Enforcing International Trade Agreements under Imperfect Private Monitoring:
Private Trigger Strategies and a Possible Role for the WTO
byJee-Hyeong Park*
First Draft: June 2004
Revised: September, 2008
Abstract
To analyze the issue of enforcing international trade agreements in the presence of potential deviationsof which countries receive imperfect and private signals, this paper analyzes a repeated bilateral traderelationship where each country can secretly raise its protection level through concealed trade barriers.
In particular, it explores the possibility that countries adopt private trigger strategies (PTS) under which
each country triggers an explicit tariff war based on its privately observed imperfect signals of thepotential use of concealed trade barriers. Focusing on simple symmetric PTS where each country
imposes its one-shot Nash tariff under any punishment phase, this paper first establishes the condition
under which symmetric countries may restrain the use of concealed trade barriers: the sensitivity ofprivate signals rises in response to an increase in concealed protection. Then, it shows that the payoff ofany equilibrium symmetric PTS should be identical to the one under simple symmetric PTS as long as
the punishment is triggered by a one-shot Nash tariff. Given this generality result of simple symmetric
PTS, the analysis reveals that it is not optimal to push down the cooperative protection level to its
minimum attainable level. The paper identifies two factors that may severely limit the effectiveness of
PTS; one is a reduction in each country’s time lag in adjusting its tariff level in response to the othercountry’s initiation of a punishment phase, and the other is asymmetry among countries. Both of thesefactors may limit the level of cooperation attainable under PTS by reducing the lengths of tariff war
phases that countries can employ. In order to analyze how an impartial third-party who receives its own
imperfect and private signals of potential deviations, such as the WTO, can facilitate enforcing
international trade agreements, this paper constructs and analyzes third-party trigger strategies (TTS)
under which each country triggers an explicit tariff war based on the WTO’s decision on whether any
deviation has occurred. A numerical analysis of symmetric TTS demonstrates that the optimal
symmetric TTS may entail a punishment phase only with an initial tariff (no tariff-war period followingit) as well as the one with a permanent Nash tariff war, depending on the sensitivity of the WTO’s
signals of potential deviations. On the one hand, the WTO may enable countries to shorten and possibly
avoid a costly tariff war, which might have been necessary to curtail the incentive to wrongly punish
other countries under PTS. On the other hand, the WTO may also enable countries to adopt a longer
punishment phase than the one under PTS, which in turn helps countries to attain higher levels of
expected payoffs.
JEL Classification Code: F020; F130
Keywords: International Trade Agreements; Trade Disputes; Concealed Trade Barriers; ImperfectPrivate Monitoring; Trigger Strategies; Repeated Game
*Address: School of Economics, Seoul National University, San 56, Sillim-Dong, Kwanak-Ku, Seoul,
151-746, Korea; tel: +82-2-880-6329; fax: +82-2-886-4231; 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, Gyu Ho Wang, Robert Staiger, and other participants of seminars at Ewha
Woman’s University, Princeton University, Seoul National University, Yale University, Yonsei
University, and Wayne State University, and of International Conference on Game Theory at Stony
Brook, Midwest International Economics and Economic Theory Meetings, and 2004 Far Eastern
Meeting of the Econometric Society. All errors are mine.

1. Introduction
Enforcing international trade agreements often involves disputes where countries present
different opinions about potential deviations from the agreements. Differences in opinions may
take various forms, such as disagreement over the existence of concealed trade barriers in
disputes between the U.S. and Japan during 1980s or disagreement over 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 deviations can be private in the sense that its true opinion is not known to othercountries. For example, when the U.S. Trade Representative (USTR) engages in a 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
To analyze the issue of enforcing international trade agreements in the presence ofpotential deviations of which countries receive imperfect and private signals, this paper
analyzes a repeated bilateral trade relationship where each country can secretly raise its
protection level through concealed trade barriers. In particular, it explores the possibility thatcountries adopt private trigger strategies (PTS) under which each country triggers an explicit
tariff war based on its privately observed imperfect signals of the potential use of concealed
trade barriers.

The analysis specifies the condition under which countries can restrain

deviations based on simple symmetric PTS where each country imposes its one-shot Nash tariff
under any punishment phase: the sensitivity of private signals rises in response to an increase inconcealed protection. This paper then establishes that the payoff of any equilibrium symmetric
PTS should be identical to the one under simple symmetric PTS, as long as the punishment is
triggered by a one-shot Nash tariff. Given this generality result, it characterizes optimal
symmetric PTS that maximize symmetric countries’ expected payoffs under PTS. This paper
also identifies factors that may severely limit the effectiveness of PTS in restraining deviations
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 companies’ private
information of which public revelation can be costly for those companies, forcing the signals to be private.

1

from trade agreements. In order to analyze how an impartial third-party who receives its own
imperfect and private signals of potential deviations, such as the WTO, can facilitate enforcing
international trade agreements, this paper constructs and analyzes third-party trigger strategies(TTS) under which each country triggers an explicit tariff war based on the WTO’s decision onwhether any deviation has occurred.
This paper differs from previous studies on the enforcement of trade agreements byfocusing on how private nature of signals about deviations may limit the use of (private) trigger
strategies. 2

Many studies have analyzed the enforcement issue of trade agreements using

trigger strategies in a repeated game setup. With regard to observable actions of imposing
tariffs, for example, Dixit (1987) established that countries can sustain the equilibrium withzero tariffs by restraining their unilateral incentive to impose tariffs based on a threat of
invoking tariff wars against imposing positive tariffs. 3 With regard to the issue of restraining
the use of concealed trade barriers, Riezman (1991) modified “trigger-price strategies” ofGreen and Porter (1984) on collusion among firms into “import trigger strategies” wherecountries start a tariff war phase when the level of imports falls below a critical level. Because
the import level is negatively correlated with countries’ protection levels, countries have an
incentive to hold down their concealed protection level to reduce the probability of triggering a
costly tariff war. In contrast to these earlier studies focusing on public (observable by all
countries) signals as a device that triggers tariff wars, this paper analyzes trigger strategies that
rely on imperfect private signals of potential deviations. The private nature of signals used
under PTS makes incentive constraints for such trigger strategies different from those undertrigger strategies relying on public signals, which in turn limits the effectiveness PTS as
discussed below.
In establishing that countries can restrict the use of concealed protection based on PTS, ananalytical challenge emerges from the necessity to check not only one-time deviations from thespecified strategy, but whole deviation paths that each country may take in order to ensure that2

Bagwell and Staiger (2005) analyzed the issue of implementing trade agreements when each government isprivately informed about its own domestic political pressure for protection. Their analysis differs from thispaper’s because it focuses on identifying the structure of trade agreements that can induce the truthful revelation ofprivate political pressure rather than analyzing the enforcement of trade agreements when countries privately
observe imperfect signals of potential deviations.
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.

2

proposed PTS can serve as a supergame equilibrium of the repeated protection-setting game. 4
To solve this issue, I provide a full characterization of an optimal (possibly deviatory)
protection sequence that each country may take under PTS in Section 2. In the same section, Iestablish that there exists a stationary protection level that each country may sustain as acooperative protection level if the sensitivity of countries’ private signals rises in response to anincrease in concealed protection.
In addition to establishing that symmetric countries may restrain the use of concealed tradebarriers with simple symmetric PTS, the analysis in Section 3.1 also establishes that the payoffof any equilibrium symmetric PTS will the same as the one under simple symmetric PTS as longas the punishment is triggered by a one-shot Nash tariff. Then, Section 3.2 characterizes
optimal symmetric PTS under which countries maximize their expected discounted payoffs.
The analysis shows that it is not optimal to push down the cooperative protection level to its
minimum level attainable under symmetric PTS due to the cost of increasing the probability of
costly tariff wars.
A potential limitation of PTS comes from the private nature of signals utilized under PTS
because it limits the lengths of tariff war phases that countries can employ. Note that eachcountry may misrepresent its private signal under PTS either by initiating a tariff war when its
private signal does not belong to the range under which it is supposed to trigger a tariff war or
by not initiating a tariff war when its private signal does belong to such a range. On the one
hand, if the tariff war phase is too long so that its expected payoff is higher when it ignores its
private signals to trigger a tariff war, then each country will never initiate a tariff war. On theother hand, each country will always initiate a tariff war by imposing its static optimal tariff if
the following war is too short. 5 In contrast to repeated games with public information wherecountries can choose any length for their tariff war phases, the private nature of signals imposes
restrictions on the lengths of tariff war phases under PTS!
This paper identifies two factors that may severely limit the effectiveness of PTS; one is areduction in each country’s time lag in adjusting its tariff level in response to the other4

As discussed in more details in Section 2, if private signals trigger tariff wars as under PTS, any deviatory actionthat each country might have taken in a previous period can influence its optimal deviatory action in a current
period. This prohibits us from applying the logic of the well-known “one-stage-deviation principle” to PTS.
5
Each country benefits in the period that it unilaterally initiates a tariff war phase because it imposes its static
optimal tariff. If there is no actual tariff war phase to be followed, then imposing the static optimal tariff becomesa dominant strategy for each country.

3

country’s initiation of a tariff war, and the other is asymmetry among countries. Both of thesefactors may limit the level of cooperation attainable under PTS by reducing the length of a tariff
war phase. One of factors that determine the length of a tariff war phase under PTS is the oneperiod gain that it can realize during the period that it initiates a tariff war phase. If there is areduction in each country’s time lag in adjusting its tariff level in response to the othercountry’s initiation of an explicit tariff war, the tariff-war-initiating country’s one-period gainfrom imposing its static optimal tariff decreases as the other country reacts more quickly with
its own static optimal tariff. Such a reduction in the gain from initiating a tariff war decreases
the length of the tariff war phase, which in turn reduces the level of cooperation sustainable
under PTS. Asymmetry among countries reduces the length of a tariff war phase that a small
country can initiate against a large country in a similar way. As asymmetry in countries’
relative market size increases, the small country’s ability to change the terms of trade in its
favor by imposing tariffs decreases, reducing the one-period gain that it can realize by
imposing its static optimal tariff. This shortens the length of a tariff war phase that the small
country can initiate against the large country’s potential use of concealed protection.
The analysis of PTS provides useful insights on the unilateral approach of the U.S. towards
enforcing international trade agreements. According to the analysis, the fact that the U.S. is thelargest trading economy in the world makes her a country that is most capable of using PTS todeter its trading partners’ use of concealed trade barriers. Indeed, the U.S. is the only country
in the world that has the legislation, Section 301, authorizing its government to invoke a tariff
war based on its own unilateral judgment of potential deviations from trade agreements. As
emphasized in the analysis, the credibility of invoking a tariff war is crucial for the success ofsuch a unilateral approach. To raise the credibility of a punitive tariff war, the U.S. Congressmodified Section 301 in 1988 by shifting the retaliation authority from the President (who islikely to care about diplomatic relationships) to the USTR and by mandating retaliation against
unjustifiable practices. Note also that the USTR has often taken its Section 301 investigation
cases to the WTO panel investigation. 6 Such an action seemingly respecting the multilateralenforcement mechanism of the WTO, can be an attempt to make the threat of invoking a tariff
war more credible by delegating the judgment of deviations to a third party panel who would

6

See Bayard and Elliott (1992) for a detailed discussion of Section 301 cases.

4presumably care less about the cost or the benefit from triggering a tariff war that may result
from the judgment.
In order to analyze how an impartial third-party who receive its own imperfect and private
signals of potential deviations, such as the WTO, can facilitate enforcing international trade
agreements, this paper constructs and analyzes third-party trigger strategies (TTS) under which
each country triggers an explicit tariff war based on the WTO’s decision on whether any
deviation has occurred. A numerical analysis of symmetric TTS demonstrates that the optimal
TTS may entail a punishment phase only with an initial tariff (no tariff-war period following it)
as well as the one with a permanent Nash tariff war, depending on the sensitivity of the WTO’s
signals of potential deviations. On the one hand, the WTO may enable countries to shorten andpossibly avoid a costly tariff war, which might have been necessary to curtail each country’s
incentive to wrongly punish the other country with its static optimal tariff. On the other hand,
the WTO may also enable countries to adopt a longer punishment phase than the one under PTS,
which in turn helps countries to attain higher levels of expected payoffs
While this paper focuses on the issue of enforcing international agreements, the analysis of
PTS is applicable to a broader range of problems. The protection-setting game analyzed in thispaper belongs to repeated games under imperfect private monitoring. It is well known in thegame theory literature that analyzing repeated games with imperfect private monitoring isdifficult because utilization of privately observed signals in determining continuation plays candestroy the recursive structure of repeated games. 7 As a way to overcome this problem in
repeated games with imperfect private monitoring, Kandori and Matsushima (1998) andCompte (1998) allow players to communicate their privately observed information regarding
potential deviations. Such communication serves as imperfect public information (in the sensethat every player can observe the communication), thus, restoring a recursive structure to therepeated game. 8 PTS considered in this paper show an alternative way to restore a recursive
structure to a repeated game with imperfect private monitoring. Under PTS, each country
initiates a tariff war phase by imposing explicit tariffs if it receives private signals that arehighly correlated with other countries’ defective behaviors. Because all trading partners can
7

Kandori (2002) discusses this point and recent developments in repeated game with private monitoring in detail.
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.

85

perfectly observe explicit tariffs, countries can avoid potential confusion between punishment
phases and non-punishment phases, which ensures a “recursive” structure of the repeated gamealong the equilibrium path. In the context of collusion among firms that can engage in secretprice cuttings, firms can employ advertised (thus public) sales to initiate a punishment phaseagainst potential defections from collusive pricing as a way to coordinate their punishments.
According to this paper’s analysis of PTS, the expansion of internet commerce can negatively
affect collusive behaviors based on such PTS because it reduces each firm’s time lag ineffectively advertising its sales over the internet in response to the other firm’s initiation ofadvertised sales.
The paper is organized as follows. Section 2.1 develops a bilateral trade model where eachcountry receives imperfect private signals of each other’s use of concealed trade barriers andspecifies simple PTS that countries employ to deter such protection. Section 2.2 describes
incentive constraints under simple PTS, providing conditions under which those incentive
constraints are satisfied. Section 3.1 proves the existence of simple symmetric PTS as asupergame equilibrium of the repeated protection-setting game. Section 3.2 proves that thepayoff of any equilibrium symmetric PTS will be the same as the one under simple symmetric
PTS as long as the punishment is triggered by a one-shot Nash tariff. Section 3.2 thencharacterizes optimal symmetric PTS under which countries maximize their joint expecteddiscounted payoffs. Section 4.1 analyzes how a reduction in the time lag in adjusting tariff
levels in response to an initiation of an explicit tariff war affects effectiveness of PTS, andSection 4.2 provides an analysis of the effect of introducing asymmetry among countries on
PTS. To analyze the role of the WTO in enforcing international trade agreements in thepresence of imperfect private monitoring, Section 5 specifies symmetric TTS, providing acomparison between optimal symmetric PTS and optimal symmetric TTS through a numericalanalysis. Section 6 concludes by providing a summary of the results together with a discussion
about a possible extension of this paper’s analysis towards understanding the workings ofdispute settlement procedure of the WTO.

2. Private Trigger Strategies
2.1. A Trade Model with Concealed Trade Barriers and Private Trigger Strategies

6

The basic bilateral trade model comes from Dixit (1987) with concealed trade barriers
being introduced in a similar way as in Riezman (1991). There exist two countries, home (H)and foreign (F), producing and trading two products, good 1 and good 2, under perfectcompetition.

H imports good 2 and F imports good 1.

In each period, each country

simultaneously chooses its explicit tariff level, ei ≥ 0 and its (total) import protection level, τi ≥0, where i = * or none, having variables with and without superscripts * respectively denoteforeign and home variables henceforth. Then, τ − e ≥ 0 and τ* − e* ≥ 0 represent the concealed
protection levels of H and F, respectively, and the local prices, p1 , p 2 , p1∗ , and p 2∗ are related
as follows: p 2 = p 2∗ (1 + τ ) and p1∗ = p1 (1 + τ ∗ ) . 9 Given the assumption of perfect competition,
I can define each country’s one-period payoff function as a function of the terms of trade,denoted by π (≡ p1 / p 2∗ ) , and its own protection level. Such a payoff function, represented byw i (π ,τ i ) with i = * or none, induces a corresponding import demand function, denoted bym i (π ,τ i ) . If there exists no uncertainty (random elements) in this world, each country’s
amount of imports is a deterministic function of each country’s protection level and the term oftrade. This implies that each country may figure out the exact level of the other country’s
protection based on the information about the terms of trade and the amount of imports, even in
the presence of concealed trade barriers.
However, when I introduce uncertainty into the model as a way of representing shocks totechnology or preferences, the exact derivation of other countries’ protection levels based on
the amount of imports and the terms of trade may become impossible. Uncertainty caused byrandom shocks can be modeled into random components in countries’ import demand functions
as follows:(1)

mti = m i (π t ,τ ti , θ ti ) with i = * or none,

where θ ti ∈ Θ i denotes each country’s random components affecting its import demand at
period t (subscript t denotes the variables determined in period t), which follow a joint density
function, f(θt,θt*) that is iid across periods. In equilibrium, the following balance of payment
condition should be satisfied:(2)9

π t ⋅ m(π t ,τ t ,θ t ) = m ∗ (π t ,τ t∗ ,θ t∗ ) ,Thus, this paper does not consider the possibility of using negative or prohibitive protection.

7

which determines the equilibrium values for π t , mt , and mt∗ as functions of τ t , τ t∗ , θ t , and θ t∗ .
With random shocks realizing after τ and τ* being determined, each country’s one-period
expected payoff, denoted by ui, is a function of both countries’ protection levels:(3) u i (τ ti , τ t j ) =∫∫ w (π (τ ,τ , θ ,θ ),τ ;θ ) f (θ , θ )dθ dθ with i, j = * or none, and i ≠ j,
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 . Regarding derivatives of u i (τ i ,τ j ) with respect to τ i and τ j , I assume
the following standard trade-theoretic results continue to hold in the presence of these randomvariables: ∂u i ∂τ i > 0 at τ i = 0 (each country has an incentive to raise its protection level
abovezero);∂u j ∂τ i > 0(suchprotection

hurts

the

other

country);∂u i ∂τ i+ ∂u j ∂τ i < 0 (such protection reduces the total payoff of H and F as it creates distortionalloss) with i, j = * or none, and i ≠ j. For analytical simplicity, I introduce the following
2

additional assumptions: ∂ 2 u i ∂τ i < 0 (the marginal gain from protection decreases as theprotection level increases); ∂ 2 u i ∂τ i ∂τ j = 0 (the marginal gain from protection is not affectedby the other country’s protection level) with i, j = * or none, and i ≠ j. 10 These additional
assumptions guarantee the existence of a unique static optimal protection level for each country
which I denote by hi (> 0) with i = * or none. The one-shot protection setting game between H
and F then generates a Nash equilibrium where (τ, τ*) = (h, h*).
At the end of period t, each country privately observes realized values of its payoff and its
own random variable, (ut, θt) by H and (ut*, θt*) by F, and both countries observe a pair of
explicit tariffs, (et, et*). Each country cannot infer the exact level of the other country’s
concealed protection because it does not know the realized value of the other’s random variable.
However, note that the privately observed information, denoted by ω ti = (u ti , θ ti ,τ ti ) ∈ Ω i withi = * or none, can serve as a measure for detecting the other country’s potential use of

10

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.

8

concealed protection. 11 More specifically, each country can choose a subset of its possiblei

i

private signals, denoted by Ω D , such that ∂Pr (ω ti ∈ Ω D ) / ∂τ t j > 0 where Pr(B) denotes theprobability of event B occurring with i, j = * or none, and i ≠ j. For example, H can assign
values of ut that are less than a certain critical value as the payoff part of ΩD. This can induce∂Pr(ωt∈ΩD)/∂τt* > 0 because ∂ut/∂τt* < 0, and the sensitivity of ut against τt* can improve once
it is properly controlled for θt and τt.
Given the stage game depicted above, I can describe an infinitely repeated protectionsetting game between H and F with their privately observed signals serving as measures todetect the potential use of concealed protection as follows. A strategy for each country isdefined by s i = ( s i (t )) t∞=1 with(4)s (t ) : A t −1 × Ω t −1 × ( E * ) t −1 → A and s ∗ (t ) : ( A ∗ ) t −1 × (Ω ∗ ) t −1 × E t −1 → A ∗where Ai denotes the set of possible actions that each country can take in a period witha i ≡ (τ i , e i ) ∈ A i , and Ei denotes the set of possible explicit tariffs that each country can impose
in a period with ei ∈ Ei , having i = * or none. s(t), the strategy of H in period t, assigns its
action (τt, et) based on the history of its own previous actions, (a1, ⋅⋅⋅, at−1) ∈ At−1, the history of
its own private information, (ω1,ω2, ⋅⋅⋅, ωt−1) ∈ Ωt−1, and the history of the other country’s
explicit tariffs, (e1∗ , e 2∗ ,⋅ ⋅ ⋅, et∗−1 ) ∈ (E*)t−1, while s*(t) assigns the action for F in a similar manner.
If each country conforms to its strategy defined in (4), then the expected discounted payoff isgiven 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 strategy profile (s, s*), and δ C ∈ [0, 1) denotes the common discount factor with i, j
= * or none, and i ≠ j. Now, I define a supergame equilibrium in this infinitely repeatedprotection setting game as follows:

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 ∗ )
⎠

11

9

A strategy profile ( s, s ∗ ) is a supergame equilibrium in the repeated game

Definition 1.

between H and F, if V ( s, s ∗ ) ≥ V ( s / , s ∗ ) and V ∗ ( s, s ∗ ) ≥ V ∗ ( s, s ∗ / ) for all s / ≠ s ands∗/ ≠ s∗ .

12

To explore the possibility of supporting cooperative protection levels, denoted by l and l*,that are lower than the one-shot Nash protection levels (h and h*) as a supergame equilibrium of
the repeated game described above, I consider following strategies under which each country
uses its private signal, ω and ω*, as a device to trigger an explicit tariff war against the othercountry’s potential use of concealed protections: 13(i) Given that period t − 1 was a “cooperative” period with (et −1 , et∗−1 ) = (0, 0), each countryikeeps cooperating by setting (τ ti , eti ) = (li, 0) if ω ti−1 ∉ Ω D , and it initiates a tariff war byisetting (τ ti , eti ) = (hi, hi) if ω ti−1 ∈ Ω D with i = * or none.
(ii) Given that a “tariff war phase” was initiated in period t − 1 with (et −1 , et∗−1 ) ≠ (0, 0), eachcountry sets (τ i , e i ) = (hi, hi) for the following (T− 2) periods and it continues to do so one
more period with probability λ if et*−1 = 0 ; each country sets (τ i , e i ) = (hi, hi) for thefollowing (T* − 2) periods and it continues to do so one more period with probability λ* ifiiSet −1 = 0 ; each country sets (τ i , e i ) = (h , h ) for the following (T − 2) periods and it

continues to do so one more period with probability λS if et −1 > 0 and et∗−1 > 0 , with T, T*,
and TS being integer numbers that are greater than or equal to 2, λ, λ*, and λS belonging to[0, 1], and i = * or none.
(iii) In period 1 and other “initial” periods right after the end of any tariff war phase, eachcountry sets (τ i , e i ) = (li, 0) with probability (1 − Pri) but initiates a tariff war by settingi(τ i , e i ) = (h , h ) with probability Pr , where Pr = Pr(ωt ∈ Ω D ) with (τt, et) = (l, 0), (τt , et )
i

i

i

i

i

*

*

= (l*, 0), and i = * or none.

12

This definition of a supergame equilibrium of a repeated game with privately observed signals of other players’
actions follows Matsushima (1991).
13
One trivial supergame equilibrium strategy profile is to assign a one-shot Nash protection level for all periods
because that would assign the static optimal behavior for each country.

10Note that the absence or presence of explicit tariffs classifies any period into either a“cooperative” period (with no explicit tariffs) or a “tariff war” 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 tariff war phases as described in (i) and (ii).
Extending a tariff war phase one more period with a certain probability as specified in (ii) is an
instrument to make the length of a tariff war phase to behave as if it were a continuous
variable. 14 Also note that the actions for period 1 and other “initial periods” described in (iii)
are designed to make them mimic those in a period that immediately follows a “cooperative”
one, which in turn simplifies the analysis of the trigger strategies defined above. 15 Finally, notethat the sets of private signals that trigger tariff wars (ΩD, ΩD*) and the lengths of different
tariff war phases (T − 1 if H triggers, T*− 1 if F triggers, and TS − 1 if H and F trigger
simultaneously) with corresponding probabilities to extend the tariff war phases one more
period (λ, λ*, λS) characterize the strategy profile defined by (i), (ii) and (iii), together with thecooperative protection levels (l, l*). Thus, I define simple private trigger strategies (PTS) as
follows:
Definition 2. If (i), (ii), and (iii) describe a strategy profile (s, s*), then, (s, s*) are simple
private trigger strategies (PTS) with (l, l*,ΩD, ΩD*, T, T*, TS, λ, λ*, λS) as characterizing
parameters.
Given this definition, I can derive the expected discounted payoff under (s, s*) with (l,l*,ΩD, ΩD*, T, T*, TS, λ, λ*, λS), denoted by V i ( s, s ) , as follows:*

For example, λ = 0 implies that H and F play the one-shot Nash tariff war for (T − 2) periods, and λ = 1 implies
that they play the one-shot Nash tariff war for (T − 1) periods, with each λ ∈ (0,1) being equivalent to a casewhere they play the one-shot Nash tariff war for some intermediate length of periods between (T − 2) and (T − 1).
This allows the expected discounted payoff from invoking a tariff war phase to vary smoothly (by varying thelength of a tariff war phase “smoothly”) so that it can be set to equal the expected discounted payoff from notinvoking a tariff war phase, an important requirement for incentive constraints considered in Section 2.2.1.
15
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.
14

11

(1 − Pr i Pr j )[u i (l i , l j ) − u i (h i , h j )]V ( s, s ) =1 − δ C + Pr (1 − Pr ∗ )(δ C − δ ) + Pr ∗ (1 − Pr )(δ C − δ ∗ ) + PrPr ∗ (δ C − δ S )
i

(6)

+

∗

Pr j (1 − Pr i )[u i (l i , h j ) − u i (l i , l j )] + Pr i (1 − Pr j )[u i (h i , l j ) − u i (l i , l j )] u i (h i , h j )+1 − δ C + Pr (1 − Pr ∗ )(δ C − δ ) + Pr ∗ (1 − Pr )(δ C − δ ∗ ) + PrPr ∗ (δ C − δ S )
1−δ C
k

k

where δ k = λk (δ C ) T + (1 − λk )(δ C ) T −1 with k = *, s, or none, i = * or none, and i ≠ j, having(δ C − δ ) , (δ C − δ ∗ ) , and (δ C − δ S ) respectively represent the relative length of the tariff war
phase initiated by H alone, by F alone, and by H and F simultaneously. Because (T, T*, TS, λ,

λ*, λS) uniquely defines (δ,δ*,δS) as shown above, I will describe PTS using (l, l*,ΩD, ΩD*, δ, δ*,δS) instead of using (l, l*,ΩD, ΩD*, T, T*, TS, λ, λ*, λS) from now on.
Note that simple PTS defined above is simple in the sense that each country imposes its
one-shot Nash tariff under any punishment phase. PTS may involve more complex punishment
phases such as imposing lower tariffs if the signal indicates weaker violations or/and
employing a stronger punishment, such as autarky, against presumably more severe violations.
As shown later, the payoff of any equilibrium symmetric PTS will be identical to the one undersimple symmetric PTS defined above with l = l*,ΩD = ΩD*, and δ = δ*, as long as thepunishment is triggered by a static Nash tariff. Henceforth, I denote simple PTS by 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 tariff wars
under PTS makes such incentive constraints different from the incentive constraints for trigger
strategies under which public signals trigger punishment phases in two distinctive ways. First,
the private nature of signals imposes restrictions on the lengths of tariff war phases under PTS,
which contrasts with the repeated game with public information where countries can choose
any length for their tariff war phases. Section 2.2.1 analyzes such limits on the lengths of tariff
war 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-time12

deviations from the specified strategy, but whole deviation paths that each country may take. 16
If private signals trigger tariff wars 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 tariff war being initiated in the current
period, which in turn influences the current-period 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 shows that H’s optimal protection
sequence can be a stationary one of setting τ at l (the cooperative protection level) in all periods
until a tariff war starts, a prerequisite for PTS to be a supergame equilibrium.
2.2.1. Constraints on Lengths of Tariff War 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 tariff war phase), each country
faces the choice of whether or not to initiate a tariff war phase by imposing its static optimal
tariff. To eliminate the incentive to misrepresent private signals in such periods, the expected
payoff from initiating a tariff war phase should be identical to the expected payoff from notinitiating it for each country. Denote such conditions that equate those expected payoffs by
ICP for H and ICP* for F. Then,
ICPi:
(7)

(1 − Pr j )[u i (l i , l j ) + δ CVCi )] + Pr j [u i (l i , h j ) + (δ C − δ j )V Ni + δ jVCi ] =(1 − Pr j )[u i (h i , l j ) + (δ C − δ i )V Ni + δ iVCi ] + Pr j [u i (h i , h j ) + (δ C − δ S )V Ni + δ S VCi ],∗where VCi ≡ V i ( s, s ) and V Ni ≡ u(hi, hj)/(1 − δC ) with i = * or none, and i ≠ j. For eachcountry in period t that follows a cooperative period or in an initial period, the left side of theequality in (7) represents the expected discounted payoff from not initiating a tariff war phase,
setting (τti, eti) = (li, 0). The right side of the equality represents the expected discounted payofffrom initiating a tariff war phase, setting (τti, eti) = (hi, hi). In calculating these expected
16

When a public signal triggers tariff wars, any deviatory actions that each country might have taken in any
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

13

discounted payoffs in (7), it is assumed that the other country initiates a tariff war phase with acertain probability that conforms PTS, Pr* by F and Pr by H.
Using ui(li, lj) − ui(li, hj) = ui(hi, lj) − ui(hi, hj) implied by ∂2ui/∂τi∂τj = 0, I simplify (7) into
(ICPi)u i (l i , l j ) − u i (h i , l j ) + (δ C − δ i )(VCi − V Ni ) = Pr j [(δ C − δ j ) − (δ i − δ S )](VCi − V Ni ),where i = * or none, and i ≠ j. For any given cooperative protection levels (l, l*) and any given
ranges of private signals that trigger tariff war phases (ΩD, ΩD*), I have three variables( δ , δ ∗ , δ S ) to be determined with two equations (ICP and ICP*), potentially having infinite
combinations of ( δ , δ ∗ , δ S ) that satisfies ICP and ICP*. To have ICP and ICP* satisfied
regardless of values that Pr* and Pr may take, however, δ C + δ S = (δ ∗ + δ ) needs to hold. 17Then, δ C − δ = [u(h, l*) − u(l, l*)]/(VC − VN) from ICP and δ C − δ ∗ = [u*(h*, l) − u*(l*, l)]/( VC∗− V N∗ ) from ICP*. The following lemma provides a sufficient condition for H and F not having
any incentive to misrepresent its private signals along the equilibrium path under PTS:
Lemma 1.
(a) If δ C − δ = [u(h, l*) − u(l, l*)]/(VC − VN), δ C − δ ∗ = [u*(h*, l) − u*(l*, l)]/( VC∗ − V N∗ ), andδ C + δ S = (δ ∗ + δ ) , then, ICP and ICP* are satisfied for any values of Pr* and Pr.
(b) If H and F value their future payoffs high enough (δC is close enough to 1) and theprobability of a tariff war being triggered along the equilibrium path is low enough (Pr andPr* are close enough to 0), then, for any given combination of (l, l*, ΩD, ΩD*) with (l, l*) <(h, h*) there exists a unique combination of δ, δ*, and δS that satisfies the sufficientcondition for ICP and ICP* defined in Lemma 1 (a).
Proof) (a) is obvious from simplified ICP and ICP*. See Appendix A for the proof of (b).

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).
17
Generally speaking, ICP and ICP* need not be satisfied for any values of Pr* and Pr because Pr = Pr(ωt∈ΩD)and Pr* = Pr(ωt*∈ΩD*) with (τt, et) = (l, 0) and (τt*, et*) = (l, 0) along the equilibrium path under PTS. As discussed
in Section 2.2.2, however, countries may misrepresent private signals along their deviation path under which Pr*
and Pr may not take the same values as those along the equilibrium path. Lemma 4 (b) in Section 2.2.2 establishes
that the sufficient condition in Lemma 1 (a) indeed guarantees countries have no strict incentives to misrepresent
their private signals even along their deviation paths.

14

For a given combination of (l, l*, ΩD, ΩD*), ICP and ICP* specify the lengths of tariff war
phases that countries can employ under PTS.
∗∗

Since δ = λ (δ C ) T + (1 − λ )(δ C ) T −1 , δ ∗ =SSλ∗ (δ C ) T + (1 − λ∗ )(δ C ) T −1 and δ S = λ S (δ C ) T + (1 − λ S )(δ C ) T −1 , ICP limits the length of atariff war phase that H can initiate (T, λ); ICP* limits the length of a tariff war phase that F caninitiate (T*, λ*); and δ C + δ S = (δ ∗ + δ ) determines the length of a tariff war phase that H andF initiate simultaneously (TS, λS). Note that the length of a tariff war phase that H can initiate( δ C − δ ) increases in its expected gain in the initial period of the tariff war phase from
imposing its static optimal tariff (u(h, l*) − u(l, l*)) and decreases in its expected loss in thetariff war periods that will follow (VC − VN ). The expected gain in the initial period provides H
with the incentive to start a tariff war phase despite the expected loss from engaging in a tariffwar. Thus, the larger the expected gain in the initial period, the longer a tariff war that H cantolerate (without violating ICP) and the larger the expected loss from a tariff war, the shorter atariff war that H can tolerate (without violating ICP). The same logic applies in determining
the length of a tariff war phase that F can initiate through ICP*.
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 for H to maximize 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*(τt) = Pr( ω t∗ ∈ΩD*) given (τt, et) = (τt, 0) and ( τ t∗ , et∗ ) = (l*, 0), and τ0 = l; andF (τ d ,τ d +1 ) = Pr ∗ (τ d )[u (τ d +1 , h ∗ ) + (δ C − δ ∗ )V N + δ ∗VCO ] + [1 − Pr ∗ (τ d )]u (τ d +1 , l ∗ ) with VCO =VC. Note that the protection sequence {τ d +1 }∞d =0 in (8) specifies the protection levels only until
F triggers an initial tariff war phase and the optimization in (8) assumes that H will follow its*

specified strategy under PTS once F triggers an initial tariff war phase with VCO = VC ≡ V ( s, s ) .
15

The full optimization problem should characterize the optimal protection sequence after the endof each tariff war 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 to be 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 tariff war
phase in the future. 18 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 tariff war phases satisfy the sufficient condition for ICP
and ICP* in Lemma 1 (a), then, H cannot increase its payoff by using 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). 19
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: 20
(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 ]18

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.
19
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.
20(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 ofH. 21 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 maximizes 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(τ0) be the supremum function that is generated by (8). Then, (i) VS satisfies (9);(ii) the solution to (9) V(τ−1) = VS(τ−1); (iii) every optimal protection sequence solving (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 A 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 generalized envelope theoremsof Milgrom and Segal (2002) and a general result on the differentiability of the value functionof Cotter and Park (2006), I can characterize V and G as follows: 22
Lemma 3.

21

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.
22
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 the 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.

17

Assume that the lengths of punishment phases satisfy the conditions in Lemma 1 (a).
(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 A for Proof)
Because a higher τ−1 (a higher protection level in the cooperative previous period) implies ahigher probability that F triggers a tariff war 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) isan act to 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 tariff war.
Figure 1 demonstrates this. Given the pervious-period protection level τ−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 ∗VC +(δ C − δ ∗ )V N ] , 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 tariff war being initiated in the current period, (δ C − δ ∗ )V N + δ ∗VC . By reducing τ
below

h,

however,Hcanincrease

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 as a balancing act between the static
incentive to raise τ closer to h and the dynamic incentive to avoid a tariff war by reducing τ, I
can explain why G(τ−1) strictly increases in τ−1 using Figure 1. When τ−1 increases from τ −/ 1 toτ −//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 , implying that the static incentive to raise τ closer to h stays the same; forexample, 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 tariff war in a future period
18because it increases the likelihood of a tariff war phase starting in the current period. Figure 1,
illustrates this by δ C [1 − Pr ∗ (τ −//1 )][V ( g (τ −/ 1 )) − V (h)] < δ C [1 − Pr ∗ (τ −/ 1 )][V ( g (τ −/ 1 ) − V (h)] withPr ∗ (τ −//1 ) > Pr ∗ (τ −/ 1 ) ; the dynamic gains from reducing τ from h to g (τ −/ 1 ) decreases as τ−1increases from τ −/ 1 to τ −//1 . As a result, a higher τ−1 moves the balance for choosing an optimal τ
towards a higher 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 entails both an increasing protection
sequence and a decreasing one as shown in Figure 2; if τ 0 = l = τ 0/ , then the optimal protection
sequence will be increasing with τ 0/ < τ 1/ < τ 2/ < ⋅ ⋅ ⋅ ; and if τ 0 = l = τ 0// , then the optimal
protection sequence will be decreasing with τ 0// > τ 1// > τ 2// > ⋅ ⋅ ⋅ . 23 If τ 0 = l = τ S , however, theresulting optimal protection sequence will be stationary with τ 0 = τ 1 = τ 2 = ⋅ ⋅ ⋅ . If there exists
such a stationary 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 tariff war 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 + δ*VC]}. 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 to a generalized differentiability result of Cotter and Park (2006). Therefore, (11) is
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 does
emerge in the context of an exporting firm’s dynamic pricing problem in the presence of antidumping policy; once
an exporting firm becomes subject to 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 overtime) depending on whether the initial export pricing is higher or lower than a stationary pricing.
23

19

indeed a necessary condition for any stationary protection level that belongs to (0, h), thus itserves as an incentive constraint (IC) for H to sustain the cooperative protection level, l, under
PTS with l = τS.
For τS to be a stationary protection level for H, its static incentive to raise its protectionlevel, ∂F(τS, τS)/∂τ > 0 in (11), needs be balanced by its dynamic incentive to avoid a costly
tariff war 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) showsthat H does not have any incentive to utilize tariffs as part of its deviation path if l = τS.
Lemma 4.
Assume that the lengths of tariff war 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). 24(b) If l = τS, then H cannot increase its discounted payoff above V ( s, s ) by taking any*(deviatory) protection sequence that involves initiating tariff wars by imposing tariffs.
(See Appendix A for Proof)
According to Lemma 4 (a), it is possible to have IC in (11) satisfy 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 tariff war, 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 tariff war, 1 − Pr*(τS), byincreasing the probability of a tariff war in the current period. If ∂2Pr*(τS)/∂τ2 > 0, an enhanced
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.
24

20

sensitivity of F’s private information in detecting a rise in H’s protection can offset such a
reduction in H’s incentive to avoid a tariff war; 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 – 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 aunique τS ∈ (0, h) that satisfies IC in (11) if ∂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
period, δC[1 − Pr*(τS)]⋅[∂V(τS)/∂τ] in (11), decreases as τS gets higher, entailing the possibilityof IC in (11) not being satisfied for any τS < h.
While Lemma 4 specifies the condition under which H (and F under an analogous
condition) would follow PTS by keeping its protection at a cooperative level until a tariff war is
triggered, note that Lemma 4 “assumes” the lengths of tariff war phases to satisfy the conditions
in Lemma 1 (a). Because such lengths of tariff phases “vary” with the cooperative protectionlevels (l, l*) to sustain under PTS, it still remains to be shown whether there exist PTS thatsatisfy ICP, ICP*, IC, and IC* simultaneously. One obvious candidate is symmetric PTS forsymmetric countries with l = l*, ΩD =ΩD*, and δ = δ*, which is the focus of Section 3.

3. Optimal Symmetric Private Trigger Strategies for Symmetric Countries
This section establishes that symmetric countries can sustain a symmetric cooperative
protection level under symmetric PTS if the sensitivity of their private information satisfies
certain conditions. In addition, this section proves that any equilibrium symmetric trigger
strategies that initiate a punishment period by imposition of a static Nash tariff based on eachcountry’s imperfect private information should generate the same payoff as the one that the
specific symmetric PTS defined in Definition 2 would yield. After proving the existence andthe 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. Symmetric Private Trigger Strategies and Uniqueness Results21

To analyze the case where H and F are symmetric, this section assumes that u(τ1,τ2) =u*(τ1,τ2) for all τ1 andτ2 ∈ [0, h] and that Pr(ωt∈ΩD) = Pr(ωt*∈ΩD*) for all (τt, et) = (τt*, et*) andΩD = ΩD*. This section focuses on symmetric PTS with l = l*, ΩD = ΩD*, and δ = δ*. Ifsymmetric PTS satisfy ICP and IC for H, such PTS will also satisfy ICP* and IC* for F bysymmetry. Without loss of generality, the following analysis focuses on proving the existenceof symmetric PTS that satisfy ICP and IC.
Assume that there exists τS that satisfies IC in (11) with τS = l. This implies 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 tariff war phase starts.
To guarantee that PTS satisfy ICP, I assume that the lengths of tariff phases are determinedby the sufficient condition for ICP in Lemma 1 (a); δ C − δ = [u(h, l*) − u(l, l*)]/(VC − VN) (=δ C − δ ∗ = [u*(h*, l) − u*(l*, l)]/( VC∗ − V N∗ ) by symmetry) and δ C + δ S = (δ ∗ + δ ) . Denoting
the symmetric cooperative protection level under symmetric PTS by lc (= l = l*), IC in (12) canthen be rewritten into the following implicit function, I(lc):(13)

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

by substituting δ C − δ ∗ with [u(h, l*) − u(l, l*)]/(VC − VN). Using I(lc), Proposition 1 provides a
sufficient condition for the existence of symmetric PTS that countries can sustain as asupergame equilibrium of their repeated protection setting game:
Proposition 1.
If ∂2Pr*(lc)/(∂lc)2 > 0 with [∂2Pr*(lc)/(∂lc)2][1 – Pr*(lc)] – {1 + δC[1 – Pr*(lc)]}[∂Pr*(lc)/∂lc]2 > 0
for all lc ∈ [0, h], ∂Pr*(lc)/∂lc ≈ 0 at lc = 0, and there exists at least one protection level, l Sc < hsuch that I( l Sc ) = 0, then, symmetric H and F can employ symmetric PTS with lc = l Sc , δ = δ *=

δC − [u(h, l Sc ) − u( l Sc , l Sc )]/(VC − VN), and δS = δ + δ* − δC as a supergame equilibrium of therepeated protection setting game. (See Appendix A 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 protection

22

level τS ∈ (0, h) with G(τS) = τS. In addition, it requires I(lc) = 0 for at least one value of lc < h,denoting it by l Sc . With lc = l Sc and the lengths of punishment phases (T = T* and TS) definedby δ = δ *= δC − [u(h, l Sc ) − u( l Sc , l Sc )]/(VC − VN), and δS = δ + δ* − δC , I( l Sc ) = 0 guarantees IC
and ICP to be simultaneously satisfied under such PTS. According to Lemma 4, l Sc is theunique stationary protection level with G( l Sc ) = l Sc and countries have no incentive to deviate
from such PTS.
The sufficient condition in Lemma 4 (a) does not necessarily imply that the second term ofI(lc) in (13), δC[∂Pr*(lc)/∂τ ][1 − Pr*(lc)][u(h, lc) – u(lc, h*)], representing H’s dynamic incentive
to avoid a tariff war, increases in response to a rise in lc. 25 Thus, one may consider the casewhere multiple values of lc satisfy I(lc) = 0 as illustrated in Figure 3; lc = lmax as well as lc = lminsatisfy I(lc) = 0. Denoting the minimum of such lc by lmin, then symmetric PTS with lc = lmin
will Pareto-dominate the others when Pr*( lc) is small enough. 26
While the above result establishes that symmetric countries may employ simple symmetric
PTS characterized by Definition 2 and Proposition 1 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 PTS as long as theinitial explicit tariff that starts a punishment phase is the static optimal tariff of each country.
The first part of the following result applies to a larger class of trigger strategies: any
symmetric PTS where each country can start the initial punishment phase with an explicit tariffof any level.
Denote the level of an explicit tariff that initiates the first punishment phase by d0 and thelevel of a cooperative protection level for the initial cooperative periods (prior to any
punishment being triggered) by l0. Then, I can represent the expected discounted payoff of H
of employing such a symmetric PTS as follows, denoting it by VP(l0; d0):25

For the proof of this claim, see the proof for Proposition 1 in Appendix A.
Note that u(lmin, lmin) > u(lmax, lmax) and Pr*(lmin) < Pr*(lmax) imply a higher cooperative-period payoff and a lower
probability of tariff wars with lc = lmin than with lc = lmax. While the lengths of tariff war phases may be longer with
lc = lmin than with lc = lmax, an increase in lc will lower the expected discounted payoff under symmetric PTS if
Pr*(lc) is close enough to 0, as shown in (15) of the following section.
26

23

(15)

V P (l 0 ; d 0 ) = (1 − Pr )(1 − Pr ∗ )[u (l 0 , l 0 ) + δ C VCP ] + (1 − Pr ) Pr ∗ [u (l 0 , d 0 ) + δ C V DP∗ ]+ Pr (1 − Pr ∗ )[u (d 0 , l 0 ) + δ C V DP ] + PrPr ∗ [u (d 0 , d 0 ) + δ C V DsP ],where Pr* ≡ Pr*(l0) = Pr( ω t∗ ∈ΩD*) given (τt, et) = (l0, 0) and ( τ t∗ , et∗ ) = (l0, 0), VCP ≡ VP(l0; d0),
and VDP, VD*P, and VDsP denote the expected discounted payoff of H after an initial punishment
phase had been by triggered, respectively by H alone, by F alone, and by H and F
simultaneously in the previous period, with Pr = Pr* and VDP = VD*P by the symmetry
assumption. The payoff function in (15) implicitly assumes that each country sets its initialcooperative explicit tariff to be zero and each country starts the initial punishment phase by
setting (τ, e) = (d0, d0). Relaxing the latter assumption by allowing e ≠ d0 (and τ = d0) would
not affect the following result because the equilibrium value of VP(l0; d0) does not depend on
the values of VDP, VD*P, and VDsP, as shown later. Relaxing the former assumption by allowinge > 0 in the initial cooperative periods would not raise the payoff in (15) as long as thesensitivity of private information of concealed trade barriers improves with a higher level ofconcealed trade barriers. 27 For the proposed PTS to be incentive-compatible (so that they canbe supported as equilibrium behaviors), the payoff in (15) needs to be equal to the following
expression:(15/)

V P (l 0 ; d 0 ) =u (l 0 , l 0 )
1−δ

C

− Pr

u ( d 0 , l 0 ) − u (l 0 , d 0 )
1−δ C

.

Note that the payoff in (15/) depends only on the values of l0 and d0. Once established, theabove result will remarkably simplify 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 need to find l0 and d0 that are incentive-compatible, which in turn maximize thepayoff in (15/).
To prove (15/), I use three incentive-compatibility conditions: (i) a generalized version of
ICP, (ii) IC for setting the initial protection level to be l0 with the action of setting τ = l0 being
followed upon the contingency of no punishment phase being initiated and (iii) IC for setting
the initial protection level to be l0 with the action of setting τ = d0 being followed upon thecontingency of no punishment phase being initiated. First, note that any symmetric PTS27

τ = τ * = 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.

24triggering the initial punishment phase with an explicit tariff of d0 should satisfy the following
ICPP:(ICPP)(1 − Pr ∗ )[u (l 0 , l 0 ) + δ CVCP ] + Pr ∗ [u (l 0 , d 0 ) + δ CVDP* ]= (1 − Pr ∗ )[u (d 0 , l 0 ) + δ CVDP ] + Pr ∗ [u (d 0 , d 0 ) + δ CVDsP ],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(d01,d01) implied by ∂2u/∂τ∂τ* = 0, once again I can simplify (ICPP) as follows:(ICPP) u (l 0 , l 0 ) − u ( d 0 , l 0 ) + δ C (VCP − V DP* ) = δ C Pr ∗ [(VCP − V DP ) + (VCP − V DP* ) − (VCP − V DsP )] .
To identify ICs for setting the initial protection level to be l0, I can write the expecteddiscounted payoff of H of choosing its initial protection level to be τ0 as
Pr ∗ [u (τ 0 , d 0 ) + δ CVDP ]+ (1 − Pr ∗ ){u (τ 0 , l 0 ) + δ C Pr ∗ (τ 0 )[u (l 0 , d 0 ) + δ CVDP* ] + δ C [1 − Pr ∗ (τ 0 )][u (l 0 , l 0 ) + δ C VCP ]},orPr ∗ [u (τ 0 , d 0 ) + δ CVDP ]+ (1 − Pr ∗ ){u (τ 0 , l 0 ) + δ C Pr ∗ (τ 0 )[u (d 0 , d 0 ) + δ CV DsP ] + δ C [1 − Pr ∗ (τ 0 )][u (d 0 , l 0 ) + δ CV DP ]}depending on whether H sets τ = l0 or τ = d0, respectively, upon the contingency of nopunishment phase being initiated after setting τ = τ0 as the initial protection level. To be able tosupport τ0 = l0 as an equilibrium behavior, the following ICP need to be satisfied(ICP)∂u (l 0 , l 0 )∂Pr ∗ (l 0 )= δ C (1 − Pr ∗ )[u (l 0 , l 0 ) − u (l 0 , d 0 ) + δ C (VCP − V DP* )],
∂τ
∂l 0

,

and
∂u (l 0 , l 0 )∂Pr ∗ (l 0 )= δ C (1 − Pr ∗ )[u (d 0 , l 0 ) − u (d 0 , d 0 ) + δ C (V DP − V DsP )].
∂τ∂l 0

which implies that u(l0, l0) – u(l0, d0) + δC(VCP − VD*P) = u(d0, l0) – u(d0, d0) + δC(VDP − VDsP).
Using this last equality together with the simplified ICPP and VDP = VD*P (by symmetry), one
can easily show that (VCP − VDP) + (VCP − VD*P) – (VCP − VDsP) = 0, thus u(d0, l0) – u(l0, l0) =δC(VCP − VDP). Given these equalities, one can rewrite VP(l0; d0) in (15) into the one in (15/)
using the following steps:

25

V P (l 0 ; d 0 ) = u (l 0 , l 0 ) + δ C VCP + Pr[u (l 0 , d 0 ) − u (l 0 , l 0 ) − δ C (VCP − V DP )]+ Pr ∗ [u (d 0 , l 0 ) − u (l 0 , l 0 ) − δ C (VCP − V DP* )](16)+ PrPr ∗ {[u (d 0 , d 0 ) − u (l 0 , d 0 )] − [u ( d 0 , l 0 ) − u (l 0 , l 0 )]}+ δ C PrPr ∗ [(VCP − V DP ) + (VCP − V DP* ) − (VCP − V DsP )]= u (l 0 , l 0 ) + δ C VCP + Pr[u (l 0 , d 0 ) − u (l 0 , l 0 ) − u (d 0 , l 0 ) + u (l 0 , l 0 )],where the second line through the forth one in (16) all take zero values and the last line in (16)generates (15/) using VP(l0;d0) = VCP. I can summarize this result in the following lemma.
Lemma 5.

The payoff of any equilibrium symmetric PTS with the initial cooperative

protection level being l0 and the level of explicit tariff that starts the initial punishment phasebeing d0, denoted by VP(l0; d0), is a function of only l0 and d0 with
V P (l 0 ; d 0 ) =u (l 0 , l 0 )
1−δ

C

− Pr

u ( d 0 , l 0 ) − u (l 0 , d 0 )
1−δ C

,

where Pr = Pr* ≡ Pr*(l0) = Pr( ω t∗ ∈ΩD*) given (τt, et) = (l0, 0) and ( τ t∗ , et∗ ) = (l0, 0).
The above lemma establishes that one can fully characterize the payoff of any equilibrium
symmetric PTS only with the information about l0 and d0 that are incentive-compatible. Givenu(d0, l0) – u(l0, l0) = δC(VCP − VDP), the necessary condition for incentive compatible choice of
l0 is
P

(IC )

∗
∂u (l 0 , l 0 )C∗ ∂Pr (l 0 )= δ (1 − Pr )[u (d 0 , l 0 ) − u (l 0 , d 0 )],∂τ∂l 0a condition that is identical to IC for simple symmetric PTS characterized in (13) if d0 = h, thestatic optimal protection level. While Proposition 1 guarantees the existence of incentivecompatible symmetric PTS with d0 = h, characterizing the necessary condition for incentive
compatible choice of d0 ≠ h is far from being a simple task. Once d0 ≠ h, then each country
would have an incentive to start an initial punishment by choosing e = d0 but τ ≠ d0,
necessitating a punishment scheme against such a deviation incentive. In fact, one can showthat the only incentive compatible choice of d0 is h, if one does not allow each country to startanother punishment phase in the second period of an initial punishment phase that it started.
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 the26expected discounted payoff in comparison with the choice of setting d0 = h. For example,
setting d0 < h would raise VP(l0; d0) = {u(l0, l0) –Pr[u(d0, l0) – u(l0, d0)]}/(1 − δC) by decreasingu(d0, l0) – u(l0, d0) with ∂[u(d0, l0) – u(l0, d0)]/∂d0 > 0. However, lowering d0 weakens the ICP
by lowering the right hand side value of ICP shown above, thus decreasing VP(l0; d0) by raising
the value for l0. The optimality of choosing d0 ≠ h, even when it is incentive-compatible,
therefore, depends on the trade-off between its direct on the payoff through changing u(d0, l0) –u(l0, d0) and its indirect effect through changing the incentive-compatible l0, which in turnrequires further characterization of the private information of concealed trade barriers.
In the following analysis of characterizing the optimal symmetric PTS, I will focus on the
optimal symmetric PTS with d0 = h. Note that this constraint of setting d0 = h allows full
flexibility over the choice of strategies that each government can take once an initialpunishment phase starts. With regard to the issue of characterizing the efficient frontier among
this subset of symmetric PTS with d0 = h, one can focus on the simple symmetric PTS
characterized in Proposition 1 as the following corollary clarifies.
Corollary to Lemma 5.

The equilibrium payoff of any symmetric PTS with the initialcooperative protection level being lc and the level of explicit tariff that starts the initialpunishment phase being h, is identical to the payoff of the simple symmetric PTS characterized
in Proposition 1 with
V P (l c ; h) = VC =u (l c , l c )u (h, l c ) − u (l c , h)
and
Pr
−
1−δ C
1−δ C

∂u (l c , l c )∂Pr ∗ (l c )= δ C (1 − Pr ∗ )[u (h, l c ) − u (l c , h)],c∂τ∂lwhere Pr = Pr* ≡ Pr*(l0) = Pr( ω t∗ ∈ΩD*) given (τt, et) = (l0, 0) and ( τ t∗ , et∗ ) = (l0, 0).
Note that one can derive H’s expected discounted payoff under symmetric PTS, VC, shown in
the above corollary, from (6), using δ C − δ = δ C − δ ∗ = [u(h, lc) − u(lc, lc)]/(VC − VN),δ C + δ S = (δ ∗ + δ ) and Pr*(lc) = Pr(lc).
Up to this point, I have assumed that the range of private signals that trigger a tariff war
phase (ΩD for H and ΩD* for F) is fixed. Symmetric countries can change the cooperative
protection level lc under symmetric PTS by changing the range of tariff-war-triggering private
27signals, ΩD = ΩD*, because it affects the probability of a tariff war being triggered. The
following section characterizes optimal symmetric PTS, focusing its analysis on the choice ofΩD (= ΩD*) that maximizes the expected discounted payoff under PTS.
3.2 Optimal Symmetric Private Trigger Strategy
The private signal ω ∈Ω has two distinctive quality dimensions as a measure that 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 a tariff war. The
other is the stability of the signal that rewards the cooperative behavior with a lower probabilityof a tariff war. I can represent the sensitivity by Pr/(τ*) ≡ ∂Pr(τ*)/∂τ* > 0 and the stability by 1
− Pr(τ*) at τ* = lc for H’s private signal, with corresponding expressions for F.
A change in the range of private signals that trigger a tariff war phase affects thesequalities of signals. In particular, H may raise the sensitivity by properly expanding the rangeof tariff-war-triggering private signals, ΩD, but at the cost of undermining the stability. By
denoting the degree of such expansion with a parameter ωD (namely, a trigger control variable),
I can formalize this trade-off that H faces in choosing ωD by assuming ∂Pr/(τ*)/∂ωD > 0 and∂Pr(τ*)/∂ωD > 0.
The analysis of optimality in this section constrains itself to the symmetric PTS identified
in Proposition 1 with the cooperative protection level being determined by a choice over ωD.
Assuming that ωD uniquely determines l Sc with I( l Sc ) = 0, I can represents l Sc as a function of

ωD; l Sc = l Sc (ωD). Then, as shown in Corollary to Lemma 5,(17)∗

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

− Pr (l Sc )u (h, l Sc ) − u (l Sc , h ∗ )1−δ CNote that the expected discounted payoff in (14) is no longer depending on the lengths of thetariff war phases. Therefore, I can describe the optimal choice for ωD (=ωD*) as ωD thatsatisfies the following first order condition:

28

∂VC ∂l Sc (ω D ) ∂VC ∂Pr (l Sc )
∂VC
= c
+
= 0, with∂Pr ∂ω D∂l S ∂ω D
∂ω D
c
c
∗
∂VC ∂u (l Sc , l Sc ) 1
c u ( h, l S ) − u (l S , h )
/
=
−
Pr
(
l
)
S
1−δ C
∂l Sc
∂l Sc
1−δ C

(18)

{∂u (l Sc , l Sc ) / ∂τ ∗ − ∂u (l Sc , l Sc ) / ∂τ }< 0 for Pr (l Sc ) being close to 0,
C
1−δ
c
D
D
∂l S (ω )∂Pr / (l Sc )∂Pr (l Sc ) / c∂I / ∂ω
c
=
−
<
0
iff
[
1
−
Pr
(
l
)]
−
Pr (l S ) > 0, andS∂ω D∂I / ∂l Sc
∂ω D
∂ω D
− Pr (l Sc )∂VC ∂Pr (l Sc )∂Pr (l Sc ) [u (h, l Sc ) − u (l Sc , h ∗ )]=−< 0,
∂ω D ∂ω D
∂ω D
1−δ C
where I = I( l Sc ) is the implicit function defined in (13). The first order condition is informative
about the trade-off that the countries face in choosing an optimal ωD. Raising the trigger
control variable (ωD) will have a positive effect on the expected discounted payoff (VC) bylowering the cooperative protection level ( l Sc ) if ∂ l Sc /∂ωD < 0 and ∂VC /∂ l Sc < 0, but it also has anegative effect on the expected payoff by increasing the probability of a tariff war phase being
invoked, as shown by ∂VC /∂ωD < 0 in (18). Thus, the optimal ωD should balance the gain fromraising the sensitivity of the private signal (thus achieving a lower l Sc ) against the loss fromreducing the stability of the cooperative equilibrium with a higher tariff war probability.
When the initial ωD is at a very low level, then, it is generally possible to lower l Sc by
raising the trigger control variable. For example, if ΩD = ∅, then l Sc = h, Pr( l Sc ) = Pr/( l Sc ) = 0,
implying ∂ l Sc /∂ωD < 0 with ∂Pr/( l Sc )/∂ωD > 0 from (18). If countries continue to raise ωD, themarginal increase 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 Sc )/∂(ωD)2 < 0 and ∂2Pr( l Sc )/∂(ωD)2 = 0, with the latter assumption making
the effect of a higher ωD on Pr( l Sc ) to be constant. Then, it is possible to have ∂2 l Sc /∂(ωD)2 > 0
and ∂ l Sc /∂ωD = 0 for a high enough ωD. While it is possible to raise ωD to such a point that thecountries would no longer be able to lower the cooperative protection level any further(∂ l Sc /∂ωD = 0), note that it is never optimal to do so. If it does, then the first order condition forthe optimal ωD in (18) will be violated as ∂VC /∂ωD = (∂VC /∂Pr)(∂Pr( l Sc )/∂ωD) < 0, implying

29

that countries can increase their payoffs by lowering the trigger control variable. I summarize
these characterizations of optimal symmetric PTS in the following proposition.
Proposition 2.
The choice over the trigger control variable ωD is a balancing act between raising the sensitivityof the private signal (thus achieving lower cooperative protection levels) and reducing thestability of the cooperative equilibrium (by a higher tariff war probability). As a result, optimal
symmetric PTS do not raise the trigger control variable to the level that pushes down thecooperative protection level to its minimum attainable level with ∂ l Sc /∂ωD = 0. 28
In applying Special Section 301 to protect the U.S. intellectual property rights (IPR) in
foreign markets, the USTR specifies not only “Priority Foreign Countries” who are “pursuing
the most onerous or egregious policies that have the greatest adverse impact on U.S. rightholders or products, and are subject to accelerated investigations and possible sanctions,” butalso “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.” 29This practice indicates the USTR’s willingness to tolerate some level of deviations fromagreements, reserving retaliatory sanctions mainly against considerable deviations. Such apractice may not lead to the maximal protection of the U.S. IPR, but may reduce the probabilityof costly tariff wars invoked by Special Section 301.

4. Limitations of Private Trigger Strategies
In this section, I consider factors that may limit the effectiveness of PTS in restraining theuse of concealed protection. Section 4.1 analyzes how a reduction in the time lag to adjust
protection levels in response to an initiation of an explicit tariff war affects the effectiveness of
PTS. Section 4.2 discusses how asymmetry among countries may limit the use of PTS.
4.1. A Faster Response to an Initiation of an Explicit Tariff War28
29

A similar characterization has been drawn for optimal cartel trigger price strategies by Porter (1983).
These quoted definitions come from the USTR website (http://www.ustr.gov).

30

The analyses in previous sections assume that H and F simultaneously set their concealed
and explicit protection levels at the beginning of each period and cannot adjust those protectionlevels until that period is over. While this is a standard assumption in the literature thatanalyzes self-enforcing trade agreements in a repeated game setup, the time lag (one period) for
countries to readjust their protection levels plays an important role in determining theeffectiveness of PTS as shown below. In particular, I analyze how reducing the time lag inadjusting protection levels in response to an initiation of an explicit tariff war affects the levelof cooperation attainable under PTS. 30
To analyze the effect of such a reduction in the time lag in adjusting protection levels
without changing the basic structure of the model, I represent one-period payoffs for a period
when F initiates a tariff war by imposing its static optimal tariff by u(l, h*)/n + (n − 1)u(h, h*)/n
for H and u*(h*, l)/n + (n − 1)u*(h*, h)/n for F, with n ∈ [1, ∞) denoting how fast each country
can readjust its protection level in response to an initiation of a tariff war phase by the othercountry. 31 Note that n = 1 implies no reduction in the time lag to adjust protection levels and n→ ∞ means instantaneous adjustment (no lag). For a period when H initiates a tariff war, oneperiod payoffs are u(h, l*)/n + (n − 1)u(h, h*)/n for H and u*(l*, h)/n + (n − 1)u*(h*, h)/n for F.
As in Section 3, I continue to focus on symmetric PTS for symmetric countries. Then, achange in the speed that each country can readjust its protection level, denoted by n, affects thecooperative protection level that countries can attain under PTS, lc, by affecting the incentive
constraint, IC in (11) and the lengths of tariff war phases that satisfy ICP and ICP* in (7).
To analyze how a faster response to an initiation of a tariff war affects the lengths of tariff
war phases under symmetric PTS, I rewrite ICP in (7) as follows:30

The reduction in the time lag considered in this section does not apply to the time lag in adjusting concealed
protection levels, but only to the time lag in adjusting explicit protection levels in response to an initiation of an
explicit tariff war. If the speed of adjusting concealed protection levels changes, it may affect countries’ optimal
protection sequence, thus, the whole optimization problem under PTS. While fixing the speed of adjusting
concealed protection levels, one may still consider a reduction in the time lag in adjusting protection levels by
allowing countries to adjust their explicit tariff levels faster. In comparison with controlling concealed protections,which may require non-explicit and customary arrangements with various players in the market, changing the
explicit tariff levels in response to a foreign initiation of a tariff war may take much less time, like issuing an
executive order.
31
This representation of one-period payoffs for a period when F initiates a tariff war implicitly assumes that H willreact to the initiation by adjusting its protection to its static optimal level (h) as soon as possible. Such a reaction
is an optimal action for H, thus PTS with such reactions would still be a supergame equilibrium as long as IC, IC*,
ICP, ICP* are satisfied with changes in the speed of such reactions being taken into account.

31u (l , l ∗ ) − u (l , h∗ ) (n − 1)[u (l , l ∗ ) − u (h, h* )]u (l , l ) + δ VC − Pr {++ (δ C − δ ∗ )(VC − VN )} =nn∗∗u (h, l ) (n − 1)u (h, h )u (h, l ∗ ) − u (h, h∗ )++ (δ C − δ )VN + δVC − Pr ∗[+ (δ − δ S )(VC − VN )].
n
n
n
∗

(19)

C

∗

To have the above ICP satisfied for all Pr*∈ [0, 1], δ = δ* and δS need to satisfy(20)( n − 1)[u (h, l ∗ ) − u ( h, h * )], andn(n − 1)[u (l , l ∗ ) − u (h, h * )]SC∗[(δ − δ ) − (δ − δ )](VC − V N ) =,n(δ C − δ )(VC − V N ) = [u ( h, l ∗ ) − u (l , l ∗ )] −which becomes the same sufficient condition for ICP and ICP* specified in Lemma 1(a) if n = 1.
Note that the length of a tariff war phase that H (or F) can initiate, represented by δC − δ (=

δC − δ* by symmetry) in (20), decreases in n because the last term on the right side of the firstequality in (20), − (n − 1)[u(h, l*) – u(h, h*)]/n decreases in n. Recall that the length of a tariff
war phase that H can initiate under PTS increases in its expected gain from imposing its static
optimal tariff in the initial period of the tariff war phase, u(h, l*) − u(l, l*), as discussed in
Section 2.2.1. Because a higher n implies a smaller expected gain in the initial period with [u(h,l*)/n + (n − 1)u(h, h*)/n] − u(l, l*) decreasing in n, the length of a tariff war phase H canemploy gets shorten with a higher n, weakening the punishment against concealed protection.
Now, using (20) and F(τd,τd+1) = Pr*(τd)[u(τd+1, h*)/n + (n − 1)u(h, h*)/n + (δC − δ*)VN +δ*VC] + [1 − Pr*(τd)]u(τd+1,l*), I can rewrite IC in (11) into: 32(21)∂u (l c ,l c ) / ∂τ = δ C [∂Pr ∗ (l c ) / ∂τ ][1 − Pr ∗ (l c )][u (h, l c ) − u (l c , h ∗ )]+ [(n − 1) / n]{Pr ∗ (l c )[∂u (l c ,h ∗ ) / ∂τ ] − δ C [∂Pr ∗ (l c ) / ∂τ ][1 − Pr ∗ (l c )][u (h, l c ) − u (l c , h ∗ )]}.
Note that IC in (18) takes the same form as the one in (13), IC under symmetry with n = 1,

except for the last term on the right-hand side of the equality. This last term on the right-handside takes a negative value when it is measured at lc = l Sc with I( l Sc ) = 0 in (13) because ∂u( l Sc ,h*)/∂τ = ∂u( l Sc , l Sc )/∂τ = δC[∂Pr*( l Sc )/∂τ ][1 − Pr*( l Sc )][u(h, l Sc ) – u( l Sc , h*)] and Pr*( l Sc ) < 1.
An increase in n raises [(n − 1)/n], making the last negative term in (21) to have a larger

32

A change in the speed that each country can readjust its protection level, n, affects VC by changing H’s payoff in
the initial period of any unilaterally-triggered tariff war phase and by affecting the lengths of punishment phases.
While the derivation of IC in (18) does not explicitly consider this effect of a change in n on VC, IC in (18)
embodies such effect because the modified sufficient condition for ICP and ICP* in (17) is assumed to be satisfied
with such effect being taken into account.

32

absolute value. Therefore, the cost associated with raising the concealed protection level,
represented by the right-hand side of the equality in (21), decreases in n, implying that eachcountry will have a higher incentive to raise its concealed protection when it can react faster tothe other country’s initiation of a tariff war phase.
I can dissect the negative effect of a rise in n on the cost associated with raising theconcealed protection level, represented by [(n − 1)/n]{Pr*(lc)[∂u(lc, h*)/∂τ] − δC[∂Pr*(lc)/∂τ ][1− Pr*(lc)][u(h, lc) – u(lc, h*)]} in (21) as follows. For the case where F initiates a tariff war in
the current period, H’s gain from raising concealed protection decreases as n increases because
H would react faster to such a case by setting its static optimal tariff during that period. This isrepresented by Pr*(lc)[∂u(lc, h)/∂τ], representing an increase in the cost of raising the concealed
protection level. This increase in the cost, however, is dominated by the negative term that
follows, − δC[∂Pr*(lc)/∂τ ][1 − Pr*( lc)][u(h, lc) – u(lc, h*)], which represents multiple factorsthat strengthen the incentive to raise concealed protection. With a higher value for n, first, H
can react faster to F’s initiation of a tariff war in the following period by imposing its static
optimal tariff, reducing the cost associated with raising concealed protection. Furthermore, anincrease in n reduces the lengths of punishment phases that countries can employ against thepossible use of concealed protection under PTS, as discussed earlier.
Finally note that it becomes impossible to employ PTS to curtail the use of concealed
protection if the speed of protection readjustment gets too high. This is because there will be
no length of a tariff war phase that satisfies ICP if n > n ≡ [u(h, l*) – u(h, h*)]/[u(l, l*) – u(h,h*)] (>1); (δC − δ)(VC − VN) < 0 in (20) if n > n , implying a NEGATIVE length for the tariff
war phase that H can initiate (δC < δ) to satisfy ICP. The following proposition summarizes theabove observation about the effect of a faster reaction to an initiation of a tariff war on thecooperative protection level attainable under symmetric PTS.
Proposition 3.
If each country can readjust its protection level faster in response to the other country’s
initiation of a tariff war phase (a higher n), the cooperative protection level attainable undersymmetric PTS increases. In particular, if n gets higher than n ≡ [u(h, l*) – u(h, h*)]/[u(l, l*) –u(h, h*)], then it becomes impossible to use PTS to sustain any level of cooperative protection
because there will be no lengths of tariff war phases that satisfy ICP and ICP*.

33

This finding on the effect of a faster response to an initiation of a tariff war on PTS
contrasts with how the same increase in the speed of protection readjustment would affect thecooperative protection level attainable under trigger strategies based on public signals. As suchtrigger strategies embody simultaneous imposition of explicit tariffs in the initial period of any
tariff war phase, a faster response to an initiation of a tariff war would not affect thecooperative protection level attainable under trigger strategies based on public signals. Whencountries can readjust their protection levels quickly, Proposition 3 implies that trigger
strategies based on public signals are more likely to be successful than PTS in restraining theuse of concealed protection.
To demonstrate that a change in the time lag in adjusting the protection level matters inenforcing international trade agreements, one may discuss the WTO’s amendment of GATT
Article XIX; Article XIX, known as the “escape clause,” allows a government to temporarilysuspend a concession agreed upon in a previous negotiation if its import-competing industry is
injured as a consequence of a temporary surge in import volume. 33 To encourage the use of
Article XIX as opposed to the use of managed-trade policies (such as voluntary export
restraints), the WTO’s amendment prohibits retaliatory responses by affected partners for athree-year period following the original imposition of Article XIX-based tariffs. Even though
the escape clause is not about imposing tariffs to punish other countries’ use of concealed
protection, this amendment does reflect the WTO member countries’ concern that the lack oftime lag in imposing retaliatory tariffs in response to the imposition of Article XIX-motivatedtariffs may discourage the use of Article XIX tariffs. One can interpret this motivation behindamending Article XIX as an (indirect) evidence for Proposition 3: a faster retaliatory reaction
to an imposition of tariffs may undermine the effectiveness of PTS.
4.2. Asymmetry among Countries
In this section, I discuss how asymmetry among countries affects the cooperative
protection levels sustainable under PTS. To save space, this section provides a summary of theanalysis, having a more complete analysis in Appendix B. Following Bond and Park (2002),
one can introduce asymmetry among countries by analyzing the following partial equilibrium
trade model where H exports good 1 and F exports good 2, with σ ∈ [1, ∞) denoting the size of34

H’s markets relative to F’s. Demand for good i in H is Di = σ ( A − Bp i ) and supply of good iin H is X i = σ (α i + βp i ) , 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 p i∗ . To ensure that H will export good
1 and import good 2 and that the countries will be symmetric when σ = 1, I assume that

α 1 − α 1∗ = α 2∗ − α 2 > 0 and α 1 = α 2∗ . By varying σ on [1, ∞), one can consider the range ofrelative country sizes from symmetric countries (σ = 1) to the case where F is a price taker inthe world market (σ → ∞).
To analyze the effect of asymmetry among countries on PTS, I focus on how theasymmetry, represented by σ, affects the following IC and IC*:(22)

IC : ∂u (l,l ∗ ) / ∂τ = δ C [∂Pr ∗ (l ) / ∂τ ][1 − Pr ∗ (l )][u (l , l ∗ ) − u (l , h ∗ ) + (δ C − δ ∗ )(VC − V N )]IC ∗ : ∂u ∗ (l ∗ ,l ) / ∂τ ∗ = δ C [∂Pr (l ∗ ) / ∂τ ∗ ][1 − Pr (l ∗ )][u ∗ (l ∗ , l ) − u ∗ (l ∗ , h) + (δ C − δ )(VC∗ − V N∗ )],where (δC – δ* ) = [u*(h*, l) – u*(l*, l)]/(VC* – VN*) and (δC – δ) = [u(h, l*) – u(l, l*)]/(VC – VN)
with δ C + δ S = (δ ∗ + δ ) to satisfy the sufficient condition for ICP and ICP* defined in Lemma
1 (a). 34 To satisfy the sufficient condition for ICP and ICP*, note that the lengths of tariff war
phases (thus, δ, δ*, and δS) will change in responses to changes in l and l*, which in turn affects
VC and VC∗ . To make this analysis more tractable, I can rewrite IC in (22) into:(23)

V − VN}IC : ∂u(l, l ∗ ) / ∂τ = δ C [∂Pr ∗ (l )][1 − Pr ∗ (l )]{[u(l, l ∗ ) − u(l , h ∗ )] + [u ∗ (h ∗ , l ) − u ∗ (l ∗ , l )] C∗
VC − VC∗
with

VC − VN[u(l , l ∗ ) − u(h, h ∗ )] + Pr(l ∗ )[u(h, l ∗ ) − u(l , l ∗ )] − Pr ∗ (l )[u(l , l ∗ ) − u(l , h ∗ )]=.
VC∗ − VN∗ [u ∗ (l ∗ , l ) − u ∗ (h ∗ , h)] + Pr ∗ (l )[u ∗ (h ∗ , l ) − u ∗ (l ∗ , l )] − Pr(l ∗ )[u ∗ (l ∗ , l ) − u(l ∗ , h)]I can now illustrate the effect of an increase in the size of H relative to F (σ > 1) byanalyzing its effect on IC in (23) and on a corresponding expression for IC*. First, note that ahigher σ leads to a higher value for the left-hand side of the equality in (23) because ∂2u(l,l*)/∂l∂σ > 0, implying a higher incentive for H to raise its protection level above thecooperative one. This reflects H’s enhanced ability to change the terms of trade in its favor by

33

Bagwell and Staiger (Section 6.2.1, 2002) provide a more detailed discussion on Article XIX and its amendment.
Given that each country sets its protection level on its imports (equivalent to a specific tariff), denoted by τ for
H and τ* for F, the partial equilibrium model described above yields each country’s one-period payoff as afunction of σ as well as τ and τ*. For analytical simplicity, I assume that uncertainties in the economy are suchthat one-period payoff functions under uncertainties are identical to those under certainty.
34

35

raising its protection level as it gets larger relative to F.35 An increase in σ also makes thevalue of the right hand side of the equality in (23) smaller, representing a decrease in the costassociated with raising the current protection level for H. Two factors contribute to thisreduction in the cost of raising the protection level. One of them is F’s reduced ability todecrease H’s one-period payoff by initiating an explicit tariff war, reflected by ∂[u(l, l*) − u(l,h*)]/∂σ < 0. As F gets smaller relative to H, its static optimal protection will entail a smallerchange in the terms of trade, thus a smaller decrease in H’s one-period payoff in the period
when F initiates a tariff war phase. The other factor comes from a decrease in the damage that
H needs to endure during the tariff war that follows F’s imposition of its static optimal tariff.
As H gets larger, the damage that H needs to endure during the tariff war that F initiates,
represented by [u*(h*, l) – u*(l*, l)](VC – VN)/(VC* – VN*) on the right-hand side of the equality in(23), decreases because ∂[u*(h*, l) – u*(l*, l)]/∂σ < 0 and ∂[(VC – VN)/(VC* – VN*)]/∂σ < 0. The
level of such damage depends on both the length of a tariff war phase that F initiates, δC – δ*,and H’s loss of its expected discounted payoff that it could have earned in a cooperative period,
VC – VN. Note that (δC – δ*) = [u*(h*, l) – u*(l*, l)]/(VC* – VN*) decreases with a lower value foru*(h*, l) – u*(l*, l) and with a higher value for (VC* – VN*). As F gets smaller relative to H, F
can obtain a less increase in its one-period payoff from imposing its static optimal tariff (alower value for u*(h*, l) – u*(l*, l)) while it can attain more gains from trading with a bigger H
in a cooperative period (a higher value for VC* – VN*). This implies that the length of a tariff
war phase that F can employ against H’s potential deviation gets shorter with a higher σ. In
addition, H’s loss of its expected discounted payoff that it could have earned in a cooperative
period, VC – VN, gets smaller as H can obtain smaller gains from trade with a smaller F. 36
In summary, an increase in σ induces H to have a higher incentive to raise its protectionlevel in a cooperative period (a larger value for the left-hand side of the equality in (23)) while35

It is well known in the trade literature that a country’s ability to change the terms of trade in its favor by
imposing tariffs strengthens as it gets relatively larger than its trading partner. Kennan and Riezman (1988),
McLaren (1997), and Park (2000) analyze how such an asymmetry in countries’ ability to influence the terms oftrade affects the cooperation attainable among asymmetric countries.
36
In a bilateral trade relationship, the terms of trade under free trade gets closer to a larger country’s autarky priceratio as the larger country become larger relative to the smaller one. This implies that the larger country canrealize smaller gains from the bilateral trade as it gets relatively larger and the reverse is true for the smaller
country. This discussion about the size of gains from free trade does not directly translate into a higher value for
VC* – VN* and a lower value for VC – VN under PTS because the changes in the lengths of tariff war phases may

36the cost associated with such a deviatory behavior gets smaller for H (a smaller value for theright-hand side of the equality in (23)). This leads to an overall increase in H’s incentive toraise its protection level under PTS. One can apply the same type of logic to explain how ahigher σ lowers F’s incentive to raise its protection level under PTS. These change in IC andIC* together imply that a larger country (H) ends up using more concealed protection than asmaller country under PTS and Proposition 4 summarizes these findings as follows:
Proposition 4.
If there is an increase in H’s market size relative to F’s with their trigger control variables, ωDand ωD*, being fixed and identical with each other, then the cooperative protection levels that H
and F sustain under PTS change into a direction where l gets higher and l* gets lower. A
decrease (increase) in the length of a tariff war phase that F (H) can initiate against H’s (F’s)
potential deviations under PTS, as well as an increase (decrease) in H’s (F’s) ability to change
the terms of trade through protection, contribute to this change. (See Appendix B for Proof)
According to Proposition 4, the asymmetry among countries may severely limit the small
country’s ability to restrain the large country’s use of concealed protection under PTS. In fact,
the U.S. is the only country in the world that has the legislation, Section 301, authorizing its
government to invoke a tariff war based on its own unilateral judgment of potential deviations
from trade agreements. In the above analysis, I assume that H and F do not change their trigger
control variables (ωD = ωD* being fixed) when the asymmetry in their size affects thecooperative protection levels they sustain under PTS. This implies that the smaller country F
will initiate tariff wars more often than the larger country H as Pr*(l) > Pr(l*) with l > l* and ωD= ωD*. However, this result may change if H and F change their trigger control variables in
response to an increase in the size of H relative to F. If H raises ωD and F lowers ωD* inresponse to a higher σ, it is possible to have a case where F initiates tariff wars less often thanH even when H uses concealed protection more intensely than F. 37 In the presence of analso affect VC* and VC. However, ∂{[u*(h*, l) – u*(l*, l)](VC – VN)/(VC* – VN*)}/∂σ < 0 ensures the damage that H
needs to endure during the tariff war that F initiates gets smaller under PTS as H gets larger relative to F.
37
When one considers the case with asymmetric countries, defining “optimal” PTS regarding the choice over thetrigger control variables becomes a less obvious matter because maximizing the simple sum of the two countries’
expected discounted payoffs under PTS may entail a gain for one country at the expense of the other. This opens
up the question of how each country will change its trigger control variable in response to an increase in the37

alternative way to enforce international trade agreements, such as dispute settlement procedureof the WTO, small countries may choose not to rely on trigger strategies based on unilateral(and private) judgments of deviations.

5. 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. As shown by the analysis of PTS in
section 4, the effectiveness of enforcement mechanism based on unilateral judgments ofpotential deviations can be severely limited because the private nature of such judgments may
undermine the credibility of strong punitive actions.

This understanding provides a newperspective in interpreting the role that the WTO may play in enforcing international trade
agreements: it generates (possibly imperfect) public signals about potential deviations fromagreements of which countries may have different opinions, which in turn enhances thecredibility of punitive actions against such deviations.38 For example, the WTO mandates aregular review on its members under the Trade Policy Review Mechanism (TPRM), generating“public” reports which consist of detailed chapters examining the trade policies and practices ofthe members. 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.”
To illustrate how and by what degree that the existence of a third-party opinion of potential
deviations, such as the opinion of the WTO, may help countries to enforce international trade
agreements, this section analyzes third-party trigger strategies under which a third-party, suchas the WTO, decides upon whether a (potential) deviation has occurred and tells each country
to initiate a punishment phase based on its decision. To make the direct comparison between
the third-party trigger strategies with (optimal) symmetric PTS characterized in section 3, I
make the following assumptions in this section.
asymmetry among countries, possibly to maximize its expected discounted payoff under PTS. While this potential
game between H and F of setting trigger control variables is an interesting issue, it remains to be analyzed,
possibly in a future work.
38
Note that countries can employ a tariff war phase of any length under trigger strategies where a public signal(which is correlated with concealed protection levels) triggers tariff wars.

38

The stage-game payoffs and action variables of H and F are the same as those described in
Section 2. In addition to these two players, there exists a third-party, namely the WTO, whoreceives its own (private) signals of each country’s potential deviations at the end of period t,
denoting it by ω t ∈ Ω regarding potential deviations of F and ω t∗ ∈ Ω ∗ regarding potential
deviations of H with Pr(ω ∈ΩD) and Pr(ω* ∈ΩD*) taking the same values as those in section 2.
This assumption implies that the WTO has no informational advantage over H and F withregard to detecting potential deviations. 39 Once again ΩD (ΩD*) denote the range of private
signals that triggers H (F) to initiate a punishment phase by imposing an explicit tariff, but it is
the WTO that tell each country to initiate such a punishment phase in third-party trigger
strategies. The infinitely repeated protection-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 tariff based on its own private signals of potential deviations.
For simplicity, I denote the WTO’s decision to tell H to initiate a punishment phase in period t
based on its signals in period t by μt-1 ∈ Μ ≡ {1, 0}, with μt-1 being 1 iff ω t ∈ΩD, 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, with

(24)s W (t ) : A t −1 × Μ t −1 × Μ ∗

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 WTO’s decision of telling H and F
to initiate a punishment phase up to period t, having i = * or none. Note that the strategies
defined in (24) allow each country to observe the WTO’s decision for the other country toinitiate a punishment phase only afterwards. This strategy specification under which eachcountry choose its current action without knowing the WTO’s current decision on the othercountry’s initiation of a punishment phase may seem unnatural. This specification, however,enables a direct comparison between the third-party-trigger strategies and the private trigger
39

This assumption on the WTO’s information of each government’s potential deviations does not necessarily
imply that its signals are perfectly correlated with those of each government’s private signals of the othergovernment’s potential deviations. What is crucial is that the quality of information that the WTO uses indetecting potential deviations is the same as that of each government’s so that any improvement of third-party
trigger strategies over the PTS does not come from the WTO’s informational advantage. I indicate that the WTO’s
information can be potentially private by putting “(private)” in front of “signals” in this sentence. The WTO canmake these private signals (or at least its decision on whether any deviation has occurs based on such signals) to be
public by announcing its signals or its decision. Different from governments, I assume that the WTO does nothave any incentive to lie about its information.

39

strategies by making these two types of strategies only differ in their ability in selecting thelengths of punishment phases. The following analysis will focus on the symmetric third-party
trigger strategies for symmetric countries 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 ) = (lc, 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 ω t −1 ∈ Ω D = Ω D and it∗

tells F to initiate a punishment phase by setting (τ t∗ , et∗ ) = (h, 0) iff ω t∗−1 ∈ Ω D = Ω D .
(iii) Given that a “tariff war phase” was initiated in period t − 1 by only one country, thecountries set (τ , e) = (h, h) and (τ ∗ , e ∗ ) = (h, h) for the following (T− 2) periods and it
continues to do so one more period with probability λ. Given that a “tariff war phase” wasinitiated in period t − 1 simultaneously by both countries, the countries set (τ , e) = (h, h)and (τ ∗ , e ∗ ) = (h, h) for the following (TS − 2) periods and they continues to do so one more
period with probability λS. T and TS are integer numbers that are greater than or equal to 2
with λ and λS belonging to [0, 1].
(iv) In period 1 and other “initial” periods right after the end of any tariff war phase, eachcountry sets (τ i , e i ) = (lc, 0) with probability (1 − Pri) but initiates a tariff war by settingi(τ i , e i ) = (h , h ) with probability Pr , where Pr = Pr(ωt ∈ Ω D ) with (τt, et) = (l , 0), (τt , et )
i

i

i

i

i

c

*

*

= (lc, 0), and i = * or none.
Definition 4. If (i), (ii), (iii), (iv) describe (sW, sW*), then, (sW, sW*) are symmetric third-party
trigger strategies (TTS) with (lc, 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 (lc,ΩD, T, TS, λ λS), denoted by VW(sW, sW*), is identical V(s, s*) in (6) as follows:

40

W

W

V (s , s
(25)
+

W*

(1 − PrPr ∗ )[u (l c , l c ) − u (h, h)])=
1 − δ C + Pr (1 − Pr ∗ )(δ C − δ ) + Pr ∗ (1 − Pr )(δ C − δ ) + PrPr ∗ (δ C − δ S )Pr ∗ (1 − Pr )[u (l c , h) − u (l c , l c )] + Pr (1 − Pr ∗ )[u (h, l c ) − u (l c , l c )]u (h, h)
+
,
∗
∗
∗
C
C
C
C
S
1 − δ + Pr (1 − Pr )(δ − δ ) + Pr (1 − Pr )(δ − δ ) + PrPr (δ − δ ) 1 − δ Ckkwhere δ k = λk (δ C ) T + (1 − λk )(δ C ) T −1 with k = s, or none, having (δ C − δ ) and (δ C − δ S )respectively represent the relative length of the tariff war phase initiated by one country and byboth countries simultaneously.
While the expression for the expected discounted payoff is same under the symmetric TTS
defined above and the symmetric PPT defined earlier in Definition 2 (with l = l* = lc, ΩD = ΩD*,

δ = δ*), there exists an important distinction between these two types of trigger strategies: theWTO does not have any incentive to lie about its private signals so that the TTS is not subject
to the ICP. This implies that one can choose any values for the lengths of punishment phaseswith (δ C − δ ) and (δ C − δ S ) ∈ [0, δ C ]. Note that the sufficient condition for satisfying the
ICP is δ C − δ = [u(h, lc) − u(lc, lc)]/(VC − VN) and δ C + δ S = 2δ . To make the comparison
between the TTS and the PTS even simpler, I continue to assume that δ C + δ S = 2δ (which
eliminates the incentive to use the initiation of punishment phases as a part of a deviation path)holds under the symmetric TTS, thus allowing full flexibility only over the choice of (T, λ), thelength of the single-country initiated punishment phase. This simplifying assumption enables
one to tell whether the length of the punishment phase under the optimal symmetric PTS is too
short or too long (compared with the optimal symmetric TTS) by comparing the endogenously
determined value for (T, λ) under the PTS with the optimal choice of (T, λ) under the TTS.
Given this assumption of δC − δS = 2δ and Pr = Pr* by symmetry, one can further simplify
VW(sW, sW*) in (25) into VCW = VW(sW, sW*) = (1 − Pr)[u(lc, lc) − u(h, h)]/[1 − δC + 2Pr(δC− δ)] +VN with VN = u(h, h)/(1 − δC).
To be able to support the symmetric TTS as an equilibrium of the repeated protectionsetting game between H and F, the symmetric TTS need to satisfy the following incentive
constraint, denoted by ICW:(ICW) IW(lc) ≡ ∂u(lc, lc)/∂τ −{δC[∂Pr*(lc)/∂τ ][1−Pr*(lc)][u(lc,lc) – u(lc,h)+(δC–δ)(VCW – VN)]} = 0.
Note that the ICW is identical to the IC in (12) under the symmetric PTS as long as δ under the
TTS is the same as the one under the PTS. This equivalence results from constructing the41

symmetric TTS in the way that it only differs from the symmetric PTS in its flexibility to havethe single-country-initiated punishment phase to last in any length. The intuition behind thisequivalence between the incentive constraint for each country to follow the specified strategies
under the PTS and the one under the TTS is quite simple: each country chooses its cooperativeperiod protection level, knowing that raising the protection level increases the probability of apunishment phase being triggered in the same manner under both trigger strategies.40
In addition to the ICW, there is one more incentive constraint that the TTS needs to satisfy:
each country does not have any incentive not to follow the WTO’s decision on initiating apunishment phase of a given length. Because the WTO’s decision becomes public information
in a period lag, one may construct a (off-equilibrium-path) punishment strategy, such as apermanent Nash tariff war, against the behavior of not following the WTO’s decision on
triggering a punishment phase. Given that the expected discounted payoff under symmetric
TTS is strictly greater than the discounted payoff of playing the static Nash tariff war forever,its is easy (and quite standard in the literature) to show that no country has any incentive not tofollow the WTO’s decision as long as the discount factor (δC) is high enough. I assume that
this standard result is valid for the following analysis with a high enough value for δC.
For the analytical simplicity (and without loss of generality), one can represent a choice ofW(T, λ) by a real number TW ∈ [1, ∞) with δ C − (δ C ) T = (δ C − δ ) : TW = 1 (equivalent to thecase of T = 2 and λ = 0) represents the case where no-tariff period is followed by an initiationof a punishment phase (the shortest possible punishment phase) and TW → ∞ represents thecase where a permanent Nash tariff war is followed by an initiation of a punishment phase (thelongest possible punishment phase). Then, the problem of finding the optimal symmetric TTS
is solving the following maximization problem:(26)⎧⎪ [1 − Pr (l c )][u (l c , l c ) − u (h, h)]
⎫⎪
+
V
⎨
N
Max 1 − δ C + 2 Pr (l c )[δ C − (δ C )T W ] ⎬⎪ subject toω D and T W ∈[1,∞ ) ⎪⎩⎭W∂u (l c , l c )∂Pr (l c )=δC[1 − Pr (l c )]{u (l c , l c ) − u (h, h) + [δ C − (δ C ) T ](VCW − V N )},
∂τ
∂τ

40

Formally proving this equivalence, however, is not totally trivial because each government needs to follows thedecision of the WTO on triggering a punishment phase after any cooperative period under the TTS, which in turncomplicates the expression for the incentive constraint under the TTS. Despite this complication, one can showthat the resulting incentive constraint is the ICW above. I can provide a formal proof of this equivalence result
upon request.

42

where ωD represents a trigger control variable, defined in the same way as in section 3.2.
Because the problem of finding the optimal symmetric PTS in section 3.2 is to choose
only ωD to maximize the same payoff function as in (26) subject to the same incentive
compatibility condition, but with TW (or equivalently the corresponding T and λ) beingW

determined by δ C − (δ C ) T = [u(h, lc) − u(lc, lc)]/(VCW − VN), it is obvious that the optimal
symmetric TTS of solving the maximization problem in (26) will yield an expected discounted
payoff that is greater (or at least equal) that the one under the optimal symmetric PTS. The
question is how and by what degree the less-constrained optimal symmetric TTS will
outperform the optimal symmetric PTS. Because providing a full analytical answer to thisquestion is difficult (if not impossible) as I explain later, I will start with a numerical analysis
that shows some interesting characteristics of the optimal symmetric TTS.
To do a numerical analysis of comparing the optimal TTS with the optimal PTS, I use thesame partial equilibrium model of trade between H and F with σ = 1 (for symmetry ofcountries). In addition, I assume that Pr(lc) takes the following functional form, an adjusted
Rayleigh distribution:(27)

Pr (l c ) = Pr (l c ω D ; ρ , χ ) =1 − exp[−(ω D l c ) 2 /(2 ρ 2 ) − χω D ],where 1/ρ ∈ (0,∞) represents the sensitivity of the signal in detecting an increase in the level ofconcealed trade barriers and χ ∈ [0, ∞) is a parameter that determines the likelihood of apunishment phase being triggers (even in the absence of any protection) with ωD ∈ [0, ∞).
Given χ = 8, α1 − α1* = 3, β + B = 1 and δC = 0.9, then Figure 4 shows the outcome of thenumerical analysis, demonstrating how (a) the expected percentage payoff gains compared withplaying the static Nash tariff war forever, (VCW − VN)/VN and (VC − VN)/VN; (b) the cooperative
protection level, lc; (c) the length of a punishment phase, TW; (d) the probability of apunishment phase being triggered, Pr(lc), changes in response to a decrease in the sensitivity ofthe signal (a higher value for ρ). The bold lines represent variables for the optimal symmetric
TTS and the thin lines depict variables for the optimal symmetric PTS. Some outcomes of theanalysis are as expected (not surprising): (i) the expected payoff gains decrease as thesensitivity of the signal decreases and the optimal TTS always provides a higher gain than the
optimal PTS, having the bold line is always located higher than the thin line in Figure 4 (a),43

except when ρ = 0.0439 where the TTS and the PTS coincides; (ii) the cooperative protectionlevels rise as the sensitivity of the signal decreases.
The numerical analysis shown in Figure 4 also reveals some interesting (at first, evensurprising) characteristics of the optimal symmetric TTS. The most notable aspect is that the
optimal symmetric TTS may require TW =1, thus no tariff-war period being followed by aninitiation of a punishment phase, when the signals are sensitive enough against concealed
protection with ρ < 0.033. Having TW =1 (or equivalently, T = 2 and λ = 0) never occur under
the PTS because it will surely violate the ICP because each country would always prefer
initiating a punishment phase to not initiating one with no costly tariff-war period(s) following
the action of setting the protection level to be equal to the static optimal protection level. Whenthe signals are relatively sensitive to an increase in concealed trade barriers, the optimal
symmetric PTS can be inferior to the optimal symmetric TTS because it requires a punishmentthat is too strong! 41
Analyzing the first order conditions of the maximization problem for the optimal
symmetric TTS in (26) can provide some insight of the factors that drive this result of no-tariffwar punishment:∂VCW / ∂l c ⎞ ∂I w (l c ) ∂VCW ∂Pr (l c )∂VCW ∂l c∂VCW ∂Pr (l c ) ⎛⎜⎟= 0,= ⎜− W c
+
=
+
c ⎟
D
∂Pr ∂ω D∂Pr ∂ω D
(
)
/
∂
∂
dω D
∂l c ∂ω D
I
l
l
⎝
⎠ ∂ω

dVCW
dVCW
dT

=
w

∂VCW ∂l c∂VCW ⎛∂VCW / ∂l c ⎞ ∂I W (l c ) ∂VCW⎜−⎟+=
+ W =0∂l c ∂T w ∂T W ⎜⎝ ∂I W (l c ) / ∂l c ⎟⎠ ∂T W
∂T

with

(28)

W

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

W

=

2 ln(δ C )(δ C ) T Pr (1 − Pr )[u (l c , l c ) − u (h, h)]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

where IW(lc) represents the implicit function defining the ICW above, ∂lc/∂ωD = − (∂IW/∂ωD)/(∂IW/∂lc) and ∂lc/∂TW = − (∂IW/∂TW)/(∂IW/∂lc) generate the second equality for dVCW/dωD and
41

This result also contrasts to the limitations of the PTS discussed in section 4 where the limitations of the PTS
largely come from a weak punishment rather than a strong one.

44

dVCW/dTW, respectively, and the expression after the second equality for ∂VCW/∂TW is obtainedusing the expression for ∂VCW/∂Pr in (28) 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 costly punishment period being longer, implying that the optimal
choice of TW also involves balancing between these counter acting forces.
Then, how one can explain the corner solution of choosing TW = 1 in Figure 4 (a) for thecase where the signals are sensitive with ρ < 0.033. Using the fact that dVCW/dωD = 0 holds
under the optimal symmetric TTS together with the second expression for ∂VCW/∂TW in (28), I
can rewrite dVCW/dTW as follows:⎛ ∂VCW ⎞
⎜
⎟⎡ W
−1
W
c
∂I
2 ln(δ C )(δ C ) T Pr (1 − Pr ) ⎛ ∂Pr (l c ) ⎞ ∂I W ⎤
⎜
⎟
∂
l
⎢
⎥
⎜
⎟
= ⎜− W ⎟+⎜ ∂ω D ⎟ ∂ω D ⎥ withCC
C TW
⎢ ∂T W∂IdT
1 − δ + 2[δ − (δ ) ] ⎝
⎠
⎦
⎜⎜
⎟⎣
c ⎟
⎝ ∂l ⎠(29)⎛ ∂VCW ⎞
⎜
⎟
−1
W
W
C
C TW
c
W
⎜ − ∂l c ⎟ < 0, ∂I < 0, ∂I < 0, and 2 ln(δ )(δ ) Pr (1 − Pr ) ⎛⎜ ∂Pr (l ) ⎞⎟ ∂I > 0.
W
D ⎟
D
⎜
⎜ ∂I W ⎟∂T W∂ω D
1 − δ C + 2[δ C − (δ C ) T ] ⎝ ∂ω
⎠ ∂ω
⎜⎜
⎟
c ⎟
⎝ ∂l ⎠dVCW
W

The above first order condition for the choice of TW, which also embodies the first ordercondition for the choice of ωD, reveals potentially a competing nature of these two choicevariables in restraining the use of concealed trade barriers. ∂IW/∂TW < 0 and ∂IW/∂ωD < 0
represent the nature of these choice variables in relaxing the ICW, which in turn enablecountries to lower the cooperative protection level, lc. If the effectiveness of ωD in relaxing the
ICW dominates that of TW, then it is possible that the optimal symmetric TTS may only involve
using ωD to support cooperation, having TW take its minimum value, 1. In fact, the second termin the second bracket of the expression on the right side of the equality in (29) is greater than
the first term (∂IW/∂TW ) in absolute values, having dVCW/dTW < 0 for TW = 1 when ρ < 0.033.
Thus, one possible explanation for TW = 1 when ρ < 0.033 is that the relatively accurate signalsof concealed trade barriers makes raising the trigger control variable (ωD) a more effective

45

means to curtail the incentive to use the concealed trade barrier than strengthening thepunishment phase by making it longer, which in turn leads to TW = 1.
One can also use the same expression and logic in explaining the other corner solution ofsetting TW → ∞ when ρ > 0.04656: the relatively inaccurate signals of concealed trade barriers
makes strengthening the punishment phase by raising TW a more effective means to curtail the
incentive to use the concealed trade barrier than raising the trigger control variable (ωD), which
in turn leads to setting TW → ∞, having dVCW/dTW > 0 for TW→ ∞ when ρ > 0.04656. This
understanding of the corner solution with regard to the choice of the punishment length is also
helpful in explaining other parts of Figure 4. For example, the probability of a punishment
phase deceases in response to a rising ρ in Figure 4 (d) when ρ ∈ (0.033, 0.04656). Because
the cooperative protection level rises in response to a higher value for ρ as shown in Figure 4, it
must be the case that ωD decreases in response to an increase in ρ. When ωD no longer
dominates TW in its effectiveness in restraining the use of concealed barriers with ρ > 0.033,
then the WTO should strengthen the punishment phase by raising TW and down-playing the roleof ωD by lowering its value in response to an increase in ρ, if such an increase in ρ makes TW
relatively more effective means to curtailing the use of concealed trade barriers than ωD.
If one can theoretically establish that the variables on the right side of the equality in (29)change as the above discussion implies in response to a reduction in the sensitivity of signals(higher values for ρ), it would worth putting such a result into a proposition. Unfortunately,
establishing such a result has not been possible (even with the specific functional form assumed
for Pr in the numerical simulation). This is because the variables on the right side of theequality in (29) are highly non-linear in ρ and the optimal choices of ωD and TW (as well as theresulting lc) also vary in response to a change in ρ, further complicating the theoretical analysis.
Therefore, I summarize the above findings into the following observation
Observation (based on the numerical analysis of the optimal TTS and the optimal PTS).
(a) When the sensitivity of signals detecting the use of concealed trade barriers is high enough,then TW can be lower under the optimal symmetric TTS than under the optimal symmetric PTS,
possibly having TW = 1.

46

(b) When the sensitivity of signals detecting the use of concealed trade barriers is low enough,then TW can be higher under the optimal symmetric TTS than under the optimal symmetric PTS,
possibly having TW → ∞.
Note that TW = 1 in Observation (a) implies that the enforcement involves highly
asymmetric actions: only the government under valid suspicion of using concealed tradebarriers will be penalized and the other government is rewarded by having the chance of
imposing punitive tariffs.

This kind of asymmetric actions is often one of importantcharacteristics of optimal strategies of repeated games under various applications, such as in
Kandori of Matsushima (1998) and Compte (1998), because such asymmetry allow player toavoid actions with (at least heavy) dead-weight losses. Thus, TW = 1 under the optimal TTS is
no longer a surprising result after having a good understanding of the reason behind it.
Observation (i) provides a potentially useful new insight regarding the role that the WTO
may play in enforcing international trade agreement: the presence of an impartial third party
that can judge (possibly with some errors) whether deviation has occurred, such as the WTO,
may enable countries to shorten and possibly avoid a costly tariff war, that might have beennecessary to curtail the incentive to wrongly punish other countries in the absence of the WTO.
While Observation (ii) seemingly confirms another role that the WTO may play inenforcing the international trade agreement that the analysis in Section 4 already emphasized:
the WTO may enable governments to employ a longer punishment phase than they could
employ in the absence of the WTO, thereby strengthening cooperation by a stronger
punishment scheme. However, TW being higher under the optimal symmetric TTS than under
the optimal symmetric PTS in Observation (ii) is driven by a reason that is slightly different
from those in Section 4 where the introduction of the WTO would have strengthened thecooperation by lowering the cooperative protection levels. Figure 4 shows that lc is higher
under the optimal TTS than under the optimal PTS, a result that contracts the prediction basedon the notion of using stronger punishments for lower levels of cooperative protection. Asemphasized in the earlier discussion using (29), the presence of the WTO may enable countries
to substitute the less effective means of curtaining the use of concealed trade barriers, namely

ωD with a more effective one, TW. If one reduces ωD enough in response to an increase TW, then47

it is possible to have lc being raised and Pr being lowered at the same time (compared withthose under the PTS), as shown in Figure 4 for ρ > 0.0439.

6. Concluding Remarks
The analysis in this paper establishes that symmetric countries may restrain the use ofconcealed trade barriers with simple symmetric PTS (Private Trigger Strategies) if thesensitivity of their private signals rises in response to an increase in such barriers. This paper
shows that the payoff of any equilibrium symmetric PTS will the same as the one under simple
symmetric PTS as long as the initial punishment is triggered by a one-shot Nash tariff,
justifying its focus on simple PTS. The analysis reveals that it is not optimal to push down thecooperative protection level to its minimum level attainable under symmetric PTS due to thecost associated with increasing the probability of costly tariff wars. This paper identifies two
factors that may limit the effectiveness of PTS. One is a reduction in each country’s time lag inadjust its tariff protection level in response to the other country’s initiation of an explicit tariff
war and the other is asymmetry among countries. Both of these factors may limit the level ofcooperation attainable under PTS by reducing the lengths of tariff war phases that countries canemploy against potential deviations.
To illustrate how and by what degree that the existence of a third-party opinion of potential
deviations, such as the opinion of the WTO, may help countries to enforce international trade
agreements, this paper analyzes TTS (Third-party Trigger Strategies) under which a third-party,
such as the WTO, decides upon whether a (potential) deviation has occurred and tells eachcountry to initiate a punishment phase based on its decision.

A numerical analysis ofsymmetric TTS demonstrates that the optimal TTS may entail a punishment phase only with an
initial tariff (no tariff-war period following it) as well as the one with a permanent Nash tariffwar, depending on the sensitivity of the WTO’s signals of potential deviations. On the one
hand, the WTO may enable countries to shorten and possibly avoid a costly tariff war, whichmight have been necessary to curtail the incentive to wrongly punish other countries in theabsence of the WTO. On the other hand, the WTO may also enable countries to adopt a longer
punishment phase than the one under PTS, which in turn helps countries to attain higher levels

48

of expected payoffs. Publicizing the WTO’s judgment of potential deviations, as in the case of
TPRM (Trade Policy Review Mechanism), may enable countries to adopt punishment phasesof optimal lengths, which in turn raises countries’ expected payoffs under international trade
agreements.
Another crucial role that the WTO plays in enforcing trade agreements is settling disputes
through its Dispute Settlement Procedure (DSP). 42

While enforcing trade agreements

ultimately relies on the threat of invoking trade sanctions against violations, the DSP of the
WTO contributes to the enforcement mechanism by generating third-party rulings on disputed
cases. If countries have perfect information about violations, such third-party rulings would
play no role in enforcing the agreements. If countries may form different opinions of potential
deviations, then publicizing the opinion of a third-party on disputed cases plays the role ofgenerating public signals about potential deviations. As shown in this paper, this publicizing a
supposedly impartial opinion may enhance the credibility of punitive actions against suchdeviations. However, the DSP of the WTO encourages settlements through consultations
among disputing parties as a preferred way to settle 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.” 43 This indicates that the DSP plays a role that goesbeyond simply generating public signals of potential deviations. Carefully analyzing the rolethat the DSP plays in the WTO regime, therefore would be a meaningful extension of this paper.

42

When countries bring a dispute case to the WTO presenting different opinions about potential deviations, theDSP first encourage them to solve disputes through a consultation stage prior to initiating a panel stage where athird-party panel provides a ruling on the disputed case. Countries can appeal the panel’s ruling to have the caseexamined by an Appellate Body. Once the case has been determined by the Appellate Body, then the losing“defendant” must comply with the ruling or face the possibility of trade sanctions by the complaining side.
43
This quote comes from the following website: http://www.wto.org/english/thewto_e/whatis_e/tif_e/disp1_e.htm.

49

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51

Appendix A
Proof for Lemma 1 (b)If Pr = Pr* = 0, then δ = δC − (1 − δC) [u(h, l*) − u(l, l*)]/[u(l, l*) − u(h, h*)] and δ* = δC − (1 − δC)[u*(h*, l) − u*(l*, l)]/[u*(l*, l) − u*(h, h*)] with such δ, δ*, and δS = δ + δ* − δC all belonging to (0, δC) if

δC is close enough to 1. This proves (b) for the case of Pr = Pr* = 0 with δC being close enough to 1.
Now, it remains to prove (b) for the case that Pr and Pr* are close to 0 with δC being close enoughto 1.

To prove this, I first rewrite the sufficient condition in (a) as δ = δ C − (1 − δ C )k ,

δ ∗ = δ C − (1 − δ C )k ∗ , and δ S = (δ ∗ + δ ) − δ C , where k = [u(h, l*) − u(l, l*)]/[(1 − δC)VC − u(h, h*)] andk* = [u*(h*, l) − u*(l*, l)]/[(1 − δC)VC* − u*(h*, h)]. Note that ∂k/∂δ = − Pr(1 − Pr*)C, ∂k/∂δ* = − Pr*(1 −Pr)C, ∂k/∂δS = − PrPr*C, ∂k*/∂δ = − Pr(1 − Pr*)C*, ∂k*/∂δ* = −Pr*(1 − Pr)C*, and ∂k*/∂δS = −PrPr*C*
with C = (1 − δC)[u(h, l*) − u(l, l*)]{(1 − PrPr*)[u(l, l*) − u(h, h*)] + Pr(1 − Pr*)[u(h, l*) − u(l, l*)] +Pr*(1 − Pr)[u(l, h*) − u(l, l*)]}/{[(1 − δC)V − u(h, h*)][1 − δC + Pr(1 − Pr*)(δC − δ) + Pr*(1 − Pr)(δC − δ*)+ PrPr*(δC − δS)]}2> 0, and C* = (1 − δC)[u*(h*, l) − u*(l*, l)]{(1 − PrPr*)[u*(l*, l) − u*(h*, h)] + Pr(1 −Pr*)[u*(l*, h) − u*(l*, l)] + Pr*(1 − Pr)[u*(h*, l) − u*(l*, l)]}/{[(1 − δC)V* − u*(h*, h)][1 − δC + Pr(1 −Pr*)(δC − δ) + Pr*(1 − Pr)(δC − δ*) + PrPr*(δC − δS)]}2> 0. Let δ (δ*) to be the implicit function from δ− δC + (1 − δC)k = 0 and δ*(δ) to be the implicit function from δ* − δC + (1 − δC)k* = 0 with δS = δ + δ*−δC. Then, ∂δ(δ*)/∂δ* = [Pr*(1− Pr)(1 − δC)C]/[1 − Pr(1− Pr*)(1 − δC)C] and ∂δ*(δ)/∂δ = [Pr(1− Pr*)(1 −δC)C*]/[1 − Pr*(1− Pr)(1 − δC)C*]. Note that ∂δ(δ*)/∂δ* ∈ (0,1) and ∂δ*(δ)/∂δ ∈ (0,1) with δ (δ* = 0) ≈
δC − (1 − δC) [u(h, l*) − u(l, l*)]/[u(l, l*) − u(h, h*)] > 0 and δ*(δ = 0) ≈ δC − (1 − δC)[u*(h*, l) − u*(l*,l)]/[u*(l*, l) − u*(h, h*)] > 0 for Pr and Pr* being close enough to 0 and δC is close enough to 1. Because∂δ(δ*)/∂δ* → 0 and ∂δ*(δ)/∂δ → 0 as δC → 1 with δ (δ* = 0) ≈ (<) δC and δ*(δ = 0) ≈ (<) δC, this implies

that there exists a unique (δ,δ*,δS) that satisfies δ = δ C − (1 − δ C ) k , δ * = δ C − (1 − δ C )k ∗ , andδ S = (δ ∗ + δ ) − δ C , with such δ, δ*, and δS all belonging to (0, δC) as δC is close enough to 1.
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.

52

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
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 and
n→∞

(

)

Lucas holds when one replaces u ( x) = F ( x0 , x1 ) + δ C u ( x ) with u ( x) = F ( x0 , x1 ) + δ C 1 − Pr ∗ (x0 ) 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)}.
y∈[ 0 , h ]T satisfies Blackwell’s sufficient condition forcontraction 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,53

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):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– δ*VC} is bounded for all τ ∈ [0, h]. This implies that f(τ−1, τ) is absolutely continuous w.r.t. τ−1 for allτ ∈ [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 – δ*VC}.
(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 – δ*VC > 0 for all m ∈ [0, h], because ∂Pr*(m)/∂m > 0 by assumption. To
show that u(g(m), l*) + δCV(g(m)) – u(g(m), h*) – (δC – δ*)VN – δ*VC > 0 for all m ∈ [0, h] > 0, I firstestablish that the inequality 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 good as 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], and the first inequality comes from the fact that [1 – Pr*(g(m))][u(l, l*) + δCVC] + Pr*(g(m))[u(l, h*)+ (δC – δ*)VN + δ*VC] 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 + δ*VC] , 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 + δ*VC if g(m) ≤ l.
Now, I will show that u(g(m), l*) + δCV(g(m)) > u(g(m), h*) + (δC – δ*)VN + δ*VC if g(m) > l.
Define K ≡ u(g(m), h*) + (δC – δ*)VN + δ*VC. 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 inequality54represents a discounted expected payoff from playing a potentially suboptimal strategy of setting thecurrent 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 + δ*VC]}/{1 – δC[1 –Pr*(g(m))]}. Note that the last term has a positive sign because u(g(m), l*)/(1 – δC) > [u(g(m), h*) + (δC– δ*)VN + δ*VC] with u(g(m), l*)/(1 – δC) > VC as g(m) > l. This implies that u(g(m), l*) + δCV(g(m)) > K.
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 allτ −//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,

55

(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 ]from (A4),contradicting ∂[ 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
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 thatτ ∈ (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 + δ* VC]} ≈ 0 at τS = 0 from theassumption of ∂Pr*(τ)/∂τ ≈ 0 at τ = 0, F(τS, τS)/∂τ > 0 at τS = 0 implies that IC in (11) will not besatisfied at τS = 0. Now, define A(τS) ≡ u(τS, l*) + δCV(τS) – [u(τS, h*) + (δC – δ* )VN + δ* VC] 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 andA(τS) > 0 as shown in the proof for Lemma 3 (a). This implies that there exists a unique τS ∈ (0, h) suchthat ∂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).

56I can show that h ∉ G(h) by contradiction. First, assume that h = G(h), implying that V(h) =∑ ⎨(δ C ) d ⋅ ⎢Π [1 − Pr ∗ (τ i )]⎥ ⋅ F (τ d ,τ d +1 )⎬ with {τ d = h}∞d =0 .
∞ ⎧⎡ d −1⎤⎫d =0 ⎩⎣ i =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 + δ* VC]} − {Pr*(h)u(h, h*) + [1 – Pr*(h)]u(h, l*) + Pr*(h)[(δC – δ*) VN + δ* VC]}+ δC[1 – Pr*(h)][Pr*(h − ε) − Pr*(h)]{u(h, h*) – u(h, l*) + [(δC – δ*) VN + δ* VC]} − δC[Pr*(h) − Pr*(h −ε)]F(h, h)δC[1 − Pr*(h)]/{1 − δC[1 − Pr*(h)]}. lim[V A ( h) − V ( h)] /(ε − 0) = −δC[∂Pr*(h)/∂τ][1 –ε →0Pr*(h)]{(1 −δC)[u(h, h*) + (δC – δ*)VN + δ*VC] − u(h, l*)}/{1 − δC[1 − Pr*(h)]}> 0 where the lastinequality comes from ∂Pr*(h)/∂τ > 0 and u(h, l*)/(1 −δC) > u(h, h*) + (δC – δ*)VN + δ* VC as shown inLemma 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 initiating
an explicit tariff war 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 tariff war phases satisfy the sufficient condition for ICP and
ICP* in Lemma 1 (a). Once I prove this result, this implies that H cannot increase its discounted
expected payoff by initiating explicit tariff wars 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 ∗ )VN + δ ∗VC ] + [1 − Pr ∗ (l / )][u (l , l ∗ ) + δ CVC ] .

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 thetariff war phase is over. Denote the discounted expected payoff from taking this potentially suboptimal
action by D(l /), then(A6)

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

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

57

(A7)

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

Now, note that C(l /) − D(l /) = [u(l, l*) − u(h, l*)] + ( δ C − 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 the sufficient 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 payofffrom choosing an optimal protection sequence of not involving an initiation of a tariff war phase, this
implies that H cannot strictly increase its discounted payoff by initiating an explicit tariff war in a periodthat follows a cooperative period during which H sets its protection level at l / ≠ l = τS.
Proof for Proposition 1

With δ = δ *= δC − [u(h, l Sc ) − u( l Sc , l Sc )]/(VC − VN), and δS = δ + δ* − δ, note that setting τS = l Sc
satisfies IC in (11), thus l Sc 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 Sc is also such a protection level for
F. If l = l* = l Sc , then PTS satisfy ICP and ICP* as well as IC and 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 lc (< h) thatsatisfies I(lc) = 0 in (13)? For example, does the condition for Lemma 4 (a) guarantee the existence ofsuch lc? To address this issue, I show that the second term of I(lc) in (13), δC[∂Pr*(lc)/∂τ ][1 −Pr*(lc)][u(h, lc) – u(lc, h*)], representing H’s dynamic incentive to avoid a tariff war, may not necessarily
increase in lc 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 lc (< h) that satisfies I(lc) = 0, validating the insertion of an additional condition to guarantee
the existence of such lc in Proposition 1.

58

Appendix B: the Analysis of the Effect of Asymmetry among Countries on PTSGiven that each country sets its protection level on its imports (equivalent to a specific tariff),
denoted by τ for H and τ* for F, domestic prices are p 2 = p2∗ + τ and p1∗ = p1 + τ ∗ , and each country’s
one-period payoff function can be expressed aspij⎡ A/ B⎤jw (τ ,τ ) = ∑ ⎢ ∫ Di (u )du + ∫ X i j (u )du ⎥ + τ j Dmj ( pmj ) − X mj ( pmj )⎥⎦i =1, 2 ⎢ p j−α i j / β
⎣ i
j

(B1)

j

(

k

)

where j, k = * or none with j ≠ k and m = 1 (2) when j = * (none). The above trade model does notincorporate uncertainties in the economy.

For analytical simplicity, however, I assume that

uncertainties in the economy are such that one-period payoff functions under uncertainties defined by(3) in Section 2 are the same as those in (B1); u(τ, τ*) = w(τ, τ*) and u*(τ*, τ) = w*(τ*, τ). Then, I canshow that all the assumptions about derivatives of u and u* with respect to τ and τ* in Section 2 aresatisfied; ∂u/∂τ > 0 at τ = 0 and ∂u*/∂τ* > 0 at τ* = 0; ∂u/∂τ * < 0, ∂u*/∂τ < 0, ∂u/∂τ + ∂u*/∂τ < 0, and∂u/∂τ* + ∂u*/∂τ* < 0 for (τ ,τ*) that are not trade-prohibitive; ∂2u/∂τ2 < 0 and ∂2u*/∂τ*2 < 0; ∂2u/∂τ∂τ* = 0

and ∂2u*/∂τ∂τ* = 0.
To analyze the effect of asymmetry on PTS, I use Figure B, which shows the combinations of (l, l*)that satisfy IC and IC*, denoted by IC and IC*. First, I establish that IC and IC* are both positively
sloped as shown in Figure B with IC being flatter than 45 degree line and IC* being steeper than 45
degree line for the case where H and F are symmetric (σ = 1; ICσ =1 and ICσ∗ =1 in Figure B), then I
explain how an increase in the relative size of H affects IC and IC* (σ > 1; ICσ >1 and ICσ∗ >1 in FigureB), thus the cooperative protection levels sustainable under PTS.

Proof for Proposition 4

Proof for Proposition 4 is composed of two parts: (i) proving that IC and IC* in Figure B are
positively sloped with IC being flatter than 45 degree line and IC* being steeper than 45 degree line;(ii) proving that a higher σ shifts IC upwards and IC* to the left in Figure B. To save the space, thefollowing proof will only provide the derived values of the variables of which I need to know signsof their derivatives to prove Proposition 4 and the resulting signs of those derivatives, without
showing the corresponding derivation processes. The work showing those derivations is available
upon request.

59

Part (i):To prove that IC is positively sloped in Figure B, I use IC in (20). First, I sketch the process ofderiving
V −V
IC : ∂u(l , l ∗ ) / ∂τ = δ C [∂Pr∗ (l )][1 − Pr∗ (l )]{[u(l, l ∗ ) − u(l , h∗ )] + [u ∗ (h∗ , l ) − u ∗ (l ∗ , l )] C∗ N∗ }VC − VC

(20)

with

VC − VN[u(l, l ∗ ) − u(h, h∗ )] + Pr(l ∗ )[u(h, l ∗ ) − u(l , l ∗ )] − Pr∗ (l )[u(l , l ∗ ) − u(l , h∗ )],=VC∗ − VN∗ [u ∗ (l ∗ , l ) − u ∗ (h∗ , h)] + Pr∗ (l )[u ∗ (h∗ , l ) − u ∗ (l ∗ , l )] − Pr(l ∗ )[u ∗ (l ∗ , l ) − u(l ∗ , h)]from ICP and ICP* in (7) together with the sufficient condition for them to hold in Lemma 1 (a). From(δC – δ*) = [u*(h*, l) – u*(l*, l)]/(VC* – VN*), (δC – δ*)(VC – VN) = [u*(h*, l) – u*(l*, l)](VC – VN) /(VC* – VN*)
as shown in (20). To derive the expression for (VC – VN)/(VC* – VN*) in (20), I starts from (VC – VN) /(VC*– VN*) = [(δC – δ*)/(δC – δ)]{[u(h, l*) – u(l, l*)]/[u*(h*, l) – u*(l*, l)]}, implied by the sufficient condition
for them to hold in Lemma 1 (a). ICP in (7) implies that {u(l, l*) – u(h, h*) – Pr*(l)[u(l, l*) – u(l,h*)]}/{u(h, l*) – u(h, h*) – Pr*(l)[u(h, l*) – u(h, h*)]} = [(1– δC) + Pr*(l)(δC – δ*)]/[(1– δ) + Pr*(l)(δC –δ*)] and ICP* in (7) implies that {u*(l*, l) – u*(h*, l) – Pr(l*)[u*(l*, l) – u*(l*, h)]}/{u*(h*, l) – u*(h*, h) –Pr(l*)[u*(h*, l) – u*(h*, h)]} = [(1– δC) + Pr(l*)(δC – δ)]/[(1– δ*) + Pr(l*)(δC – δ)]. From these two
equalities, I can derive an expression for [(δC – δ*)/(δC – δ)], which in turn being plugged into (VC – VN)/(VC* – VN*) = [(δC – δ*)/(δC – δ)]{[u(h, l*) – u(l, l*)]/[u*(h*, l) – u*(l*, l)]} to generate the corresponding
expression in (20). While I omit the complete derivation process to save the space, it is available uponrequest.
Having derived (20), now I provide the closed-form solutions (in terms of parameters of the model)
for the variables in (20), from which one can derive the signs of derivatives of variables that are relevant
for the analysis to follow:σ[σ (α 2∗ − α 2 ) − ( 2σ + 1)( β + B )l ],2(1 + σ )σu (l , l ∗ ) − u (l , h ∗ ) =[( β + B )(l * + h ∗ ) + 2(α 1∗ − α 1 )](l ∗ − h ∗ ),2
2(1 + σ )σu ∗ (h ∗ , l ) − u ∗ (l ∗ , l ) = −[2(α 1 − α 1∗ ) − (2 + σ )( β + B)(l ∗ − h ∗ )](h ∗ − l ∗ )2(1 + σ ) 2∂u (l , l * ) / ∂τ =(B2)[u (l , l ∗ ) − u (h, h ∗ )] + Pr (l ∗ )[u (h, l ∗ ) − u (l , l ∗ )] − Pr ∗ (l )[u (l , l ∗ ) − u (l , h ∗ )]= [1 − Pr ∗ (l )]σ2(1 + σ )+ [1 − Pr (l ∗ )]2[( β + B)(l * + h ∗ ) + 2(α 1∗ − α 1 )](l ∗ − h ∗ )σ2(1 + σ )2[σ (α 2∗ − α 2 ) − (2σ + 1)( β + B )l )](l − h),60[u ∗ (l ∗ , l ) − u ∗ (h ∗ , h)] + Pr ∗ (l )[u ∗ (h ∗ , l ) − u ∗ (l ∗ , l )] − Pr (l ∗ )[u ∗ (l ∗ , l ) − u (l ∗ , h)]= −[1 − Pr ∗ (l )]+ [1 − Pr (l ∗ )]σ2(1 + σ )2

σ2(1 + σ )2[ 2(α 1 − α 1∗ ) − (2 + σ )( β + B )(l ∗ + h ∗ )](h ∗ − l * )[−2(α 2∗ − α 2 ) + ( β + B)(l + h))](l − h),σ[(α 1 − α 1∗ ) − (σ + 2)( β + B)l ∗ ],(1 + σ ) 2σu ∗ (l ∗ , l ) − u ∗ (l ∗ , h) =[−2(α 2∗ − α 2 ) + ( β + B )(l + h))](l − h), and2(1 + σ ) 2σu (h, l ∗ ) − u (l , l ∗ ) = −[σ (α 2∗ − α 2 ) − (2σ + 1)( β + B )l )](l − h),2(1 + σ ) 2∂u ∗ (l ∗ , l ) / ∂τ ∗ =where h = σ (α 2∗ − α 2 ) /[(2σ + 1)( β + B)] , h∗ = (α1 − α1∗ ) /[(2 + σ )( β + B )] , and the last three equations
provide closed-form solutions for the variables in IC* defined in the similar way as in (20).
Given the above expressions in (B2), it is easy to show that IC is positively sloped. Note that anincrease in l* will reduce the term on the right-hand side of the equality in (20) because ∂u(l, l*)/∂l* < 0,− ∂u*(l*, l)/∂l* < 0, and ∂[(VC − VN)/(VC* − VN*)]/∂l* < 0 with the last inequality can be checked bydifferentiating the corresponding expression in (B2), implying a reduced incentive for H keep its
protection level down. Also note that an increase in l will reduce the left-hand side of the equality in(20) as ∂2u(l, l*)/(∂l)2 < 0 and will increase the right-hand side of the equality in (20) because∂{δC[∂Pr*(l)/∂l][1 − Pr*(l)]}/∂l > 0, ∂[(VC − VN)/(VC* − VN*)]/∂l > 0. Therefore, dl dl ∗prove that dl dl ∗

IC ∗

IC

> 0 , and I can> 0 in a similar way.

Now, I need to prove that IC is sloped flatter than the 45 degree line and IC* is sloped steeper than
45 degree line as shown in Figure B. For σ = 1, I prove that IC is sloped flatter than the 45 degree line
once again using IC in (20). First note that IC cannot be sloped as the 45 degree line because there
exists a unique lc (= l = l*) that satisfies IC in (13), or equivalently IC in (20) with σ = 1, given theassumption of Proposition 1. For lc (= l = l*) > lσ =1 = lσ∗ =1 , one can easily check that IC in (20) is
violated because the right-hand side of the equality in (20) gets greater than the left-hand side. Notealso that a decrease in l will increase the left-hand side of the equality in (20) and will decrease theright-hand side of it as shown above. Therefore, the equality in (20) can be restored for l* (= lc = l) >lσ =1 = lσ∗ =1 , only by lowering l from l = l*, implying that IC should be sloped less than 45 degree line.
By symmetry, I can use the same argument for proving that IC* is sloped steeper than 45 degree line for

σ = 1. One can link this fact to the stability of the cooperative equilibrium under PTS in the following

61

way. Even when the cooperative protection combination, E σ =1 is perturbed by some shocks (possibly
random errors in setting protection levels), the relative slopes of IC and IC* ensure that countries’ selfcorrection incentives (to change its protection level back to its payoff-maximizing level) move theprotection combination back to E σ =1 .
Part (ii):

To prove that a higher σ shifts IC upwards in Figure B, I analyze how an increase in σ will affectIC in (20). Using (B2), I can show that ∂2u(l, l*)/∂l∂σ > 0, ∂[u(l, l*) – u(l, h*)]/∂σ < 0, ∂[u*(h*, l) – u*(l*,l)]/∂σ < 0, and ∂[(VC – VN)/(VC* – VN*)]/∂σ < 0, implying that an increase in σ will increase the lefthand side of the equality in (20) and will decrease the right-hand side of it. To satisfy IC in (20),therefore, an increase in σ requires an increase in l (which will decrease the left-hand side of theequality in (20) and will increase the right-hand side of it, as shown above) for any given level of l*.
This implies that a higher σ shifts IC upward in Figure B. I can prove that a higher σ shifts IC* to theleft in Figure B in a similar way.

ICσ∗ >1l

ICσ∗ =1

ICσ >1

σ↑
σ↑

Eσ >1

ICσ =1

lσ >1
lσ =1

Eσ =1

45 0

0

lσ∗ >1lσ∗ =1l∗Figure B. The Effect of Asymmetry among Countries on IC and IC*
62

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 τ63

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

64

δC[∂Pr*(lc)/∂τ ][1 − Pr*(lc)][u(h, lc) – u(lc, h*)]∂u(lc, lc)/∂τ

0

lmin

lmax

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

65

h

lc

Figure 4. A numerical analysis of optimal symmetric TTS and optimal symmetric PTS
for different values of ρ (rho) with χ = 8, α1 − α1* = 3, β + B = 1 (so, h = 1), and δC = 0.966