dispute settlement with second-order uncertainty: the case of...

Dispute Settlement with Second-OrderUncertainty: The Case of International Trade

Disputes

Mostafa BeshkarIndiana University

Jee-Hyeong Park∗

Seoul National University

October, 2017

Abstract

The literature on pretrial dispute settlement has studied the effect of first-order uncertainty on pre-trial settlement bargaining while assuming awayany uncertainty about higher-order beliefs. We propose a settlement bargain-ing model in which one player receives a private and noisy signal of anotherplayer’s private type, thereby generating second-order uncertainty. We findthat a private signal improves the efficiency of settlement bargaining. How-ever, if the noisy signal of types is publicly observable, thereby eliminatingthe second-order uncertainty, the informational value associated with the sig-nal of types completely disappears.

∗Beshkar: Assistant Professor of Economics, Indiana University ([email protected]). Park: Professor of Eco-

nomics, Seoul National University ([email protected]). For their useful comments, we would like to thank Treb Allen,

Pol Antras, Kyle Bagwell, Andrew Bernard, Eric Bond, Andrew Daughety, Taiji Furusawa, Jota Ishikawa, Gita Gopinath,

Jihong Lee, Marc Melitz, Hiroshi Mukunoki, Jennifer Reinganum, Joel Rodrigue, Kenneth Rogoff, Kathryn Spier, Christo-

pher Snyder, Robert Staiger, Alexander Wolitzky and the seminar participants at Dartmouth College, Harvard, Stanford,

Vanderbilt, the Asia Pacific Trade Seminar, and the Midwest Theory Conference. We also thank Jun-Tae Park and Sang Min

Lee for their research assistance. Park would like to thank the people at the economics department of Harvard for their

hospitality during his visit (2016-2017) as a Fulbright scholar, conducting this research. Park acknowledges the financial

support from National Research Foundation of Korea (NRF-2014-S1A3A2043505), Seoul National University Asia Center

(Asia Research Foundation Grant #SNUAC-2017-008), and ’Overseas Training Expenses for Humanities & Social Sciences’

through Seoul National University.

1

1 Introduction

Pre-trial dispute settlement has been often studied in the economics literature asa bargaining game under asymmetric information. The information asymmetryconsidered in the literature is limited to the uncertainty about the first-order be-liefs, while higher-order beliefs are assumed to be common knowledge. In partic-ular, it is commonly assumed that each disputing party possesses private informa-tion about its type (determining its belief of court outcomes) while the distributionof types is commonly known to both parties.1

Our objective in this paper is to propose a model of pre-trial dispute settlementbargaining under higher-order uncertainty. To this end, we analyze a settlementbargaining game in which one player receives a private and noisy signal of anotherplayer’s private type, thereby generating second-order uncertainty. In particular,we study a game of bilateral bargaining over actions in which (i) the outcome oflitigation depends on the private type of one the parties and (ii) the uninformedparty receives a noisy private signal about the informed party’s type.2

The Dispute Settlement Process (DSP) of the World Trade Organization is anexample of bargaining under asymmetric information with higher-order uncer-tainty. The obligations of an importing country under the WTO are contingent onits domestic political economy conditions, which are likely to be the private in-formation of the importing government. Other governments, however, could alsoconduct their own investigations and receive informative signals about the im-porting country’s political economy conditions. These signals, which are poten-tially the private information of the investigating governments, creates second-order uncertainty in the pre-trial dispute settlement game.

We model the pretrial settlement negotiation of a WTO trade dispute as a sig-naling game between a complaining government and a defending government

1There is a growing literature, including Bergemann and Morris (2009), Chen et al. (2017),Morris et al. (2016), that analyzes the effect of higher order beliefs and associated uncertainty ongames and mechanism design problems. The literature on pretrial settlement, however, has notexplored such an issue. For a comprehensive review of the literature on litigation and pretrialsettlement see Daughety and Reinganum (2017) and Spier (2007).

2Our study is distinct from the literature on two-sided private information. Schweizer (1989)assumes that each disputing party receives an independent signal about the probability of its suc-cess in the court. Daughety and Reinganum (1994) also propose a dispute settlement model withtwo-sided imperfect information under which the complainant is privately informed about theextent of damages incurred, and the defendant is privately informed about the likelihood of beingfound liable for damages in the court. In both of these studies, the distribution of types is commonknowledge and thus no second-order uncertainty exists.

2

with the following structure.3 At the beginning of the game the defendant ob-serves its type (i.e., facing either high or low protectionist pressure) and thecomplainant receives a noisy signal about the defendant’s type. The defendantthen proposes a settlement offer, which may be accepted or rejected by the com-plainant. In the case of rejection, the case is litigated, with the outcome of courtruling being uncertain: The defendant with high protectionist pressure would ex-pect a more favorable outcome than the one with low protectionist pressure butthe court outcome is still uncertain as the court ruling is based on its own noisysignal of the defendant’s protectionist pressure.

We follow the literature on trade agreements by assuming that intergovern-mental transfers are usually in the form of policy adjustments, such as a bilateralchange in the level of protection, rather than cash transfers. This assumption re-flects some realities about international relations. First, transferring cash amonggovernments involves both political and public finance costs, which reduces itsappeal as a compensation mechanism. Moreover, in response to violation of theirrights in an international agreement, governments normally seek compensationsthrough withdrawal of concessions previously granted to the defecting govern-ment.4 This self-help method of receiving compensation may reflect the fact thatthe defecting country, being a sovereign state, cannot be coerced to compensatethe injured government.5

This paper is closely related to the recent literature on dispute settlement inthe WTO. In particular, Beshkar (2010b, 2016), Park (2011), and Maggi and Staiger(2015, ming, 2017) study the role of the WTO as a public signaling device that re-veals some useful -–albeit imperfect— information about the type or action ofthe defending party.6 A question has been hovering over these studies: Howwould the value of the court as a public signaling device change if we consider

3We also model the settlement negotiations as a screening game under which the uninformedparty (i.e., the complainant in our model) makes a settlement offer. We find that in a screeninggame, the second-order uncertainty does not play any role. The best offer strategy of the com-plainant depends only on the probability that the defendant is a strong type conditional on itssignal, which is unaffected by whether its signal is private or public.

4In section 2, we discuss a case from the WTO in which a party withdrew its concessions inresponse to an initial violation by a trading partner.

5Bagwell and Staiger (2005) show that when cash transfer is possible, the governments couldimplement the first-best outcome by requiring a proper amount of cash transfers as a price forimposing a contingent tariff.

6Another related paper is Maggi and Staiger (2011) in which the DSB is modeled as an arbi-trator that interprets ambiguous obligations, fills gaps in the agreement, and modifies rigid obli-gations. See Park (2016) for a comprehensive review of the recent literature on trade disputes andsettlement.

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the ability of the uninformed parties to conduct their own investigations to obtainan independent signal about the state of the world or the other party’s actions?This question is particularly interesting given that the disputing parties are mostlikely better equipped than the WTO arbitrators to monitor and extract informa-tion about the private type or actions of each other. Our study advances this lineof research by shedding light on the impact of private monitoring conducted bythe disputing parties.

Our main finding is related to the role of uncertainty about second-order be-liefs on the likelihood and efficiency of settlement. Uncertainty about second-order beliefs depends on whether the noisy signal that the complaining party re-ceives is publicly observable or not. If the signal is public—implying no uncer-tainty in second-order beliefs—then the signal has no bearing on the equilibriumof the game. The noisy signal becomes useful (by way of reducing the likelihoodof litigation) if it is privately observed by the complaining party. This result isreminiscent of the anti-transparency result of Morris and Shin (2002), which willbe discussed below.

For a general intuition for this result, note the difference between litigationstrategy of the complaining party under private and public signals: If the signalis public, the defending country perceives the same risk of litigation regardlessof its type. Put differently, a public signal only changes the common prior of theparties and, thus, have no impact on the set of separating equilibria. In contrast,if the defending country does not observe the signal received by the complainingparty, then a low-type defending country faces a higher likelihood of litigationthan a high-type defending country. This is precisely why a private signal couldchange (and improve) the equilibrium while a public signal has no effect on theequilibrium.

Our anti-transparency (i.e., anti-publicization of signals) result on the settle-ment of trade disputes contrasts with Park (2011)’s pro-public-monitoring resulton the enforcement of international trade agreements. Using a repeated-gameframework with imperfect monitoring of the potential use of concealed trade bar-riers, Park (2011) demonstrates that publicizing the imperfect private signal ofpotential deviations may facilitate a higher level of cooperation by relaxing theincentive constraint associated with utilizing imperfect private signals in invok-ing punishments.7

7Using a repeated game framework with incomplete information of potentially persistent po-litical pressure for protection, Bagwell (2009) analyzes enforcement issues in trade agreements,demonstrating that a government facing a low political pressure may “pool” and apply its tariff atthe bound rate, which is inefficiently high for her.

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We find that a government with high protectionist pressure will propose tarifflevels that are higher than the Pareto-efficient protection levels. Choosing such aninefficient action combination is more costly for a weak type than for a strong one,which enables the strong type to signal its type through such an offer, generating afully separating equilibrium.8 The proposed tariff levels are possibly even higherthan the ones that the Dispute Settlement Body (DSB) of the WTO would recom-mend when it finds evidence in favor of the defending government. This findingprovides a new perspective on the observation that the DSB often rules againsta defending government, recommending reduction or removal of the contingentprotection.9 Our analysis suggests that even if the DSB finds that the prevailingsituations legitimizes an increase in protection, it would order a cut in the protec-tion level proposed by the defendant.

Our analysis also predicts that an improvement in the quality of the signal re-ceived by the complaining country will reduce the probability of litigation. In theextreme case where the complainant’s signal is completely accurate, pretrial bar-gaining always leads to a settlement. This theoretical finding is consistent withthe evidence provided by Ahn, Lee, and Park (2014) who find a positive correla-tion between a proxy for information asymmetry and rate of litigation. The factthat the rate of the WTO disputes have decreased over time may also reflect a re-duction in information asymmetry between the parties (i.e., an improved signal)after years of partnership.10

Outside the literature on disputes and settlement, our anti-transparency resultis related to that of Morris and Shin (2002) who show that an increase in the pre-cision of public information may generate a detrimental effect on the overall wel-fare of the participants in a coordination game when each participant has accessto private information. The public information in Morris and Shin (2002) servesas a coordination device among participants, creating the possibility of inducinga weight on the public information that is higher than the socially optimal level.In our signaling game of settlement bargaining, the public information eliminates

8As the receiver’s private signal of the sender’s type becomes increasingly accurate, the strongtype’s equilibrium offer will approach a Pareto efficient action combination.

9Sykes (2003) points out that the DSB has always ruled against the defending party in litigationregarding safeguard measures.

10The number of WTO dispute cases decreased from 335 during its first 10 years (1995-2005) to165 during the next 10 years (2006-2015). This decrease in the WTO disputes is even more surpris-ing once we consider the steady expansion of the WTO membership from to 123 countries in 1995to 162 countries in 2015, including major ones, such as China (2001) and Russia (2012). As trad-ing partners interact for a longer period, informational asymmetry between them will naturallydecrease.

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second-order uncertainty that enables the receiver to make its rejection threat con-tingent upon the information about fundamentals (i.e., the sender’s type), whichin turn completely eliminates its informational value.

Our result that a public signal has no impact on the equilibrium is related tothe analysis of Bagwell (1995). He shows that any level of noise in a follower’sobservation of a first mover’s action can induce the follower to completely ignoreits imperfect information in a pure strategy equilibrium, which in turn eliminatesthe first mover advantage.11 Public information without any noise will induce theplayers to utilize such information in our settlement bargaining game, eliminat-ing the need for inefficient litigation in the equilibrium. In contrast to the gameanalyzed by Bagwell (1995), the noisy information of the follower (settlement of-fer recipient) retains its informational value as long as it generates second-orderuncertainty to the first mover (settlement offer maker).

Applying mechanism design to international conflict resolution,Hörner, Morelli, and Squintani (2015) demonstrate how a mediator withoutenforcement power can replicate the welfare outcome of an optimal settlementmechanism that utilizes an arbitrator with enforcement power. Under theirmodel, the mediator overcomes its lack of enforcement power by choosing arecommendation strategy that does not reveal the type of a weak player to astrong player. This restrains the strong player’s incentive for fighting against theweak one. One could interpret the optimality of uncertainty in the mediator’srecommendation as optimality of second order uncertainty as we find in thispaper.

In terms of informational structure, Feinberg and Skrzypacz (2005) analyze asimilar bargaining game in which a seller has private information about his be-liefs about the buyer’s private valuation (i.e., type), thus entailing second-orderuncertainty. This second-order uncertainty creates a surprising result that delayin bargaining occurs even when a rational seller makes frequent offers to a ratio-nal buyer with common knowledge of gains from trade, thus generating a resultthat does not follow the “Coase property.”

Section 2 provides a brief description of the WTO’s dispute settlement sys-tem and a WTO dispute case. Section 3 describes the basic setup of our model.

11Maggi (1999) demonstrates that the strategic value of commitment (e.g., moving first) is re-stored even with imperfect observability of commitment when a leader has private type informa-tion. As the optimal leader action depends on her type, the follower has an incentive to utilizeeven its imperfect information of the leader’s action, restoring at least a certain value of commit-ment. Although our settlement bargaining game also analyzes the situation in which the firstmover (a defendant who makes a take-or-leave offer) has private type information, the follower’snoisy information is not about the first mover’s action but about her type.

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In Section 4, we analyze the pretrial settlement bargaining as a signaling gamewith private noisy signals. In Section 5, we analyze the signaling game underthe assumption that the noisy signal received by the complaining party is pub-licly observable. In Section 6, we analyzes the pretrial settlement bargaining as ascreening game. We provide some concluding remarks in Section 7.

2 WTO Dispute Settlement System and a Dispute

Case

WTO gives its member governments the right to raise their protection level con-tingent upon realizing a certain condition that warrants protection.12 With regardto the condition that justifies the need for protection, however, trading partnersmay disagree, thus, leading to a trade dispute. If disputing parties fail to settle,then the DSB of WTO provides its ruling on the disputed case and possibly au-thorizes a complainant’s compensatory/retaliatory protection against a defendantwho refuses to follow its ruling. When a defending government has private infor-mation of her condition, i.e., her type, and a complaining government receives anoisy private signal of the defendant’s type, then their trade dispute becomes thesettlement bargaining game with second-order uncertainty described above.

WTO underscores the rule of law by requiring unanimous voting of all mem-ber countries to overturn the recommendation of a third-party panel (and possiblythe report of the Appellate Body on the appealed recommendation of a panel). Bycontrast, WTO also strongly emphasizes settling disputes through consultationsas its priority. As stated in the WTO’s official website, “..., the point is not topass judgment. The priority is to settle disputes, through consultations if pos-sible. By January 2008, only about 136 of the nearly 369 cases had reached thefull panel process. Most of the rest have either been notified as settled ’out ofcourt’ or remain in a prolonged consultation phase - some since 1995.” Of theeconomic importance of trade disputes filed to the WTO dispute settlement pro-cedure, Bown and Reynolds (2015) find that the value of imported goods subject

12Various types of contingent protection are allowed under the WTO regime. They can be safe-guard measures, anti-dumping measures, countervailing duties against export subsidies, or evengeneral exceptions to General Agreement of Tariffs and Trade (GATT, the agreement prior to theWTO, that remains a central part of the WTO regime) obligations. These general exceptions aregiven based on two articles, Article XX to protect public morals, human, animal, or plant life orhealth, international intellectual property rights, and etc., and Article XXI to protect national secu-rity. Sykes (2016) provides a comprehensive description of these contingent protection measuresand other aspects of legal obligations of WTO.

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to the WTO disputes from 1995 to 2011 is almost $1 trillion, an average of $55 bil-lion per year, or equivalently about 0.5 percent of world imports in 2011. Giventhat only a small portion of trade disputes ends up being filed to WTO, a muchbigger percentage of world imports must be under trade disputes, implying thattrade disputes and settlements affect a sizable portion of the world trade.

For concrete understanding of trade disputes and settlements, we provide adetailed discussion of the WTO dispute case number 235. (For empirical analy-ses of the WTO dispute settlement process, see Beshkar and Majbouri 2016; Bown2004; Bown and Reynolds 2017.) Poland filed a consultation request on July 11,2001, claiming that Slovakia had imposed a safeguard measure on imports ofsugar in a manner inconsistent with the obligations under the Agreement on Safe-guards, with the following statement as one of her key claims in the consultationrequest document:

“No document presented, including notifications to the SG Committee underArticle 12, contained analyses on causal link between increased imports and serious in-jury to the domestic industry or factor other than imports which might have causedinjury.”

A safeguard measure is justifiable if a causal link between increased imports andserious injury to the domestic industry exists, upon which disputing parties maydisagree.13 With regard to the strength of this case, the defending governmentis likely to have some private information, such as assessment of the degree ofdamages to her domestic firms’ profitability caused by increased imports. Publicrevelation of such private information can be costly, as discussed in the introduc-tion. A complaining government may also have a noisy and private signal of thecausal link that is provided by her own domestic exporting firms to persuade herto file a dispute case to the WTO. As reported by the WTO, the outcome of thisdispute was a mutually agreed solution as follows:

“On 11 January 2002, the parties notified the DSB that they have reached amutually agreed solution within the meaning of Article 3.6 of the DSU. Accord-ingly, Slovakia agreed to a progressive increase of the level of its quota of importsof sugar from Poland between 2002 and 2004, and Poland agreed to remove itsquantitative restriction on imports of butter and margarine. Both parties agreedto implement the above by 1 January 2002.”

This report demonstrates that a settlement is achieved by exchanging desired

13Beshkar and Bond (2016) provide a literature review of studies on safeguard measures in theWTO and other trade agreements.

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trade policies among disputing parties. We may interpret this settlement as anexchange of temporary protection policies.

3 Basic Setup

To analyze a settlement bargaining game between a defendant (D) and a com-plaint (C) that occurs before proceeding to litigation with uncertain outcomes,this section describes general properties of payoff functions, distribution of D’stypes (θ), and C’s noisy signal of D′s types (θC).

Payoff FunctionsDefendant’s payoff function, WD((τ , r);θ), is increasing (decreasing) in τ (r) atτ = 0 (r = 0), and is concave in τ . Moreover, WD

τθ > 0.

Complainant’s payoff function, WC((τ , r)) is decreasing (increasing) in τ (r) atτ = 0 (r = 0), and is concave in r.The joint payoff, W J((τ , r);θ) = WD((τ , r);θ) + WC((τ , r)), is increasing in τ at

τ = 0 iff θ > 0. For θ = 0, W Jτ ((τ , r);θ) < 0. Moreover, W J

τθ > 0.

In the context of trade disputes, the above payoff functions reflect a situation inwhich D and C bilaterally trade goods in two competitive sectors, of which eachcountry has a comparative advantage in one of two sectors. Each government canimpose an import tariff to manipulate its terms of trade in its favor and/or to pro-tect her import-competing sector due to her domestic pressure for protection. Insuch a case, τ and r denote D’s and C’s import tariff, respectively. θ represents arandom domestic for protection that D faces, of which C receives a noisy signal,θC.14 WD

τθ > 0 reflects that that D’s incentive to raise τ increases as it is subjectto a higher pressure protection (a higher θ). Although raising τ is a costly way totransfer payoff from D to C (by changing the term of trade with some distortionary

14An increase in domestic pressure for protection may come from various random factors, suchas public concerns about safety of imported products and a sudden import increase that threatensinjury to domestic import-competing firms. In modeling such random pressure for protection,existing studies on contingent protection often assume, à la Baldwin (1987), that each governmentmaximizes a weighted sum of its producer surplus, consumer surplus, and tariff revenues, possi-bly with a higher weight on the surplus of its import-competing sector. Grossman and Helpman(1994) demonstrate that the higher weight given to the import-competing sector may be the resultof political pressure, through lobbying for example, that a government faces. θ may denote thepolitical weight on the import-competing sector. Although we utilize such a political economymodel in our numerical analysis of Section 4, θ can be any factor representing domestic pressurefor protection that may necessitate contingent protection, of which a foreign government a noisysignal, θC.

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losses) with W Jτ ((τ , r);θ = 0) < 0, W J

τθ > 0 reflects the desirability of allowinga higher import tariff in response to a higher pressure for protection, even fromthe joint-payoff point of view. The absence of non-costly transfers, which we as-sume here, is common in inter-governmental agreements or settlement. This non-transferable utility assumption naturally leads to incentives to settle a dispute dueto risk aversion against uncertain outcomes of litigation, which we assume below.

Regarding a possible protectionist pressure that may affect C’s import tariffchoice, r, we assume that it is constant and known across governments. This as-sumption enables us to focus on uncertain types of D as the source of informa-tional asymmetry across countries, which in turn may block governments fromchoosing collectively efficient actions. r may play a role in settlement bargainingbecause it enables C to affect D’s payoff as well as its own, thus possibly affectingthe settlement outcome, especially based on her noisy signal of D’s type.

Distribution of θ and the noisy signal received by Cθ can take one of two values, low (l) and high (h) with probability of 1 − ρ and ρ,respectively. Let tN(θ) and tE(θ) represent a Nash tariff pair

(

τN (θ) , rN (θ))

and

an efficient (joint-payoff-maximizing) tariff pair(

τE (θ) , rE (θ))

given θ, respec-

tively. We assume that τN (l) > τE (h).15 Dθ denotes D with θ ∈ {l, h}.C receives a noisy signal of θ, denoted by θC, which is accurate with a proba-

bility of γ, namely,

Pr(

θC = l |θ = l)

= Pr(

θC = h |θ = h)

= γ.

CθC denotes C with θC ∈ {l, h}.

DSBWe assume that the disputing parties can resort to arbitration by the DSB if theyfail to reach a mutually accepted solution in the consultation stage. We treat theDSB as a black-box that, if used, will result in outcomes that satisfy a set of con-ditions to be laid out below. We let Pθ denote the set of Pareto efficient tariffpairs, and Wi

L(θ) denotes the expected welfare of country i = {D, C} from litiga-

tion if the true state of the world is θ. Moreover, we define tminl , tmax

l ∈ Pl and

15This assumption simplifies the analysis by eliminating the possibility of tariff binding over-hang under an optimal agreement. For an analysis of tariff binding overhang, see Beshkar et al.(2015); Beshkar and Bond (2017).

10

tminh , tmax

h ∈ Ph such that

WD(

tminl ; l

)

≡ WDL (l) ,

WC (tmaxl ) ≡ WC

L (l) ,

WD(

tminh ; h

)

≡ WDL (h) ,

WC (tmaxh ) ≡ WC

L (h) .

We assume that the DSB rulings satisfy the following conditions:

1. C will strictly prefer an expected DSB ruling when θ = l to an expected DSBruling when θ = h. WC

L (l) > WCL (h) or

tmaxl ≻

Ctmaxh . (1)

2. D of any type will strictly prefer an expected court ruling when θ = h to anexpected court ruling when θ = l.

tminl ≺

Dh

tminh , (2)

tminl ≺

Dl

tminh .

3. Consider any incentive-compatible mechanism, t(θ) such that t(l) ≈Dl

t(h) in

the absence of C’s information of D’s type with γ = 0.5. The expected jointpayoff is greater under litigation than under any such mechanism. Namely,

(1 − ρ)[

WCL (l) + WD

L (l)]

+ ρ[

WCL (h) + WD

L (h)]

>

(1 − ρ)[

WC(t(l)) + WD(t(l); l)]

+ ρ[

WC(t(h)) + WD(t(h); h)]

. (3)

4. For any realization of θ, there are mutual gains from settlement in lieu oflitigation, that is,

(

tminl ≺

Dl

tmaxl

)

∧

(

tminl ≻

Ctmaxl

)

, (4)

and,(

tminh ≺

Dh

tmaxh

)

∧

(

tminh ≻

Ctmaxh

)

. (5)

11

Figure 1, in which TC(t) and TDθ(t) denote the indifference curves of C andDθ that cross t respectively, demonstrates the nature of settlement bargaining thatwe analyze. First, the efficient contractual arrangement is to assign t = tE(l)(=tE(h)) if θ = l (= h). However, such arrangement is not incentive compati-ble because Dl has an incentive to mimic Dh with WD(tE(h); l) > WD(tE(l); l).Figure 1 also demonstrates players’ incentives for settlement to avoid uncertainlitigation outcomes. If C knows that θ = h, then any settlement that belongs toConv

(

TDh(

tminh

)

, TC(

tmaxh

))

is preferred by both C and D, and similarly they pre-

fer settlement in Conv(

TDl(

tminl

)

, TC(

tmaxl

))

if C knows thatθ = l. Despite theseincentives for settlement, the equilibrium of a pretrial settlement game may entaila positive probability of litigation due to the existence of asymmetric information,as demonstrated in the following analysis.

4 Signaling Game with a Private Noisy Signal

This section, along with Section 5, analyzes the settlement bargaining betweenC and D prior to litigation as a signaling game by assuming that D proposes asettlement offer to C. Specifically, the sequence of events in the pretrial bargainingis as follows:

Sequence of Events

1. State of the world, θ, is realized and observed privately by D.

2. C receives a private noisy signal, θC, such that Pr(

θC = θ|θ)

= γ. D may nothave access to this information (private noisy signal) or may have (publicnoisy signal).

3. D proposes a tariff pair, tS ∈ (R+ , R+), for settlement.

4. C either accepts tS, in which case a settlement is achieved and tS is imple-mented, or rejects tSand the dispute escalates to the DSB.

A strategy for Dθ prescribes a probability distribution αθ(tS) over action tS for

each type θ ∈ {l, h}. A strategy for CθC prescribes the probability that C assignsfor litigation in response to the offered action tS of D, denoted by βθC(tS) ∈ [0, 1]with θC ∈ {l, h}. Given these notations for a strategy profile, we can define aPerfect Bayesian equilibrium (PBE) of this pretrial settlement game as follow:

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Definition 1. The strategy profile (αl (.) ,αh (.) ,βl (.) ,βh (.)) is a PBE iff

1. [Incentives of C j]

(a) If 0 < β j

(

tS)

< 1, then C j is indifferent between settlement at tS andlitigation.

0 < β j

(

tS)

< 1

⇒WC(

tS)

= Pr(

θ = l|θC = j, tS)

WCL (l) + Pr

(

θ = h|θC = j, tS)

WCL (h) .

(b) If β j

(

tS)

= 1(

β j

(

tS)

= 0)

, then C j at least weakly prefers litigation

(settlement at tS).

2. [Incentives of D j] For any tS for which 0 < α j

(

tS)

< 1, we should have

tS = arg maxt

([

γ(

1 −β j (t))

+ (1 − γ) (1 −βi (t))]

WD (t;θ = j)+

[

γβ j (t) + (1 − γ)βi (t)]

WDL ( j)

)

,

where i 6= j ∈ {l, h}.

3. [Consistency of beliefs] Ifαl

(

tS)

> 0 or αh

(

tS)

> 0, then

Pr(

θ = l|θC = l, tS)

=Pr

(

tS|θC = l,θ = l)

Pr(

θC = l,θ = l)

Pr (θC = l, tS)

=αl

(

tS)

γ (1 − ρ)

αl (tS)γ (1 − ρ) +αh (tS) (1 − γ)ρ

As in other signaling games, a set of PBEs of our pretrial bargaining game exist.We adopt Universal Divinity of Banks and Sobel (1987) as the refinement concept.

Definition 2. Denote Dh’s (Dl’s) expected equilibrium payoff by EWDh (EWDl ).Define Bh(t

′) and Bl(t′) as a subset of best-response litigation strategies,

(βl (t′) ,βh (t

′)) of C on D’s off-the-equilibrium settlement proposal, t′, that re-spectively satisfies the following inequalities:

[

γβh(t′) + (1 − γ)βl(t

′)]

WDL (h) +

{

γ[

1 −βh(t′)]

+ (1 − γ)[

1 −βl(t′)]}

W(

t′, h)

> EWDh ,[

(1 − γ)βh(t′) + γβl(t

′)]

WDL (l) +

[

(1 − γ)[

1 −βh(t′)]

+ γ[

1 −βh(t′)]]

W(

t′, l)

> EWDl ,

13

with B̄h(t′) and B̄l(t

′) defined as a subset of best-response strategies of C on D’sproposal, t′, that respectively satisfies the above inequalities with weak inequal-ities. For any off-the-equilibrium settlement proposal, t′, C believes that Dh (Dl)proposed t′ iff Bh(t

′) ⊃ B̄l(t′) (Bl(t

′) ⊃ B̄h(t′)). A PBE of our dispute settle-

ment game is universally divine iff it is supported by such belief of C about anyoff-equilibrium proposal.

Although universal divinity defined above is a stronger refinement conceptthan divinity, we will refer a universally divine equilibrium as a divine equilibriumfor simplicity.

In this section we assume that C receives a noisy private signal of D’s type(thus, C’s signal remains private). Section 4.1 proves the existence of a Divine PBEof the pretrial bargaining game, in which a fully separating behavior by D ariseswith Dh’s expected payoff being maximized. Then, Section 4.2 characterizes howthe divine PBE changes in response to a change in the accuracy of C’s signal, γ.

4.1 Divine PBE

Prior to characterizing a Divine PBE, the two following lemmas limit the possiblePBE strategies of D and C.

Lemma 1. Under any separating PBE, i.e., ∃tS with αl

(

tS)

> 0 and αh

(

tS)

= 0,

tS = tmaxl and βl

(

tmaxl

)

= βh

(

tmaxl

)

= 0.

Proof. See Appendix.

Lemma 2. Under any separating PBE, any tS( 6= tmaxl ) with αh

(

tS)

> 0 and a posi-

tive settlement probability (i.e., βl

(

tS)

or βh

(

tS)

< 1) belongs to Conv(

TDh(

tminh

)

,

TC(

tmaxh

))

.


Among all possible separating PBEs, Lemmas 1 and 2 enable us to focuson separating PBEs in which tS = tmax

l with αl

(

tmaxl

)

> 0, αh

(

tmaxl

)

= 0,

and βl

(

tmaxl

)

= βh

(

tmaxl

)

= 0, and tS( 6= tmaxl ) with αh

(

tS)

> 0 belongs to

Conv(

TDh(

tminh

)

, TC(

tmaxh

))

in Figure 1.16

16Although Lemma 2 does not rule out the possibility of having a separating PBE in which

14

In any separating PBE, Dl weakly prefers proposing tmaxl over proposing tS ∈

Conv(

TDh(

tminh

)

, TC(

tmaxh

))

with αh

(

tS)

> 0 according to Lemma 1; otherwise,

Dl engages in a pooling behavior with αl

(

tmaxl

)

= 0. Under a separating PBE, thefollowing incentive constraint of Dl should hold:

WD (tmaxl ; l) ≥

(6)[

γβl

(

tS)

+ (1 − γ)βh

(

tS)]

WDL (l) +

[

1 − γβl

(

tS)

− (1 − γ)βh

(

tS)]

WD(

tS; l)

.

In addition, βl

(

tS)

∈ [0, 1) implies βh

(

tS)

> 0, and βh

(

tS)

∈ (0, 1] implies

βl

(

tS)

= 1 in any PBE equilibrium.17 This relationship between βl

(

tS)

and

βh

(

tS)

comes from the following incentive conditions for C: for any tS with

αh

(

tS)

> 0, if Cl weakly prefers settlement over litigation, then Ch will prefersettlement; if Ch weakly prefers litigation over settlement, then Cl will prefer liti-gation.

A divine PBE is a separating equilibrium that maximizes D’s expected payoff,as shown later. The following lemma narrows down the set of tS( 6= tmax

l ) with

αh

(

tS)

> 0 under such a separating PBE:

Lemma 3. Under a separating PBE that maximizes Dh’s expected payoff, any tS( 6= tmaxl )

with αh

(

tS)

> 0 belongs to TC(

tmaxh

)

.


With Lemmas 2 and 3, finding tS( 6= tmaxl ) with αh

(

tS)

> 0 under a separatingPBE that maximizes the expected payoff of Dh is reduced to solving the following

tS /∈ Conv(

TDh(

tminh

)

, TC(

tmaxh

))

with αh

(

tS)

> 0 and a zero settlement probability (βl

(

tS)

=

βh

(

tS)

= 1), a separating PBE exists, yielding Dh’s expected payoff that is strictly higher than

WDL (h), as shown later. This implies that tS /∈ Conv

(

TDh(

tminh

)

, TC(

tmaxh

))

with αh

(

tS)

> 0 and

βl

(

tS)

= βh

(

tS)

= 1 will not be a part of a separating PBE strategy of Dh that maximizes itsexpected payoff. Among PBEs, we are interested in characterizing a Divine PBE, and it is the onethat maximizes Dh’s expected payoff among separating PBEs, as shown later.

17This statement is not valid for tS ∈ TC(

tmaxh

)

with αh

(

tS)

> 0 and αl

(

tS)

= 0 because C is

indifferent between litigation and settlement for such tSregardless of her private signal. However,there is no loss of generality in focusing on βl

(

tS)

and βh

(

tS)

that satisfy these conditions forour analysis of a divine PBE. As shown below, a Divine PBE is a PBE that maximizes the expectedpayoff of Dh, and such a Divine PBE needs to satisfy these conditions even for tS ∈ TC

(

tmaxh

)

with

αh

(

tS)

> 0 andαl

(

tS)

= 0.

15

maximization problem:

tS ∈Argmaxt∈TC(tmax

h )

⟨

{γ [1 −βh(t)] + (1 − γ) [1 −βl(t)]}WD (t; h) (7)

+ [γβh(t) + (1 − γ)βl(t)] WDL (h)

⟩

s.t.

WD (tmaxl ; l) = [γβl(t) + (1 − γ)βh(t)] WD

L (l) + [1 − γβl(t)− (1 − γ)βh(t)] WD (t; l)(8)

where βl (t) ∈ [0, 1) implies βh (t) = 0, and βh (t) ∈ (0, 1] implies βl (t) = 1.Although this maximization problem involves choosing over a pair of tariffs

(τ , r) instead a single variable, we can (and will) treat t ≡ (τ , r) ∈ TC(tmaxh ) as

if it is a single variable in the following use of derivatives for solving the maxi-mization problem. Given that any change in the choice of t occurs along a line,TC(tmax

h ), let ∂t represent a change in t such that ∂τ = ∂t with ∂r being defined by

(τ + ∂τ , r + ∂r) ∈ TC(tmaxh ).18 Then, the first order derivative associated with the

maximization problem in (7) is

−

[

γ∂βh(t)

∂t+ (1 − γ)

∂βl(t)

∂t

]

[

WD (t; h)− WDL (h)

]

+ (9)

{γ [1 −βh(t)] + (1 − γ) [1 −βl(t)]}∂WD (t; h)

∂t.

Definition 3. tmax ≡ max{

TC(tmaxh ) ∩ TDh(tmin

h

}

) and tb ∈ TC(tmaxh ) denotes tS

that solves the maximization problem in (7) subject to (8).

Given the first order condition in (9) with tmax and tb being defined as above,we can show that tb ∈ (tmax

h , tmax) as follows. The first multiplicative part in (9)

is strictly positive, except on the highest possible value of t ∈ TC(tmaxh ), denoted

by tmax, on which it equals 0. The first bracketed term is strictly negative, which

equals either γ ∂βh(t)∂t or (1−γ) ∂βl(t)

∂t respectively depending on whetherβh (t) ≥ 0(with βl (t) = 1) or βl (t) < 1; the second bracketed term is strictly positive anddecreases in t, except becoming 0 at t = tmax. The second multiplicative part in(9) is continuous and strictly negative (positive) for all possible values of t > tmax

h

18Without loss of generality, we can focus on the range of TC(tmaxh ) that is positively sloped in

the space of (τ , r).

16

(t < tmaxh ) and t ∈ TC(tmax

h ), except at t = tmaxh on which it equals 0. The first

(and curly) bracketed term is strictly positive, which equals either γ [1 −βh(t)]or γ + (1 − γ) [1 −βl(t)], respectively, depending on whether βh (t) ≥ 0 (withβl (t) = 1) orβl (t) < 1; the second term is strictly positive (negative) for t < tmax

h(t > tmax

h ) and 0 when t = tmaxh . Given these characteristics of the first and the

second multiplicative parts in (9) , tb ∈ (tmaxh , tmax).

The following lemma then characterizes the strategy profile of a separatingPBE that maximizes Dh’s expected payoff:

Lemma 4. There exists a separating PBE that maximizes the expected payoff of Dh, inwhich having tS = tb(> tmax

h ) ∈ TC(

tmaxh

)

with αh (tb) = 1, αl

(

tmaxl

)

= 1, βl (tb)and βh (tb) being uniquely determined by 8.

Proof. See Appendix

Although Lemma 4 allows the possibility of having multiple separating PBEs

that maximize the expected payoff of Dh, we will assume the uniqueness of such aPBE in the following analysis.19 In proving that the separating PBE characterizedin Lemma 4 is the unique Divine PBE equilibrium, the following lemma abouta pooling PBE (i.e., αl

(

tmaxl

)

= 0 and ∃tS ∈ Conv(

TDh(

tminh

)

, TC(

tmaxh

))

with

αl

(

tS)

> 0 andαh

(

tS)

> 0), is helpful.

Lemma 5. (i) Under a pooling PBE,αl

(

tS)

> 0 only ifαh

(

tS)

> 0. (ii) Under a pooling

PBE if tS ∈ TC(

tmaxh

)

, then αl

(

tS)

= 0. (iii) All pooling equilibrium actions for D j

generate the same expected payoff for D j, for j = l or h.


Proposition 1. A PBE is divine iff it is a separating equilibrium that maximizes Dh’sexpected payoff.


19We believe that this assumption is not a strong one. This is because no systematic reasonexists for the payoff functions (characterized in the preceding section) to yield multiple valuesof tb that generate an identical maximized expected payoff of Dh under a separating PBE. Evenin the presence of multiple separating PBEs that maximize the expected payoff of Dh, the prooffor characterizing a divine PBE in Proposition 1 remains the same, implying that only a separatingPBE that maximizes the expected payoff of Dh can be supported as a divine PBE. In this subsection,the uniqueness assumption is mainly for expositional simplicity.

17

Proposition 1 enables us to focus on the separating PBE—characterized inLemma 4— that maximizes Dh’s expected payoff . Using Figure 2, we can ex-plain such an equilibrium bargaining outcome with αl

(

tmaxl

)

= 1 and αh (tb) = 1as follows. To maximize Dh’s expected payoff, her equilibrium tariff proposal, tb,needs to be on TC

(

tmaxh

)

, as shown in Figure 2, with αh (tb) = 1. Comparing with

tb ∈ TC(

tmaxh

)

, any tariff proposal t′ ∈ TDl (tb) that is not on TC(

tmaxh

)

yields astrictly lower expected payoff to Dh under a separating PBE. This outcome is be-cause it entails the same litigation probability as the one for tS = tb, with βl (t

′)and βh (t

′) being uniquely determined by 8, and a strictly lower settlement payoffto Dh with WD (t′; h) < WD (tb; h). Since C is indifferent between settlement andlitigation given that tS = tb ∈ TC

(

tmaxh

)

and θ = h, αl (tb) = 0 is necessary forC to be indifferent between settlement and litigation. Under a separating PBE, Dl

proposes tS = tmaxl with αl

(

tmaxl

)

= 1 because it is Dl’s settlement proposal thatmaximizes her settlement payoff without invoking C’s litigation when she revealsher type by proposing tS 6= tb. Finally, tb > tmax

h because the marginal benefit oflowering tb is smaller than the marginal cost of doing so at tb = tmax

h . On one hand,the marginal benefit associated with lowering tb toward tmax

h (which reflects an in-crease in Dh’s settlement payoff) decreases to 0 as tb approaches tmax

h as implied by

the fact that TDh(

tmaxh

)

is tangent to TC(

tmaxh

)

. On the other hand, the marginalcost associated with lowering tb toward tmax

h (which reflects an increase in theequilibrium litigation probability) remains positive even when tb approaches tmax

h

as implied by TDl(

tmaxh

)

not tangent to TC(

tmaxh

)

.The result that tb > tmax

h demonstrates the signaling nature of this pretrialsettlement game. To signal her type, Dh offers a tariff pair, tb, that is strictly higherthan the Pareto efficient tariff combination, tmax

h (which maximizes her settlementpayoff given the constraint that C is indifferent between litigation and settlementunder a fully separating equilibrium). The reason for Dh to offer such a tariffpair comes from her stronger preference toward more import protection (i.e., herwillingness to accept C’s even higher protection against her export) than Dl’s,which in turn enable Dh to signal her type by such an offer .

As discussed in the introduction, existing studies on trade dispute have notidentified this signaling nature of pretrial settlement.20 tb > tmax

h leads to the fol-lowing two new predictions. First, the settlement tariff pair may entail the protec-

20Although Feenstra and Lewis (1991) emphasize the signaling nature of export restraint agree-ments, their analysis does not analyze the trade dispute and settlement under the WTO regime thatprohibits the use of export restraint agreements as a contingent protection policy.

18

tion levels that are higher than the certainty equivalent (and Pareto-efficient) pro-tection levels that are expected under the litigation for each party, with tb > tmin

h

for Dh and tb > tmaxh for C. In addition, tb can be even greater than tE(h) ≡

(

τE (h) , rE (h))

, as demonstrated in Figure 2. Given that the DSB authorizes C toretaliate against D′s imposition of high protection by imposing the same protec-tion level only when it rules against D′s claim of facing high protectionist pressurebased on its own imperfect public signal of D’s pressure, Beshkar (2010b) showsthat the joint-payoff maximizing DSB will recommend D’s protection level that islower than τE(h) even when it rules that D is subject to high protectionist pres-sure.21 If tb is greater than tE(h) ≡

(

τE (h) , rE (h))

and the DSB rulings followthe prediction of Beshkar (2010b), then the DSB will always recommend reducingcontingent protection. Legal scholars identified such free trade bias in the WTO’sDSB rulings, as discussed in footnote ??.

The second prediction is about what will happen to C’s access into D’s marketwhen they settle their dispute. Given that Dh sets her contingent protection levelat τb with (τb, rb) ≡ tb prior to a trade dispute being filed to the WTO, allowingC to raise her own protection level up to rb, their settlement of a dispute will notlead to any increase in C’s access to D’s market of the product under dispute.This hypothesis can be tested, possibly using trade flow data related to mutuallyagreed settlements of the WTO trade disputes.

As emphasized by existing studies on trade dispute and settlement, govern-ments can renegotiate their protection levels, potentially inducing them to rene-gotiate tb(> tmax

h ) into lower and Pareto-efficient levels (which would increaseboth governments’ settlement payoffs). If such renegotiation is expected to takeplace immediately after settlement, then the “real” settlement should be Paretoefficient, and tmax

h is the renegotiation-proof offer that maximizes Dh’s expectedpayoff under a fully separating equilibrium via Lemma 4. As discussed above,however, the renegotiation-proof settlement offer, tmax

h , entails a higher litigationprobability, yielding a strictly lower expected payoff to Dh than tb does. Thisgives Dh a strict incentive to commit to the settlement offer of tb, if possible. Inpractice, once a government explicitly sets her contingent protection level after

21Although Beshkar (2010b) analyzes the issue of optimal safeguard protection in a model sim-ilar to ours, his model assumes that C has no information about D’s type except the prior distribu-tion of it. Thus, how the presence of C’s informative signal of D’s type may affect the DSB’s rulingis a possible future research topic, of which the conclusion provides a brief discussion. However,it is reasonable to conjecture that the joint-payoff maximizing DSB will recommend D’s protectionlevel that is lower than τE(h) even in the presence of C’s signal of D’s type because recommend-ing D’s protection level be higher than τE(h) will reduce the joint-payoff of governments withoutrelaxing D’s incentive to exaggerate her pressure for protection.

19

going through a domestic process to justify such protection, readjusting it will be(politically) costly in the absence of the DSB’s formal ruling against it because thedomestic import-competing firms are entitled to such protection. This may renderthe initial contingent protection level fixed under settlement negotiation.

4.2 Signal’s Accuracy and Divine PBE

The Divinity refinement narrows PBEs of the pretrial settlement game down to theseparating PBE that maximizes Dh’s expected payoff. This section characterizeshow the Divine PBE changes as the accuracy of C’s signal (γ) improves. Prior toproviding such characterization, we define some critical values of γ as follows.

Definition 4. γ I I I ≡ [WD(

tmaxb ; l

)

− WD(

tmaxl ; l

)

]/[WD(

tmaxb ; l

)

− WDL (l)], with

tmaxb being implicitly defined by t solving

WD (t; l)− WDL (l)


=∂WD(t;l)

∂t∂WD(t;h)

∂t

,

γ I I is the minimum value of γ that satisfies the following equality:

(1 − γ)

γ2=

∂WD(

t̂S(γ);h)

∂t

∂WD(

t̂S(γ);l)

∂t

[

WD(

t̂S(γ); l)

− WDL (l)

]2

[

WD(

tmaxl ; l

)

− WDL (l)

]

[

WD(

t̂S(γ); h)

− WDL (h)

] ,

with t̂S(γ) being defined by t that induces βh(t) = 0 and βl(t) = 1 under (8),and

γ I ≡ [WD(

tmaxh ; l

)

− WD(

tmaxl ; l

)

]/[WD(

tmaxh ; l

)

− WDL (l)] < 1.

γ I I I ∈ (0, 1) is defined so that Dl is indifferent between proposing tmaxl (thus,

revealing her type) and proposing tmaxb given that βl(t

maxb ) = 1 and βh(t

maxb ) = 0.

If γ < γ I I I, then βl(tmaxb ) = 1 and βh(t

maxb ) > 0 are necessary to make Dl be

indifferent between proposing tmaxl and proposing tmax

b . Similarly, γ I ∈ (0, 1) isdefined so that Dl is indifferent between proposing tmax

l and proposing tmaxh given

that βl(tmaxh ) = 1 and βh(t

maxh ) = 0. If γ ≥ γ I, then βl(t

S) < 1 and βh(tS) = 0

are sufficient to make Dl be indifferent between proposing tmaxl and proposing

20

tS > tmaxh . tmax

b is the maximum Divine equilibrium tariff offer, as shown in the

following proposition. tmaxb > tmax

h implies that γ I > γ I I I, and the following

proposition also shows that γ I I ∈ (γ I ,γ I I I).

Proposition 2. The Divine PBE has one of the following three types of litigation strategieson Dh’s settlement proposal, tb, with distinctive comparative statics, depending on theaccuracy of C’s private signal, γ:

(a) If γ < γ I I I, tb = tmaxb , βl (tb) = 1 and βh (tb) > 0, with ∂tb

∂γ= 0 and

∂βh(tb)∂γ

< 0;

(b) If γ I I I ≤ γ < γ I I, tb = t̂S(γ), βl (tb) = 1 and βh (tb) = 0, with ∂tb∂γ

< 0 and∂βl(tb)

∂γ= 0;

(c) If γ ≥ γ I I, tb ≤ t̂S(γ I I), βl (tb) ∈ (0, 1] and βh (tb) = 0, with ∂tb∂γ

< 0,

and if γ ≥ γ I(> γ I I), βl (tb) ∈ (0, 1) with limγ→1

tb = tmaxh and lim

γ→1βl (tb) > 0.


Figure 3, a graphical representation of the first order condition of the max-imization problem (7), is useful in explaining the results in Proposition 2. Thelitigation probabilities are defined by Dl’s incentive constraint for truth-telling in(8). Dh’s marginal benefit from raising tS, denoted by MB(tS ;γ), is the first mul-tiplicative part in (9), representing the benefit from reducing the litigation proba-bility with a higher tS. Dh’s marginal cost from raising tS, denoted by MC(tS;γ),is the second multiplicative part in (9), representing the cost of offering tS that isfurther away from tmax

h , reducing Dh’s settlement payoff.

For any level of γ, MB(tS;γ) and MC(tS ;γ) are divided into two parts in Fig-ure 3: one denoted by a bold line and the other by a thin line. The bold-line partcorresponds to the case with βl

(

tS)

= 1 and βh

(

tS)

∈ (0, 1), and the thin-line

part corresponds to the case with βl

(

tS)

∈ (0, 1) and βh

(

tS)

= 0, with t̂S(γ)denoting t that induces βh(t) = 0 and βl(t) = 1 under the requirement of (8).When γ = 0.5, for example, reducing tS from tmax toward tmax

h requires the liti-

gation probability that satisfies (8) to rise, having βh(tS) = 0 and βl(t

S) = 1 at

tS = t̂S(0.5), and βl

(

tS)

= 1 and βh

(

tS)

∈ (0, 1) for tS < t̂S(0.5). In addition,

t̂S(γ) decreases as γ increases, having t̂S(γ I) = tmaxh , thus βl

(

tS)

∈ (0, 1) and

βh

(

tS)

= 0 for all γ ∈ [tmaxh , tmax] if γ > γ I.

21

Using Figure 3, we can explain the result (a) as follows: The value of tS thatmaximizes the expected payoff of Dh given γ = 0.5, tb (γ = 0.5), is equal to tmax

b ,

with βl

(

tS)

= 1 and βh

(

tS)

∈ (0, 1). When γ increases, the bold-line parts of

MB(tS ;γ) and MC(tS ;γ) shift up by the same amount. As a result, tb (γ) remainsat tmax

b as long as βh

(

tS)

∈ (0, 1) is required to satisfy the incentive constraint of

Dl for not imitating Dh’s proposal of tS = tmaxb , i.e. as long as γ < γ I I I.

With regard to the result (b), if γ = γ I I I, then t̂S(γ) = tmaxb , thus tb (γ) contin-

ues to be equal to tmaxb , but βl

(

tS)

= 1 and βh

(

tS)

= 0. An increase in γ shifts

down the thin-line part of MB(tS ;γ) although it shifts up the thin-line part of

MC(tS ;γ). This creates discontinuity in MB(tS ;γ) on tS = t̂S(γ) for any γ > 0.5,as shown in Figure 3. If βh

(

tS)

> 0, then an increase in γ raises MB(tS ;γ), be-

cause a higher γ raises the weight on the marginal benefit from a higher tS with−γ

[

∂βh

(

tS)

/∂tS] [


]

in (9). If βl

(

tS)

< 1, then an increase in

γ reduces MB(tS;γ), because a higher γ decreases the weight on the marginalbenefit from a higher tS with − (1 − γ)

[

∂βl

(

tS)

/∂tS] [


]

in

(9). This discontinuity in MB(tS ;γ) at tS = t̂S(γ) implies that an increase in

γ from γ = γ I I I leads to having MB(tS ;γ) > MC(tS ;γ) for tS < t̂S(γ) and

MB(tS ;γ) < MC(tS ;γ) for tS > t̂S(γ), which in turn implies tb (γ) = t̂S(γ) for

γ < γ I I(

> γ I I I)

. Thus, βl

(

tS)

= 1 and βh

(

tS)

= 0, and tb (γ) = t̂S(γ) decreases

as γ increases for γ ∈[

γ I I I,γ I I)

.

Regarding the result (c), an increase in γ from γ = γ I I leads to either havingβl

(

tS)

< 1 and βh

(

tS)

= 0 or having βh

(

tS)

= 1 and βh

(

tS)

= 0 with tb (γ)

continuing to be equal to t̂S(γ). In both cases, an increase in γ leads to a decreasein tb (γ). In the former case, MB(tS ;γ) gets smaller and MC(tS ;γ) gets bigger,creating an incentive to reduce tS. If γ ≥ γ I, then βl

(

tS)

∈ (0, 1) and βh

(

tS)

= 0for all γ ∈ [tmax

h , tmax] as discussed earlier, generating the former case. In the latter

case in which tb (γ) = t̂S(γ), t̂S(γ) decreases as γ increases.The graphs on the left side in Figure 4 demonstrate the results in Proposition

2 based on a numerical analysis.22 Three distinctive regions of γ exist, and theydiffer from each other on how an increase in γ affects Dh’s equilibrium tariff offerand C’s litigation probabilities. Although we cannot theoretically rule out the pos-sibility of having βl (tb) = 1 for γ ∈ (γ I I ,γ I), the numerical analysis does showthat having βl (tb) < 1 is possible for γ > γ I I. The analysis also demonstrates thatβl (tb) may strictly decrease as γ increases for γ > γ I I.

22In the Appendix we discuss how we constructed the numerical analysis shown in Figure 4.

22

The top graphs on the right side in Figure 4 demonstrates how the litigationlikelihood (i.e., probability of litigation) against Dl’s potential choice of mimick-ing Dh’s equilibrium tariff offer, denoted by LL (l), changes as the quality of C’sinformation improves. As shown in Figure 4, this probability of litigation staysconstant until γ reaches γ I I I then monotonically increases in response to a furtherincrease in γ. Such a change in the potential (not observed in the equilibrium) lit-igation likelihood is necessary to deter Dl from mimicking Dh’s equilibrium tariffoffer (tb) that is constant until γ reachesγ I I I, then monotonically decreases towardtmaxh in response to a further increase in γ. In contrast to this potential litigation

likelihood, LL (h) represents the equilibrium litigation likelihood (i.e. probabil-ity of litigation) against Dh’s equilibrium tariff offer, tb. As the theory predicts,LL (h) monotonically decreases in response to an increase in γ up to γ I I. For afurther increase in γ beyond γ I I, the numerical simulation shows that LL (h) maymonotonically decrease toward 0 as C’s signal gets almost perfect.

The bottom graphs on the right side in Figure 4 demonstrate how each party’sexpected payoff from the pretrial bargaining game changes as the quality of C’ssignal improves. As the theory predicts, all the informational rent from an im-provement in C’s signal accrues to Dh as she takes all the surplus from the pretrialsettlement bargaining game (through her take-or-leave offer), having the expectedpayoffs of Dl and C be constant in response to an increase in γ.

5 Signaling Game with a Noisy Public Signal

In this section, we study the pretrial settlement game under the assumption thatthe complaining party receives a public signal about the type of the defendingparty. All other assumptions remain the same as the game under private signalsdiscussed in the previous section.

Assuming that C’s signal is public rather than private changes the incentivecompatibility constraint for the low-type D. Recall that when C’s signal is private,Dl perceives the likelihood of litigation given tS to be γβl

(

tS)

+ (1 − γ)βh

(

tS)

.In contrast, when C’s signal is public, Dl’s perceived likelihood of litigation giventS is βl

(

tS)

and βh

(

tS)

if θC = l and θC = h, respectively. Therefore, under the

public signal, the incentive compatibility constraint for Dl given tS and θC may bewritten as

WD (tmaxl ; l) ≥ βθC

(

tS)

WDL (l) +

(

1 −βθC

(

tS))

WD(

tS; l)

for θC = l, h.

The left-hand side of this condition is the welfare of Dl if it reveals its type by

23

proposing tmaxl . The right-hand side of this condition is the Dl’s expected welfare

if it mimics Dh by proposing tS given the realized public signal θC. However, thepublic signal has no effect on this condition because these incentive compatibilityconstraints imply that βl

(

tS)

= βh

(

tS)

. Intuitively, to make Dl indifferent be-tween revealing its true type (by proposing tmax

l ) and mimicking Dh (by propos-

ing tS), C has to choose the same rate of litigation regardless of the realized publicsignal. Therefore,

Proposition 3. The separating PBEs of the settlement bargaining game are independentof a noisy public signal about D’s private information.

This proposition implies that if we focus on separating PBEs, the introductionof a noisy public signal is completely innocuous.

Pooling EquilibriaWe now investigate the impact of a public signal on pooling equilibria. With a

public signal at the pretrial stage, the common prior belief of the parties is updated. Let ρ∗ denote the updated common belief of the parties about the likelihood ofa high type after observing the public signal. Moreover, let tpool (ρ

∗) ∈ Ph denotea tariff pair that makes C indifferent between settlement and litigation when alltypes of D pool, namely,

(1 − ρ∗)WCL (l) + ρ∗WC

L (h) ≡ WC(

tpool (ρ∗))

. (10)

tpool (ρ) ≺Dh

tminh by our assumption about the quality of the court.23 Moreover, as

ρ∗ increases, the expression on the left-hand side of 10 decreases, which impliesthat tpool (ρ

∗) moves monotonically toward tmaxh on Ph. If ρ∗ is sufficiently high

such that tpool (ρ∗) ≻

Dh

tminh , then tpool (ρ

∗) is part of a pooling PBE in which both

types of D propose tpool (ρ∗) and C accepts with certainty. Therefore,

Lemma 6. Let tpool (ρ∗) and ρ̂1 be defined by 10 and the following condition, respectively:

(1 − ρ̂1)WCL (l) + ρ̂1WC

L (h) ≡ WC(

tminh

)

.

If ρ∗ ≥ ρ̂1, then any tS ∈ Conv(

tminh , tpool (ρ

∗))

constitutes a pooling PBE where both

types of D propose tS for settlement and C accepts the proposal.

23By assumption, if the common prior belief about the likelihood of high type is ρ, full settle-ment cannot be an equilibrium. The court is assumed to be “good enough” so that if the commonprior belief of the parties is the same as the belief based on which the court is designed, the partiescannot jointly improve their welfare by adopting an alternative mechanism that exclude the court.

24

Therefore, although a public signal has no impact on the set of separating PBEs,a pooling PBE can arise with a sufficiently informative public signal. Nevertheless,

Lemma 7. Pooling PBEs are not universally divine.


Moreover, applying the Universal Divinity criterion to separating PBEs killsall but one equilibrium:

Lemma 8. Under public signaling, a PBE is universally divine iff it is a separating PBEthat maximizes the expected payoff of Dh.


As we showed that a private signal improves the expected payoff of the high-type D as well as the expected joint welfare of the parties, this proposition impliesthat

Corollary 1. An informative signal about D’s private information improves the expectedpayoffs of the parties in the signaling game iff it is privately observed by C.

The following thought experiment provides some perspective on the re-sult that a signal is useful only if it is private. Consider a tariff pair tS ∈Conv(tmin

h , tmaxh ), and the equilibrium strategies that support it as a separating

PBE under public signaling and private signaling respectively. Letting β denotethe likelihood of litigation if tS is proposed under public signaling, the incentivecompatibility constraint for Dl is given by

WD (tmaxl ; l) ≥ βWD

L (l) + (1 −β)WD(

tS; l)

.

Solving for β that makes Dl indifferent yields

β =WD

(

tS; l)

− WD(

tmaxl ; l

)

WD (tS; l)− WDL (l)

.

The value of β indicates the likelihood that Dh, as well as untruthful Dl, will facelitigation in the equilibrium of a public signaling game.

In the case of private signaling, the probability of litigation for an untruthfulDl is given by γβl + (1 − γ)βh, and the incentive compatibility constraint is givenby

25

WD (tmaxl ; l) ≥ (γβl + (1 − γ)βh)WD

L (l) + (1 − (γβl + (1 − γ)βh))WD(

tS; l)

.

Solving for γβl + (1 − γ)βh yields

γβl + (1 − γ)βh =WD

(

tS; l)

− WD(

tmaxl ; l

)

WD (tS; l)− WDL (l)

.

Therefore, the incentive compatibility constraint implies the same likelihood oflitigation for the low type under private and public signaling.

Now consider the likelihood of litigation for Dh under private signaling, whichis given by γβh + (1 − γ)βl. To check that for γ > 1

2 , we have

γβh + (1 − γ)βl < γβl + (1 − γ)βh,

which implies that Dh faces a lower likelihood of litigation under private signalingthan under public signaling.

This comparison clarifies the source of efficiency improvement under privatesignaling: when the signal is unobservable to D, C can condition its litigationstrategy on the private signal and litigate different types with different probabil-ities. In particular, under private signaling, Dh is less likely to be litigated thanuntruthful Dl.

6 Settlement Bargaining as a Screening Game

In this section, we reformulate our settlement bargaining game as a screening gamein which the uninformed party (i.e., the complainant) makes a settlement pro-posal and the informed party (i.e., the defendant) decides whether to accept theproposal and settle, or reject the proposal and litigate. We assume the followingsequence of events. First, C receives a signal of D’s type and proposes a tariff pairfor settlement. D can either drop the case and adopt the status quo, tmin

l , acceptthe proposal and settle, or litigate.

D will accept the proposal if W(

tS;θ)

≥ WDL (θ). D will drop the case and

implement the status quo, tminl , iff θ = l and WD

(

tS; l)

< WDL (l). Finally, D will

litigate if D’s expected payoff under litigation is greater than the payoff under thestatus quo and the proposed settlement, i.e., WD

L (θ) > WD(

tminl ;θ

)

and WDL (θ) >

WD(

tS;θ)

.

26

Proposition 4. i) For Pr(θ = h|θC) >WC(tmin

l )−WC(tminh )

WC(tminl )−WC

L (h), the unique PBE is a pooling

equilibrium in which C proposes tminh for settlement and both types of D will accept this

proposal.

ii) For Pr(θ = h|θC) <WC(tmin

l )−WC(tminh )

WC(tminl )−WC

L (h), the unique PBE is a separating equilib-

rium in which C proposes tminl for settlement, which will be accepted (rejected) by Dl(Dh).


Proposition 4 does not depend on whether C’s signal is private or public. Theabove result shows that the player with her private type information should bethe one to make an offer for the second-order uncertainty to drastically affect theequilibrium of a settlement bargaining game. C’s best offer strategy depends onlyon the probability of D being a strong type conditional on her signal, which isunaffected by whether C’s signal remains private or becomes public, and D’s de-cision to accept the offer depends only on her type and the offer.

With regard to predicting the role that second order uncertainty may play in asettlement bargaining game, one needs to determine whether the bargaining gameshould be modeled as a signaling or a screening game. For the imposition of con-tingent protection under the WTO regime, a defending government has the rightto impose such protection as long as she goes through a domestic process to deter-mine its legitimacy. A complaining government then has the choice over whetherto accept such protection (possibly together with her own protection offered to betolerated by the defendant) or litigate it. Therefore, our pretrial settlement gameover contingent protection should be analyzed as a signaling game.

7 Conclusion

We consider a game of bilateral bargaining over actions in which (i) the cost ofdisagreement depends on the private type of one of the parties and (ii) the un-informed party receives a noisy signal about this private type before the actualbargaining. We analyze the role of uncertainty about higher-order beliefs in thisbargaining game and establish an anti-transparency result. That is the efficiencyof the settlement bargaining outcome is higher if the signal that the uninformedparty receives is private rather than public.

The anti-transparency result can be generalized by considering an extension ofour model in which the signal is partially public, i.e., if public but noisy information

27

about the signal is available. In that case, we can show that the more public is thesignal that the un-informed party receives, the lower is the efficiency of the set-tlement bargaining outcome. Therefore, our result is not a feature of the extremeassumption of public vs. private signaling.

Our anti-transparency result in pretrial negotiations may be contrasted to thepro-transparency result in arbitration models proposed by Beshkar (2010b) andPark (2011). In these studies, publicizing the private information of the defendingparty relaxes the self-enforceability condition and reduces the level of inefficientpunishment that is required to induce truthfulness. Beshkar (2010b) and Park(2011) find the pro-transparency result in a repeated-game framework in whichenforceability of the agreement is improved by introducing an informative publicsignal.24

Understanding caveats associated with our anti-transparency result is impor-tant. First, the noisy signal of a defendant’s type is “secondary information” inthe sense that it does not affect the players’ expectation about the court ruling (orthe outcome of hostile engagement resulting from failing to settle) for a given typeof the defendant. If the signal can change the expected court ruling conditionalon a specific type of the defendant, thus having its own informational value in-dependent from the defendant’s type, then our anti-transparency result may nothold.25 Second, we do not explore the possibility that the court utilizes such sec-ondary information once it is revealed, possibly affecting the ruling. As reviewedby Spier (2007), litigation literature on “evidence” studies how various institu-tional requirements on evidence affect pretrial settlement and ruling.26 Given thatthe nature of noisy signals that we analyze here is different from those exploredby existing studies, extending our paper toward this direction may lead to newresults.

24The main difference between Beshkar (2010b) and Park (2011) lies in the analysis of no publicsignal case. The former assumes no private signal, but the latter analyzes the case with a noisyprivate signal of potentially deviatory actions.

25The studies that explore how two-sided private information affects litigation in the law andeconomics literature assume that both sides of private information have independent informa-tional value on the expected court outcome (Schweizer, 1989; Daughety and Reinganum, 1994).

26The institutional requirements include the burden of proof, disclosure and discovery as wellas admissibility of settlement negotiations at trial. See Spier (2007) for the review of studies onthese issues.

28

Appendix

Proof. for Lemma 1) Consider a tariff proposal under a separating PBE, tS 6= tmaxl

with αl

(

tS)

> 0 and αh

(

tS)

= 0: if tS is located to the left of TC(

tmaxl

)

in the

space of (τ , r), βl

(

tS)

= βh

(

tS)

= 0; if tS ∈ TC(

tmaxl

)

, βl

(

tS)

,βh

(

tS)

∈ [0, 1]; if

tS is located to the right of TC(

tmaxl

)

in the space of (τ , r), βl

(

tS)

= βh

(

tS)

= 1.

Note that the expected payoff of Dl from proposing such tS 6= tmaxl is strictly

smaller that its payoff from proposing tS = tmaxl with βl

(

tmaxl

)

= βh

(

tmaxl

)

= 0

because WD(

tmaxl ; l

)

> WDL (l) and WD

(

tmaxl ; l

)

> WD(

tS; l)

for tS( 6= tmaxl )∈

TC(

tmaxl

)

. Then, there exists t′( 6= tS) ∈ Pl located to the left of TC(

tmaxl

)

and thatis sufficiently close to tmax

l such that the expected payoff of Dl from proposing t′

with βl (t′) = βh (t

′) = 0 is strictly greater than the expected payoff of Dl fromproposing tS( 6= tmax

l ). Because C strictly prefers settling with t′ to litigating itregardless of D’s types, βl (t

′) = βh (t′) = 0 will result from proposing t′. This

implies that a tariff proposal, tS 6= tmaxl with αl

(

tS)

> 0 and αh

(

tS)

= 0 cannot

be a part of a PBE. For a tariff proposal, tS = tmaxl with αl

(

tS)

> 0, αh

(

tS)

= 0,and βl

(

tmaxl

)

> 0 or βh

(

tmaxl

)

> 0, the same logic can be applied to show that itcannot be a part of a separating PBE.

Proof. for Lemma 2) If tS does not belong to Conv(

TDh(

tminh

)

, TC(

tmaxh

))

, then

either Dh or C will strictly prefer litigating over settlement with tS. If Dh strictlyprefers litigating over settlement with tS that entails a positive probability of set-tlement, then proposing such tS is strictly dominated by a proposal that will surelyinvoke litigation. If C strictly prefers litigation over settlement with tS, then suchtS will entail zero probability of settlement.

Proof. for Lemma 3) Under a separating PBE, consider tS( 6= tmaxl ) withαh

(

tS)

> 0

that does not belong to TC(

tmaxh

)

. For such tS, first note thatβl

(

tS)

and βh

(

tS)

(with βl

(

tS)

> 0) need to satisfy Dl’s incentive constraint in (6) with equality:

if Dl’s incentive constraint in (6) holds with a strictly inequality, then αl

(

tS)

=

0, implying that C, who correctly believes that only Dh would propose such tS,will settle for sure with WC

(

tS)

> WCL (h), which in turn contradicts with Dl’s

incentive constraint in (6) holding with a strictly inequality.

If βl

(

t/)

= βl

(

tS)

and βh

(

t/)

= βh

(

tS)

, Dh’s proposing t/ ∈ TDl(

tS)

con-

tinues to satisfy Dl’s incentive constraint in (6) with an equality, and t/ ∈ TDl(

tS)

29

∩TC(

tmaxh

)

yields the highest (and strictly higher than other t/ ∈ TDl(

tS)

) ex-

pected payoff to Dh, and a strategy profile with αh

(

t/)

= 1, βl

(

t/)

= βl

(

tS)

and βh

(

t/)

= βh

(

tS)

is a separating PBE as long asαl

(

t/)

= 0. This proves that

any tS( 6= tmaxl ) with αh

(

tS)

> 0 belongs to TC(

tmaxh

)

.

Proof. for Lemma 4) First, note that (8) uniquely defines a pair of βl

(

tS)

and

βh

(

tS)

for any tS ∈ TC(tmaxh ). Because WD

(

tS; l)

> WD(

tmaxl ; l

)

> WDL (l) for

tS ∈ TC(

tmaxh

)

and ∂WD(

tS; l)

/∂tS < 0, an increase in tS requires a correspond-

ing decrease in γβl(tS) + (1 − γ)βh(t

S). Since βl (t) ∈ [0, 1) implies βh (t) = 0,and βh (t) ∈ (0, 1] implies βl (t) = 1, as discussed earlier in relation with C’s in-centive compatibility condition associated with (6), this means that either only adecrease in βl(t

S) or only a decrease in βh(tS), which in turn proves that βl

(

tS)

and βh

(

tS)

are uniquely determined by (8).If there is a unique value of tb, then a separating PBE that maximizes the ex-

pected payoff of Dh should entail αh (tb) = 1. Even if there exist multiple valuesof tb, a separating PBE with αh (tb) = 1 is one of such PBEs that maximizes theexpected payoff of Dh. It remains to show thatαl

(

tmaxl

)

= 1. This results from the

fact that tb ∈ TC(

tmaxh

)

, implying that WC (tb) = WCL (h) . This means that C is in-

different between accepting tb and litigating with the belief of D being high-typewith probability 1. Any (equilibrium) behavior of αl

(

tmaxl

)

< 1 with αl (tb) > 0

will invoke litigation against tb ∈ TC(

tmaxh

)

with probability one. Thus, a sep-

arating PBE having having tS = tb(> tmaxh ) ∈ TC

(

tmaxh

)

with αh (tb) = 1 and

αl

(

tmaxl

)

= 1 is a PBE that maximizes the expected payoff of Dh.

Proof. for Lemma 5) First, consider a pooling PBE does entail tS withαl

(

tS)

> 0

and αh

(

tS)

= 0 in the equilibrium strategy profile. Once such tS is played, then

C will litigate it with probability one because WC(

tS)

< WCL (l) for all tS ∈

Conv(

TDh(

tminh

)

, TC(

tmaxh

))

. Given that Dl’s expected payoff from playing the

pooling PBE is greater than WDL (l), then, αl

(

tS)

> 0 cannot be a part of the pool-ing PBE because it will violate the incentive constraint of Dl of the PBE.

Second, assume ∃tS ∈ TC(

tmaxh

)

withαl

(

tS)

> 0 andαh

(

tS)

> 0. Once such tS

is played, then C will litigate it with probability one because WC(

tS)

= WCL (h) is

strictly smaller than C’s expected payoff from litigation. Given that Dl’s expected

30

payoff from playing the pooling PBE is greater than WDL (l), then, αl

(

tS)

> 0cannot be a part of the pooling PBE because it will violate the incentive constraintof Dl of the PBE.

Third, D j’s expected payoff from offering a specific tariff proposal, tS with

α j

(

tS)

> 0, is identical to D j’s expected payoff from the pooling equilibriumbecause otherwise, it will violate the incentive constraint of D j of the PBE.

Proof. for Proposition 1) The proof is divided into two parts: first, proving that theonly separating PBE (including semi-separating ones) that satisfies the universaldivinity criterion is the fully separating equilibrium that maximizes the expectedpayoff of Dh (characterized in Lemma 4); second, proving that no pooling PBEsatisfies the universal divinity criterion.

First, consider a separating PBE that is not maximizing Dh’s expected payoff.The expected payoff of Dl under such a PBE (or any separating PBE), denoted byEWDl , is equal to WD

(

tmaxl ; l

)

: if EWDl > WD(

tmaxl ; l

)

, Dl has a strict incentive to

assign αl

(

tmaxl

)

= 0, thus invalidating the separating behavior in such an equilib-

rium; if EWDl < WD(

tmaxl ; l

)

, Dl has a strict incentive to assignαl

(

tmaxl

)

= 1, thus

having EWDl = WD(

tmaxl ; l

)

. Denote the expected payoff of Dh from playing such

a separating PBE by EWDh(< EWDh (tb)), where EWDh (tb) represents the maxi-mized expected payoff of Dh. Now consider a possible off-the-equilibrium devi-

ation in which tS = t/ ≡(

τ/, r/)

∈ TDl (tb) with τ/ < τb , where τb is defined

by (τb, rb) ≡ tb. In particular, t/ is close enough to tb so that the expected payoff

of Dh from playingαh

(

t/b

)

= 1 with βl

(

t/)

and βh

(

t/)

being respectively equal

to β∗l

(

t/)

and β∗h

(

t/)

that are uniquely determined by 8, which we denote by

EWDh

(

t/)

, satisfies the following inequalities: EWDh < EWDh

(

t/)

< EWDh (tb).

Recall that βl (t) ∈ [0, 1) implies βh (t) = 0, and βh (t) ∈ (0, 1] implies βl (t) =1. This enables us to represent the strategy of C by a single variable, βS(t

S) ≡

βl(tS) +βh(t

S) ∈ [0, 2]. In a similar manner, define β∗S

(

t/)

≡ β∗l

(

t/)

+β∗h

(

t/)

.

Also, define β̂S

(

t/)

≡ β̂l

(

t/)

+ β̂h

(

t/)

as the value of βS

(

t/)

that makes

the expected payoff of Dh from playing αh

(

t/)

= 1 with β̂S

(

t/)

be equal to

EWDh . Thus, Bh

(

t/)

= {βS

(

t/)

< β̂S

(

t/)

}. B̄l

(

t/)

= {βS

(

t/)

≤ β∗S

(

t/)

}

31

because the expected payoff of Dl from playingαl

(

t/b

)

= 1 with βS = β∗S(t

/), de-

noted by EWDl

(

t/)

, is equal to WD(

tmaxl ; l

)

, thus EWDl

(

t/)

= EWDl . Because

β̂S

(

t/b

)

> β∗S(t

/), Bh

(

t/)

⊃ B̄l

(

t/)

, inducing C to believe that Dh has deviated to

proposing t/. Because WC(t/) > WCL (h), C will accept such a deviation proposal

t/, creating a deviation incentive for D. This proves that a separating PBE that isnot maximizing Dh’s expected payoff is not a Divine equilibrium.

For the first part, it remains to show that the fully separating equilibriumthat maximizes the expected payoff of Dh, characterized in Lemma 4, does sat-

isfy the Divinity criterion against any off-the-equilibrium action of tS = t/( 6=tb) ∈Conv

(

TDh(

tminh

)

, TC(

tmaxh

))

. With respect to an off-the-equilibrium ac-

tion of tS = t/( 6= tb) ∈Conv(

TDh(

tminh

)

, TC(

tmaxh

))

, define β∗S

(

t/)

be the

value of βS

(

t/)

that makes the expected payoff of Dl from tS = t/ being equal

to WD(

tmaxl ; l

)

. Note that β∗S

(

t/)

is uniquely determined by 8 with t = t/.

Now, define β̂S

(

t/)

as the value of βS

(

t/)

that makes the expected payoff of

Dh from playing αh

(

t/)

= 1 with βS

(

t/)

= β̂S

(

t/)

be equal to EWDh (tb).

Note that β̂S

(

t/)

< β∗S

(

t/)

because the expected payoff of Dh from playing

αh

(

t/)

= 1 with βS

(

t/)

= β∗S

(

t/)

is smaller that EWDh (tb). This implies

Bl

(

t/)

= {βS

(

t/)

< β∗S

(

t/)

} ⊃ B̄h

(

t/)

= {βS

(

t/)

≤ β̂S

(

t/)

}, which in

turn induces C to believe that Dl deviates by proposing t/. Thus, C will litigate

against t/with βS

(

t/)

= 2, eliminating any incentive for D to deviate from the

fully separating equilibrium that maximizes the expected payoff of Dh.For the second part, we need to prove that no pooling PBE does satisfies

the universal divinity criterion (not even divinity criterion nor intuitive crite-rion). Given Lemma 5, we can consider a deviation from a pooling equilib-rium as if it is a deviation from a specific tariff proposal, tS withαl

(

tS)

> 0 and

αh

(

tS)

> 0, because D j’s expected payoff from offering a specific tariff proposal,

tS with α j

(

tS)

> 0, is identical to D j’s expected payoff from the pooling equi-

librium with j = l or h. Denote such tS by tpool and denote the expected pay-

offs of Dl and Dh in a pooling PBE by EWDlpool and EW

Dhpool , respectively. Now

consider a possible off-the-equilibrium deviation in which tS = t/ ≡(

τ/, r/)

32

∈ TDl(

tpool

)

with τ/ < τmaxpool , where τmax

pool is defined by(

τmaxpool , rmax

pool

)

≡ tmaxpool

≡ TDl(

tpool

)

∩ TC(

tmaxh

)

. In particular, τ/ is close enough to τmaxpool so that the ex-

pected payoff of Dh from playing αh

(

t/)

= 1 with βS

(

t/)

= β∗S

(

tpool

)

(≥ 0),

the probability of litigation associated with tpool in the pooling equilibrium, de-

noted by EWDhpool

(

t/)

, satisfies the following inequality: EWDhpool < EW

Dhpool

(

t/)

.

Finally, define β̂S

(

t/)

as the value of βS

(

t/)

that makes the expected payoff of

Dh from playing αh

(

t/)

= 1 with βS

(

t/)

= β̂S

(

t/)

be equal to EWDhpool . Note

that Bh

(

t/)

= {βS

(

t/pool

)

< β̂S

(

t/pool

)

} and B̄l

(

t/)

= {βS

(

t/)

≤ β∗S

(

t/)

}.

Because β̂S

(

t/)

> β∗S

(

t/)

, Bh

(

t/)

⊃ B̄l

(

t/)

inducing C to believe that Dh has

deviated to proposing t/. Because WC(t/) > WCL (h), C will accept such a devia-

tion proposal t/, creating a deviation incentive for D. This proves that any poolingPBE is not a Divine equilibrium.

Proof. for Proposition 2)(a) We show this result in the following two steps: first, assume that the equi-

librium litigation strategy profile embodies βl (tb) = 1 and βh (tb) > 0, establish-ing the associated comparative static results; second, the equilibrium litigationstrategy profile embodies βl (tb) = 1 and βh (tb) > 0, if γ < γ I I I. If the divinePBE embodies βl (tb) = 1 and βh (tb) > 0, it implies that the first order deriva-tive associated with the maximization problem in (7) takes the following form andequals to zero:

−γ∂βh(t)

∂t

[


]

+ γ [1 −βh(t)]∂WD (t; h)

∂t= 0. (11)

33

with

∂βh(t)

∂t=

1

(1 − γ)

∂WD (t; l)

∂t

[

WD(

tmaxl ; l

)

− WDL (l)

]

[


]2,

1 −βh(t) =1

(1 − γ)

[

WD(

tmaxl ; l

)

− WDL (l)

]

[


] , (12)

∂βh(t)

∂γ= −

[

WD(

tmaxl ; l

)

− WDL (l)

]

(1 − γ)2[


] < 0.

The first two equalities in (12) implies that (11) is not affected by γ, which in turn

implies that ∂tb∂γ

= 0 . In fact, (11) implicitly define tmaxb in Definition 4, having tb =

tmaxb . Given this result, the last inequality in (12) implies

∂βh(tb)∂γ

< 0. It remains

to show that the divine PBE embodies βl (tb) = 1 and βh (tb) > 0 if γ < γ I I I.If γ = γ I I I, then βl(t

S) = 1 and βh(tS) = 0 to make Dl is indifferent between

proposing tmaxl and proposing tS = tmax

b , which in turn implies that βl(tS) = 1

and βh(tS) > 0 are required to make Dl is indifferent between proposing tmax

l and

tS = tmaxb if γ < γ I I I.

(b) Given γ = γ I I I, the first order derivative associated with the maximizationproblem in (7) takes the following form, being strictly smaller than zero for t ≥ tb

in the neighborhood of tmaxb if γ I I I > 0.5:

−(1 − γ)∂βl(t)

∂t

[


]

+ [1 − (1 − γ)βl(t)]∂WD (t; h)

∂t< 0. (13)

with

∂βl(t)

∂t=

1

γ

∂WD (t; l)

∂t

[

WD(

tmaxl ; l

)

− WDL (l)

]

[


]2< 0,

1 − (1 − γ)βl(t) = 1 −(1 − γ)

γ

[

WD (t; l)− WD(

tmaxl ; l

)]

[


] , (14)

∂βl(t)

∂γ= −

[

WD (t; l)− WD(

tmaxl ; l

)]

γ2[


] < 0.

First, note that (8) requires βh(tmaxb ) = 0 and βl(t

maxb ) = 1 by definition of γ I I I,

and βh(t) = 0 and βl(t) < 1 for t > tmaxb . The strict inequality in (13) for t > tmax

bcomes from the definition of tmax

b (being the unique value of t that maximizes the

34

expected payoff of Dh with γ = γ I I I). The strict inequality in (13) holds even att = tmax

b because the first term in (13) is strictly smaller than the first term in (11)

at t = tmaxb with γ = γ I I I > 0.5.

This strict inequality in (13) at t = tmaxb implies βl(tb) = 1 and βh(tb) = 0

with∂βl(tb)

∂γ= 0 for γ > γ I I I in the neighborhood of γ I I I. On the one hand, an

increase in γ raises the absolute value of the second term both in (11) and (13),thus raising the negative effect from raising t in maximizing the expected payoffof Dh. An increase in γ also decreases t̂b(γ) with . On the other hand, an increasein γ decreases the first term in (13) but increases the first term in (11). Thesecomparative static results implies that a small increase in γ at γ = γ I I I will leadto a decrease in tb (thus, ∂tb/∂γ < 0 ) so that tb= t̂b(γ). This is because the firstorder derivative associated with the maximization problem in (7) is positive fortb< t̂b(γ) and it is negative for tb> t̂b(γ). For γ ≥ γ I I I, therefore, βl (tb) = 1 and

βh (tb) = 0 with ∂tb/∂γ < 0 and∂βl(tb)

∂γ= 0 in the neighborhood of γ I I I.

Recall that we define γ I I as the minimum value of γ such that the left hand

side of the inequality in (13) becomes zero with tb= t̂S(γ), thus γ I I > γ I I I . Note

that such a value of γ exists and it is strictly smaller than γ I. Because t̂S(γ) = tmaxh

if γ = γ I, having the left hand side of the inequality in (13) is greater than zero at

t = t̂S(γ) = tmaxh , γ I I does exist and it is smaller than γ I.

(c) Ifγ ≥ γ I I, thenβh (tb) = 0 because the first order derivative associated with

the maximization problem in (7) is strictly greater than zero for any t < t̂S(γ). This

implies that βl (tb) = 1 with tb = t̂S(γ) or βl (tb) < 1 with tb > t̂S(γ), if γ ≥ γ I I,which in turn implies that ∂tb/∂γ < 0: ∂tb/∂γ < 0 for the case in which βl (tb) < 1is shown in what follows.

If γ ≥ γ I, first note that βh(t) = 0 and βl(t) < 1 under the requirementof (8) for all t ∈ TC

(

tmaxh

)

with tS > tmaxh by the definition of γ I. This implies

that the first derivative associated with the maximization problem in (7) takes thesame form as the one in (13), and it equals to zero at tb. When γ increases, thefirst term in (13) decreases, reflecting a reduced benefit from raising t, and the(negative) second term γ decreases, reflecting an increased cost from raising t.These changes in (13) in response to an increase in γ implies ∂tb/∂γ < 0 . Toshow that ∂tb/∂γ < 0 for any divine PBE with βl (tb) ∈ (0, 1) and βh (tb) = 0,we can apply the same logic. This is because the first derivative associated withthe maximization problem in (7) takes the same form as the one in (13) in theneighborhood of t = tb if βl (tb) ∈ (0, 1) and βh (tb) = 0.

limγ→1

(tS) = tmaxh because the limit of the firm term in (13) approaches

35

to zero with γ → 1 while the second term in (13) remains negative,except at t = tb. Finally, the requirement of (8) implies βl (t) =[

WD(

tS; l)

− WD(

tmaxl ; l

)]

/γ[

WD(

tS; l)

− WDL (l)

]

for βl(t) < 1, thus

limγ→1

βl

(

tS)

=WD

(

tmaxh ; l

)

− WD(

tmaxl ; l

)

WD(

tmaxh ; l

)

− WDL (l)

> 0

Numerical Analysis in Figure 4

The numerical analysis in Figure 4 uses the following political-economy model indefining governments’ payoff functions.27 It is a two-good (m and x) two-country(D and C) model with linear demand and supply. The demand functions of eachcountry are as follows:

DDm(pD

m) = 1− pDm , DD

x (pDx ) = 1− pD

x , DCm(pC

m) = 1− pCm, DC

x (pCx ) = 1− pC

x , (15)

where pig denotes the price of goods g in country i. Specific import tariffs τ and

r are chosen by D and C, respectively. These are assumed to be the only tradepolicy instruments. In particular, pD

m = pCm + τ and pD

x = pCx − r.

D and C produce m and x using the following supply functions:

QDm(pD

m) = pDm , QD

x (px) = bpDx , QC

m(pCm) = bpC

m , QCx (px) = pC

x . (16)

Assuming b > 1, D becomes a natural importer of m and a natural exporter of x.Under this specification, the politically-weighted government payoff from the

importing sector in D is given by

u(τ ;θD) =1

(3 + b)2

{

1

2(1 + b)2 + 2θD + [2θD(1 + b)− 4]τ

+

[

1 +θD

2(1 + b)2 − 2(3 + b)(1 + b)

]

τ2

}

, (17)

27Many studies use this simple political-economy model to analyze various issues of tradeagreements that arise from private political pressure for protection, including Bagwell and Staiger(2005) and Beshkar (2010a, 2016). The following representation of such a political-economy modeldirectly comes from Beshkar (2016).

36

where θD ≥ 1 denotes D’s political pressure from the import competing industry.D’s payoff from the exporting sector is a function of C’s import tariff r:

v(r) =1

(3 + b)2

{

(1 + b)2

2+ 2b + 2(1 − b)r + 2(1 + b)r2

}

. (18)

For the derivation of equations (17) and (18), see Appendix A of Beshkar (2016).C′s politically-weighted payoff from the importing industry u(r,θC) and her

payoff from the exporting industry v(τ) can be defined symmetrically.Using u and v constructed above, the payoff function of each government can

be defined as follows:

WD(τ , r;θ = θD) = u(τ ;θ) + v(r), WC(τ , r;θC) = u(r;θC) + v(τ). (19)

The parameter values used for the numerical analysis are as follows: b =15, l = 1.05, h = 1.35. θ = h if the political pressure is high, and θ = l, oth-erwise. θC is fixed at l throughout the analysis.

Proof. for Lemma 7) Consider a pooling PBE under which both types of D proposetpool and C accepts it. Now consider an alternative tariff pair, t′, such that t′ ≺

Dl

tpool ≺Dh

t′ and WCL (h) < WC (t′) . Then a defection to t′ will certainly hurt Dl

compared to the equilibrium outcome regardless of C’s reaction. Therefore, Cwould believe that the defection to t′ has been committed by Dh, in which case Cwill accept the proposal. Knowing this, Dh will defect to t′ with certainty. Thus,tpool cannot be supported by reasonable off-equilibrium belief.

Proof. for Lemma 8) First, we show that a PBE that involves β0 and t0 ∈TC

(

tmaxh

)

, t0 6= tb is not universally divine. To this end, note that there must

exist t′′ ∈ TDl (tb) such that t′′ ≺ tb

Dh

and

(1 −β0)WD (t0, h) +β0WDL (h) = (1 −βb)W

D(

t′′; h)

+βbWDL (l) .

Given continuity of welfare functions, there must also exist t′ ∈ TDl (tb), t′′ ≺

Dh

t′ ≺Dh

tb, and β′ > βb such that

β′WDL (l) + (1 −β′)WD

(

t′; h)

= (1 −βb)WD(

t′′; h)

+βbWDL (l) .

37

Now suppose that under the PBE that involves β0 and t0, D defects to t′. This de-fection will strictly reduce Dl’s expected welfare if C responds with β > βb. How-ever, this defection will strictly increases Dh’s expected welfare if C responds withany β < β′. Therefore, there for any β ∈ (βb,β′), Dh (Dl) benefits (loses) froma defection to t′. Therefore, if a defection to t′ is observed (instead of observingone of possible equilibrium outcomes, t0 or tmax

l ) , C will believe it is committedby Dh. Since t′ is strictly preferred by C to litigation, C cannot reject this proposalunder any reasonable belief about D’s type.

To complete the proof, we need to show that the PBE that involves tb and βb isuniversally divine. To this end, note that (tb,βb) is not universally divine if andonly if there exist (t′,β′) such that

(

t′,β′)

≻Dh

(tb,βb) ,

(

t′,β′)

≺Dl

(tb,βb) ≈Dl

tmaxl .

However, this is not possible because these preference orderings imply that thereexists a PBE involving t′ that is preferred to (tb,βb) by Dh. Therefore, (tb,βb) isuniversally divine.

Proof. for Proposition 4) First note that regardless of its information, D’s settle-ment proposal, tS, must be either tmin

l or tminh . To see this, first note that tS ≻

Dh

tminh

is a suboptimal proposal for C because proposing tminh or tS ≻

Dh

tminh will draw

the same response from D, which is acceptance, but tminwill generate a higherwelfare for C. Second, note that it is not optimal for C to propose tS such thattminl ≺

Dl

tS ≺Dh

tminh . That is because the Dl (Dh) will settle (litigate) whether tmin

l or

tS that satisfies tminl ≺

Dl

tS ≺Dh

tminh is proposed. But C prefers tmin

l to tS that satisfies

tminl ≺

Dl

tS ≺Dh

tminh . Therefore, the equilibrium proposal is either tmin

l or tminh .

If tS = tminl , then Dl will settle and Dh will litigate. Therefore, given θC, C’s

expected payoff from proposing tS = tminl is

Pr(θ = h|θC)WCL (h) +

[

1 − Pr(θ = h|θC)]

WC(

tminl

)

.

38

If tS = tminh , then both types of D will accept the proposal in which case the payoff

of C is given by WC(

tminh

)

. Therefore, C will propose tS = tminh if and only if

WC(

tminh

)

≥ Pr(θ = h|θC)WCL (h) +

[

1 − Pr(θ = h|θC)]

WC(

tminl

)

,

or, equivalently, iff

Pr(θ = h|θC) ≥WC

(

tminl

)

− WC(

tminh

)

WC(

tminl

)

− WCL (h)

.

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42

Figure 1. Potential Gains from Settlement

lPthPt

τ

r

maxhtmax

ltminlt

)(hτ E)(lτ E

Er )(lt E )(ht E

)(:)( max lWtT CLl

C

)(:)( min lWtT DLl

Dl

minht

)(:)( max hWtT CLh

C

)(:)( min hWtT DLh

Dh

Figure 2. Divine E

quilibrium Settlem

ent Offer

lP

th

Pt

r

max

h tm

axl t

bS

tt

Sr

b

)(

bD

tT

h

min

h t

)(

:)(

max

hW

tT

CLh

C

)(

:)(

min

hW

tT

DLh

Dh

)(

/t

Tl

D

/t

tS

)(

max

hD

tT

h

max

max

h

)(h

rE

)(h E

)(:)

(m

axl

Wt

TCL

lC

)(

max

lD

tT

l

max

r

MB

Figure 3. Effect of on the Divine Equilibrium

MBMB

MB

MBMB

MC

MC

MC

MC

Figure 4. Num

erical Analysis w

ith Linear Dem

ands and Supplies

dispute settlement with second-order uncertainty: the case of...

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