade
Organization (WTO).

These disagreements reflect imperfectness of information about

deviations from trade agreements. In addition to being imperfect, each country’s opinion ofpotential violations can be private in the sense that the country’s true opinion is not known toother countries. For example, when the United States Trade Representative (USTR) engages ina negotiation with China to curtail piracy and counterfeiting that impede the U.S. intellectual
property rights, China and the USTR may not know each other’s true belief regarding the
Chinese government’s effort level to curtail such practices, which in turn may contribute to abreakdown in the negotiation. 1
Trade disputes typically involve the WTO as a third party that generates impartial opinionsof potential violations when countries receive imperfect private signals of violations. 2 To
identify the role that the WTO plays in facilitating the enforcement of trade agreements, thispaper first assumes away the presence of the WTO and characterizes what countries canachieve alone in a repeated bilateral trade relationship where each country can secretly raise its
protection level through concealed trade barriers.

In particular, this paper explores thepossibility that countries adopt “private trigger strategies (PTS)” under which each country
triggers a punishment phase by imposing an explicit tariff based on its privately observed
imperfect signals of such barriers. The analysis identifies the condition under which countries

1

The signals that the USTR receives regarding potential deviations from trade agreements often come from the
U.S. companies whose interests are affected by deviations. Such signals may involve the companies’ private
information. Public revelation of the private information can be costly for those companies, forcing the signals tobe private. There exist many U.S. antidumping cases in which foreign companies under investigation decide notto provide costs and sales related “private” information despite the fact that such nondisclosure often leads toexcessive dumping duties based on “best information available.”2
When countries bring a disputed case to the WTO presenting different opinions about potential violations, theDispute Settlement Procedure of the WTO encourages them to solve disputes through a consultation stage prior toinitiating a panel stage where a third-party panel provides a ruling on the disputed case. Countries can appeal thepanel’s ruling to have the case examined by an Appellate Body. Once the case has been determined by theAppellate Body, the losing “defendant” must comply with the ruling or face the possibility of trade sanctions by
the complaining side.

1

can restrain the use of concealed trade barriers based on simple PTS where each country
imposes its static optimal tariff in all periods under any punishment phase. The condition isthat the sensitivity of private signals rises in response to an increase in concealed protection.
This paper then establishes that the equilibrium payoff of any almost strongly symmetric PTS
will be identical to the one under simple PTS, as long as the initial punishment is triggered by astatic optimal tariff.

Given this generality result, it characterizes the optimal PTS thatmaximize symmetric countries’ expected payoffs under simple PTS. The analysis shows that it
is not optimal to push down the cooperative protection level to its minimum attainable level,thus not setting it to the free trade level even when it is attainable.
To analyze a possible role that the WTO can play in enforcing trade agreements, this paper
analyzes “third-party trigger strategies (TTS)” under which the WTO decides upon whether aviolation has occurred and tells each country to initiate a punishment phase based on its
decision as an impartial third party. The WTO under TTS changes the nature of punishmenttriggering signals from private into public, enabling countries to employ punishment phases ofany length, which in turn can help countries to attain a better cooperative equilibrium. The
comparison between the optimal TTS and optimal PTS illustrates how and to what degree that
the WTO can help countries to enforce international trade agreements beyond what countries
can achieve alone under PTS. The analysis establishes that the optimal TTS will involve an
asymmetric and minimum punishment if the probability of a punishment phase being triggered
is low enough, but it will entail punishments involving a permanent Nash tariff war if theprobability of a punishment being triggered is high enough. A numerical comparison of the
optimal TTS and optimal PTS indicates that the contribution of the WTO is likely to besignificant when the signals of potential violations are relatively accurate.

Under suchcircumstances, the WTO enables countries to adopt a more efficient punishment, such as theasymmetric and minimum punishment, which in turn enables countries to be less tolerant ofpotential violations. As a result, a higher level of cooperation is attained compared to thesituation without the WTO.
This paper contributes to the literature on two levels. First, it provides a new way ofunderstanding the role that the WTO plays in enforcing international trade agreements in thepresence of potential violations of which countries receive imperfect and private signals.
Because the enforcement of trade agreements ultimately rely on the threat of invoking trade2sanctions against violations, previous studies have also analyzed the enforcement issue using
trigger strategies in a repeated game setup. 3 Earlier models on this issue, such as Dixit (1987),
Bagwell and Staiger (1990), and Riezman (1991) suggest that the WTO may serve the role ofhelping countries coordinate on more efficient equilibria among multiple equilibria thattypically arise in a repeated game setup. To model a more explicit role of the WTO, Kovenoch
and Thursby (1993) assume that the Dispute Settlement Procedure (DSP) of the WTO has an
informational superiority (over trading countries) of distinguishing between true violations andmistaken perceptions, which in turn assists the workings of a reputation mechanism to support
cooperation. 4 In a multilateral trading environment, Maggi (1999) shows that the WTO may
facilitate cooperation enhancing third-country retaliations by disseminating information about
deviations. 5

While these models introduce more specific roles for the WTO to play incoordinating a cooperative equilibrium, the literature has not resolved the question of why the
WTO is necessary for coordination because these previous studies offer no theory of why
countries could not coordinate a cooperative equilibrium in the no-WTO environment. 6
This paper represents the emergence of the WTO as a change in the observation structureof the repeated game. The presence of the WTO changes the nature of punishment-triggering
signals from private into public. This enables countries to employ punishment phases of any
length, and as a result countries can attain a better cooperative equilibrium. As emphasized in
the analysis of PTS, the private nature of signals of potential violations limits the flexibility ofpunishment phases that countries can use in the absence of the WTO because they need toprovide countries with the incentive for truthful revelation of private signals in triggeringpunishment phases. The WTO can publicize its opinions of potential violations, which relaxes

3

Bagwell and Staiger (2002) provide a comprehensive review of studies analyzing international trade agreements
as a subgame perfect equilibrium in a repeated trade relationship.
4
Hungerford (1991) develops a model where the WTO plays a negative role in enforcing trade agreements
because the model assumes that the DSP of the WTO involves uninformative and costly investigation.
5
As pointed out by a referee, third-party retaliation is rarely observed, and Maggi (1999) does not model
information transmission directly and offers no theory as to why information could not be shared in the absence ofthe WTO.
6
Bagwell and Staiger (2005) and more recently Bagwell (2008) analyze the issue of implementing trade
agreements when each government is privately informed about its own domestic political pressure for protection.
Their analysis differs from this paper’s because it focuses on identifying the structure of trade agreements that can
induce the truthful revelation of private political pressure rather than analyzing the enforcement of trade
agreements when countries privately observe imperfect signals of potential deviations.

3

such a constraint in developing an optimal punishment mechanism, enabling a better
cooperative equilibrium even in the absence of any informational superiority of the WTO. 7
The other contribution of this paper is more generally toward the literature on repeatedgames with imperfect private monitoring. It is well known that analyzing repeated games withimperfect private monitoring is difficult because utilization of privately observed signals in
determining continuation plays can destroy the recursive structure of repeated games. 8 Kandori
and Matsushima (1998) and Compte (1998) demonstrate that communication can serve as apublic signal that restores the recursive structure and enables players to achieve cooperation insuch a repeated game. 9

In the absence of communication, PTS in this paper show analternative way to restore a recursive structure to repeated games with imperfect private
monitoring. 10 If players can choose explicit actions as well as concealed actions as in the caseof governments’ choosing their protection levels, then players can avoid confusion between
punishment phases and non-punishment phases by requiring players to signal an initiation ofpunishments by their explicit, thus public actions.11 This can ensure a recursive structure of the7

Ludema (2001) emphasizes that the DSP of the WTO may require trade agreements to be renegotiation-proof bypromoting communication among countries prior to starting punishments. This negatively affects cooperation by
forcing countries to rely on weaker (renegotiation-proof) punishments. In contrast to his analysis in a repeatedgame with perfect monitoring, an optimal trade agreement with imperfect monitoring would not typically involvethe lowest levels of protection with the most severe credible threat because punishments do occur. With imperfectprivate monitoring, the WTO can help countries to achieve better cooperation by enabling countries to adopt
weaker punishments, as shown in this paper.
8
Kandori (2002) discusses this point and recent developments in repeated games with private monitoring in detail.
9
In these studies, the communication among players entails no cost (so that it is “cheap talk”) and each country’s
revealed private information does not affect its own continuation payoff in order to ensure truthful revelation ofprivate information. As pointed out by a referee, however, they are unable to show what communication “does”
though, since they were unable to show what would happen in the no-communication setting.
10
A referee points out that communication is not illegal in the context of international trade agreements, different
from communication in the context of price-fixing oligopolists. It suggests the possibility of using communicationto achieve cooperation in the absence of the WTO. For example, one may consider applying the communication
mechanism developed by Kandori and Matsushima (1998) to sustain international trade agreements. There are
two reasons why such mechanism may not work well among countries. First, in the context of an international
relationship, it is not easy to allege potential violations when violations do not affect the alleging country,
especially when such allegations will negatively affect the alleged country. In fact, the DSP of the WTO reduces
the burden for countries of playing the third-party role of “alleging” potential wrong doings of another country bymaking the DSP to be a kind of legal procedure primarily run by experts. Second, the use of transfers is rarely
observed between countries, especially as compensation for potential violations of international trade agreements.
If countries need to rely on imposition of tariffs in punishing potential deviations as they do in practice, then“communication” will face a similar incentive constraint as the one under PTS because “alleging your trading
partner’s wrong doings” needs to be supported by the “action” of punishing such behavior with tariffs.
11
In the context of collusion among firms engaging in secret-price cuttings, for example, firms can employ
advertised (thus public) sales to initiate a punishment phase against potential defections from collusive pricing.
Similar to Green and Porter (1984), then occasional “explicit” price wars will occur as dynamic equilibrium
behaviors to sustain collusion overtime. Different from the model of Green and Porter where firms would always

4repeated game along the equilibrium path. The analysis of PTS specifies the condition underwhich such trigger strategies can restrain the deviatory use of concealed actions.
The use of explicit actions to initiate punishment phases, however, does not simplify
everything. If each player triggers a punishment phase based on its private signals as it does
under PTS, then in any period after a cooperative one, players need to choose their actions
knowing only the probability of a punishment phase being triggered by other players. Because
an action taken by each player in a current (cooperative) period affects the probability of apunishment being triggered in a next period, an optimal action in the next period depends on anaction taken in the current period, and an optimal action in a period after the next perioddepends on an action taken in the next period, and so on until a punishment phase is triggered.
Therefore, the use of PTS necessitates a complete characterization of optimal and potentiallydeviatory action sequences that each player may take in checking incentive compatibility forsuch strategies. 12 Using a dynamic programming method, this paper establishes that countries
can use simple PTS in achieving cooperation as long as the private signals satisfy somesensitivity constraints. With regard to the possibility of proving a folk theorem result under
PTS, this paper generates yet another anti-folk theorem result within a class of private trigger
strategies, namely almost strongly symmetric PTS, when private monitoring is far from being
perfect. 13
The paper is organized as follows. Section 2.1 develops a bilateral trade model where eachcountry receives imperfect private signals of the other country’s use of concealed trade barriers
and specifies simple PTS.

Section 2.2 describes incentive constraints under simple PTS,

providing conditions under which those incentive constraints are satisfied. Section 3.1 shows
start a price war concurrently, each firm may unilaterally initiate a price war phase by lowering its explicit price(and gains from it in that period) under such private trigger strategies and the lengths of price war phases will be
endogenously determined.
12
This aspect of PTS does not allow one to apply dynamic programming techniques develops by Abreu et al.
(1986) to characterize the set of equilibrium payoffs under PTS because those techniques rely on the “one-stage
deviation principle.” For further discussion of the “one-stage deviation principle,” see footnote 21 in Section 2.2.
13
Ely and Välimäki (2002) provide a concise discussion of why many of the strategies to prove folk theorems withpublic monitoring fail when monitoring is private and conditionally independent. This paper also analyses thecase where monitoring is private and conditionally independent and shows that countries cannot attain thesymmetric efficient frontier under almost strongly symmetric PTS if the monitoring is far from perfect (Corollary
1 to Proposition 2). This anti-folk theorem result, however, may rely on the countries’ use of distortional
measures like tariffs to punish potential violations. For example, Horner and Jamison (2007) show that fullcollusion can be approximated under minimal information in private strategies where punishment phases arecarefully designed so that no loss occurs (collectively) for colluding firms. Such punishments are possible because5that countries can support simple PTS in the repeated protection-setting game, achieving acertain level of cooperation. It also establishes that the equilibrium payoff under any almost
strongly symmetric PTS will be identical to the payoff under simple PTS as long as eachcountry starts the initial punishment phase by imposing its static optimal tariff. Section 3.2
then characterizes optimal simple PTS under which countries maximize their joint expecteddiscounted payoffs. To demonstrate a role that the WTO may play in enforcing international
trade agreements, Section 4 characterizes optimal TTS and provides a numerical comparison
between the optimal PTS and optimal TTS. Section 5 discusses some additional factors that
may severely limit the use of PTS and summarizes results. It concludes with a discussion of a
possible extension of this paper’s analysis towards further understanding of the Dispute
Settlement Procedure of the WTO.

2. Private Trigger Strategies
2.1. A Trade Model with Concealed Trade Barriers and Private Trigger Strategies
The basic bilateral trade model comes from Dixit (1987) with concealed trade barriers
being introduced in a way similar to Riezman (1991). There exist two countries, home (H) andforeign (F), producing and trading two products, good 1 and good 2, under perfect competition.
H imports good 2 and F imports good 1. In each period each country simultaneously chooses
its action, a i ≡ (τ i , e i ) ∈ A i , where both elements of A i may take any non-negative real
number.

Total import protection level and explicit tariff level are given by τ i and e i ,respectively, with i = * or none. Variables with and without superscripts * denote foreign andhome variables, respectively.

I assume that τ − e ≥ 0 and τ ∗ − e ∗ ≥ 0 , representing theconcealed protection levels of H and F, respectively. The local prices p1 , p 2 , p1∗ , and p 2∗ arerelated as follows: p2 = p2∗ (1 + τ ) and p1∗ = p1 (1 + τ ∗ ) . 14

Given the assumption of perfectcompetition, I can define each country’s one-period payoff function as a function of the terms

firms can avoid collective losses as long as any low-cost firm ends up selling its product at a monopoly price, aspecial feature that countries in a trade relationship may not replicate easily in their punishment phases.
14
Thus, this paper does not consider the possibility of using negative or prohibitive protection.

6

of trade, represented by π (≡ p1 / p 2∗ ) , and its own total protection level, τ i . Such a payoff
function, denoted by wi (π ,τ i ) , induces a corresponding import demand function, m i (π ,τ i ) .
In the absence of uncertainty (no random element) in this world, each country’s amount ofimports is a deterministic function of its own total protection level and the terms of trade. Thisimplies that each country may figure out the exact level of the other country’s protection basedon the information about the terms of trade and the amount of imports, even in the presence ofconcealed trade barriers. However, when I introduce uncertainty into the model as a way ofrepresenting shocks to technology or preferences, exact derivation of other countries’
protection levels based on the amount of imports and the terms of trade may become
impossible. Uncertainty caused by random shocks can be modeled into random components in
the import demand functions as follows:(1)

mti = m i (π t ,τ ti , θ ti ) ,where θ ti ∈ Θ i denotes each country’s random components affecting its import demand at
period t, which follow a joint density function, f( θ t , θ t∗ ) that is iid across periods. Subscript t
denotes the variables determined in period t. In equilibrium, the following balance of payment
condition should be satisfied:(2)

π t ⋅ m(π t ,τ t ,θ t ) = m ∗ (π t ,τ t∗ ,θ t∗ ) .

This determines the equilibrium values for π t , mt , and mt∗ as functions of τ t , τ t∗ , θ t , and θ t∗ .
Given that each country sets its total protection level prior to the realization of random
shocks, each country’s one-period expected payoff, denoted by ui, is a function of bothcountries’ total protection levels:(3)u i (τ ti , τ t j ) =∫∫ w (π (τ ,τ , θ ,θ ),τ ;θ ) f (θ , θ )dθ dθ ,
i

t

t

∗
t

t

∗
t

i
t

i
t

t

∗
t

t

∗
t

(θ t ,θ t∗ )∈( Θ , Θ ∗ )where w i (π , τ i ;θ i ) represents each country’s one-period payoff function that is affected byrandom shocks, θ i , and where i ≠ j.
This paper focuses on the analysis of symmetric equilibria of a repeated protection-setting
game between symmetric countries. Thus, I assume that u (τ 1 ,τ 2 ) = u ∗ (τ 1 ,τ 2 ) for all nonnegative, real values of τ 1 and τ 2 .

Regarding derivatives of u(τ ,τ ∗ ) and u ∗ (τ ∗ ,τ ) withrespect to τ and τ ∗ , I assume that the following standard trade-theoretic results continue to7

hold in the presence of random variables: ∂u ∂τ > 0 at τ = 0 (each country has an incentive toraise its protection level above zero); ∂u ∗ ∂τ < 0 (such protection hurts the other country);∂u ∂τ + ∂u ∗ ∂τ < 0 (such protection also reduces the total payoff to H and F as it creates

distortional losses). For analytical simplicity, I introduce the following additional assumptions:∂ 2 u ∂τ < 0 (the marginal gain from protection decreases as the protection level increases);2∂ 2 u ∂τ ∂τ ∗ = 0 (the marginal gain from protection is not affected by the other country’s
protection level). 15 These additional assumptions guarantee the existence of a unique static
optimal protection level for H, which I denote by h (> 0). The one-shot protection-setting gamebetween H and F then generates a Nash equilibrium where (τ ,τ ∗ ) = (h, h*) with h = h* bysymmetry.
Private monitoring is specified as follows. At the end of period t, each country privately
observes realized values of its payoff and own random variable, (u ti , θ ti ) , and both countries
observe a pair of explicit tariffs, (et , et∗ ) . Denote the privately observed signal by ω ti = (u ti ,θ ti )∈ Ω i . I assume that the probability distribution of private signal profile conditional on action
profile has full support, that is Pr (ω t , ω t∗ at , at* ) > 0 for each ωt ∈ Ω , ωt∗ ∈ Ω ∗ , at ∈ A andat∗ ∈ A∗ . Note that while each country cannot infer the exact level of the other country’s
concealed protection even after observing its private signal (because it does not know therealized value of the other random variable), the privately observed information can serve as ameasure for detecting the other country’s potential use of concealed protection. 16

More

specifically, H can choose a subset of its private signals, Ω D ∈ Ω , so that ∂Pr (ωt ∈ Ω D ) / ∂τ t∗> 0, with Pr (ωt ∈ Ω D ) ≡ Pr (ωt ∈ Ω D at , at∗ ) denoting the probability that H’s private signal
belongs to Ω D conditional on an action profile. For example, H can assign values of u t thatare less than a critical value as the payoff part of Ω D . This can induce ∂Pr (ωt ∈ Ω D ) / ∂τ t∗ > 0

15

These properties of a social utility function can be derived from a two good, partial equilibrium model of tradewith linear demand and supply curves. See Bond and Park (2002) for derivation of such properties.
16
Once H observes ut, θt, and τt, for example, H can calculate the probability of τt* ≤ l (a certain protection level)l⎛⎞by Pr(τt*≤ l | ut,θt,τt) = ∫ ⎜f (θ t , θ ∗ )dθ ∗ ⎟dτ ∗ where Θ*(ut,θt,τt,τ*) = {θ* ∈ Θ* | u(τt,τ*,θt,θ*) = ut}.
∫⎟⎜
0 ⎝ θ ∗ ∈Θ ∗ (u t ,θ t ,τ tτ ∗ )⎠8

because ∂ut / ∂τ t∗ < 0 , and the sensitivity of u t against τ t∗ can improve once it is properly
controlled for θ t . With regard to the relationship between ωt and ω t∗ , I assume conditional
independence, meaning that for each action profile, countries’ private signals are independently
distributed of one another. This implies that each country cannot infer the other country’s
private signal based on its own private signal. 17 For symmetry between H and F, I also assume
that Pr (ωt ∈ Ω D ) = Pr (ωt* ∈ Ω D ) for all (τ t , et ) = (τ t∗ , et∗ ) ∈ A = A* and Ω D ∈ Ω = Ω ∗ .
Given the stage game and associated private monitoring depicted as above, I can describe
an infinitely repeated protection-setting game between H and F as follows. A strategy for eachcountry is defined by s i = ( s i (t )) t∞=1 with(4)s i (t ) : ( Ai ) t −1 × (Ω i ) t −1 × ( E j ) t −1 → Ai ,where E j denotes the set of possible explicit tariffs that each country can impose in a periodwith e j ∈ E j and j ≠ i. s i (t ) assigns each country’s current action (τ ti , eti ) based on the historyof its own previous actions, (a i ) t −1 ≡ (a1i , a 2i ⋅ ⋅⋅, ati−1 ) ∈ ( Ai ) t −1 , the history of its own private
information, (ω i ) t −1 ≡ (ω1i , ω 2i ⋅ ⋅⋅, ωti−1 ) ∈ (Ω i ) t −1 , and the history of the other country’s explicit
tariffs, (e j ) t −1 ≡ (e1j , e2j ,⋅ ⋅ ⋅, etj−1 ) ∈ ( E j ) t −1 with j ≠ i. If each country conforms to its strategy
defined in (4), then the expected discounted payoff is given by:
(5)

⎡∞
⎤
V i ( s i , s j ) = E ⎢∑ u i (τ ti ,τ t j )(δ C ) t −1 ( s, s * )⎥ ,⎣ t =1⎦

where E[⋅⏐(s, s*)] is the expectation with respect to the probability measure on histories
induced by the strategy profile (s, s*), and where δ C ∈ [0, 1) denotes the common discountfactor with i ≠ j. Now, I define a supergame equilibrium in this infinitely repeated protectionsetting game as follows:

17

Matsushima (1991) analyzes repeated play of stage games with a unique static Nash equilibrium andconditionally independent private signals, a problem that is similar to the repeated protection-setting game of thispaper, and shows that any pure-strategy equilibrium other than the static Nash equilibrium should involve
conditioning on payoff-irrelevant history. As discussed by Ely and Välimäki (2002), repeated games withimperfect private monitoring, especially with conditionally independent private signals, limit the use of strategies
that are often useful for repeated games with public monitoring under which each player typically has a strict
incentive to follow her equilibrium strategy after every history. Private trigger strategies considered in this paper
will be subject to similar constraints, but differ from previous works by considering the use of explicit actions, liketariffs, as a punishment coordination device.

9

Definition 1. 18 A strategy profile ( s, s ∗ ) is a supergame equilibrium in the repeated gamebetween H and F, if V ( s, s ∗ ) ≥ V ( s / , s ∗ ) and V ∗ ( s ∗ , s) ≥ V ∗ ( s ∗ / , s) for all s / ≠ s and s ∗ / ≠ s ∗ .
To explore the possibility of supporting a cooperative protection level, denoted by l, that islower than the one-shot Nash protection level (h > l) as a symmetric supergame equilibrium of
the repeated game described above, I consider “private trigger strategies” under which eachcountry uses its private signal, ω and ω*, as a device to trigger a punishment phase against theother country’s potential use of concealed protections. 19 Focusing on symmetric strategies
with s(t ) = s ∗ (t ) for all a t −1 × ω t −1 × (e ∗ ) t −1 = (a ∗ ) t −1 × (ω ∗ ) t −1 × e t −1 and t ≥ 1, I describe H’s
strategy s (and accordingly, F’s strategy s ∗ ) as follows:(i) Given that period t − 1 was a “cooperative” period with (et −1 , et∗−1 ) = (0, 0), H continues
cooperating by setting (τ t ,et ) = (l, 0) if ω t −1 ∉ Ω D , but it initiates a punishment phase by
setting (τ t ,et ) = (h, h) if ω t −1 ∈ Ω D .
(ii) Given that a “punishment” phase was initiated in period t − 1 with (et −1 , et∗−1 ) ≠ (0, 0), H sets

(τ ,e) = (h, h) for the following (T − 2) periods and it continues to do so one more periodwith probability λ if et −1 > 0 and et∗−1 = 0 ; H sets (τ ,e) = (h, h) for the following (TS − 2)
periods and it continues to do so one more period with probability λS if et −1 > 0 and et∗−1 > 0 ,where T and TS are integer numbers that are greater than or equal to 2, and λ and λS belong
to [0, 1]. H knows these variables (T, TS, λ, λS) when it initiates a punishment phase. The
actual length of a punishment phase is determined by some public randomizing device(determining λ and λS) after the punishment phase has been initiated.
(iii) In period 1 and other “initial” periods right after the end of any punishment phase, H sets

(τ ,e) = (l, 0) with probability (1 − Pr) but initiates a punishment phase by setting (τ i , e i ) =(h, h) with probability Pr, where Pr ≡ Pr( ωt ∈ Ω D ) with (τ t , et ) = (l, 0) and (τ t∗ , et∗ ) = (l, 0).

This definition of a supergame equilibrium of a repeated game with privately observed signals of other players’
actions follows Matsushima (1991).
19
One trivial supergame equilibrium strategy profile is to assign the one-shot Nash protection level for all periods
because that would assign the static optimal behavior for each country.
18

10Note that the absence or presence of explicit tariffs classifies any period into either a“cooperative” period (with no explicit tariffs) or a “punishment” period (with some positive
tariffs). While H and F cannot observe each other’s concealed protection levels, they use their
explicit tariffs as public signals to coordinate punishment phases as described in (i) and (ii).
Extending a punishment phase one more period with a certain probability as specified in (ii) is
an instrument to make the expected discounted payoff from invoking a punishment phase varysmoothly so that it can be set to equal the expected discounted payoff from not invoking apunishment phase, which is an important requirement for incentive constraints considered in
the following section. Also note that the actions for period 1 and other “initial” periods
described in (iii) are designed to mimic those in a period that immediately follows a“cooperative” one, which in turn simplifies the analysis of the trigger strategies defined
above. 20 Finally, note that the set of private signals that trigger a punishment phase ( Ω D ), thelengths of different punishment phases (T − 1 if a single country triggers and TS − 1 if H and F
trigger simultaneously), and the corresponding probabilities of extending the punishment
phases (λ, λS) characterize the strategy profile defined by (i), (ii) and (iii), together with thecooperative protection level, l. I define simple private trigger strategies as follows:∗

Definition 2. If (i), (ii), and (iii) describe a symmetric strategy profile (s, s*) with s (t ) = s (t )for all a t −1 × ω t −1 × (e ∗ ) t −1 = (a ∗ ) t −1 × (ω ∗ ) t −1 × e t −1 and t ≥ 1, then (s, s*) are simple private trigger
strategies (simple PTS) with (l , Ω D , T , T S , λ , λS ) as characterizing parameters.

Given this definition, I can derive H’s expected discounted payoff under (s, s*) with∗(l , Ω D , T , T S , λ , λS ) , denoted by V ( s , s ) , as follows:∗(1 − Pr 2 )[u (l , l ) − u (h, h)]
1 − δ C + 2 Pr (1 − Pr )(δ C − δ ) + Pr 2 (δ C − δ S )Pr (1 − Pr )[u (l , h) − u (l , l )] + Pr (1 − Pr )[u (h, l ) − u (l , l )] u (h, h)+,+
1 − δ C + 2 Pr (1 − Pr )(δ C − δ ) + Pr 2 (δ C − δ S )1−δ C

V ( s, s ) =(6)

If, for example, Pr = 0 ≠ Pr(ωt∈ΩD) with (τt, et) = (l, 0) and (τt*, et*) = (l, 0), then the expected one-period
payoffs for period 1 and other “initial” periods will be different from those for any period immediately following acooperative one, making the expected discounted payoffs along the equilibrium path more complicated than thosein (6). Furthermore, having actions in period 1 and in other “initial” periods different from those in periods
immediately following a cooperative period will make deviation incentives different across these periods, which inturn complicates characterization of the optimal protection sequence in Section 2.2.2.
20

11

K

K

where δ K = λ K (δ C ) T + (1 − λ K )(δ C ) T −1 with K = s or none.

(δ C − δ ) and (δ C − δ S )respectively represent the relative length of the punishment phase initiated by H or F alone andby H and F simultaneously. Because (T , T S , λ , λS ) uniquely defines (δ , δ S ) as shown above, I
will describe simple PTS using (l , Ω D , δ , δ S ) instead of using (l , Ω D , T , T S , λ , λS ) henceforth.
Note that simple PTS defined above is simple in the sense that each country imposes its
static optimal tariff in all periods under any punishment phase. More generally, PTS may
involve more complex punishment phases such as imposing lower tariffs if the signal indicatesweaker violations or/and employing a stronger punishment, such as autarky, againstpresumably more severe violations. As shown later, the equilibrium payoff of any (“almost
strongly”) symmetric PTS will be identical to the one under simple PTS defined above, as longas the initial punishment is triggered by a static optimal tariff. From now on, I abbreviate
simple PTS to PTS unless it is necessary to distinguish them.
2.2. Incentive Constraints under Private Trigger Strategies

In this section, I analyze incentive constraints for PTS to be a supergame equilibrium inthe repeated game defined in Section 2.1.

The private nature of signals that trigger

punishments under PTS makes such incentive constraints different from the incentive
constraints for trigger strategies under which public signals trigger punishments in two
distinctive ways. First, the private nature of signals imposes restrictions on the lengths ofpunishment phases under PTS, which contrasts with the repeated game with public information
where countries can choose any length for their punishment phases. Section 2.2.1 analyzes
such limits on the lengths of punishment phases under PTS. Second, to check the absence of∗∗s / (≠ s) or s ∗ / (≠ s ) such that V ( s / , s ) > V ( s, s ) and V ∗ ( s ∗ / , s) > V ∗ ( s , s) , one needs to**check not only one-time deviations from the specified strategy, but whole deviation paths thateach country may take. 21 If private signals trigger punishments as under PTS, any deviatory
action that each country might have taken in a previous period can influence its optimal
deviatory action in a current period: The previous period defection affects the probability of a
21

When a public signal triggers a punishment phase, any deviatory actions that each country might have taken inany previous periods will not affect its optimal deviatory action in the current period for a given history of public
signals up to the current period. This is because one country’s defections in the previous periods affect the othercountry’s current and future actions only through affecting the history of public signals. Therefore, we can apply

12punishment phase being initiated in the current period, which in turn influences the currentperiod optimal action. This necessitates characterization of an optimal (potentially deviatory)∗

protection sequence that each country may take against ( s, s ) in analyzing the incentive
constraints for PTS. Section 2.2.2 characterizes such a sequence for H under PTS, and showsthat H’s optimal protection sequence can be a stationary one of setting τ at l (the cooperative
protection level) in all periods until a punishment phase starts, which is a prerequisite for PTS
to be a supergame equilibrium.
2.2.1. Constraints on Lengths of Punishment Phases

In any period that immediately follows a cooperative period with (e, e*) = (0, 0) and in any
initial periods (period 1 and a period right after the end of any punishment phase), each country
faces the choice of whether or not to initiate a punishment phase by imposing its static optimal
tariff. To eliminate the incentive to misrepresent private signals in such periods, the expected
payoff from initiating a punishment phase should be identical to the expected payoff from notinitiating it for each country. Denote the condition that equates those expected payoffs by ICP
for H (with the same condition applying for F by symmetry). Then,
ICP:(1 − Pr )[u (l , l ) + δ CVC )] + Pr[u (l , h) + (δ C − δ )VN + δVC ] =(7)(1 − Pr )[u (h, l ) + (δ C − δ )VN + δVC ] + Pr[u (h, h) + (δ C − δ S )VN + δ S VC ],∗where VC ≡ V ( s, s ) and VN ≡ u (h, h) /(1 − δ C ) . The left side of the equality in (7) represents
the expected discounted payoff from not initiating a punishment phase but continuing to set (τ,e) = (l, 0). The right side of the equality represents the expected discounted payoff frominitiating a punishment phase, setting (τ, e) = (h, h). In calculating these expected discounted
payoffs in (7), it is assumed that the other country initiates a punishment phase with aprobability that conforms PTS, denoted by Pr.
Using u(l, l) − u(l, h) = u(h, l) − u(h, h) implied by ∂u/∂τ∂τ∗ = 0, I simplify (7) into(ICP)u (l , l ) − u (h, l ) + (δ C − δ )(VC − VN ) = Pr[(δ C − δ ) − (δ − δ S )](VC − VN ) .

the logic of one-stage-deviation principle for the subgame perfect equilibrium with observable actions (Theorem
4.1. and Theorem 4.2 in Fudenberg and Tirole, 1991) to the perfect public equilibrium (with unobservable actions).

13

For any given cooperative protection level (l) and any given range of private signals that trigger
punishment phases ( Ω D ), I have two variables ( δ , δ S ) to be determined with one equation(ICP), potentially having infinite combinations of ( δ , δ S ) that satisfies ICP. However, Lemma
1 (a) below establishes that δ C − δ S = 2(δ C − δ ) and ICP are the necessary conditions for eachcountry to truthfully represent its private signals under PTS.
Lemma 1.

(a) δ C − δ = [u(h, l) − u(l, l)]/(VC − VN) and δ C − δ S = 2(δ C − δ ) are necessary conditions foreach country to truthfully represent its private signals under PTS, triggering a punishment
phase iff its private signal belongs to Ω D .
(b) If H and F value their future payoffs high enough ( δ C ≈ 1 ) and the probability of apunishment phase being triggered along the equilibrium path is low enough (Pr ≈ 0), then,for any given combination of (l, Ω D ) with l < h, there exists a unique combination of( δ , δ S ) that satisfies the necessary condition for truthful revelation of private signals in
Lemma 1 (a). (See Appendix for proof.)
Recall that δ C − δ and δ C − δ S respectively represent the length of a punishment phasethat H or F can initiate alone (T , λ ) and the length of a punishment phase that H and F initiate
concurrently (T S , λS ) as δ = λ (δ C ) T + (1 − λ )(δ C ) T −1 and δ S = λ S (δ C ) T + (1 − λ S )(δ C ) T −1 .
SSThus, for a given combination of (l, Ω D ), ICP with δ C − δ S = 2(δ C − δ ) determines δ C − δ =[u(h, l) − u(l, l)]/(VC − VN). Note that the length of a punishment phase that each country
initiates by itself ( δ C − δ ) increases in its expected gain in the initial period of the punishment
phase by imposing its static optimal tariff unilaterally (u(h, l) − u(l, l)) but decreases in its
expected loss in the tariff-war periods that will follow (VC − VN). The expected gain in theinitial period of a punishment phase provides each country with the incentive to start apunishment phase despite the expected loss from engaging in a tariff war that follows under apunishment phase.

Thus, the larger the expected gain in the initial period, the longer apunishment phase that H can tolerate (without violating ICP) and the larger the expected loss
from a tariff war, the shorter a punishment phase that H can tolerate (without violating ICP).

14

Even when ICP is satisfied so that each country has no (strict) incentive to untruthfully
represent its private signal after a “real” cooperative period, it may still have an incentive to
misrepresent its private signal after a “pseudo” cooperative period under which it deviates by
setting τ ≠ l (or τ ∗ ≠ l ) with its explicit tariff being zero. The proof for Lemma 1 (a) in the
Appendix shows that δ C − δ S = 2(δ C − δ ) is indeed a necessary condition for each country not
to misrepresent its private signals in a period following a pseudo cooperative period. For
example, if δ C − δ S < 2(δ C − δ ) so that the length of a punishment phase that H and F initiate
concurrently is shorter than what it is supposed to be, then each country will have an incentive
to set its protection level higher than l in a cooperative period and then initiate a punishment
phase in the following period regardless of its private signal. Such a deviation strategy may
pay off because an increase in the protection level in a cooperative period raises the probabilityof a punishment phase being triggered by the other country in the following period, which
would then lead to a short punishment phase ( δ C − δ S being small) when the deviating country
initiates a punishment regardless of its private signals.
2.2.2. Optimal Protection Sequence and Existence of a Stationary Protection Level

To characterize the optimal protection sequence, I analyze the dynamic optimization
problem in which H maximizes its expected discounted payoff by choosing a protection
sequence {τ d +1}∞d =0 , given that F follows its specified strategy under PTS.

The dynamic

optimization problem for H is∞⎧⎫
⎡ d −1⎤Sup ∑ ⎨(δ C ) d ⋅ ⎢Π [1 − Pr (τ t )]⎥ ⋅ F (τ d ,τ d +1 )⎬{τ d +1 }∞d = 0 d = 0 ⎩⎣ t =0
⎦
⎭

(8)

d −1

−1

t =0

t =0

where Π [1 − Pr (τ t )] = [1 − Pr (τ 0 )] × [1 − Pr (τ 1 )] × ⋅ ⋅ ⋅ × [1 − Pr (τ d −1 )] with Π [1 − Pr (τ t )] = 1;
Pr (τ t ) ≡ Pr (ωt∗ ∈ Ω D ) given ( τ t , et ) = (l, 0) and ( τ t∗ , et∗ ) = (l, 0), and τ 0 = l; andF (τ d ,τ d +1 ) ≡ Pr (τ d )[u (τ d +1 , h) + (δ C − δ )VN + δVCO ] + [1 − Pr (τ d )]u (τ d +1 , l ) with VCO = VC. Notethat the protection sequence {τ d +1 }∞d =0 in (8) specifies protection levels only until F triggers an
initial punishment phase. The optimization in (8) assumes that H will follow its specified
strategy under PTS once F triggers an initial punishment phase with VCO = VC ≡ V ( s, s ) . The*

15

full optimization problem should characterize the optimal protection sequence after the end ofeach punishment phase that may occur in the future periods. Characteristics of the optimal
protection sequence derived from (8), however, will be qualitatively identical to those of thefull optimization problem. This is because the optimal sequence resulting from (8) will beidentical to the one from the full optimization problem if VCO in (8) is set equal to the
maximized expected discounted payoff of the full problem, having H face an identicaloptimization problem in determining the protection sequence after the end of each punishment
phase in the future.22 Also note that the optimal protection sequence considered in (8) excludes
the possibility of using explicit tariffs as a part of its path. As shown in Lemma 4 (b) of thissection, however, once the lengths of punishment phases satisfy the necessary conditions fortruthful revelation of private signals given in Lemma 1 (a), then H cannot increase its payoff byusing explicit tariffs along its deviation path. Hence, there is no loss of generality in analyzing
the optimal protection sequence for H through the optimization problem defined in (8). 23
Even though the optimization problem in (8) does not take a standard form for which a
dynamic programming method is typically applied, Lemma 2 (a) below establishes equivalency
between (8) and the following (non-standard) dynamic programming problem: 24
(9)

{

}

V (τ −1 ) = Sup F (τ −1 ,τ ) + δ C [1 − Pr (τ −1 )]V (τ ) for all τ−1 ∈ [0, h],τ ∈[ 0 ,h ]22

The discounted payoff of the full optimization problem can be obtained by applying the following iterative
process to the optimization problem in (8). Initially set VCO in (8) to be VC defined in (6) and solve theoptimization in (8), obtaining a discounted payoff as an outcome of this initial optimization problem. Then, set thevalue of VCO in (8) to have the value of this initially generated discounted payoff, supposedly higher than (or equalto) the initial VCO (= VC), which redefines the optimization problem in (8). This redefined optimization problem
will generate another discounted payoff as an outcome of this second optimization problem. Then, set VCO in (8)to have the value of this newly generated discounted payoff and continue this iterative process until the discounted
payoff generated through this process reaches its limit. As the sequence of the discounted payoffs generated
through this process is monotonically increasing and bounded, there exists such a limit. This limit will be equal tothe discounted payoff of the full optimization problem.
23
While I focus on characterizing the optimal protection sequence for H under PTS in this section, the samecharacterization can be applied to the optimal protection sequence for F.
24(8) is not a standard problem in the sense that the component that corresponds to the return function of ad −1standard problem, [Π [1 − Pr (τ t )]]F (τ d , τ d +1 ) , depends not only on the current choice variable and the choicet =0made in the immediate prior period (as in the case of a usual return function of a typical dynamic programming
problem) but also on all the choices made since the initial period. The dynamic programming problem in (9) is nota standard form because the current state variable, τ−1, affects not only the current return function part, F(τ−1,τ),but also the future discounted payoff part through [1 – Pr(τ−1)].

16where τ−1 and τ, respectively denote a previous-period and a current-period protection level of
H. 25 Given a solution V(⋅) to (9), the optimal policy correspondence G: [0, h] → [0, h] isdefined by:G(τ−1) = {τ ∈ [0, h]: V(τ−1) = F(τ−1, τ) + δC[1 – Pr(τ−1)]⋅V(τ)},(10)

which contains values of τ that maximize V(τ−1) for each τ−1 ∈ [0, h]. Despite the fact that the
dynamic optimization problem in (8) and the corresponding dynamic programming problem in(9) take non-standard forms, Lemma 2 establishes the following standard results on V and G:
Lemma 2.

(a) Define VS (τ 0 ) be the supremum function that is generated by (8). Then, (i) VS satisfies(9); (ii) the solution to (9) is V (τ −1 ) = VS (τ −1 ) ; (iii) every optimal protection sequencesolving (8) is generated from G in (10); (iv) any protection sequence generated by G in(10) is an optimal protection sequence that solves (8).
(b) There exists a unique continuous function V that satisfies (9).
(c) The optimal policy correspondence G defined by (10) is compact-valued and upper hemicontinuous. (See Appendix for Proof)
Given Lemma 2, I can characterize the optimal protection sequence of H by characterizingG(⋅) in (10) because any protection sequence generated by G with the initial τ−1 being set at l is
an optimal protection sequence that solves (8). Utilizing one of the generalized envelope
theorems of Milgrom and Segal (2002) and a general result on the differentiability of the valuefunction of Cotter and Park (2006), I can characterize V and G as follows: 26
Lemma 3. Assume that the lengths of punishment phases satisfy the conditions in Lemma 1 (a).
Note that limiting H’s protection choice to be equal or less than h as in (9) does not affect the generality of the
optimization problem because H has no incentive to raise its protection level above its static optimal protectionlevel, h.
26
In characterizing V and G, I cannot use the well-known result of Benveniste and Scheinkman (1979) on the
differentiability of the value function. While Benveniste and Scheinkman established that concavity of the return
function on the state and choice variables is sufficient to guarantee the differentiability of the resulting valuefunction of a typical dynamic programming problem, the dynamic problem of choosing an optimal protection
sequence analyzed in this paper does not belong to the typical dynamic programming problem, as explained earlier.
Recently, Milgrom and Segal (2002) developed generalized envelope theorems for arbitrary choice sets, andCotter and Park (2006) established differentiability of the value function on the range of the optimal policy
correspondence, regardless of the curvature of the return function. I apply these results in characterizing V and G,
as shown in the proof of Lemma 3.
25

17

(a) V(τ−1) is strictly decreasing in τ−1 ∈ [0, h].
(b) G(τ−1) is strictly increasing in τ−1 in the sense that g( τ −//1 ) > g( τ −/ 1 ) for all τ −//1 > τ −/ 1 ∈ [0, h]with g( τ −//1 ) ∈ G( τ −//1 ) and g( τ −/ 1 ) ∈ G( τ −/ 1 ). (See Appendix for Proof)
Because a higher τ−1 (a higher protection level in the cooperative previous period) implies ahigher probability that F triggers a punishment phase in the current period, a higher τ−1 also
implies a more hostile environment for H to maximize its discounted payoff. Therefore, theoutcome of the maximization problem, V(τ−1), will get smaller as τ−1 increases (Lemma 3 (a)).
To understand Lemma 3 (b), first note that choosing τ (a current-period protection level)
must balance the current period’s loss from setting the protection level below h (the static
optimal one) against the future periods’ gain from reducing the probability of a punishment
phase being triggered. Figure 1 demonstrates this. Given the pervious-period protection levelτ−1 is equal to τ −/ 1 , setting τ = h maximizes F (τ −/ 1 ,τ ) = Pr (τ −/ 1 )u (τ , h) + [1 − Pr (τ −/ 1 )]u (τ , l ) +Pr (τ −/ 1 )[(δ C − δ )VN + δVC ] because it maximizes the expected current period payoff,

Pr (τ −/ 1 )u (τ , h) + [1 − Pr (τ −/ 1 )]u (τ , l ) and τ does not affect the future expected discounted payoff
contingent upon a punishment phase being initiated in the current period, (δ C − δ )VN + δVC .
By reducing τ below h, however, H can increase its expected discounted payoff, F (τ −/ 1 ,τ ) +δ C [1 − Pr (τ −/ 1 )]V (τ ) because V(τ) strictly decreases in τ by Lemma 3 (a). As shown in Figure 1,
if H lowers τ from h, δ C [1 − Pr (τ −/ 1 )]V (τ ) strictly increases. Therefore, g( τ −/ 1 ), the optimalcurrent-period protection with τ −/ 1 being the previous-period protection level, is lower than h.
Given this understanding of the optimal choice over τ, I can explain why G(τ−1) strictlyincreases in τ−1 using Figure 1. When τ−1 increases from τ −/ 1 to τ −//1 , it may shift F (τ −1 ,τ )upwards as shown in Figure 1 but it will not affect ∂F (τ −1 ,τ ) / ∂τ = ∂u (τ , l ∗ ) / ∂τ , implyingthat the static incentive to raise τ closer to h stays the same; for example,

F (τ −//1 , h) − F (τ −//1 , g (τ −/ 1 )) = F (τ −/ 1 , h) − F (τ −/ 1 , g (τ −/ 1 )) in Figure 1. An increase in τ−1, however,
weakens the dynamic incentive for lowering τ to avoid a punishment phase in a future period
because it increases the likelihood of a punishment phase starting in the current period. Figure
1 illustrates this: [1 − Pr (τ −//1 )][V ( g (τ −/ 1 )) − V (h)] < [1 − Pr(τ −/ 1 )][V ( g (τ −/ 1 ) − V (h)] with Pr (τ −//1 ) >18

Pr (τ −/ 1 ) . The dynamic gains from reducing τ from h to g (τ −/ 1 ) decreases as τ−1 increases fromτ −/ 1 to τ −//1 . As a result, a higher τ−1 moves the balance for choosing an optimal τ towards ahigher current period protection level so that g( τ −//1 ) > g( τ −/ 1 ) as shown in Figure 1.
The fact that G(τ−1) is strictly increasing in τ−1 may entail both an increasing protection
sequence and a decreasing one as shown in Figure 2; if τ 0 = τ 0/ , then the optimal protection
sequence will be increasing with τ 0/ < τ 1/ < τ 2/ < ⋅ ⋅ ⋅ ; and if τ 0 = τ 0// , then the optimal protection
sequence will be decreasing with τ 0// > τ 1// > τ 2// > ⋅ ⋅ ⋅ . 27

If τ 0 = τ S , however, the resulting

optimal protection sequence will be stationary with τ 0 = τ 1 = τ 2 = ⋅ ⋅ ⋅ . If there exists such astationary protection level, τS ∈ [0, h) under PTS with G(τS) = τS and l = τS, then H would
continue to set its protection level at l until a punishment phase begins.

Therefore, theexistence of such a stationary protection level, τS, is a prerequisite for PTS to be a supergame
equilibrium of the repeated game. An increasing optimal policy correspondence (Lemma 3 (b))
itself, however, does not rule out the possibility that the only stationary protection level of the
dynamic problem in (9) is h, as demonstrated by G/(τ−1) in Figure 2.
To address the existence issue of a stationary protection level τS ∈ [0, h) with G(τS) = τS, I
analyze a necessary condition for such τS. If V(τ) is differentiable with respect to τ, then τSshould satisfy the following first order condition for a stationary equilibrium, denoted by IC:
(11)

IC:

∂F(τS, τS)/∂τ + δC[1 − Pr(τS)]⋅[∂V(τS)/∂τ] = 0,

where ∂F(τS, τS)/∂τ = ∂u(τS, l)/∂τ and ∂V(τS)/∂τ = –[∂Pr(τS)/∂τ ]{u(τS, l) + δCV(τS) – [u(τS, h) +(δC – δ)VN + δVC]}. While I cannot assume differentiability of V(τ) on τ ∈ [0, h] as explained
earlier, V(τ) is differentiable on any τ ∈ G(τ−1) and τ ∈ (0, h) for each τ−1∈ [0, h], according toa generalized differentiability result of Cotter and Park (2006). Therefore, (11) is indeed anecessary condition for any stationary protection level that belongs to (0, h). Thus it serves as
an incentive constraint (IC) for H to sustain the cooperative protection level, l = τS under PTS.
If the cooperative protection level is set too low under PTS with l = τ0/, then H would keep raising the protectionlevel above the cooperative one until it reaches a stationary level, τS, and the opposite is true if the cooperative
protection level is too high with l = τ0//. Blonigan and Park (2004) identify that a similar dynamic behavior
emerges in the context of an exporting firm’s dynamic pricing problem in the presence of antidumping policy;
once an exporting firm becomes subject to an antidumping duty, it would either continue to decrease its export
price (thus, having the duty increase over time) or continue to increase its export price (thus, having the duty
lowered over time) depending on whether the initial export pricing is higher or lower than a stationary pricing.
27

19

For τS to be a stationary protection level for H, the static incentive to raise the protectionlevel, ∂F(τS, τS)/∂τ > 0 in (11), needs be balanced by the dynamic incentive to avoid a costly
punishment phase in the future, δC[1 − Pr(τS)]⋅[∂V(τS)/∂τ] < 0 in (11). Lemma 4 (a) below
provides a sufficient condition for the existence of such τS ∈ (0, h) with G(τS) = τS, and Lemma
4 (b) shows that H does not have any incentive to utilize explicit tariffs as part of its deviation
path if l = τS.
Lemma 4. Assume that the lengths of punishment phases satisfy the conditions in Lemma 1 (a).

(a) If ∂2Pr(τ)/(∂τ)2 > 0 with [∂2Pr(τ)/(∂τ)2][1 – Pr(τ)] – {1 + δC[1 – Pr(τ)]}[∂Pr(τ)/∂τ ]2 > 0
for all τ ∈ [0, h] and ∂Pr(τ)/∂τ ≈ 0 at τ = 0, then there exists a unique stationary
equilibrium protection level τS ∈ (0, h) with G(τS) = τS. τS is also a globally stableequilibrium with G(τ) > τ for τ ∈ [0, τS) and G(τ) < τ for τ ∈ (τS, h). 28(b) If l = τS, then H cannot increase its discounted payoff above V ( s, s ) by taking any*(deviatory) protection sequence that involves initiating punishment phases by imposing
explicit tariffs. (See Appendix for Proof)
According to Lemma 4 (a), it is possible to have IC in (11) satisfied for some τS < h if thesensitivity of F’s private information in detecting a rise in H’s concealed protection, ∂Pr(τS)/∂τ,
increases as H’s concealed protection level rises with ∂2Pr(τ)/(∂τ)2 > 0. On the one hand, H’sstatic incentive to raise its protection level, ∂F(τS, τS)/∂τ = ∂u(τS, l)/∂τ in (11), diminishes as τS
increases with ∂2u(τS, l)/∂τ2 < 0, reaching zero atτS = h. On the other hand, H’s dynamic
incentive to avoid a future punishment phase, δC[1 − Pr(τS)]⋅[∂V(τS)/∂τ] in (11), may diminish
or intensify in response to an increase in τS, depending on the value that ∂2Pr(τS)/∂τ2 takes. A
higher τS reduces H’s weight on its dynamic incentive to avoid a punishment phase, 1 − Pr(τS),
by increasing the probability of a punishment phase being triggered in the current period. If∂2Pr(τS)/∂τ2 > 0, an enhanced sensitivity of F’s private information in detecting a rise in H’sprotection can offset such a reduction in H’s incentive to avoid a punishment phase. The
absolute value of δC[1 − Pr(τS)]⋅[∂V(τS)/∂τ] rises in response to a rise in τS if [∂2Pr(τ)/(∂τ)2][1 –τS being a globally stable protection level is a contributing factor to the stability of PTS as an equilibrium of therepeated game. This is because H will eventually return to its globally stable behavior of setting τ = τS (= l) after
any arbitrary perturbations (possibly caused by errors) in its protection level choices.
28

20

Pr(τ)] – {1 + δC[1 – Pr(τ)]}[∂Pr(τ)/∂τ ]2 > 0 for all τ ∈ [0, h], as assumed in Lemma 4 (a).
This in turn guarantees the existence of a unique τS ∈ (0, h) that satisfies IC in (11) with∂Pr(τ)/∂τ ≈ 0 at τ = 0.
Having the sensitivity of private information rise against increasing concealed protection
can be crucial in discouraging the use of concealed protection under PTS. If ∂2Pr(τ)/(∂τ)2 = 0,
for example, the dynamic incentive for lowering τ below h to avoid a tariff war in a future
punishment phase, δC[1 − Pr(τS)]⋅[∂V(τS)/∂τ] in (11), decreases as τS gets higher, entailing thepossibility of IC in (11) not being satisfied for any τS < h.
While Lemma 4 specifies the condition under which H (and F) would follow PTS bykeeping its protection at a cooperative level until a punishment phase is triggered, note that
Lemma 4 “assumes” that the lengths of punishment phases satisfy the conditions in Lemma 1(a). Because such lengths of punishment phases “vary” with the cooperative protection level tosustain under PTS, it still remains to be shown whether there exist PTS that satisfy theconditions in Lemma 1 (a) and IC simultaneously.

The following section provides anaffirmative answer.

3. Optimal Private Trigger Strategies

This section establishes that symmetric countries can sustain a symmetric cooperative
protection level under simple PTS defined in the previous section if the sensitivity of theirprivate information satisfies certain conditions.

In addition, this section proves that any

equilibrium payoff under (“almost strongly”) symmetric trigger strategies that start an initialpunishment phase by imposition of a static optimal tariff based on each country’s imperfectprivate signal should be identical to the payoff under simple PTS. 29

After proving theexistence and the uniqueness (in terms of payoffs, at least among a certain class of trigger
strategies) of symmetric PTS as a supergame equilibrium in Section 3.1, I characterize optimal
symmetric PTS under which H and F maximize their joint expected discounted payoffs in
Section 3.2.
3.1. Private Trigger Strategies and Uniqueness Results

21This section first proves the existence of simple PTS that satisfy the conditions in Lemma 1(a) and IC simultaneously. Assume that there exists τS that satisfies IC in (11) with τS = l. Thisimplies that V(τS) = VC, and I can rewrite IC in (11) as follows:(12)∂u(τS, l)/∂τ = δC[∂Pr(τS)/∂τ ][1 − Pr(τS)][u(τS, l) – u(τS, h) + (δC – δ )(VC – VN)].

As discussed in the previous section, (12) is a necessary condition for H to have no incentive tochange its protection level away from the cooperative one until a punishment phase starts. I
also assume that the lengths of punishment phases are determined by the conditions in Lemma
1 (a); δ C − δ = [u(h, l) − u(l, l)]/(VC − VN) and δ C − δ S = 2(δ C − δ ) . IC in (12) then can be
rewritten into the following implicit function, I(l):(13)

I(l) ≡ ∂u(l, l)/∂τ − δC[∂Pr(l)/∂τ ][1 − Pr(l)][u(h, l) – u(l, h)] = 0,

by substituting δ C − δ with [u(h, l) − u(l, l)]/(VC − VN). Using I(l), Proposition 1 provides a
sufficient condition for the existence of simple PTS that countries can sustain as a supergame
equilibrium of their repeated protection-setting game:
Proposition 1. If ∂2Pr(l)/(∂l)2 > 0 with [∂2Pr(l)/(∂l)2][1 – Pr(l)] – {1 + δC[1 – Pr(l)]}[∂Pr(l)/∂l]2> 0 for all l ∈ [0, h], ∂Pr(l)/∂l ≈ 0 at l = 0, and there exists at least one protection level, l S < hsuch that I( l S ) = 0, then, H and F can employ simple PTS with l = l S , δ C − δ = [u(h, l S ) −u( l S , l S )]/(VC − VN), and δ C − δ S = 2(δ C − δ ) as a supergame equilibrium of the repeatedprotection-setting game. (See Appendix for Proof)
Proposition 1 assumes the same condition regarding the sensitivity of private information
as the one in Lemma 4 (a), ensuring that there exists a unique stationary equilibrium protectionlevel τS ∈ (0, h) with G(τS) = τS. In addition, it requires I(l) = 0 for at least one value of l < h,denoting it by l S . With l = l S , δ C − δ = [u(h, l S ) − u( l S , l S )]/(VC − VN) and δ C − δ S =2(δ C − δ ) , I( l S ) = 0 guarantees that IC and the conditions in Lemma 1 (a) are simultaneously
satisfied under such PTS. According to Lemma 4, l S is the unique stationary protection levelwith G( l S ) = l S and countries have no incentive to deviate from such PTS.

29

Definition 3 below provides a formal definition of “almost strongly symmetric private trigger strategies.”

22

The sufficient condition in Lemma 4 (a) does not necessarily imply that the second term ofI(l) in (13), δC[∂Pr(l)/∂τ ][1 − Pr(l)][u(h, l) – u(l, h)], representing H’s dynamic incentive to
avoid a tariff war, increases in response to a rise in l. 30 Thus, one may consider the case wheremultiple values of l satisfy I(l) = 0 as illustrated in Figure 3; l = lmax as well as l = lmin satisfy I(l)= 0. Denoting the minimum of such l by lmin, then simple PTS with l = lmin will Paretodominate the others when Pr(l) is small enough. 31
While the above result establishes that symmetric countries may employ simple PTS
characterized by Proposition 1 (and Definition 2) in restraining the use of concealed tradebarriers, one may wonder whether there exist other (symmetric) private trigger strategies that
may outperform this simple one. Surprisingly, the following result shows that there is no lossof generality in focusing on this simple PTS to characterize the optimal symmetric private
trigger strategies as long as the explicit tariff that starts an initial punishment phase is the static
optimal tariff of each country. The first part of the following result applies to a larger class oftrigger strategies: any symmetric private trigger strategies where each country can start an
initial punishment phase with an explicit tariff of any level.
Denote the level of τ (total protection) that initiates the first (or initial) punishment phasewith e (explicit tariff) > 0 by d0 and the cooperative protection level for the initial cooperative
periods (prior to any punishment being triggered) by l0, thus focusing on the symmetric private
trigger strategies where the cooperative protection level and the protection level that starts an
initial punishment phase are stationary at least prior to an initial punishment phase. Then, I canrepresent the expected discounted payoff of H of employing such symmetric private trigger
strategies as follows, denoting it by V(l0; d0):(14)

V (l0 ; d 0 ) = (1 − Pr )(1 − Pr )[u (l0 , l0 ) + δ CVC ] + Pr (1 − Pr )[u (l0 , d 0 ) + δ CVD∗ ]+ Pr (1 − Pr )[u (d 0 , l0 ) + δ CVD ] + Pr 2 [u (d 0 , d 0 ) + δ CVD S ],where Pr ≡ Pr(l0) = Pr( ω ∈ Ω D ) given (τ, e) = (l0, 0) and (τ*, e*) = (l0, 0), and VC ≡ V(l0; d0).
V D , VD∗ and VD S denote the expected discounted payoff of H after an initial punishment phasehas been by triggered, respectively by H alone, by F alone, and by H and F simultaneously in a
For the proof of this claim, see the proof for Proposition 1 in Appendix.
Note that u(lmin, lmin) > u(lmax, lmax) and Pr(lmin) < Pr(lmax) imply a higher cooperative-period payoff and a lower
probability of punishment phases with l = lmin than with l = lmax. While the lengths of punishment phases may belonger with l = lmin than with l = lmax, an increase in l will lower the expected discounted payoff under simple PTS
if Pr(l) is close enough to 0, as shown in (18) of the following section.
30
31

23

previous period. Given these notations, the following definition defines “almost strongly
symmetric PTS”:
Definition

3. Among the set of symmetric strategies withs(t ) = s ∗ (t ) for alla t −1 × ω t −1 × (e ∗ ) t −1 = (a ∗ ) t −1 × (ω ∗ ) t −1 × e t −1 and t ≥ 1, almost strongly symmetric PTS with (l0;d0) are private trigger strategies under which each country starts its initial punishment phase by
imposing an explicit tariff (e > 0) with its τ = d0, and the initial cooperative protection level is
l0 with V D = VD∗ .
It is “almost strongly symmetric” strategies rather than “strongly symmetric” strategies
under which s (t ) = s ∗ (t ) for all t ≥ 1 because s(t ) ≠ s ∗ (t ) may occur when ωt −1 ∈ Ω D or

ωt∗−1 ∈ Ω D , and V D = VD does not necessarily entail s (t ) = s ∗ (t ) after such contingencies.
∗

The payoff function in (14) implicitly assumes that each country sets its initial cooperative
explicit tariff to be zero. Relaxing this assumption by allowing e > 0 in the initial cooperative
periods would not raise the payoff in (14) as long as the sensitivity of private information ofconcealed trade barriers improves with a higher level of concealed trade barriers. 32

Forsimplicity, I will abbreviate almost strongly symmetric PTS by symmetric PTS henceforward.
For symmetric PTS defined by Definition 3 to be incentive-compatible (so that they can besupported as equilibrium behaviors), the following analysis establishes that the payoff in (14)needs to be equal to the following expression:(15)

V (l 0 ; d 0 ) =u (l 0 , l 0 )u (d 0 , l0 ) − u (l 0 , d 0 ).
− Pr (l 0 )
C
1−δ
1−δ C

Note that the payoff in (15) depends only on the values of l0 and d0. Once established, theabove result remarkably simplifies the job of characterizing the payoff frontier attainable under
any symmetric PTS that rely on triggering a punishment phase with some explicit tariffs: one
only needs to find l0 and d0 that are incentive-compatible, which in turn maximize the payoff in(15).

e = e * = 0 forces each country to raise its protection level all through concealed trade barriers. If the sensitivityof private information improves with a higher level of such barriers, the effectiveness of private trigger strategies
against the incentive to raise protection levels should improve with such constraints of setting e = e * = 0 in the(initial) cooperative periods.
32

24

To prove (15), I use three incentive-compatibility conditions: (i) a generalized version of
ICP, (ii) IC for setting the initial cooperative protection level to be l0 and continuing to set τ =l0 in the following period upon the contingency of no punishment phase being initiated, and (iii)
IC for setting the initial cooperative protection level to be l0 and starting an initial punishment
phase by setting τ = d0 in the following period upon the contingency of no punishment phasebeing initiated. First, note that any symmetric PTS triggering the initial punishment phasewithτ = d0 and e > 0 should satisfy the following ICPG:G(ICP )(1 − Pr )[u (l0 , l0 ) + δ CVC ] + Pr[u (l0 , d 0 ) + δ CVD∗ ]= (1 − Pr )[u (d 0 , l0 ) + δ CVD ] + Pr ∗ [u (d 0 , d 0 ) + δ CVD S ],which equalizes the payoff of initiating the (initial) punishment phase with the payoff of notinitiating it, similarly to ICP in the previous section. Using u(l0, l0) − u(l0, d0) = u(d0, l0) − u(d0,d0) implied by ∂2u/∂τ∂τ* = 0, I can simplify (ICPG) as follows:(ICPG)u (l 0 , l 0 ) − u ( d 0 , l 0 ) + δ C (VC − VD∗ ) = δ C Pr[(VC − VD ) + (VC − V D∗ ) − (VC − V D S )] .

To identify incentive constraints for setting the initial cooperative protection level to be l0, I
can write the expected discounted payoff of H setting τ to be τ I in an initial period (prior to any
punishment phase being triggered) as

Pr[u (τ I , d 0 ) + δ CVD ]+ (1 − Pr ){u (τ I , l0 ) + δ C Pr (τ I )[u (l0 , d 0 ) + δ CVD∗ ] + δ C [1 − Pr (τ I )][u (l0 , l0 ) + δ CVC ]},or
Pr[u (τ I , d 0 ) + δ CVD ]+ (1 − Pr ){u (τ I , l0 ) + δ C Pr (τ I )[u (d 0 , d 0 ) + δ CVD S ] + δ C [1 − Pr (τ I )][u (d 0 , l 0 ) + δ CVD ]}depending on whether H sets τ = l0 or τ = d0, respectively, in the following period upon thecontingency of no punishment phase being initiated after setting τ = τ I as the initialcooperative protection level. To be able to support τ I = l0 as an equilibrium behavior, thefollowing ICG need to be satisfied:∂Pr (l 0 )∂u (l 0 , l 0 )= δ C (1 − Pr )[u (l 0 , l 0 ) − u (l 0 , d 0 ) + δ C (VC − V D* )],
∂l 0
∂τ

(ICG)

and
∂u (l 0 , l 0 )∂Pr (l 0 )[u ( d 0 , l 0 ) − u (d 0 , d 0 ) + δ C (V D − V D S )],= δ C (1 − Pr )
∂τ
∂l 0

25

implying that u (l 0 , l 0 ) − u ( d 0 , l 0 ) + δ C (VC − VD∗ ) = u (d 0 , l0 ) − u ( d 0 , d 0 ) + δ C (VD − VD S ).

Using

this last equality together with the simplified ICPG and VD = VD∗ , one can show thatP

(VC − VD ) + (VC − VD∗ ) − (VC − VD S ) = 0, thus u (d 0 , l0 ) − u (l0 , l0 ) = δ C (VC − VD ).

Given theseequalities, one can rewrite V(l0; d0) in (14) into the one in (15) using the following steps:
V (l0 ; d 0 ) = u (l0 , l0 ) + δ CVC + Pr[u (l0 , d 0 ) − u (l0 , l0 ) − δ C (VC − VD )]+ Pr[u (d 0 , l0 ) − u (l0 , l0 ) − δ C (VC − VD∗ )]+ Pr 2 {[u (d 0 , d 0 ) − u (l0 , d 0 )] − [u (d 0 , l0 ) − u (l0 , l0 )]}(16)+ δ C Pr 2 [(VC − VD ) + (VC − VD∗ ) − (VC − VD S )]= u (l0 ,l0 ) + δ CVC + Pr[u (l0 , d 0 ) − u (d 0 , d 0 ) − u (d 0 , l0 ) + u (l0 , l0 )],where the second line through the forth in (16) all take zero values and the last line in (16)generates (15) using V (l0 ; d 0 ) = VC . I can summarize this result in the following proposition:
Proposition 2. The equilibrium payoff of any symmetric PTS (defined by Definition 3) with theinitial cooperative protection level being l0 and the level of total protection that starts the initialpunishment phase being d0, denoted by V(l0; d0), is a function of only l0 and d0 with
V (l 0 ; d 0 ) =u (l 0 , l 0 )u (d 0 , l0 ) − u (l 0 , d 0 ),− Pr (l 0 )
C
1−δ
1−δ C

where Pr(l0) = Pr( ω t∗ ∈ Ω D ) given ( τ t , et ) = (l0, 0) and ( τ t∗ , et∗ ) = (l0, 0).
The above proposition establishes that one can fully characterize the equilibrium payoff ofany symmetric PTS only with the information about l0 and d0 that are incentive-compatible.
Given u (d 0 , l0 ) − u (l0 , l0 ) = δ C (VC − VD ) , the necessary condition for an incentive-compatible
choice of l0 is(ICG)∂u (l 0 , l 0 )∂Pr (l 0 )= δ C [1 − Pr (l 0 )][u (d 0 , l 0 ) − u (l 0 , d 0 )] .
∂τ∂l 0This condition is identical to IC for simple PTS characterized in (13) if d0 = h, the static optimal
protection level.

ICG and Proposition 2 together imply that countries cannot attain thesymmetric efficient frontier where VC = u(l0, l0)/(1 − δC) with l0 = 0 as their equilibrium payoffs
under any symmetric PTS if their private signals entail non-negligible errors in detecting the useof concealed trade barriers.

If l0 = 0, then ∂u (l 0 , l 0 ) / ∂τ > 0 , which in turn requires26u (d 0 , l0 ) − u (l0 , d 0 ) > 0 to satisfy ICG. Given that Pr(l0) > 0 due to non-negligible errors in theprivate information, Proposition 2 implies that u (l 0 , l 0 ) /(1 − δ C ) − V (l 0 ; d 0 ) = Pr (l 0 )[u (d 0 , l 0 ) −u (l 0 , d 0 )] /(1 − δ C ) > 0 with l0 = 0. The following corollary to Proposition 2 states this finding:Corollary 1 to Proposition 2. Under any symmetric PTS, countries cannot attain the symmetricefficient frontier where VC = u(l0, l0)/(1 − δC) with l0 = 0 as their equilibrium payoffs if theirprivate signals entail non-negligible errors in detecting concealed trade barriers with Pr(l0) > 0.
This anti-folk theorem result under symmetric PTS demonstrates an aspect of private
trigger strategies considered in this paper: To have each country properly trigger a punishment
phase under which it may gain in the initial punishment period, such a punishment-initiating
country needs to be penalized later in the punishment phase, a costly process for all countries
involved. Also note that this anti-folk theorem result is attained under symmetric PTS, a subset
of private trigger strategies that countries can employ, thus it is still an open question whether
one can obtain a folk theorem result under a more general private trigger strategies.
Beyond proving the above anti-folk theorem result under symmetric PTS, a further
characterization of symmetric PTS is not a simple matter. While Proposition 1 guarantees theexistence of incentive-compatible symmetric PTS with d0 = h = e, characterizing the necessary
condition for an incentive compatible choice of d0 ≠ h is far from being an easy task. Once d0 ≠h, each country would have an incentive to start an initial punishment phase by choosing τ ≠ d0
such as τ = h, necessitating a punishment scheme against such a deviation incentive. Note thatthe punishment scheme against a deviatory initiation of an initial punishment phase generates
yet another private monitoring problem, which can be different from the one for the initialperiods. This process of having an additional and different private monitoring issue against
deviatory uses of punishment phases may continue forever, making a general characterizationof it a very difficult task. 33

If PTS are strongly symmetric with s(t) = s*(t) for all t ≥ 1, except for unilateral initiations of punishment phases
against potential violations, and if PTS do not allow each country to start another (new) punishment phase rightafter its initiation of an initial punishment phase, then one can show that the only incentive compatible choice of d0
is h. This is possibly another way of justifying the paper’s focus on simple PTS, but this approach seems toimpose rather stringent constraints on PTS.
33

27

Even when one ignores the issue of finding an incentive-compatible d0 and pretends that
one can choose any value for d0, it is not clear whether choosing d0 < h would increase theexpected discounted payoff in comparison with the choice of setting d0 = h. For example,
setting d0 < h would raise V(l0; d0) by decreasing u(d0, l0) – u(l0, d0) with ∂[u(d0, l0) – u(l0,d0)]/∂d0 > 0. However, lowering d0 weakens the ICG by lowering the right hand side value of
ICG shown above, thus decreasing V(l0; d0) by raising the value for l0. The optimality ofchoosing d0 ≠ h, even when it is incentive-compatible, therefore, depends on the trade-off
between its direct effect on the payoff through changing u(d0, l0) – u(l0, d0) and its indirect
effect through changing the incentive-compatible l0, which in turn requires further
characterization of the private information of concealed trade barriers.
In the following characterization of the optimal symmetric PTS, I will focus on the optimal
symmetric PTS with d0 = h. Note that this constraint of setting d0 = h still allows full flexibility
over the choice of strategies that each government can take once an initial punishment phasestarts. With regard to the issue of characterizing the efficient frontier among this subset ofsymmetric PTS with d0 = h, one can focus on simple PTS characterized in Proposition 1 as thefollowing corollary clarifies.
Corollary 2 to Proposition 2. The equilibrium payoff of any symmetric PTS (defined by

Definition 3) with the initial cooperative protection level being l and the level of total
protection that starts the initial punishment phase being h, is identical to the payoff of simple
PTS characterized in Proposition 1 with
V (l ; h) =u (l , l )u ( h , l ) − u (l , h ),− Pr (l )
C
1− δ
1−δ C

where Pr(l) = Pr( ω t∗ ∈ Ω D ) given ( τ t , et ) = (l, 0) and ( τ t∗ , et∗ ) = (l, 0). 34
3.2 Optimal Private Trigger Strategies

Up to this point, I have assumed that the range of private signals that trigger a punishment
phase, Ω D , is fixed.

Countries can change the (initial) cooperative protection level bychanging the range of punishment-phase-triggering private signals, Ω D , because it affects theprobability of a punishment phase being triggered against the potential use of concealed trade28

barriers.

This section characterizes the optimal simple PTS, or equivalently the optimal

symmetric PTS with l0 = l and d0 = h (Corollary 2 to Proposition 2), focusing its analysis on the
choice of Ω D that maximizes the expected discounted payoffs of countries. Once again I
abbreviate optimal simple PTS by optimal PTS hereafter, unless it is necessary to distinguishthem.
The private signal ω ∈Ω has two distinctive yet related quality dimensions as a measurethat detects the potential use of concealed protection. One is the sensitivity of the signal in
detecting possible defections, which links a higher protection to a higher probability of apunishment phase being triggered.

The other is the stability of the signal that rewards

cooperative behaviors with a lower probability of a punishment phase. I can represent thesensitivity by Pr/(τ) ≡ ∂Pr(τ)/∂τ > 0 and the stability by 1 − Pr(τ) measured at τ = l .
A change in the range of private signals that trigger a punishment phase may affect thesequalities of signals in different directions. In particular, countries may raise the sensitivity byproperly expanding the range of punishment-phase-triggering private signals, Ω D , but at thecost of undermining the stability. By denoting the degree of such expansion with a parameter

ωD, to be termed “a trigger control variable,” I can formalize this trade-off that countries facein choosing ωD by assuming ∂Pr/(τ)/∂ωD > 0 and ∂Pr(τ)/∂ωD > 0.
The analysis of optimality in this section focuses on simple PTS identified in Proposition 1,
with the cooperative protection level being determined by a choice over ωD. Assuming that ωD
uniquely determines l with I(l) = 0, I can represents l as a function of ωD; l = l(ωD). Then, as
shown by Corollary 2 to Proposition 2,(17)∗

VC ≡ V ( s, s ) =u (l , l )u (h, l ) − u (l , h),
− Pr (l )
C
1−δ C
1− δ

where (s, s*) are simple PTS defined in Definition 2. Note that the expected discounted payoffin (17) is no longer depending on the lengths of punishment phases. Therefore, I can describe
the optimal choice for ωD using the following first order condition:(18)∂VC ∂VC ∂l (ω D ) ∂VC ∂Pr (l )=+= 0, with∂l ∂ω D∂Pr ∂ω D∂ω D

One can derive H’s expected discounted payoff under symmetric PTS shown in the above corollary, from (6),using δC − δ = [u(h, l) − u(l, l)]/(VC − VN) and δC − δS = 2(δC − δ).
34

29

∂VC ∂u (l , l ) 1
u (h, l ) − u (l , h)=
− Pr / (l )C∂l∂l 1 − δ
1− δ C{∂u (l , l ) / ∂τ ∗ − ∂u (l , l ) / ∂τ }− Pr (l )< 0 for Pr (l ) being close to 0,
1− δ C∂l (ω D )∂I / ∂ω D∂Pr / (l )∂Pr (l ) /
=
−
<
0
iff
[1 − Pr (l )] −Pr (l ) > 0, andD
D∂I / ∂l∂ω∂ω∂ω D∂VC ∂Pr (l )∂Pr (l ) [u (h, l ) − u (l , h)]
=−
< 0,
D
D
∂ω ∂ω
∂ω D
1− δ C

where I = I(l) is the implicit function defined in (13). The first order condition is informative
about the trade-off that countries face in choosing an optimal ωD. Raising the trigger control
variable (ωD) will have a positive effect on the expected discounted payoff (VC) by lowering thecooperative protection level (l) since ∂l/∂ωD < 0 and ∂VC/∂l < 0, but it also has a negative effecton the expected payoff by raising the probability of a punishment phase being invoked, as
shown by ∂VC /∂ωD < 0 in (18). Thus, the optimal ωD should balance the gain from raising thesensitivity of the private signal (thus achieving a lower l) against the loss from reducing thestability of the cooperative equilibrium with a higher punishment phase probability.
When the initial ωD is at a very low level, then, it is generally possible to lower l by raising
the trigger control variable. For example, if Ω D = ∅, then l = h and Pr(l) = Pr/(l) = 0, implying∂l/∂ωD < 0 with ∂Pr/(l)/∂ωD > 0 from (18). If countries continue to raise ωD, the marginalincrease in the sensitivity of private signals in response to an increase in ωD is likely to get
smaller. To formalize this decreasing return to raising the trigger control variable, I assume
that ∂2Pr/(l)/∂(ωD)2 < 0 and ∂2Pr(l)/∂(ωD)2 = 0, with the latter assumption making the effect of ahigher ωD on Pr(l) to be constant. Then, it is possible to have ∂2l/∂(ωD)2 > 0 and ∂l/∂ωD = 0 for
a high enough ωD.
Even when it is possible to raise ωD to such a point that the countries would no longer beable to lower the cooperative protection level any further (∂l/∂ωD = 0), note that it is never
optimal to do so. If countries were to raise ωD in this way, then the first order condition for the
optimal ωD in (18) will be violated with ∂VC /∂ωD = (∂VC /∂Pr)(∂Pr(l)/∂ωD) < 0, implying thatcountries can increase their payoffs by lowering the trigger control variable. One can use asimilar argument to show that setting l = 0 cannot be optimal when ∂u(l, l) /∂l = 0 at l = 0 and30∂Pr(l)/∂l ≈ 0 at l = 0, as assumed in Proposition 1. I summarize these characterizations of
optimal simple PTS in the following proposition.
Proposition 3. Assume that the sufficient conditions for the existence of equilibrium simple

PTS in Proposition 1 are satisfied. In addition, assume that ∂Pr/(l)/∂ωD > 0, ∂Pr(τ)/∂ωD > 0,
∂2Pr/(l)/∂(ωD)2 < 0, and ∂2Pr(l)/∂(ωD)2 = 0 where ωD denotes the trigger control variableassociated with an expansion of Ω D . Then, under the optimal PTS, countries do not raise thetrigger control variable to the level that pushes down the cooperative protection level to its
minimum attainable level where ∂l/∂ωD = 0. In particular, the optimal PTS will not set l = 0
with ∂u(l, l) /∂l = 0 at l = 0. 35
The characterization of optimal PTS in Proposition 3 emphasizes the need for toleratingsome level of concealed trade barriers under PTS. For example, setting the concealed tradebarriers to zero in the cooperative period is not optimal: a slightly higher cooperative protectionlevel (by choosing a slightly lower ωD) would cause no first order loss as free trade is efficientwith ∂u(l, l) /∂l = 0 at l = 0 and would decrease the likelihood of a costly punishment phasebeing triggered. One cannot directly apply PTS for understanding the working of Section 301
of the U.S. under which the United States Trade Representative (USTR) follows an elaborate
procedure prior initiating a punishment against potential deviatory actions of other countries.
However, the following practice of Special Section 301 to protect U.S. intellectual property
rights (IPR) in foreign markets does illustrate the U.S. government’s willingness to toleratesome level of deviations from agreements, reserving retaliatory sanctions mainly against
considerable deviations.

In applying Special Section 301, the USTR specifies not only

“Priority Foreign Countries” who are “pursuing the most onerous or egregious policies thathave the greatest adverse impact on U.S. right holders or products, and are subject toaccelerated investigations and possible sanctions,” but also “Priority Watch List” of countries“who do not provide an adequate level of IPR protection or enforcement, or market access for
persons relying on intellectual property protection.” 36 Such a practice may not lead to the
maximal protection of the U.S. IPR, but may reduce the probability of costly tariff wars
invoked by Special Section 301.
35

A similar characterization has been drawn for optimal cartel trigger price strategies by Porter (1983).

31

4. A Possible Role for the WTO: Optimal Third Party Trigger Strategies

Regarding the issue of enforcing international trade agreements, this paper focuses on aphenomenon that the trade literature has not fully explored; countries may form different
opinions about potential violations of trade agreements. In the absence of a third party like the
WTO that can generate supposedly impartial opinions about such violations, Section 2 and 3 ofthis paper explore the possibility of countries’ adopting private trigger strategies, under which
each country initiates punishment phases based on its own imperfect private signals of the othercountry’s potential use of concealed trade barriers. In particular, this paper characterizes the
optimal PTS as an attempt to describe what countries can achieve with regard to trade policy
coordination in the absence of the WTO, a prerequisite for analyzing how the WTO canfacilitate improved coordination, especially when the WTO can simply generate its opinion ofpotential violations without any coercive power to impose its opinions upon countries.
To understand a possible role that the WTO can play under imperfect private monitoring ofpotential violations of trade agreements, this section analyzes “third-party trigger strategies”
under which a third party, such as the WTO, decides upon whether a violation has occurred andallows each country to initiate a punishment phase based on its decision.

Given thecharacterization of optimal PTS of the previous section, the comparison between the optimal
third-party trigger strategies and optimal PTS will illustrate how and to what degree the WTO
can help countries to enforce international trade agreements beyond what countries can doalone.
This paper, however, does not attempt to build a model that can proxy the actual operationof the WTO in dealing with potential violations and associated trade disputes: Though as I
discuss in the conclusion, this in itself would be a meaningful research direction. Instead, thissection will consider third-party trigger strategies under which the only role that the WTO
plays is providing an impartial third-party (thus, public) opinion of violations so that trigger
strategies are no longer subject to constraints imposed by the private nature of countries’
signals of violations under private triggers strategies, such as ICP.

36

These quoted definitions come from the USTR website (http://www.ustr.gov).

32

This analysis thus

illustrates the minimum role that the WTO can play in facilitating countries to improve theirtrade policy coordination.
To make a direct comparison between third-party trigger strategies and PTS characterized
in Section 3, I make the following assumptions in this section. The stage-game payoffs andaction variables of H and F are the same as those described in Section 2. In addition to thesetwo players, there exists the WTO, a third party supposedly neutral with regard to the issue of
enforcing international trade agreements. At the end of period t, the WTO obtains ω t ∈ Ωand ω t∗ ∈ Ω ∗ , the same private signals that each country receives of the other country’s
potential violations. One may model a mechanism under which each country truthfully reports
its private signals to the WTO in a non-public manner if the WTO can verify the reported
signals. For simplicity, this section simply assumes that the WTO has an access to such signals.
Given the setup of Section 2, then the WTO would have complete information of τ t and τ t∗
because the WTO knows all the random components of the model. Even when one introduces
additional random components into the model, the WTO may still have an informational
superiority over countries given the access to private signals of both countries. The analysis of
how the WTO may utilize such an informational superiority, which itself is attributable to the
WTO’s neutrality, is an interesting topic. As mentioned earlier, this paper assumes away sucha possibility, simply focusing on the possible role of the WTO in relaxing the constraints on thelengths of punishment phases imposed by the private nature of signals that trigger punishments,
namely the conditions specified in Lemma 1 (a).

Therefore, the following analysis will

characterize how changing “private” trigger strategies into “third-party” ones through the
WTO may improve the enforcement of international trade agreements, controlling the qualityof available information about potential deviations.
Once again Ω D denotes the range of private signals that triggers H (F) to initiate apunishment phase by imposing an explicit tariff, but it is the WTO that tells each country toinitiate such a punishment phase in third-party trigger strategies. The infinitely repeatedprotection-setting game between H and F stays the same as before, except that now the WTO
tells or does not tell each country to initiate a punishment phase by imposing an explicit tariffbased on its own (the WTO’s) signals of potential deviations. Note that these signals remain“not public” unless the WTO decides to make them “public.” For simplicity, I denote the33

WTO’s decision to tell H to initiate a punishment phase in period t based on its signals received
at the end of period t − 1 by μt-1 ∈ Μ ≡ {1, 0}, with μt-1 being 1 iff ω t-1 ∈ Ω D , denoting its
similar decision for F by μ*t-1 ∈ Μ ∗ ≡ {1, 0}. Then, a strategy for each country is defined byiis W = ( s W (t )) t∞=1 , similarly to the ones in Section 2, withs W (t ) : A t −1 × Μ t −1 × Μ ∗

(19)

t −2

× E*

t −1

∗

→ A and s W (t ) : A∗

t −1

×Μ ∗

t −1

× Μ t − 2 × E t −1 → A∗t −1

where Μ t −1 and Μ ∗ , respectively denote the history of the WTO’s decision of telling H and
F to initiate a punishment phase up to period t − 1. Note that strategies defined in (19) allow
each country to observe the WTO’s decision for the other country to initiate a punishment
phase only afterwards. This strategy specification under which each country chooses its current
action without knowing the WTO’s current decision on the other country’s initiation of apunishment phase may seem unnatural.

This specification, however, enables a directcomparison between third-party trigger strategies and PTS of Section 2 by making these two
types of strategies differ only in their ability in selecting the lengths of punishment phases.
Henceforth, the analysis will focus on third-party trigger strategies defined in Definition 4
below.
(i) Given that period t − 1 was a “cooperative” period with (et −1 , et∗−1 ) = (0, 0), each country
keeps cooperating by setting (τ ti , eti ) = (l, 0) as long as the WTO does not tell it to initiate
to a punishment phase by having μt-1i = 0 with i = * or none.
(ii) Given that period t − 1 was a “cooperative” period with (et −1 , et∗−1 ) = (0, 0), the WTO tells
H to initiate a punishment phase by setting (τ t , et ) = (h, 0) iff ωt −1 ∈ Ω D and it tells F toinitiate a punishment phase by setting (τ t∗ , et∗ ) = (h, 0) iff ωt∗−1 ∈ Ω D .
(iii) Given that a “punishment phase” was initiated in period t − 1 by only one country,
countries set (τ , e) = (h, h) and (τ ∗ , e ∗ ) = (h, h) for the following (T− 2) periods and they
continue to do so one more period with probability λ. Given that a “punishment phase”
was initiated in period t − 1 simultaneously by both countries, countries set (τ , e) = (h, h)and (τ ∗ , e ∗ ) = (h, h) for the following (TS − 2) periods and they continue to do so one more
period with probability λS. T and TS are integers that are greater than or equal to 2 with λand λS belonging to [0, 1]. Each country knows these variables (T, TS, λ, λS) when it
34

initiates a punishment phase and the actual length of a punishment phase is determined bysome public randomizing device (determining λ and λS) after a punishment phase being
initiated.
(iv) In period 1 and other “initial” periods right after the end of any punishment phase, withprobability Pr the WTO tells each country to initiate a punishment phase by setting(τ i , e i ) = (h, h), and with probability (1 − Pr) the WTO does not tell each country to initiate

a punishment phase so that it sets (τ i , e i ) = (l, 0), where Pr = Pr( ωti ∈ Ω D ) with (τ t , et ) = (l,0), (τ t∗ , et∗ ) = (l, 0), and i = * or none.
Definition 4. If (i), (ii), (iii), (iv) describe (sW, sW*), then (sW, sW*) are third-party trigger
strategies (TTS) with (l, ΩD, T, TS, λ, λS) as characterizing parameters.

Given this definition, it is easy to check that the expected discounted payoff under (sW, sW*)with (l,ΩD, T, TS, λ λS), denoted by VW(sW, sW*), is identical to V(s, s*) in (6). Once again, I
have (δ C − δ ) and (δ C − δ S ) respectively represent the (relative) length of the punishment
phase initiated by one country and by both countries simultaneously.
While the expression for the expected discounted payoff is same under TTS defined above
and under PTS defined in Definition 2, there exists an important distinction between these two
types of trigger strategies: The WTO has no incentive to lie about its private signals so that TTS
are not subject to the ICP. This implies that one can choose any values for the lengths ofpunishment phases, (δ C − δ ) and (δ C − δ S ) ∈ [0, δ C ]. Recall that δ C − δ = [u(h, lc) − u(lc,lc)]/(VC − VN) and (δ C − δ S ) = 2(δ C − δ ) under PTS. To make the comparison between the TTS
and the PTS even simpler, I make one more assumption that (δ C − δ S ) = 2(δ C − δ ) holds under
TTS, thus allowing full flexibility only over the choice of (δ C − δ ) , the length of a singlecountry-initiated punishment phase. This assumption enables one to tell whether the lengths ofpunishment phases under the optimal PTS are too short or too long (compared with the optimalTTS) by comparing the endogenously determined value for (δ C − δ ) under the optimal PTS
with the optimal choice of (δ C − δ ) under the TTS. Given this assumption of (δ C − δ S ) =35

2(δ C − δ ) , one can simplify VW(sW, sW*) into VCW ≡ VW(sW, sW*) = (1 − Pr)[u(l, l) − u(h, h)]/[1− δC + 2Pr(δC− δ)] + VN with VN = u(h, h)/(1 − δC).
To be able to support TTS as an equilibrium of the repeated protection-setting gamebetween H and F, TTS need to satisfy the following incentive constraint, denoted by ICW:
(ICW)

IW(l) ≡ ∂u(l, l)/∂τ −{δC[∂Pr(l)/∂τ ][1−Pr(l)][u(l, l) – u(l, h)+(δC–δ)(VCW – VN)]} = 0.

Note that ICW is identical to IC in (12) under PTS as long as δ under TTS is the same as under
PTS. This equivalence results from constructing TTS in the way that it may only differ fromPTS in its flexibility to choose the single-country-initiated punishment phase to last for any
length. The intuition behind this equivalence between IC under PTS and ICW under TTS isquite simple: Each country chooses its cooperative-period protection level, knowing that raising
the protection level increases the probability of a punishment phase being triggered in the samemanner under both trigger strategies.
In addition to ICW, there is one more incentive constraint that TTS needs to satisfy: Eachcountry has an incentive to follow the WTO’s decision on initiating a punishment phase.
Because the WTO’s decision becomes public (known to all players) with a one period lag, one
may construct a (off-equilibrium-path) punishment strategy, such as a permanent Nash tariffwar, against the behavior of not following the WTO’s decision on triggering a punishment
phase.

Given that the expected discounted payoff under TTS is strictly greater than the

discounted payoff of playing the static Nash tariff war forever, it is easy (and standard in theliterature) to show that each country has an incentive to follow the WTO’s decision as long as
the discount factor (δC) is high enough. I assume that this standard result is valid for thefollowing analysis with a high enough value for δC.
For the analytical simplicity, one can represent a choice of (T, λ) by a real number TW ∈ [1,∞) with δ C − (δ C ) T = (δ C − δ ) . TW = 1 (equivalent to the case of T = 2 and λ = 0) is the caseW

where any country’s initiation of a punishment phase by imposing its static optimal tariff is notfollowed by any punishment period where countries play a Nash tariff war of setting theirtariffs to be the static optimal ones, representing the shortest possible punishment phase. TW →∞ is the case where a permanent Nash tariff war is followed by an initiation of a punishment36

phase, representing the longest possible punishment phase. 37 Then, the problem of finding the
optimal TTS is solving the following maximization problem:⎧⎪ [1 − Pr (l )][u (l , l ) − u (h, h)]

(20)

⎫⎪

Max ⎨⎪1 − δ + 2Pr (l )[δ − (δ ) ] + V ⎬⎪ subject toωDand T W ∈[1,∞ )

⎩

C

c

C

C T

W

N

⎭

W
∂u (l , l )∂Pr (l )=δC[1 − Pr (l )]{u (l , l ) − u (h, h) + [δ C − (δ C ) T ](VCW − VN )},∂τ∂τwhere ωD represents a trigger control variable, defined in the same way as in Section 3.2.
Because the problem of finding the optimal PTS in Section 3.2 is to choose only ωD to
maximize the same payoff function as in (20) subject to the same incentive compatibility
condition, but with TW (or equivalently, corresponding T and λ) being determined byWδ C − (δ C ) T = [u(h, lc) − u(lc, lc)]/(VCW − VN), it is obvious that the optimal TTS of solving the
maximization problem in (20) will yield an expected discounted payoff that is greater than (orat least equal to) that under the optimal PTS. The question is how and to what degree the lessconstrained optimal TTS will outperform the optimal PTS.
Analyzing the first order conditions of the maximization problem for the optimal TTS in(20) can provide some insight into the factors that determine the optimal choice of ωD and TW:∂VCW ∂Pr (l ) ⎛ ∂VCW / ∂l ⎞ ∂I w (l ) ∂VCW ∂Pr (l )dVCW ∂VCW ∂l⎟⎟=+= ⎜⎜ − W+= 0,D∂l ∂ω D ∂Pr ∂ω D
∂Pr ∂ω Ddω D⎝ ∂I (l ) / ∂l ⎠ ∂ωdVCW ∂VCW ∂l∂VCW ⎛ ∂VCW / ∂l ⎞ ∂I W (l ) ∂VCW
⎟
=
+
= ⎜−
+ W =0∂l ∂T w ∂T W ⎜⎝ ∂I W (l ) / ∂l ⎟⎠ ∂T W
∂T
dT w
with

(21)

W

∂VCW
{[1 − δ C + 2[δ C − (δ C ) T ]}[u (l , l ) − u (h, h)]=−< 0 andW∂Pr{1 − δ C + 2 Pr[δ C − (δ C ) T ]}2
W∂VCW 2 ln(δ C )(δ C ) T Pr (1 − Pr )[u (l , l ) − u (h, h)]=W∂T w{1 − δ C + 2 Pr[δ C − (δ C ) T ]}2W− 2 ln(δ C )(δ C ) T Pr (1 − Pr ) ∂VCW=
< 0,W
1 − δ C + 2[δ C − (δ C ) T ] ∂Pr

Under TTS, it is not impossible to choose TW ∈ (0, 1) by setting T = 1 and λ ∈ (0, 1). For example, the WTO
uses its own randomizing device in determining whether to tell each country to impose its static optimal tariff for
one period or not with probability λ if ω ∈ ΩD or ω* ∈ ΩD. To make a direct comparison between PTS and TTS,
once again I limit the choices of TW with TW ∈ [1, ∞).
3737

where IW(l) represents the implicit function defining the ICW above, ∂l/∂ωD = − (∂IW/∂ωD)/(∂IW/∂l) and ∂l/∂TW = − (∂IW/∂TW)/(∂IW/∂l) generate the second equality for dVCW/dωD anddVCW/dTW, respectively. The expression after the second equality for ∂VCW/∂TW is obtainedusing the expression for ∂VCW/∂Pr in (21). As explained in Section 3.2, the optimal choice ofωD involves the balance between its positive effect of lowering the cooperative protection andits negative effect of increasing the probability of costly punishment phases.

Similarly,

increasing the length of a punishment phase has a positive effect of lowering the cooperative
protection by strengthening the punishment but also entails a negative effect of increasing thecost of punishment with the costly punishment phase being longer. The optimal choice of TW
also involves balancing between these counteracting forces.
This section focuses on the analysis of an optimal choice of TW because Section 3.2
provides an analysis of the optimal choice over ωD and a similar characterization should apply
to the one under TTS. 38 For further characterization of an optimal choice of TW, I assume thatthe optimal ωD is an interior solution, thus dVCW/dωD = 0. It is reasonable to assume that
dVCW/dωD = 0 for any TTS that attains improvement over one-shot Nash equilibrium because acorner solution for ωD implies either no punishment for any contingency ( Ω D = ∅) orpunishment for all contingencies ( Ω D = Ω ). Using dVCW/dωD = 0 together with the second
expression for ∂VCW/∂TW in (21), I can rewrite dVCW/dTW as follows:⎛ ∂VCW ⎞W⎟⎜−1⎡ ∂I W
dVCW ⎜
2 ln(δ C )(δ C ) T Pr (1 − Pr ) ⎛ ∂Pr (l ) ⎞∂I W ⎤l∂⎟,+ A D ⎥ , where A =
= −
⎜
W
⎢
D ⎟
dT W ⎜ ∂I W ⎟ ⎣ ∂T W
∂ω ⎦1 − δ C + 2[δ C − (δ C ) T ] ⎝ ∂ω ⎠
⎟
⎜
∂l ⎠
⎝
(22)
⎛ ∂VCW ⎞
⎟
⎜
∂I W
∂I W
∂I W
l
∂
⎟
⎜
with − W < 0, W < 0,
< 0, and A D > 0.
⎜ ∂I ⎟
∂ω
∂ω D
∂T
⎟
⎜
∂l ⎠
⎝

The above first order condition for an optimal choice of TW, which also embodies the first ordercondition for the choice of ωD, reveals a potentially “competing” nature of these two choicevariables in restraining the use of concealed trade barriers. ∂IW/∂TW < 0 and ∂IW/∂ωD < 0
For any given level of TW, the optimal choice over ωD under TTS should be the same kind of balancing choice as
the one under PTS. See the above discussion on the choice of ωD in relation with (21). Therefore, thecharacterization of an optimal ωD of Proposition 2 should apply to the optimal ωD under TTS.

38

38

demonstrate that both of these choice variables can relax ICW, which in turn enable countries tolower the cooperative protection level, l. For example, if the effectiveness of ωD in relaxing
ICW rises so that the absolute value of ∂IW/∂ωD (and A∂IW/∂ωD) increases, then the optimal
choice of TW may involve a decrease in TW and an increase in ωD to sustain dVCW/dTW = 0 if
∂2IW/(∂TW)2 > 0 and ∂(A∂IW/∂ωD)/∂ωD < 0. 39 In fact, the following result establishes that the
optimal TW may take corner solutions depending on the probability of a punishment being
triggered in the equilibrium, which in turn may depend on the accuracy of information about
potential deviations, as shown through a numerical analysis that follows this analytical result:
Proposition 4. Given that ∂Pr/(l)/∂ωD > 0, ∂Pr(τ)/∂ωD > 0, ∂2Pr/(l)/∂(ωD)2 < 0, and∂2Pr(l)/∂(ωD)2 = 0 as assumed in Proposition 3 for the characterization of optimal PTS,(a) the length of a single-country initiated punishment phase, TW, equals 1 under the optimal
TTS if Pr(l) < Pr , wherePr =− 3(1 − δ C ) + [3(1 − δ C )] 2 + 16δ C (1 − δ C )8δ C,with ∂ Pr / ∂δ C < 0 and lim
Pr / ∂δ C = 1 / 3 so that Pr ∈ (0,1 / 3) for δ C ∈ (0,1) , andCδ →0(b) the length of a single-country initiated punishment phase, TW, goes to ∞ under the optimal
TTS if Pr(l) > Pr , wherePr =2 − [u (l , l ) − u (h, h)] /[u (l , l ) − u (l , h)],
4 − [u (l , l ) − u ( h, h)] /[(u (l , l ) − u (l , h)]with [u (l , l ) − u ( h, h)] /[u (l , l ) − u (l , h)] ∈ (0,1) for l ∈ [0, h) so that Pr ∈ (1 / 3,1 / 2) .
(See Appendix for Proof)
According to Proposition 4, the length of a single-country initiated punishment phase
under the optimal TTS takes its minimum value of TW = 1 if the probability of a punishment
phase being triggered is below a critical level, denoted by Pr . With TW = 1, note that no tariff
war period (under which both countries impose their static optimal tariffs) will follow an
initiation of any punishment phase. This implies an asymmetric (in the sense that only thepotential deviator is punished with the punishing country being rewarded by imposing its static
39

One can show that ∂2IW/(∂TW)2 > 0 but it seems to be very difficult prove that ∂(A∂IW/∂ωD)/∂ωD < 0 given the39

optimal tariff) and minimum (in the sense that the punishment length is taking its minimum
value) punishment against potential violations. Note also that countries cannot use such an
asymmetric and minimum punishment under PTS because countries will have an incentive toinitiate such a punishment phase regardless of their private signals. The presence of the WTO,
a third party who impartially judges whether a country (might have) violated a trade agreement
allows countries to use an asymmetric and minimum punishment, facilitating countries torealize higher expected payoffs beyond what they can do by themselves under PTS. 40
This asymmetric and minimum punishment (TW = 1) is optimal when the probability of apunishment phase being triggered, Pr(l) is less than a critical level, Pr . As briefly discussedwith regard to the first order condition for choosing TW in (22), one can understand this(sufficient) condition for TW = 1 by looking at how a change in Pr(l) affects the relative
effectiveness of TW and ωD in relaxing the incentive constraint, ICW. In fact, one can show thatthe (relative) effectiveness of ωD increases faster than the effectiveness of TW in response to adecrease in Pr(l) so that the effectiveness of ωD is greater than that of TW for all values of TW
when Pr(l) < Pr , thus having dVCW/dTW < 0 for all TW ∈ [1, ∞). The effectiveness of ωD
relative to TW is measured by the absolute value of A(∂IW/∂ωD) relative to the absolute value of∂IW/∂TW in the second bracket of (22).
∂ Pr / ∂δ C < 0 in Proposition 4 implies that the optimal TTS is less likely to involve TW = 1

when countries’ relative valuations of future payoffs increase with higher values of δC. Onceagain, one can understand this result by examining how a change in δC affects the relative
effectiveness of TW and ωD in relaxing the incentive constraint, ICW: One can show that theeffectiveness of TW increases faster than that of ωD in response to an increase in δC so that the
optimal TTS is less likely to set TW = 1 when δC is higher. If δC = 1/2, for example, Pr = 1/4,
implying that TW = 1 is optimal under TTS if the probability of a punishment being triggered isless than 1/4, and Pr decreases toward zero as δC approaches 1.
Proposition 4 (b) shows that TW → ∞ may also emerge as an optimal punishment length
choice under TTS if the probability of a punishment phase being triggered is above a criticalhighly non-linear nature of A in ωD, unless one introduces stringent assumptions on Pr.
40
This kind of asymmetric action is often one of important characteristics of optimal strategies of repeated games
under various applications, such as in Kandori and Matsushima (1998), Compte (1998), and Athey and Bagwell(2001), because such asymmetry allows players to avoid actions with (at least heavy) dead-weight losses.

40

level, denoted by Pr . This maximum punishment of playing the Nash tariff war forever once apunishment is triggered is a surprising result because the main reason for countries tocoordinate their trade policies is to avoid playing the Nash tariff war, and because they canchoose any length for their punishment phase under TTS. Again, it is possible to understandthis sufficient condition for TW → ∞ by looking at how a change in Pr(l) affects the relative
effectiveness of TW and ωD in relaxing the incentive constraint, ICW. The effectiveness of ωD
decreases faster than that of TW in response to an increase in Pr(l) so that the effectiveness ofωD is smaller than that of TW even when TW → ∞ if Pr(l) > Pr , thus having dVCW/dTW > 0 evenwhen TW → ∞.
Proposition 4 provides a characterization of the optimal TTS, which depends on theprobability of a punishment phase being triggered right after a cooperative period. One mayfind that such a characterization is not satisfactory because the characterization relies on Pr(l),a variable that countries choose indirectly by choosing ωD. 41 One may also wonder about thepossibility of more directly comparing the optimal PTS and the TTS, thus finding when they
will differ from each other and how they will differ.42 In response to such demands, one may
try to introduce more structures to the private signals, thus making Pr(l) depend on someaccuracy measure of private signals, then characterize the optimal TTS (and the optimal PTS)
depending on such a fundamental variable. Because of the highly non-linear nature of the
maximization problem involving two choice variables (TW and ωD) as shown through the firstorder condition in (22), pursuing such a characterization is extremely difficult, if not
infeasible. 43
While it might not be possible to derive complete analytical results regarding thecharacterization of the optimal TTS and optimal PTS in the way the preceding paragraphdiscusses, one can conduct a numerical analysis for such characterization. The following
numerical analysis does just that and reveals several interesting (numerical) results. To conduct

A positive side of the characterization of optimal TTS in Proposition 4 is that it imposes relatively weak
assumptions on private signals and is still able to drive a relatively sharp prediction of when the corner solutions
will emerge as an optimal choice for TW, depending on the equilibrium values of Pr(l).
42Proposition 4 does provide results that show how and when the optimal TTS would differ from the optimal PTS
because neither TW = 1 nor TW → ∞ occur under PTS. What is missing is a more continuous comparison of thetwo strategies, possibly depending on some fundamental variables, such as a measure for accuracy of signals.
43
As shown in the proof of Proposition 4 in the Appendix, proving the results in Proposition 4 itself is not a trivial
exercise given the highly nonlinear nature of the optimization problem to solve.
41

41

a numerical analysis, I use the same partial equilibrium trade model as the one in Bond andPark (2002) where H exports good 1 and F exports good 2, with σ ∈ [1, ∞) denoting the size of
H’s markets relative to F’s. 44 Demand for good i in H is Di = σ ( A − Bpi ) and supply of good iin H is X i = σ (α i + βpi ) , where pi is the price of good i in H with i = 1 or 2. For F, demandand supply are given by Di∗ = A − Bp i∗ and X i∗ = α i∗ + β pi∗ . To ensure that H will export good
1 and import good 2 and that the countries will be symmetric when σ = 1,

α1 − α1∗ = α 2∗ − α 2 > 0 and α 1 = α 2∗ . In addition, I assume that Pr(l) takes the following
functional form:

Pr (l ) = Pr (l ω D ; ρ , χ ) =ω D [l 2 /(2 χ ) + ρ ] for l ≤ l / 2,(23)

= ω D [(l × l ) / χ − l 2 /(2 χ ) + ρ − l /(4 χ )] for l / 2 < l ≤ l ,2= 1 for l > l ,where l ≡ 2 χ / ω D − χρ , 1/χ∈ (0,∞) represents the sensitivity of the signal in detecting an
increase in the level of concealed trade barriers, and ρ ∈ [0, ∞) represents the level of errors in
detecting concealed trade barriers (thus, the “in-”stability of the signals), making Pr(l) > 0
even when l = 0 with ρ > 0 and ωD (∈[0, 1/ρ)) > 0. While the complicated expression for Pr(l)with l > l/2 is used to make the probability density function to be symmetric around l/2 andPr(l) = 1 when l = l, the equilibrium values for l are all less than l/2 in the following numericalanalysis, thus making this part of the probability definition be redundant. Pr(l) defined in (23)is one of simplest functional forms for Pr(l) with parameters representing both the sensitivityand (in)stability of private signal and also having ∂Pr(l)/∂l > 0, ∂2Pr(l)/∂(l)2 > 0, ∂Pr/(l)/∂ωD >0, ∂Pr(τ)/∂ωD > 0, ∂2Pr/(l)/∂(ωD)2 < 0, and ∂2Pr(l)/∂(ωD)2 = 0 for l ≤ l/2, as assumed in
Proposition 4. 45
In a previous version of this paper, there was a section that provides an analysis of PTS in the presence ofasymmetry in the size of trading countries. The following concluding section briefly discusses the effect of
introducing such asymmetry on PTS as a factor that may limit the use of PTS in restraining concealed tradebarriers.
45
One may find Pr(l) = 1 for l > l not satisfying, especially when l < h. Thus, one can consider using an adjusted
Rayleigh distribution, Pr (l ) = Pr (l ω D ; ρ , χ ) =1 − exp[−(ω D l ) 2 /(2 χ 2 ) − ρω D ] for the numerical analysis because
44

Pr(l) < 1 for all l ∈ [0, ∞). The problem associated with using this Rayleigh distribution is that ∂2Pr(l)/∂(ωD)2 = 0
is no longer true, and this assumption is what enables simplification of the first order condition for TW in (22),which in turn leads to the analytical results in Proposition 4. As a robustness check, I have done a numericalanalysis using this probability function and found that characteristics of the optimal PTS and optimal TTS are
qualitatively identical to those shown in the numerical analysis of this section using Pr(l) in (23).

42

I assume that σ = 1 to analyze the case of symmetric countries, and also assume that α1 −α1* = 3, β + B = 1, which induces h = 1 for simplicity. 46 To illustrate how the optimal TTS
change as the instability of the private signal (measured by ρ) changes, Figure 4 shows theoutcome of the numerical analysis with χ = 1 and δC = 0.5. It indicates how each of thefollowing changes in response to an increase in the instability of the signal, ρ, from80(×0.00005) to 130.2(×0.00005): (i) the expected percentage payoff gain under the optimal
TTS compared with playing the static Nash tariff war forever, (VCW − VN)/VN ; (ii) the
cooperative protection level, l; (iii) the probability of a punishment phase being triggered, Pr(l);(iv) the length of a punishment phase, TW; and (v) the trigger control variable choice, ωD.
As predicted by Proposition 4, TW = 1 when Pr(l) < Pr = 1/4 (using δC = 0.5) and TW → ∞when Pr(l) > 4/9, using the fact that the maximum value that Pr can take is 4/9 as [u(l, l) − u(h,h)]/[u(l, l) − u(l, h)] reaches its minimum at 2/5 with l = 0 given the parameter values of the
trade model under consideration. It also confirms the conjecture that the probability of apunishment phase being triggered in the equilibrium would depend on the accuracy of
information about potential deviations (at least in the limits), thus having TW = 1 for low
enough values of ρ and TW → ∞ for high enough values of ρ. Another notable aspect of thisnumerical result is that Pr(l) decreases in response to an increase in ρ, the instability (orinaccuracy) measure of private signals, when optimal TTS utilize both ωD and TW (> 1). A
possible explanation for this phenomenon once again can be based on the relative effectivenessof ωD and TW in relaxing ICW: If the effectiveness of TW relative to ωD improves as ρ increases,
then countries will substitute ωD with TW, implying a lower ωD and a higher TW as shown in thebottom two graphs in Figure 4, which in turn may lead to a decrease in Pr(l) because∂Pr(l)/∂ωD > 0. 47

In deriving this result, I assume that each country’s welfare function (as a function of τ and τ*) derived fromdemand and supply functions with no uncertainties is identical to the ones derived with uncertainties described in
Section 2.1. This is a strong assumption but justifiable given the fact that what one really needs are u(τ, τ*) andu*(τ*, τ) with ∂u(τ, τ*)/∂τ > 0 at τ = 0, ∂u*(τ*, τ)/∂τ < 0, ∂[u(τ, τ*) + u*(τ*, τ)]/∂τ < 0, ∂2u(τ, τ*)/∂τ2 < 0, and ∂2u(τ,τ*)/∂τ∂τ* = 0, properties of welfare functions of the trade model of Bond and Park (2002).
47
This explanation of ∂Pr(l)/∂ρ < 0 for internal values of TW seams to be in conflict with the following explanation
for Proposition 4 (a) given earlier, “the (relative) effectiveness of ωD increases faster than that of TW in response toa decrease in Pr(l),” because ωD decreases and TW increases when Pr(l) decreases in response to an increase in ρ inthe bottom 3 graphs of Figure 4. However, these are not contradictory explanations because the explanation for
Proposition 4 (a) is explaining how the corner solution of TW = 1 may rise for small values of Pr(l) by changing
46

43

Another interesting exercise one can do with this numerical analysis is to compare theoptimal TTS with optimal PTS. Continuing to assume the same parameter values, except for χ
being 100 instead of being 1 (thus, the sensitivity of private signals being lower), Figure 5
compares the optimal TTS and the optimal PTS in all the same 5 variables as in Figure 4 when

ρ increases from 30(×0.000005) to 61.9(×0.000005).

Note that the bold lines represent

variables for the optimal TTS and the dotted lines depict variables for the optimal PTS. The
graphs on the right column in Figure 5 provide zoomed graphs of the same 5 variables for highvalues of ρ, from 59(×0.000005) to 61.9(×0.000005) because the variable for the TTS and the
PTS are very similar for these high values of ρ. One obvious result is that the gains fromcooperation under the TTS are higher than those under PTS (being identical only when ρ = 60.5
in Figure 5 with all other variables being identical as well, as they should be). One less obvious
but potentially important result is that the gains from moving from the optimal PTS to the
optimal TTS are significant when the signals are relatively accurate with low values for ρ. As
one can easily tell from the top graphs in Figure 5, such gains can become negligible for highvalues of ρ. It is important to note that the significant gains from moving from the optimal PTS
to the optimal TTS come from countries’ ability to reduce the length of punishment phase andsubstitute it with a higher value for ωD under TTS. The probability of a punishment phasebeing triggered is higher under TTS than under PTS for all ρ < 60.5 due to a higher value for ωD.
This higher value for ωD enables to countries to support a lower protection level under TTS thanunder PTS, as shown in Figure 5 for ρ < 60.5.
Given the analytical results in Proposition 4 as well as the numerical ones shown in Figure
4 and 5, I can highlight the main potential benefit of the WTO’s presence in enforcing
international trade agreements as follows. Even when the (private) signals of violations are
relatively accurate, it might be hard for countries to be responsive against potential violations(choosing a higher value for ωD) under PTS because initiating a punishment should and willaccompany a rather long and costly tariff-war phase between countries (to eliminate the
incentive to abuse the punishment). Once countries can utilize opinions of an impartial third
party, such as the WTO, then countries can employ a more effective punishment, possibly thePr(l) as if it is an exogenous variable, even though it is in fact an endogenous variable affected by the optimal
choices of ωD and TW.

44

asymmetric and minimum punishment with TW = 1, which in turn enables countries to be lesstolerant of potential violations, attaining a higher level of cooperation!

5. Concluding Remarks

In the presence of concealed trade barriers of which each country has imperfect private
signals, the WTO can facilitate a better cooperative equilibrium in the repeated traderelationship. This is established by comparing the optimal PTS (private trigger strategies) inwhich each country triggers a punishment phase based on its own private signals with the
optimal TTS (third-party trigger strategies) in which the WTO tells who should start apunishment phase based on its (the WTO’s) signals, abstracting away from any informational
advantage or disadvantage of the WTO over trading countries. Prior to discussing the role ofthe WTO, the analysis first establishes that symmetric countries may restrain the use ofconcealed trade barriers under simple PTS if the sensitivity of their private signals rises in
response to an increase in such barriers. It also shows that any equilibrium payoff under almost
strongly symmetric PTS will be identical to the one under simple PTS as long as the initialpunishment is triggered by a static optimal tariff, justifying the focus on simple PTS. The
analysis of optimal (simple) PTS reveals that it is not optimal to push down the cooperative
protection level to its minimum attainable level (such as free trade) due to the cost associatedwith increasing the probability of costly punishments.
To illustrate how and by what degree the WTO may facilitate countries in enforcing
international trade agreements beyond what they can achieve alone under PTS, this paper
conducts both an analytical analysis of the optimal TTS and a numerical comparison of the
optimal PTS and optimal TTS. If the probability of a punishment phase being triggered is low
enough, possibly because of accurate enough signals of potential violations, the analyticalanalysis establishes that the optimal TTS entail an asymmetric and minimum punishment. The
punishment is asymmetric in the sense that only the potential deviator is punished with thepunishing country being rewarded by imposing its static optimal tariff and minimum in thesense that the punishment length is taking its minimum value. Just the opposite result of using
a punishment involving a permanent Nash tariff war will emerge under the optimal TTS if theprobability of a punishment being triggered is high enough, possibly because of inaccurate

45

signals of violations. The presence of the WTO under TTS changes the nature of signals thattrigger punishments from private into public, enabling countries to employ punishment phasesof any length, which in turn can help countries to attain a better cooperative equilibrium. The
numerical analysis illustrates that the WTO’s contribution is likely to be more significant whenits private signals are relatively accurate so that the lengths of punishment phases are shorter
than those under the optimal PTS, possibly involving the asymmetric and minimum punishment.
With regard to the effectiveness of PTS, there exist other factors that may severely limit theuse of PTS so that countries cannot support any level of cooperation, as analyzed in a previous
version of this paper. One is a reduction in each country’s time lag in readjusting its tariff
protection level in response to the other country’s initiation of a punishment phase by imposing
an explicit tariff. The other is asymmetry among countries. Both of these factors may limit thelevel of cooperation attainable under PTS by reducing the lengths of punishment phases thatcountries can employ against potential deviations.
Recall that each country is willing to initiate a punishment phase involving costly tariff war
periods under PTS because it can realize some gains in the initial period of a punishment phaseby imposing its static optimal tariff unilaterally. If countries can readjust their tariff levels
faster so that countries play the static Nash tariff war (almost) instantaneously in response to an
initiation of a punishment, then no length of a punishment phase would satisfy the incentive
compatibility condition for truthful revelation of private information (ICP). 48 This is because
countries will only lose from initiating a punishment, thus making it impossible to support any
cooperation under PTS. 49 If there exists a large enough asymmetry among trading countries, asimilar problem will rise under PTS. When one of two trading countries gets very small
compared to the other one, then the small country’s static optimal tariff goes to zero because its
48

It is sometimes argued that enforcement constraints cannot be relevant in the trade policy setting, since a
government can retaliate almost immediately whenever another government defects. This result suggests that suchan argument is based on a public-action model and requires substantial modification in a private monitoring setting,as pointed out by a referee of this paper.
49
Abreu, Milgrom and Pearce (1991) and more recently Sannikov and Skrzypacz (2007) show that shortening theperiod over which actions are held fixed can hurt the possibilities for cooperation under imperfect public
monitoring, possibly leading to the impossibility of cooperation. While their impossibility of cooperation outcome
from shortening the period over which actions are held fixed is similar to the one under PTS, the driving forces
behind these impossibility results are different. Under imperfect public monitoring, shorter periods of fixed action
multiply the ways that player can deviate from the equilibrium, leading to the impossibility. Under PTS, theimpossibility of cooperation arises not because countries can deviate more effectively (the period over which
concealed trade barriers are held fixed remains constant and only the period of readjusting tariff levels in response46ability to change the terms of trade by imposing tariff becomes negligible. 50 This implies that
there will be no length of a punishment phase satisfying the incentive compatibility condition
for the small country, thus eliminating the possibility of supporting any cooperation under
PTS. 51
In the presence of factors that may limit the credibility of initiating strong punishments
against potential deviations under PTS, once again the WTO may facilitate cooperation bychanging the nature of information that triggers punishments from private into public, which inturn restores the credibility of punishments. For example, the WTO mandates a regular review
of its members under the Trade Policy Review Mechanism (TPRM), generating “public”
reports which consist of detailed chapters examining the trade policies and practices of themembers. According to the WTO’s website, “Surveillance of national trade policies is afundamentally important activity running throughout the work of the WTO. At the centre ofthis work is the TPRM.”
Another activity that the WTO does in enforcing trade agreements is settling disputes
through its Dispute Settlement Procedure (DSP). When countries form different opinions ofpotential violations based on their imperfect and private information, the DSP of the WTO may
generate third-party rulings on disputed cases, thus public signals about potential deviations.
As emphasized in this paper through the analysis of the optimal TTS, the availability of an
impartial third party’s opinion may enable countries to adopt a more efficient punishment, suchas the asymmetric and minimum punishment. This in turn enables countries to be more

to initiations of punishment phases shortens) but because the punishments that countries can use against deviations
weaken.
50
McLaren (1999) and Park (2000) analyze trade agreements between countries of asymmetric size where a small
country has no ability to change the terms of trade by its tariff so that its static optimal tariff is zero.
51
Formal proofs for these results can be found in an earlier version of this paper, “Private Trigger Strategies in the
Presence of Concealed Trade Barriers.” As correctly pointed out by one of referees of this paper, a proper way tointroduce a change in the speed of readjusting tariff protection levels is to make the model into one in which
information arrives continuously over time and to shorten the period under which tariff levels are held fixed. The
ad-hoc approach of changing the payoff function to some convex combination of the payoff before and after thereadjustment of tariffs is adopted to introduce a change in the readjustment speed of tariffs without any change inthe basic structure of the model and without any change in the readjustment speed of concealed trade barriers.
This reflects that the readjustment of concealed trade barriers may take longer than readjusting tariffs because
concealed trade barriers often rely on customary practices or implicit agreements but each country may readjust its
tariff level by simply issuing an executive order. Given the logic of the proof, however, the impossibility ofcooperation result should be still valid under a proper modeling of a change in the readjustment speed of tariffs. A
referee’s questioning the focus on the symmetry in the triggering event (ΩD = ΩD*) in the presence of asymmetry
among countries is also legitimate, but the impossibility of cooperation result under a large enough asymmetry
among countries should be valid even when one considers asymmetric triggering events with ΩD ≠ ΩD*.

47

responsive of potential violations and as a result attain a higher level of cooperation comparedto the situation with no DSP.
While this paper provides a new way of understanding the role that the WTO plays inenforcing international trade agreements, there is still much to be done for a more complete
understanding of its role in dispute settlements. 52

For example, the DSP of the WTO

encourages settlements through consultations among disputing parties as a preferred way tosettle trade disputes. According to the official website of the WTO, “The priority is to settle
disputes, through consultations if possible. By July 2005, only 130 of the nearly 332 WTO’s
dispute cases had reached the full panel process. Most of the rest have either been notified as
settled “out of court” or remain in a prolonged consultation phase — some since 1995.” 53 This
indicates that the DSP plays a role that goes beyond simply generating public signals ofpotential deviations. Carefully analyzing the role that the DSP of the WTO plays in the context
of imperfect private monitoring of potential violations, especially regarding settlements throughconsultations, would be a meaningful extension of this paper.

52

Maggi and Stagier (2008) analyze the possible role that the DSP of the WTO plays in completing an incomplete
contract, characterizing the optimal choice of contractual incompleteness and the DSP design.
53
This quote comes from the following website: http://www.wto.org/english/thewto_e/whatis_e/tif_e/disp1_e.htm.

48

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50Appendix
Proof for Lemma 1 (a)It is obvious that ICP is a necessary condition and ICP becomes δ C − δ = [u(h, l) − u(l, l)]/(VC −VN)] if δ C − δ S = 2(δ C − δ ) .

Therefore, I only need to show that δ C − δ S = 2(δ C − δ ) is also anecessary condition for each country to truthfully represent its private signals under PTS. Note that ICP
only provides the incentive for each country to truthfully initiate a punishment phase given that it was
following the equilibrium strategy of setting τ = l in a pervious (cooperative) period. Even when ICP issatisfied, there is a deviation possibility of settingτ ≠ l in a current period and starting a punishment
phase in a following period regardless of its private signal, upon the contingency of no punishment
phase being initiated in that current period. In an equilibrium of the repeated game, there should be nosuch deviation incentive and the following argument will prove that δ C − δ S = 2(δ C − δ ) is necessaryfor eliminating such an incentive.
For PTS defined in Definition 2 to be equilibrium strategies, each country should have no incentive
to set τ ≠ l in any period following a cooperative one (or in any “initial” period) unless it desires toinitiate a punishment phase by setting τ = e = h, regardless of whether it would initiate a punishment orcontinue cooperating in a following period, upon the contingency of no punishment phase being
initiated. To derive the (necessary) condition for such an equilibrium behavior, first note that the
expected discounted payoff of setting its total protection level to equal τ in any period following acooperative one isPr[u (τ , h) + (δ C − δ )V N + δVC ]+ (1 − Pr ){u (τ , l ) + δ C Pr (τ )[u (l , h) + (δ C − δ )V N + δVC ] + δ C [1 − Pr (τ )][u (l , l ) + δ CVC ]},or

Pr[u (τ , h) + (δ C − δ )V N + δVC ]+ (1 − Pr ){u (τ , l ) + δ C Pr (τ )[u (h, h) + (δ C − δ S )V N + δ S VC ] + δ C [1 − Pr (τ )][u ( h, l ) + (δ C − δ )V N + δVC ]},depending on whether H continues to cooperate (by setting its total protection level to equal l) or initiate
a punishment phase (by setting its total protection level to equal h), respectively, in the following period
upon the contingency of no punishment phase being initiated after setting its total protection level toequal τ. To be able to support τ = l, the following first order conditions need to be satisfied for each of
the above expected discounted payoff expressions: ∂u(l, l)/∂τ = δ C (1 − Pr)[∂Pr(l)/∂l][u(l, l) − u(l, h) +( δ C − δ )(VC − VN)] for the first expression, and ∂u(l, l)/∂τ = δ C (1 − Pr)[∂Pr(l)/∂l][u(h, l) − u(h, h) +( δ − δ S )(VC − VN)] for the second one. Using u(l, l) − u(l, h) = u(h, l) − u(h, h), these two first orderconditions imply that δ C − δ = δ − δ S , or equivalently δ C − δ S = 2(δ C − δ ) .

51

Proof for Lemma 1 (b)I will prove Lemma 1 (b) in the following way. First, I will assume the existence of ( δ , δ S ) that∗satisfies δ C − δ = [u(h, l) − u(l, l)]/(VC − VN)] and δ C − δ S = 2(δ C − δ ) so that VC ≡ V ( s, s ) in (6) canbe rewritten into a simpler form. Given δ C ≈ 1 and Pr ≈ 0, then I will show that there exists indeed aunique combination of ( δ , δ S ) that satisfies these necessary conditions.
∗

Using δ C − δ S = 2(δ C − δ ) , I can simplify V ( s, s ) in (6) into VC = k /[1 − δ C + 2 Pr (δ C − δ )] + V N
with k = (1 − Pr 2 )[u (l , l ) − u (h, h)] + Pr (1 − Pr )[u (l , h) − u (l , l )] + Pr (1 − Pr )[u (h, l ) − u (l , l )] . To denotethe value of δ that satisfies ICP with VC = VC (δ 0 ) ≡ k /[1 − δ C + 2 Pr (δ C − δ 0 )] + V N , define δ e (δ 0 ) ≡δ C − (1 − δ C )[u ( h, l ) − u (l , l )] /[(1 − δ C )VC (δ 0 ) − u (h, h)] = δ C − [u (h, l ) − u (l , l )][1 − δ C + 2 Pr (δ C − δ 0 )] / k .
If there exists a unique value of δ 0 ∈ (0, δ C ) such that δ e (δ 0 ) = δ 0 and δ S = 2δ 0 − δ C ∈ (0, δ C ) when
δ C ≈ 1 and Pr ≈ 0, then proof is done for Lemma 1 (b).

First, note that ∂δ e (δ 0 ) / ∂δ 0

= 2 Pr[u (h, l ) − u (l , l )] / k > 0 approaches zero if Pr ≈ 0. Second, note that δ e (δ 0 ) approaches δ C withδ e (δ 0 ) < δ C when δ C ≈ 1 and Pr ≈ 0, including the case with δ 0 = 0 . These two facts together imply
that there exists a unique value of δ 0 ∈ (0, δ C ) such that δ e (δ 0 ) = δ 0 when δ C ≈ 1 and Pr ≈ 0.
Because δ 0 < δ C and δ 0 ≈ δ C for δ 0 satisfying δ e (δ 0 ) = δ 0 when δ C ≈ 1 and Pr ≈ 0, δ S = 2δ 0 − δ C∈ (0, δ C ) .

Proof for Lemma 2

Proofs for the results in Lemma 2 follow the same logics as the proofs for the corresponding resultsin Stokey and Lucas (1989). More specifically, Theorem 4.2, 4.3, 4.4, and 4.5 in Stokey and Lucas
correspond to (i), (ii), (iii), and (iv) of Lemma 2 (a), respectively. One may also find corresponding
proofs for Lemma 2 (b) and Lemma 2 (c) in Theorem 4.6 in Stokey and Lucus. To save the space, I
discuss how one can adjust the corresponding proofs in Stockey and Lucas to prove the results inLemma 2. A complete proof for Lemma 2 is available upon request.

For Lemma 2 (a):Let Γ: X → X denote the correspondence describing the feasibility constraints with X =[0, h]. Givenx0 ∈ X, let Π(x0) ={ {xt }t∞=0 : xt+1 ∈ Γ(xt), t = 0, 1, …} be the set of plan that are feasible from x0. DefineF(xt, xt+1) as F(⋅) in (8). Then, Assumption 4.1 in Stokey and Lucas is satisfied. I modify Assumption

52

t −1

4.2 with lim ∑t = 0 (δ C ) t [∏ (1 − Pr (xi ))]F ( xt , xt +1 ) existing for all x0 ∈ X and x ∈ ∏( x0 ) , then it is also
n

n→∞

i =0

t −1

satisfied. For each n = 0, 1, ⋅⋅⋅, define un: Π(x0) → R by u n ( x ) = ∑t = 0 (δ C ) t [∏ (1 − Pr ( xi ))]F ( xt , xt +1 ) .
n

i =0Define u: ∏( x0 ) → R by u ( x) = lim u n ( x) . Then, it is easy to show that Lemma 4.1 in Stocky andn→∞Lucas holds when one replaces u ( x) = F ( x0 , x1 ) + δ C u ( x ) with u ( x ) = F ( x 0 , x1 ) + δ C (1 − Pr (x 0 ))u ( x ) .
//Having v* and v in Stocky and Lucas representing VS and V in Lemma 2, I can also show that Theorem
4.2, 4.3, 4.4, and 4.5 hold for these newly defined variables, replacing v ∗ ( xt∗ ) = F ( xt∗ , xt∗+1 ) + δ C v ( xt∗+1 )of (9) in Stocky and Lucas with v ∗ ( xt∗ ) = F ( xt∗ , xt∗+1 ) + δ C (1 − Pr (xi ))v( xt∗+1 ) . While one needs to modify
some lines of proofs in Stocky and Lucas, it is a pretty straightforward extension of the logics of their
proofs, as mentioned earlier.
For Lemma 2 (b) and (c):First note that Lemma 2 (b) and Lemma 2 (c) correspond to Theorem 4.6 of Stocky and Lucas.
Also note that Theorem 4.6 basically uses the Contraction Mapping Theorem (Theorem 3.2) and theTheorem of Maximum (Theorem 3.6) to prove the results. To show that the proof in Theorem 4.6
works for proving Lemma 2 (b) and Lemma 2 (c), I establish the following result. Define an operator T
by (Tv)(x) = Max {F(x, y) + δC[1 − Pr(x)]v(y)}. T satisfies Blackwell’s sufficient condition fory∈[ 0 , h ]contraction mapping as it satisfies both “Monotonicity” and “Discounting” criteria:(Monotonicity)
If v(y) ≤ w(y) for all values of y, then Tv(y) ≤ Tw(y) because [1 − Pr(x)] ≥ 0 by definition.
(Discounting)T(v + a)(x) = Max {F(x, y) + δC[1 − Pr(x)][v(y) + a]} = Max {F(x, y) + δC[1 − Pr(x)]v(y) + δC[1 −y∈[ 0 , h ]y∈[ 0 , h ]Pr(x)]a}} = (Tv)(x) + δC[1 − Pr(x)]a ≤ (Tv)(x) + δCa because [1 − Pr(x)] ∈ [0,1].
In addition, T: C(X) → C(X) from the Theorem of Maximum with C(X) denoting the set of bounded
continuous functions f: X → R. Thus, T: C(X) → C(X) is a contraction mapping with modulus δC,
implying that I can apply the Contraction Mapping Theorem to T. Thus, I can show that Lemma 2 (b)and (c) hold using the Theorem of Maximum as in Theorem 4.6.
Proof for Lemma 3
For Lemma 3 (a):53

Define f(τ−1, τ) ≡ F(τ−1, τ) + δC[1 – Pr(τ−1)]V(τ). Note that f(τ−1, τ) is everywhere differentiablew.r.t. τ−1 for all τ ∈ [0, h] and ∂f(τ−1, τ)/∂τ−1 = – [∂Pr(τ−1)/∂τ−1]{u(τ, l) + δCV(τ) – u(τ, h) – (δC – δ) VN –

δVC} is bounded for all τ ∈ [0, h]. This implies that f(τ−1, τ) is absolutely continuous w.r.t. τ−1 for all τ∈ [0, h]. Therefore, I can use Theorem 2 of Milgrom and Segal (2002) in deriving the following
expression(A1)

V (τ −1 ) = V (0) + ∫0 [∂f (m, g ( m) ) ∂m]dm ,τ −1

where g(m) ∈ G(m) and ∂f(m,g(m))/∂m = – [∂Pr*(m)/∂m]{u(g(m), l) + δCV(g(m)) – u(g(m), h) – (δC –δ)VN – δVC}.
(A1) implies that V(τ−1) will be strictly decreasing in τ−1 ∈ [0, h], if u[g(m), l] + δCV(g(m)) – u(g(m),h) – (δC – δ)VN – δVC > 0 for all m ∈ [0, h], because ∂Pr(m)/∂m > 0 by assumption. To show thatu(g(m), l) + δCV(g(m)) – u(g(m), h) – (δC – δ)VN – δVC > 0 for all m ∈ [0, h] > 0, I first establish that theinequality holds for any g(m) ≤ l, and then show that the inequality holds for any g(m) > l.
First, assume that g(m) ≤ l. To have u(g(m), l) + δCV(g(m)) ≤ u(g(m), h) + (δC – δ) VN + δ VC, VC >V(g(m)) because u(g(m), l) > u(g(m), h) with l < h and V(g(m)) ≥ VN. The last inequality is obvious
because the strategy of always setting τ = h will generate a discounted expected payoff at least as goodas VN, regardless of g(m) taking any feasible values. V(g(m)) ≥ [1 – Pr(g(m))][u(l, l) + δCVC] +Pr(g(m))[u(l, h) + (δC – δ)VN + δ*VC] ≥ [1 – Pr(l)][u(l, l) + δCVC] + Pr(l)[u(l, h) + (δC – δ)VN + δ VC],
where the last inequality comes from g(m) ≤ l and [u(l, l) + δCVC] ≥ [u(l, h) + (δC – δ)VN + δ VC], andthe first inequality comes from the fact that [1 – Pr(g(m))][u(l, l) + δCVC] + Pr(g(m))[u(l, h) + (δC – δ)VN
+ δVC] represents a discounted expected payoff of playing a potentially suboptimal strategy of setting τ= l with τ−1 = g(m). From ICP, VC = [1 – Pr(l)][u(l, l) + δCVC] + Pr(l)[u(l, h) + (δC – δ)VN + δVC],
which implies that VC ≤ V(g(m)), thus a contradiction. Therefore, u(g(m), l) + δCV(g(m)) > u(g(m), h) +(δC – δ)VN + δVC if g(m) ≤ l.
Now, I will show that u(g(m), l) + δCV(g(m)) > u(g(m), h) + (δC – δ)VN + δVC if g(m) > l. DefineK ≡ u(g(m), h) + (δC – δ)VN + δVC. Then, V(g(m)) ≥ [1 – Pr(g(m))]u(g(m), l)/{1 – δC[1 – Pr(g(m))]} +Pr(g(m))K/{1 – δC[1 – Pr(g(m))]} because the right-hand side of the inequality represents a discounted
expected payoff from playing a potentially suboptimal strategy of setting the current and all the future
protection level at g(m) with τ−1 = g(m). This implies that u(g(m), l) + δCV(g(m)) – K ≥ u(g(m), l) + δC[1
– Pr(g(m))]u(g(m), l)/{1 – δC[1 – Pr(g(m))]} + δCPr(g(m))K/{1 – δC[1 – Pr(g(m))]} – K = (1 –δC){u(g(m), l)/(1 – δC) – [u(g(m), h) + (δC – δ)VN + δVC]}/{1 – δC[1 – Pr(g(m))]}. Note that the lastterm has a positive sign because u(g(m), l)/(1 – δC) > [u(g(m), h) + (δC – δ)VN + δVC] with u(g(m), l)/(1– δC) > VC as g(m) > l. This implies that u(g(m), l) + δCV(g(m)) > K.

54

For Lemma 3 (b):To prove that G(τ−1) is strictly increasing in τ−1, I first show that τ // ≥ τ / for all τ −//1 > τ −/ 1 ∈ [0, h]with τ // ∈ G( τ −//1 ) and τ / ∈ G( τ −/ 1 ). Then, I show that τ // = τ / will lead to a contradiction using a resultin Cotter and Park (2006). Consider τ −//1 > τ −/ 1 , having V( τ −/ 1 ) = F( τ −/ 1 , τ / ) + δC[1 – Pr( τ −/ 1 )]V( τ / ) andV( τ −//1 ) = F( τ −//1 , τ // ) + δC[1 – Pr( τ −//1 )]V( τ // ). Then, F( τ −/ 1 , τ / ) + δC[1 – Pr( τ −/ 1 )]V( τ / ) ≥ F( τ −/ 1 , τ // ) +δC[1 – Pr( τ −/ 1 )]V( τ // ) and F( τ −//1 , τ // ) + δC[1 – Pr( τ −//1 )]V( τ // ) ≥ F( τ −//1 , τ / ) + δC[1 – Pr( τ −//1 )]V( τ / )because the terms of the right-hand sides of these inequalities represent discounted expected payoffs
from playing potentially suboptimal strategies. These two inequalities together imply that
(A2) [F( τ −/ 1 , τ / )–F( τ −//1 , τ / )] – [F( τ −/ 1 , τ // )–F( τ −//1 , τ // )] ≥ δC[Pr( τ −//1 )–Pr( τ −/ 1 )][V( τ // )–V( τ / )]. DefineE( τ ; τ −/ 1 , τ −//1 ) = F( τ −/ 1 , τ ) – F( τ −//1 , τ ). According to the mean value theorem (using the fact thatE( τ ; τ −/ 1 , τ −//1 ) is continuous and differentiable w.r.t. τ , then ∃ τ ∈ [Min( τ / , τ // ), Max( τ / , τ // )] such
that
(A3)

E (τ / ;τ −/ 1 ,τ −//1 ) − E (τ // ;τ −/ 1 ,τ −//1 ) = (τ / − τ // )[∂E (τ ;τ −/ 1 ,τ −//1 ) / ∂τ ]≥ δ C [ Pr (τ −//1 ) − Pr (τ −/ 1 )][V (τ // ) − V (τ / )]with the inequality coming from (A2). Note that [∂E( τ ; τ −/ 1 , τ −//1 )/∂ τ ] = [∂u( τ , l)/∂ τ – ∂u( τ , l)/∂ τ ] = 0
as ∂2u( τ , τ ∗ )/∂ τ ∂ τ ∗ = 0. Now, I will show that τ // < τ / leads to a contradiction. If τ // < τ / ,δC[Pr( τ −//1 )–Pr( τ −/ 1 )][V( τ // )–V( τ / )] > 0 because Pr( τ −//1 )–Pr( τ −/ 1 )) > 0 and [V( τ // )–V( τ / )] > 0 from
Lemma 3 (a). This contradicts δC[Pr( τ −//1 )–Pr( τ −/ 1 )][V( τ // )–V( τ / )] ≤ 0 in (A3), thus τ // ≥ τ / for all τ −//1
> τ −/ 1 ∈ [0, h].
Now, it remains to prove that τ // = τ / leads to a contraction. From Theorem 2 of Cotter and Park
(2006), V( τ ) is differentiable for τ ∈ G(τ−1) for all τ−1 ∈ [0, h]. Therefore,(A4)∂F (τ −/ 1 ,τ / ) / ∂τ + δ C [1 − Pr (τ −/ 1 )][∂V (τ / ) / ∂τ ] = 0 and∂F (τ −//1 ,τ // ) / ∂τ + δ C [1 − Pr (τ −//1 )][∂V (τ // ) / ∂τ ] = 0.

If τ // = τ / ,∂F (τ −/ 1 ,τ / ) / ∂τ − ∂F (τ −//1 ,τ / ) / ∂τ = −δ C [ Pr (τ −//1 ) − Pr (τ −/ 1 )][∂V (τ / ) / ∂τ ] fromcontradicting ∂[ F (τ −/ 1 ,τ / ) − F (τ −//1 ,τ / )] / ∂τ = 0 , [ Pr (τ −//1 ) − Pr (τ −/ 1 )] > 0 , and ∂V (τ / ) / ∂τ > 0 .
Proof for Lemma 4

55

(A4),

For Lemma 4 (a):In proving Lemma 4 (a), I use Theorem 4 in Cotter and Park (2006). According to the theorem, if
there exists a unique τS ∈ (0, h) that satisfies IC defined in (11): ∂F(τS, τS)/∂τ+ δC[1 −Pr(τS)][∂V(τS)/∂τ] = 0 and τ ∈ (0, h) for every τ −1 ∈ [0, h] and τ ∈ G( τ −1 ), then G(τS) = {τS} and τS is
a strongly stable protection level in the sense that for every τ −1 > τS and τ ∈ G( τ −1 ), τ < τ −1 , and forevery τ −1 < τS and τ ∈ G( τ −1 ), τ > τ −1 . To prove Lemma 4 (a), therefore, I first show that there exists
a unique τS ∈ (0, h) such that ∂F(τS, τS)/∂τ + δC[1 − Pr(τS)][∂V(τS)/∂τ] = 0 if [∂2Pr(τ)/(∂τ)2][1 – Pr(τ)] –{1 + δC[1 – Pr(τ)]}[∂Pr(τ)/∂τ ]2 > 0 for all τ ∈ [0, h] and ∂Pr(τ)/∂τ ≈ 0 at τ = 0, then establish that τ ∈(0, h) for every τ −1 ∈ [0, h] and τ ∈ G( τ −1 ).
First note that ∂F(τS, τS)/∂τ = ∂u(τS, l)/∂τ > 0 at τS = 0 and ∂2F(τS, τS)/∂τ2 < 0 with ∂F(τS, τS)/∂τ =∂u(τS, l)/∂τ = 0 at τS = h from the assumptions on the derivatives of u(τ, τ) w.r.t. τ . Because ∂V(τS)/∂τ= – [∂Pr(τS)/∂τ ]{u(τS, l) + δCV(τS) – [u(τS, h) + (δC – δ )VN + δVC]} ≈ 0 at τS = 0 from the assumption of∂Pr(τ)/∂τ ≈ 0 at τ = 0, F(τS, τS)/∂τ > 0 at τS = 0 implies that IC in (11) will not be satisfied at τS = 0.
Now, define A(τS) ≡ u(τS, l) + δCV(τS) – [u(τS, h) + (δC – δ)VN + δVC] and B(τS) ≡ δC[1 − Pr(τS)][∂Pr(τS)/∂τ]A(τS), thus δC[1 − Pr(τS)][∂V(τS)/∂τ] = −B(τS). Then, ∂B(τS)/τS = δCA(τS)〈[∂2Pr(τ)/(∂τ)2][1 –Pr(τ)] – {1 + δC[1 – Pr(τ)]}[∂Pr(τ)/∂τ ]2〉 > 0 for all τS ∈ [0, h] because [∂2Pr(τ)/(∂τ)2][1 – Pr(τ)] – {1+ δC[1 – Pr(τ)]}[∂Pr(τ)/∂τ ]2 > 0 for all τS ∈ [0, h] by assumption and A(τS) > 0 as shown in the proof
for Lemma 3 (a). This implies that there exists a unique τS ∈ (0, h) such that ∂F(τS, τS)/∂τ + δC[1 −Pr(τS)][∂V(τS)/∂τ] = 0.
Now, I only need to prove that τ ∈ (0, h) for every τ −1 ∈ [0, h] and τ ∈ G( τ −1 ). Because G( τ −1 )is strictly increasing in τ −1 as proved in Lemma 3 (b), it suffices to prove that 0 ∉ G(0) and h ∉ G(h).
Note that 0 ∉ G(0) is already proven above: “IC in (11) will not be satisfied at τS = 0.” Because IC in(11) is a necessary condition for any stationary protection level, IC in (11) being not satisfied at τS = 0
implies that 0 ∉ G(0).
I can show that h ∉ G(h) by contradiction. First, assume that h = G(h), implying that V(h) =∞⎧⎡ d −1⎤∑ ⎨(δ ) ⎢⎣Π [1 − Pr (τ )]⎥⎦ F (τ ,τ
d =0

⎩

C

d

i =0

i

d

d +1

⎫
)⎬ with {τ d = h}∞d =0 . Consider an alternative protection sequence⎭with τ0 = h, τ1 = h − ε, and {τ d = h}∞d = 2 , which defines a corresponding discounted expected payoff,denoted by VA(h). Then, I can show that VA(h) – V(h) = {Pr(h)u(h − ε, h) + [1 – Pr(h)]u(h − ε, l) +Pr(h)[(δC – δ) VN + δVC]} − {Pr(h)u(h, h) + [1 – Pr(h)]u(h, l) + Pr(h)[(δC – δ) VN + δVC]} + δC[1 –Pr(h)][Pr(h − ε) − Pr(h)]{u(h, h) – u(h, l) + [(δC – δ) VN + δVC]} − δC[Pr(h) − Pr(h − ε)]F(h, h)δC[1 −56

Pr(h)]/{1 − δC[1 − Pr(h)]}. lim[V A ( h) − V ( h)] /(ε − 0) = −δC[∂Pr(h)/∂τ][1 – Pr(h)]{(1 −δC)[u(h, h) + (δCε →0– δ)VN + δVC] − u(h, l)}/{1 − δC[1 − Pr(h)]}> 0 where the last inequality comes from ∂Pr(h)/∂τ > 0 andu(h, l)/(1 −δC) > u(h, h) + (δC – δ)VN + δVC as shown in Lemma 3 (a). This implies that h ∉ G(h).
For Lemma 4 (b):To prove Lemma 4 (b), I will show that H cannot strictly increase its discounted payoff by initiatinga punishment phase in a period that that follows a cooperative period during which H set its protectionlevel at l / ≠ l = τS, as long as the lengths of punishment phases satisfy the necessary conditions inLemma 1 (a). Once I prove this result, this implies that H cannot increase its discounted expected
payoff by initiating a punishment phase along any (deviatory) protection sequence, thus Lemma 4 (b).
Suppose that H sets its protection level at l in a period that follows a cooperative period during
which H sets its protection level at l / ≠ l = τS, then chooses its optimal protection sequence from thenext period on. Denote the discounted expected payoff from taking this potentially suboptimal actionby C( l / ), then(A5)

C (l / ) = Pr (l / )[u (l , h) + (δ C − δ )V N + δVC ] + [1 − Pr (l / )][u (l , l ) + δ C VC ] .

Now suppose that H initiates a tariff war phase by setting tariff level at h in a period that follows acooperative period where H set its protection level l / ≠ l = τS, then follows its specified strategy once
the tariff war phase is over. Denote the discounted expected payoff from taking this potentiallysuboptimal action by D( l / ), then(A6)

D(l / ) = Pr (l / )[u (h, h) + (δ C − δ S )V N + δ S VC ] +[1 − Pr (l / )][u (h, l ) + (δ C − δ )V N + δVC ].

I can rewrite C( l / ) and D( l / ) into
(A7)

C (l / ) = u (l , l ) + δ CVC − Pr (l / )[u (l , l ) − u (l , h) + (δ C − δ )(VC − V N )]D(l / ) = u (h, l ) + (δ C − δ )V N + δVC − Pr (l / )[u (h, l ) − u (h, h) + (δ − δ S )(VC − V N )].

Now, note that C( l / ) − D( l / ) = [u(l, l) − u(h, l)] + ( δC− δ )(VC − VN) − Pr( l / ){[u(l, l) − u(l, h)] − [u(h,l) − u(h, h)] +[(δC – δ) − (δ – δS)](VC − VN)} = 0 from [u(l, l) − u(l, h)] = [u(h, l) − u(h, h)] and thesufficient condition for ICP and ICP* in Lemma 1 (a): δ C − δ = [u(h, l) − u(l, l)]/(VC − VN) and δ C + δ S= (δ + δ ) . Because C( l / ) is equal or possibly lower than a discounted expected payoff from choosing
an optimal protection sequence of not involving an initiation of a punishment phase, this implies that H
cannot strictly increase its discounted payoff by initiating a punishment phase in a period that follows acooperative period during which H sets its protection level at l / ≠ l = τS.

57

Proof for Proposition 1

With δ = δC − [u(h, l S ) − u( l S , l S )]/(VC − VN), and δ C − δ S = 2(δ C − δ ) , note that setting τS = l S
satisfies IC in (11), thus l S is the unique stationary protection level from which H does not have any
incentive to deviate from, as described in Lemma 4. By symmetry, l S is also such a protection level for
F. If l = l S , then PTS satisfy ICP and δ C − δ S = 2(δ C − δ ) as well as IC, thus becoming a supergame
equilibrium of the protection setting game between H and F from which no country has any unilateralincentive to change its specified strategy.
What is the relationship between the condition for Lemma 4 (a) and the existence of l (< h) thatsatisfies I(l) = 0 in (13)? For example, does the condition for Lemma 4 (a) guarantee the existence ofsuch l? To address this issue, I show that the second term of I(l) in (13), δC[∂Pr(l)/∂τ ][1 − Pr(l)][u(h, l)– u(l, h)], representing H’s dynamic incentive to avoid a punishment phase, may not necessarily increasein l when the condition for Lemma 4(a) is satisfied. ∂{[∂Pr(l)/∂l][1 − Pr(l)][u(h, l) – u(l, h)]}/∂l =〈[∂2Pr(l)/(∂l)2][1 – Pr(l)] – [∂Pr(l)/∂l]2〉[u(h, l) – u(l, h)] + [∂Pr(l)/∂l][1 − Pr(l)]{∂[u(h, l) – u(l, h)]/∂l} =〈[∂2Pr(l)/(∂l)2][1 – Pr(l)] – {1 + δC[1 – Pr(l)]}[∂Pr(l)/∂l]2〉[u(h, l) – u(l, h)] + 〈{δC[1 – Pr(l)]}[∂Pr(l)/∂l]2[u(h, l) – u(l, h)] + [∂Pr(l)/∂l][1 − Pr(l)]{∂[u(h, l) – u(l, h)]/∂l}〉. Because [∂Pr(l)/∂l][1−Pr(l)]{∂[u(h, l) – u(l, h)]/∂l} < 0, once cannot rule out the possibility of having {δC[1 –Pr(l)]}[∂Pr(l)/∂l]2[u(h, l) – u(l, h)] + [∂Pr(l)/∂l][1 − Pr(l)]{∂[u(h, l) – u(l, h)]/∂l} < 0, thus∂{[∂Pr(l)/∂l][1 − Pr(l)][u(h, l) – u(l, h)]}/∂l < 0 even when [∂2Pr(l)/(∂l)2][1 – Pr(l)] –{1 + δC[1 –Pr(l)]}[∂Pr(l)/∂l]2 > 0. Therefore, the condition for Lemma 4 (a) does not necessarily guarantee theexistence of l (< h) that satisfies I(l) = 0, validating the insertion of an additional condition to guarantee
the existence of such l in Proposition 1.

Proof for Proposition 4
For (a): It is sufficient to show that dVCW/dTW in (22) is less than 0 for all values of TW ≥ 1 if Pr(l)< Pr . Using
T⎧⎪ W
2(δ C − δ C ) Pr (1 − Pr )[u (l , l ) − u ( h, h)] ⎫⎪
∂I W
C CT
C ∂Pr
δ
δ
ln(
δ
)
(
1
Pr
)
(
V
V
)
=
−
−
−
⎨ C
⎬, and
N
T
∂T W
∂l
⎪⎩
⎪⎭
[1 − δ C + 2 Pr (δ C − δ C )]2
W

W

W

(A8)

[

dI W
T
∂ 2 Pr∂Pr ∂Pr ⎤C ⎡δu (l , l ) − u (l , h) + (δ C − δ C )(VCW − V N )(1 − Pr ) +
=
−
⎢
D
D
D ⎥
dω
∂l ∂ω ⎦⎣ ∂l∂ωWTW]

δ C (δ C − δ C )(∂Pr / ∂l )(1 − Pr ) ∂PrT+[1 − δ C + 2(δ C − δ C )][u (l , l ) − u (h, h)],DT[1 − δ C + 2 Pr (δ C − δ C )]2 ∂ωWW

I can rewrite dVCW/dTW in (22) into58

⎫
⎛ ∂VCW ⎞⎧δ C δ C T ln(δ C ) ∂Pr (1 − Pr )(VCW − V N )
⎪
⎜
⎟⎪
W
l
∂
dVC
⎪
⎪
l
∂
⎜
⎟
= − W ⎨⎬, with
2
W
u
l
l
u
l
h
(
,
)
(
,
)
−
⎡
⎤
⎜ ∂I ⎟⎪ C ⎡ ∂ Pr
∂Pr ∂Pr ⎤dT(1 − Pr ) +⎥⎪⎜⎟ + δ A⎢−
D
D ⎥⎢
C
CT
W
l
ω
ω
l
∂
∂
∂
∂
∂l ⎠⎪⎩
⎝
δ
δ
V
V
(
)(
)
+
−
−
⎣
⎦ ⎣⎢
⎥ ⎪⎭
C
N ⎦
W

W

⎡ ∂ 2 Pr∂Pr ∂Pr ⎤(1 − Pr )[u (l , l ) − u (h, h)]WDW,⎢− ∂l∂ω D (1 − Pr ) + ∂l ∂ω D ⎥ < 0, to have dVC /dω = 0. Using VC − V N =T⎣⎦
1 − δ C + 2 Pr (δ C − δ C )
W

dVCW
=
dT W
∂Pr (1 − Pr )[u (l , l ) − u (h, h)]+∂l 1 − δ C + 2 Pr (δ C − δ C T )W

T⎧⎫[1 − δ C + 2 Pr (δ C − δ C )][u (l , l ) − u (l , h)]
⎪
⎪
T
−1
C
C
CT
2
)][1 − δ C + 2(δ C − δ C )] ⎪⎪∂Pr ∂Pr ⎤ ⎪⎪[1 − δ + 2 Pr (δ − δ⎛ ∂Pr ⎞ ⎡ ∂ Pr−
2 Pr ⎜(1 − Pr ) +
⎨
⎬
D ⎟ ⎢
D
T
∂l ∂ω D ⎥⎦ ⎪⎝ ∂ω ⎠ ⎣ ∂l∂ω⎪
(1 − Pr )(δ C − δ C )[u (l , l ) − u (h, h)]
⎪+
⎪
C
C
CT
C
C
CT
⎪⎩ [1 − δ + 2 Pr (δ − δ)][1 − δ + 2(δ − δ
)] ⎪⎭
W

B

W

W

W

W

W

⎛ ∂VCW ⎞
⎜
⎟
T
∂
l
⎜
with B ≡ − W ⎟δ C δ C ln(δ C )(1 − Pr ).
⎜ ∂I ⎟⎜⎟∂l ⎠⎝By replacing u(l, l) − u(l, h) with u(l, l) − u(h, h) in the above expression and using u(l, l) − u(l, h) > u(l,Wl) − u(h, h), I obtain the first inequality in the following expressions:
dVCW
dT W
T
−1
2
⎧⎪ ∂Pr∂Pr ∂Pr ⎤ ⎡1 − δ C + (1 + Pr )(δ C − δ C ) ⎤ ⎫⎪⎛ ∂Pr ⎞ ⎡ ∂ Pr
< BC ⎨−(1 − Pr ) + 2 Pr ⎜(1 − Pr ) +
⎢
⎥⎬
D ⎟ ⎢
D
∂l ∂ω D ⎥⎦ ⎢ 1 − δ C + 2(δ C − δ C T ) ⎥ ⎪⎝ ∂ω ⎠ ⎣ ∂l∂ω
⎪⎩ ∂l
⎣
⎦⎭
W

W

−1
2
⎧⎪ ∂Pr∂Pr ∂Pr ⎤⎛ 1 + Prδ C ⎞⎫⎪⎛ ∂Pr ⎞ ⎡ ∂ Pr⎜⎟⎬
< BC ⎨−−+(1 − Pr ) + 2 Pr ⎜
(
1
)
Pr
⎟
⎢
D
D
∂l ∂ω D ⎥⎦⎜⎝ 1 + δ C ⎟⎠⎪⎭⎪⎩ ∂l⎝ ∂ω ⎠ ⎣ ∂l∂ω
−1D⎛ ∂Pr ⎞ ∂ ( Pr / ω ) Pr
= BC ⎜(1 − Pr )(1 + δ C ) + 2(2 Pr − 1)(1 + Prδ C ) ,
D ⎟
C
∂
ω
∂
+
δ
1
l
⎝
⎠
u (l , l ) − u (h, h)with C ≡,TC
1 − δ + 2 Pr (δ C − δ C )

[

]

W

where ∂{[1 − δ C + (1 + Pr )(δ C − δ CTW)] /[1 − δ C + 2(δ C − δ Clast bracketed term, [1 − δ C + (1 + Pr )(δ C − δ C

TWTW)]} / ∂T W < 0 and setting T W → ∞ for the)] /[1 − δ C + 2(δ C − δ C

TW)] in the preceding expression

to the second inequality are used to obtain the second inequality. To obtain the (last) equality in the59

above expressions, I use the assumption of ∂2Pr(l)/∂(ωD)2 = 0, thus Pr(l) being linear in ωD. With thisassumption, I can rewrite Pr(l) = ωDPrD(l), which in turn implies that ∂Pr(l)/∂l = ωD[∂PrD(l)/∂l],∂Pr(l)/∂ωD = PrD(l), ∂2Pr(l)/∂l∂ωD = ∂PrD(l)/∂l, and [∂Pr(l)/∂l][∂Pr(l)/∂ωD] = Pr[∂PrD(l)/∂l]. Once I
rewrite the corresponding terms in the expression preceding to the last equality in this way, I can obtainthe last expression (or equivalently, the last equality). As a result of these transformations, I obtain thefollowing inequalities:
−1

D
dVCW
⎛ ∂Pr ⎞ ∂ ( Pr / ω ) Pr<BC
(1 − Pr )(1 + δ C ) + 2(2 Pr − 1)(1 + Prδ C ) < 0
⎜
⎟
D
C
dT W
∂
ω
∂
l
1
+
δ
⎝
⎠

(A9)

[

]

with the last inequality holding for all values of TW ≥ 1 if (1 − Pr )(1 + δ C ) + 2( 2 Pr − 1)(1 + Prδ C ) < 0,which in turn holds if Pr(l) < Pr .
For (b): It is sufficient to show that dVCW/dTW in (22) is greater than 0 for all values of TW ≥ 1 if

Pr(l) > Pr . As shown above, I can rewrite dVCW/dTW (22) into the following expression, using (A8):

dVCW
=
dT W
∂Pr (1 − Pr )[u (l , l ) − u ( h, h)]+∂l 1 − δ C + 2 Pr (δ C − δ C T )W

T⎧⎫[1 − δ C + 2 Pr (δ C − δ C )][u (l , l ) − u (l , h)]
⎪
⎪
B
T
T
−1
C
C
C
C
C
C
2
⎪
−
+
−
−
+
−
[
1
2
(
)][
1
2
(
)]
δ
Pr
δ
δ
δ
δ
δ
⎡
⎤
∂
∂
∂
∂
Pr
Pr
Pr
Pr
⎪
⎪⎪
⎛
⎞
−
2 Pr ⎜(1 − Pr ) +
⎬
D ⎟ ⎢
D
D ⎥⎨
T
∂l ∂ω ⎦ ⎪⎝ ∂ω ⎠ ⎣ ∂l∂ω⎪
(1 − Pr )(δ C − δ C )[u (l , l ) − u (h, h)]⎪+⎪TT⎪⎩ [1 − δ C + 2 Pr (δ C − δ C )][1 − δ C + 2(δ C − δ C )] ⎪⎭By replacing u(l, l) − u(h, h) with u(l, l) − u(l, h) in the above expression and using u(l, l) − u(l, h) > u(l,
W

W

.

W

W

W

W

l) − u(h, h), I obtain the first inequality in the following expressions
dVCW
dT W
T
⎧⎪ ∂Pr ∂Pr⎡ ∂ 2 Pru (l , l ) − u (h, h)∂Pr ∂Pr ⎤ ⎡1 − δ C + (1 + Pr )(δ C − δ C ) ⎤ ⎫⎪Pr
Pr
> BD ⎨
Pr
−
+
−
−
+
(
1
)
2
(
1
)
⎢
⎥⎬
⎢ ∂l∂ω DDu (l , l ) − u (l , h)∂l ∂ω D ⎥⎦ ⎢ 1 − δ C + 2(δ C − δ C T ) ⎥ ⎪⎣⎪⎩ ∂l ∂ω⎣⎦⎭2
⎧ ∂Pr ∂Pr
⎡ ∂ Pru (l , l ) − u (h, h)∂Pr ∂Pr ⎤ ⎫> BD ⎨+ 2 Pr ⎢−(1 − Pr )(1 − Pr ) +⎬DDu (l , l ) − u (l , h)∂l ∂ω D ⎥⎦ ⎭⎣ ∂l∂ω⎩ ∂l ∂ω
W

W

= BD

⎤
∂ ( Pr / ω D ) ⎡u (l , l ) − u (h, h)Pr ⎢(1 − Pr )+ 2(2 Pr − 1) ⎥,∂lu (l , l ) − u (l , h)⎣⎦with D ≡−1u (l , l ) − u (l , h)

⎛ ∂Pr ⎞
,
⎜
D ⎟
C
C
CT
1 − δ + 2 Pr (δ − δ) ⎝ ∂ω ⎠W

where [1 − δ C + (1 + Pr )(δ C − δ CTW)] /[1 − δ C + 2(δ C − δ C

60TW)] < 1 for the last bracketed term in thepreceding expression to the second inequality is used to obtain the second inequality. To obtain the(last) equality in the above expressions, once again I use the assumption of ∂2Pr(l)/∂(ωD)2 = 0 in thesame manner that I used it to obtain the last equality in the corresponding expressions in the proof for
Proposition 4 (a). As a result of these transformations, I obtain the following inequalities:
(A10)

⎤
dVCW
∂ ( Pr / ω D ) ⎡u (l , l ) − u ( h , h )>BDPr ⎢(1 − Pr )+ 2(2 Pr − 1)⎥ > 0WdT∂lu (l , l ) − u (l , h)⎣⎦with the last inequality holding for all values of TW ≥ 1 if (1 − Pr)[u(l, l) − u(h, h)]/[u(l, l) − u(l, h)] +2(2Pr − 1) > 0, which in turn holds if Pr(l) > Pr .

61

F (τ −1 ,τ )F (τ −//1 , h) −⎫⎪⎬F (τ , g (τ ))⎪⎭
//
−1

/
−1

F (τ −/ 1 , h) −⎫⎪⎬F (τ −/ 1 , g (τ −/ 1 ))⎪⎭F (τ −//1 ,τ )F (τ −/ 1 ,τ )0g (τ −/ 1 )τhg (τ −//1 )δ C [1 − Pr (τ −1 )]V (τ )

δ C [1 − Pr (τ −/ 1 )]V ( g (τ −/ 1 ))⎫⎪− δ C [1 − Pr (τ −/ 1 )]V (h)⎬⎪⎭δ C [1 − Pr (τ −//1 )]V ( g (τ −/ 1 ))⎫⎪− δ [1 − Pr (τ )]V (h)
C

//
−1

δ C [1 − Pr (τ −/ 1 )]V (τ )⎬⎪⎭

δ C [1 − Pr (τ −//1 )]V (τ )0g (τ −/ 1 )τhg (τ )//−1

Figure 1. The Effect of a Higher τ −1 on the Optimal Choice of τ62

G(τ−1)G/(τ−1)hG(τ−1)

G(τS) = τS

45 0
0

τ 0/

τ 1/ τ 2/

τS τ 2// τ 1//

τ 0//

h

τ −1

Figure 2. The Existence of a Stationary Protection Sequence at τS

63δC[∂Pr(l)/∂τ ][1 − Pr(l)][u(h, l) – u(l, h)]∂u(l, l)/∂τ

0

lmin

lmax

Figure 3. Multiple l satisfying I(l) = 0

64

h

l

Figure 4. A numerical analysis of the optimal TTS for different values of ρ (rho) with χ = 1,
α1 − α1* = 3, β + B = 1 (so, h = 1), and δC = 0.565Figure 5. A numerical analysis of the optimal TTS and optimal PTS for different values ofρ (rho) with χ = 100, α1 − α1* = 3, β + B = 1 (so, h = 1), and δC = 0.566