Communication and Cooperation in Markets
aa r X i v : . [ ec on . T H ] M a y Communication and Cooperation in Markets
S. Nageeb Ali and David A. Miller ∗ May 21, 2020
Abstract
Many markets rely on traders truthfully communicating who has cheated in the pastand ostracizing those traders from future trade. This paper investigates when truthfulcommunication is incentive compatible. We find that if each side has a myopic incen-tive to deviate, then communication incentives are satisfied only when the volume oftrade is low. By contrast, if only one side has a myopic incentive to deviate, then com-munication incentives do not constrain the volume of supportable trade. Accordingly,there are strong gains from structuring trade so that one side either moves first or hasits cooperation guaranteed by external enforcement. ∗ Ali: Pennsylvania State University. Miller: University of Michigan. We thank Ben Golub for valuablecomments and an insightful discussion of this paper. This research was financially supported by NSF grantSES–1127643.
Introduction
In many markets, buyers and sellers can renege on their promises without suffering legalconsequences, but defectors are punished by the loss of future business. If a seller tradeswith many buyers, losing business with a single buyer may not be enough of a threat todeter her from deviating. But if cheating a single buyer results in her losing business withmany buyers, then she is more inclined to cooperate. Such schemes, where actions witha single player affect cooperation with others, are at the core of multilateral enforcementor “third party” punishment. Multilateral enforcement schemes often employ personalizedpunishment, where traders will work with those who are untainted but sever their ties tothose who have deviated in the past.For personalized punishment to work, traders need to be able to communicate with eachother about their past experiences. Scholars have noted how information-sharing institutionswere critical to medieval trade and trust (Milgrom, North and Weingast 1990; Greif 2006).Today, online markets rely on ratings and reviews to collect and disseminate informationabout the behavior of market participants. Credit markets through the ages have benefitedfrom sharing information about borrower histories.We view information sharing not as a mechanical process, but as a voluntary choice. Iftraders are unwilling to communicate truthfully about their past experiences, informationwill not flow from one relationship to another, making personalized punishment impossible.Thus, we ask: when do buyers and sellers have a motive to tell the truth to other traders?
Model and Results:
We pose this question in a networked market of buyers and sellers,wherein each buyer-seller pair has a long-term trading relationship. Interactions within eachrelationship are not directly observed by third parties. When a pair interacts, the sellerchooses how much quality (or quantity) to deliver to the buyer and the buyer chooses howmuch to pay the seller. The seller faces an increasing cost function, and thus may have anincentive to shirk; the buyer analogously may have an incentive to shortchange the seller. Inaddition to these economic interactions, sellers also randomly meet other sellers, and buyersrandomly meet other buyers, just to share information about their past experiences.We emphasize that parties “may” have an incentive to deviate because whether a party actually has an incentive to do so depends subtly on the timing of trade. If the buyer andseller act simultaneously, then each party has a myopic incentive to shirk. But if the rules ofthe marketplace direct the buyer to first make a payment, and the seller then to choose thequantity to trade, then the buyer gains nothing by paying less than the proposed amount,1ince the seller could then withhold the product. Only the seller has an incentive to shirk.Or the timing might be reversed so that the buyer submits payment only after receipt of theproduct, in which case only he has an incentive to deviate. Thus, the trading interactionmay feature two-sided or one-sided moral hazard, depending on the timing of trade.It is readily apparent that for two-player repeated games, cooperation is easier to supportwith one-sided moral hazard than with two-sided moral hazard, because the latter has anadditional incentive constraint. We find that the difference is amplified by multilateralenforcement for a new and different reason: it permits some players to share informationabout others without having to worry about its consequences. We exposit this logic using theclass of permanent ostracism equilibria, and study how these equilibria perform at a fixeddiscount rate. We find that permanent ostracism supports substantially more cooperationwith one-sided moral hazard than with two-sided moral hazard.What is permanent ostracism? It embodies the idea that a trader, Ann, ceases to tradewith another, Bob, if she comes to learn that Bob has cheated in the past; however, Anncontinues trading with all partners whose reputations are untainted from her perspective. Westudy these equilibria for two reasons. First, its description matches market behavior wherepunishments are targeted towards a defector without making the entire market unravel. Second, permanent ostracism offers the simplest scheme in which traders’ reputations andrecords are used to punish or reward them. Thus, it has been the focus of many prior papers,most of which abstract from communication incentives. Our focus is on the effectiveness ofthese equilibria when traders strategically communicate about who is guilty and innocent.We study permanent ostracism equilibria at a fixed discount rate, and compare it totwo benchmarks. The first benchmark is the lower bound of bilateral enforcement , whichis the most a buyer-seller pair could credibly trade without any third-party punishment.The second benchmark is the upper bound of naive communication , which is the highestlevel of trade achievable if all non-defectors were forced to tell the whole truth, regardless ofincentives. Our main results are the following:
If each buyer-seller pair faces one-sided moral hazard, then permanent ostracismcan achieve the benchmark of naive communication. In contrast, if each buyer-seller pair faces two-sided moral hazard, no permanent ostracism equilibrium sup-ports more trade than bilateral enforcement.
This result has a clear strategic intuition. In multilateral enforcement, each trader takes This targeting of punishments towards defectors distinguishes permanent ostracism from contagion(Kandori 1992), where innocent players shirk on all others once cheated.
2n the role of monitoring each other by letting other market participants know if they observeany defections. However, a trader can be trusted only to the extent that he has more tolose in the future than he can gain by defecting. This raises the classic question of “whoguards the guardians?” With two-sided moral hazard, each side guards the other, so atrader is unwilling to reveal that she has been cheated because it reduces the degree towhich she herself can be trusted. With one-sided moral hazard, first-movers have no myopicincentive to shirk. Hence, they become guards who themselves need not be guarded, andtheir guardianship secures the cooperation of others. This difference is sufficiently stark thatour negative result holds even when traders obtain verifiable evidence and our positive resultobtains even when traders’ communication is cheap talk.While this result is simple, it elucidates an important point for the design and operationof markets: if word-of-mouth communication is to play a role in supporting trade, thereare significant gains from structuring trade (or externally enforcing cooperation) so that oneside of the market lacks an incentive to deviate. Doing so amplifies the level of supportabletrade, because that side of the market can be relied on to spread news and information.
Related Literature:
The role of word-of-mouth communication in trading relationshipshas been studied broadly. Many of these studies document the importance of communication(Greif 1993), or highlight how its speed and dynamics influence cooperation (Raub andWeesie 1990; Klein 1992; Ahn and Suominen 2001; Dixit 2003), but abstract from whethertraders have the motive to communicate truthfully. A vast literature, surveyed in Tadelis(2016), discusses the role of ratings and feedback in peer-to-peer and online markets. In thisliterature, the willingness of market participants to disclose their past experiences is oftenassumed rather than derived.Our work builds on the study of community enforcement, pioneered by Kandori (1992),Ellison (1994), Harrington (1995), and Okuno-Fujiwara and Postlewaite (1995). One strandof this work envisions that players have “reputational labels” that are updated based ontheir actions, implicitly assuming that players are sharing information about their past ex-periences. Our work offers a foundation for these analyses when the interactions involveone-sided moral hazard, and indicates a challenge when the moral hazard is two-sided. Other Another strand of community enforcement studies folk theorems that obtain in anonymous settings. Deb(2020) studies general games where players can announce names and authenticate them with their behavior.Deb and Gonz´alez-D´ıaz (2019) study how some cooperative outcomes are achievable in some games withoutcommunication. Most recently, Deb, Sugaya and Wolitzky (2020) prove a general folk theorem for anonymousrandom matching. Our work differs from this strand in that we study behavior with a fixed discount rateand focus explicitly on the role of communication in a non-anonymous environment.
Society comprises buyers N B ≡ { , . . . , B } and sellers N S ≡ { , . . . , S } ; let N ≡ N B ∪ N S .A generic buyer (“he”) is denoted by b ; a generic seller (“she”) is denoted by s . The networkof relationships in society features both trading links and communication links. Meetingsbetween buyers and sellers occur on trading links : each buyer-seller pair bs meets at randomtimes in [0 , ∞ ) at Poisson rate λ BS >
0. During these meetings, communication and tradeoccur.
Communication links involve meetings between players on the same side: each pairof buyers, bb ′ , meets at Poisson rate λ BB >
0, and each pair of sellers, ss ′ , meets at Poissonrate λ SS >
0, to communicate but not to trade. All meeting times are independent acrossthe network. Players share a common discount rate of r > t ,4nly those two players observe the timing of their meeting and what transpires. Below wedescribe the extensive form in each of these meetings.A buyer-seller interaction spans two stages: first the communication stage, in which theyexchange messages, and then the trading stage, in which they trade at a price p ≥ q ∈ [0 , ¯ q ]. A buyer-buyer interaction or a seller-seller interaction has only thecommunication phase. We describe the trading stage first. Trading Stage:
At this stage, the buyer chooses a payment p ≥ q ∈ [0 , ¯ q ]. When the buyer pays p and receives quality q , the buyer’spayoff is q − p and the seller’s payoff is p − c ( q ). We assume that c is strictly increasingand strictly convex, that c (0) = c ′ (0) = 0, and that both q − c ( q ) and c ( q ) /q are strictlyincreasing. We consider three distinct extensive forms for the trading stage:1. Simultaneous protocol : the buyer and seller make their choices simultaneously.2.
Buyer-first protocol : the buyer pays first, and upon receiving payment, the sellerchooses how much to deliver to the buyer.3.
Seller-first protocol : the seller delivers first, and upon receiving delivery, the buyerchooses how much to pay to the seller.The simultaneous protocol exhibits two-sided moral hazard, since each partner has a myopicgain from deviating. The latter two protocols exhibits one-sided moral hazard, because theparty moving first can be immediately punished.
Communication Stage:
When a pair interacts, along either a trading link or a communi-cation link, they engage in “polite cheap talk” where one of them is randomly selected withprobability to speak first, and then after her speech, her partner speaks. The messagespace enables them to exchange information about which players have deviated. Specifically,the message space is M ≡ N ; when player i sends message m in the equilibria we willconstruct, the interpretation is that player i is stating that m is the set of players who are“guilty” because they have deviated. Talk is cheap, so player i is free to send any messageregardless of his or her history. We study a class of equilibria that we call permanent ostracism equilibria. We describethe idea intuitively here, relegating formal details to the Appendix. A permanent ostracism The upper bound ensures that continuation payoffs remain bounded. We assume the bound is sufficientlyhigh that the constraint never binds. i ’s accounting, all such players begin as innocent. Player i mustdeem that partner j is innocent as long as player i has not obtained any indication to thecontrary. If player j has a myopic gain from deviating, then player i should reclassify player j as guilty if either of the following occur: (1) player j fails to exert expected effort or submitexpected payment when interacting with player i , or (2) any player k , when interacting withplayer i , sends a message m ∋ j . If player i deems partner j guilty, she permanently ceasestrading with partner j . Guilty or innocent, each player communicates truthfully about thebehavior of others, but not about herself. Off the equilibrium path, players’ beliefs reflecta correct understanding of the stochastic process governing how information about guiltdiffuses through the network given equilibrium behavior. We first describe a negative result for trading games exhibiting two-sided moral hazard.The benchmark for this result is bilateral enforcement with a simultaneous protocol . Underbilateral enforcement, the behavior within each buyer-seller relationship depends only onpast interactions between them, independently of others. In this benchmark strategy profile,each time they meet, the buyer pays p and the seller chooses quality q ; if either deviates, thepair responds by setting prices and quantities to zero in all future interactions. This grimtrigger punishment leads to the incentive constraints q ≤ q − p + Z ∞ e − rt λ BS ( q − p ) dt , (Buyer’s Bilateral IC) p ≤ p − c ( q ) + Z ∞ e − rt λ BS ( p − c ( q )) dt . (Seller’s Bilateral IC)Setting both inequalities to bind leads to the highest level of trade supportable by bilateralenforcement, q , which solves c ( q ) q = (cid:18) λ BS r + λ BS (cid:19) . Proposition 1.
With two-sided moral hazard, in every permanent ostracism equilibrium,the level of trade never exceeds q in any equilibrium path history. Here is the logic (the proof is in the Supplementary Appendix): if the level of tradeexceeds q , then Ann trusts Bob if she believes others are innocent and available to punishBob. But if Bob knows that others have deviated, and divulging this information to Alicewill make her trust him less, he has no incentive to do so. He is better off concealing it andshirking on Ann. Since all he needs to do is conceal, the conclusion holds even if Bob hasverifiable evidence that others have deviated. This section presents our main contribution: communication is incentive compatible at highlevels of trade if the stage game exhibits one-sided moral hazard. For concreteness, weconsider a buyer-first protocol; a similar result obtains for a seller-first protocol.Before proving our result, we first describe a naive communication benchmark that wouldbe relevant if innocent traders were forced to communicate truthfully in all of their interac-tions. A player who is guilty will be punished by every partner he meets who deems himguilty. The best a guilty player can hope is to cheat every unsuspecting partner he meets.This logic leads to the incentive constraints:0 ≤ q − p + S Z ∞ e − rt λ BS ( q − p ) dt , (Buyer’s IC) p + ( B − p v S B,S ≤ p − c ( q ) + B Z ∞ e − rt λ BS ( p − c ( q )) dt . (Seller’s IC)The Buyer’s IC reflects that so long as he is gaining from trade ( q ≥ p ), he must be betteroff from cooperating both myopically and in terms of discounted continuation value.By contrast, the Seller’s IC reflects that the seller has a myopic incentive to deviatewhenever c ( q ) >
0, and her motive for not doing so is to maintain cooperation with futurebuyers. In Seller’s IC, v S B,S describes the (discounted) probability that when the seller nextmeets a given buyer, he has not yet learned of her guilt; naturally, v S B,S depends on the rateat which information about a guilty seller diffuses in the market. There are several ways We show in Footnote 10 how to compute v S B,S recursively from primitives. i may cheat on another unsuspecting buyer, or a buyeror seller who knows seller i is guilty may pass on that information. It is only in the firstcase that seller i accrues payoffs from others learning about her guilt, and v S B,S captures thediscounted probability of her obtaining a payoff from these future defections. Accordingly,the term ( B − pv S B,S measures the discounted value of seller i ’s future opportunities tocheat other buyers in the future, before they have learned that she is guilty.Setting these constraints to bind and simplifying yields the maximum level of cooperationwith naive communication under a buyer-first protocol, q ∗ BF , which solves c ( q ) q = Bλ BS − ( B − rv S B,S r + Bλ BS . (1)This is our benchmark for high cooperation. All of the “social collateral” here is put on theside of the sellers: because buyers have no myopic incentive to deviate, they can be deterredfrom deviating within the stage game and don’t need to be rewarded or punished throughcontinuation play. By contrast, the seller is deterred by multilateral enforcement, where sheputs several relationships at risk if she cheats in any one relationship. Thus, she is willingto produce higher quality in each relationship than under bilateral enforcement. But this benchmark is an unsatisfactory description of behavior because it assumes thatinnocent players are simply forced to reveal the truth. Our positive result is that this bench-mark is attainable, even if players can strategically choose what to disclose: we construct apermanent ostracism equilibrium that attains this naive communication benchmark q ∗ BF .In this equilibrium, only sellers can be considered guilty; because a buyer has no incentiveto deviate, there is no reason to ostracize him. When a buyer meets a seller he believes isinnocent, he first pays p (regardless of whether he has deviated in the past). Then theseller produces quality q if she is innocent and price p was paid; she produces zero qualityotherwise. If the seller deviates in the trading stage, she becomes guilty, and thereafteraims to shirk on all future buyers too. Buyers, in turn, aim to ostracize guilty sellers whilecontinuing to trade at equilibrium-path levels with innocent sellers.Players communicate truthfully, with the exception that each seller never communicatesabout herself. Consequently, an innocent seller’s incentive constraint is Seller’s IC fromabove, enabling cooperation at price and quality p = q = q ∗ BF both on and off the equilibrium This benchmark uses both buyers and sellers as conduits of information about sellers’ behavior, so asto maximize the level of cooperation. Later, we discuss equilibria in which only buyers communicate. This holds for both bilateral enforcement under a simultaneous protocol, which yields quality q , and forbilateral enforcement under a seller-first protocol, which yields quality that solves c ( q ) q = λ BS r + λ BS . In this equilibrium, buyers have no incentive to deviate at the tradingstage. Moreover, neither sellers nor buyers have any incentive to lie. These observations leadto our main result:
Proposition 2.
With a buyer-first protocol, there exists a permanent ostracism equilibriumthat attains the naive communication benchmark.
The logic is that with a buyer-first protocol, an innocent seller meeting a buyer doesnot care whether that buyer has valuable relationships with other sellers—the buyer canbe trusted to pay in any case. Thus, we can make a buyer’s payoff from each interactionindependent of his message, which makes him willing to truthfully reveal whether othersellers have defected. Because buyers themselves are guarded—by having to move first—their communication guards the cooperation of others.We describe several attractive properties of this equilibrium. First, consistent with thetheme of personalized punishment, it ensures that a few rotten apples do not spoil collectivecooperation. Second, the equilibrium does not require coordination on a common calendar orstart date. Third, it may be that an intricate construction is needed to increase the averagelevel of trade beyond this equilibrium; standard constructions such as contagion equilibria(Kandori 1992; Ellison 1994) cannot.Before we prove this result, we make two further remarks. First, the same approachalso works for a seller-first protocol: a seller who deviates can be punished immediatelyby the buyer, while buyers need to be disciplined by permanent ostracism. A similar lineof reasoning leads to an expression comparable to (1) for the highest level of supportabletrade. One could investigate which of seller-first and buyer-first protocols is better; such acomparison subtly depends on the model’s primitives.Second, our model assumes that sellers impose no externalities on each other. In manysettings, sellers may generate externalities, and hence cannot be trusted to report truthfullyabout each other. Buyers can still be relied on, however, so one could construct an analo-gous equilibrium that uses only buyer-to-buyer communication. Because information wouldspread more slowly, the supportable level of trade would be lower than q ∗ BF . This argument is similar to how in contagion, contagion phase incentive constraints are necessarilysatisfied if cooperation phase incentive constraints hold with equality, as in Ellison (1994). We note that while in our construction, the buyer obtains zero payoffs in each relationship, this isunnecessary for communication incentives. All that is needed is that his payoff in any interaction with aninnocent or guilty seller is independent of his report. Clark, Fudenberg and Wolitzky (2020) also view this as an attractive property for multilateral enforce-ment and develop schemes in anonymous environments that do not require such coordination. roof of Proposition 2. We pair the strategy profile described above with a system of beliefsin which buyers believe that sellers are innocent until revealed to be guilty, either by theirown actions or via communication from other players. We consider the incentives of buyers,innocent sellers, and guilty sellers in turn.
Buyer Incentives:
At every interaction, the equilibrium prescribes that a buyer commu-nicate truthfully; following communication, the buyer pays p = q ∗ BF to a seller he deemsinnocent, and pays zero to a seller he deems guilty. Because the buyer’s payment does notaffect continuation play (with either the same seller or any other), his incentive to pay per-tains only to the current period. When facing an innocent seller, he has no myopic incentiveto deviate, because any deviation leads the seller to subsequently deliver zero quality. Whenhe meets a seller he deems guilty, he expects the seller to deliver zero quality regardless ofhis payment, so there is no incentive to pay more than zero. Finally, the buyer does not gainby deviating in the communication stage of any meeting, because, regardless of his message,he obtains the same payoff in every interaction. Innocent-Seller Incentives:
Both on and off the equilibrium path, an innocent seller’s in-centive constraint to produce quality q = q ∗ BF is Seller’s IC, which is satisfied with equality:in each case, she expects all buyers to continue communicating truthfully and cooperat-ing with her regardless of the guilt or innocence of any other seller and regardless of thepast deviations of any buyer. Similarly she expects all buyers and all other sellers, guiltyor innocent, to communicate truthfully about her guilt or innocence. Finally, there is nocontingency where a deviation from truthful communication improves her payoff. Guilty-Seller Incentives:
Equilibrium strategies prescribe that a guilty seller communi-cates truthfully about other sellers and produces zero quality. A guilty seller has no incentiveto misreport about other players, because her report doesn’t affect what happens in her cur-rent or subsequent interactions.We now prove that a guilty seller finds it incentive compatible to produce zero quality.Suppose, without loss of generality, that seller s ′ and buyer b ′ meet at time 0, and seller s ′ deviates and produces zero quality. Suppose that at time t , the seller meets buyer b ′′ . Ifseller s ′ already knows that buyer b ′′ deems her to be guilty—either because she has alreadydeviated in a prior meeting with b ′′ or because b ′′ told her so in the communication stage—then she has no incentive to deviate to higher quality. If instead b ′′ deems her innocentand pays the equilibrium-path price p = q ∗ BF , then the seller could deviate once to choosingquality q ∗ BF (so as to delay buyer 1 from learning of her guilt). A sufficient condition for the10eller to choose zero quality rather than deviate to q ∗ BF is if, for every k b ≥ k s ≥ q ∗ BF + q ∗ BF V ( k b + 1 , k s ) ≥ q ∗ BF − c ( q ∗ BF ) + q ∗ BF V ( k b , k s ), (2)where q ∗ BF V ( k b , k s ) is her expected continuation payoff when k b buyers and k s sellers (in-cluding herself) deem her guilty. Although seller s is uncertain about ( k b , k s ), if (2) holdspointwise for every realization of k b ∈ { , . . . , B } and k s ∈ { , . . . , S } , then it is sequentiallyrational for her to produce zero quality. Observe that (2) can be re-arranged to c ( q ∗ BF ) q ∗ BF ≥ V ( k b , k s ) − V ( k b + 1 , k s ) . We verify that this inequality is satisfied for every k b ≥ k s ≥
1. Because q ∗ BF bindsthe equilibrium path incentive constraint, a seller is just indifferent between the equilibriumpath and producing zero quality in every trading stage. Let q ∗ BF V (0 ,
1) be her continuationpayoff if she has been on the equilibrium path until now but plans to shirk on the next buyershe meets. Then her binding Seller’s IC can be re-written as q ∗ BF + q ∗ BF V (1 ,
1) = q ∗ BF − c ( q ∗ BF ) + q ∗ BF V (0 , . (4)Re-arranging (4) implies that c ( q ∗ BF ) q ∗ BF = V (0 , − V (1 , V ( k b , k s ) − V ( k b + 1 , k s ) ≤ V (0 , − V (1 , . (5) The value of V ( k b , k s ) is computed from a recursive system of equations: for k b ∈ { , . . . , B } and k s ∈ { , . . . , S } , let V ( k b , k s ) = Z ∞ e − rt e − ( λ BB k b ( B − k b )+ λ SS ( k s − S − k s )+ λ BS k b ( S − k s )+ λ BS k s ( B − k b )) t · λ BB k b ( B − k b ) V ( k b + 1 , k s ) + λ SS ( k s − S − k s ) V ( k b , k s + 1)+ λ BS k b ( S − k s ) V ( k b , k s + 1) + λ BS k s ( B − k b ) V ( k b + 1 , k s ) + λ BS ( B − k b ) ! dt ,(3)where V ( B, k s ) = 0 for each k s . Finally, let v SB,S ≡ V (1 , / ( B − λ BS ( B − k b ).
11e prove that (5) is satisfied for all k b = 0 , . . . , B and k s = 1 , . . . , S ,, adapting the argumentof Lemma 1 of Ellison (1994). We consider every sequence of link recognitions in which notwo links meet simultaneously, and then take expectations over them. Let ξ = ( τ z , ℓ z ) ∞ z =1 be asequence of link recognitions that take place in time span [0 , ∞ ), where ( τ z ) ∞ z =1 is the orderedlist of link recognition times and ( ℓ z ) ∞ z =1 is the list of links in their order of recognition. (Wedefine a link ℓ between player i and player j as a set { i, j } . But with some abuse of notation,we also say that ℓ ∈ A × B if either ( i, j ) ∈ A × B or ( j, i ) ∈ A × B .)Let K = ( K b , K s ) be the initial “ s -state”—the sets of buyers and sellers (respectively)who deem seller s guilty, at a start time normalized to zero. Then, if the sequence of linkrealizations is ξ , the s -state immediately following the interaction at time τ z is (cid:0) κ zb ( K , ξ ) , κ zs ( K , ξ ) (cid:1) = ( K b , K s ) if z = 0, (cid:0) κ z − b ∪ ( ℓ z ∩ N B ) , κ z − s (cid:1) if z > ℓ z ∈ ( N B \ κ z − b ) × ( κ z − b ∪ κ z − s ), (cid:0) κ z − b , κ z − s ∪ ( ℓ z ∩ N S ) (cid:1) if z > ℓ z ∈ ( N S \ κ z − s ) × (cid:0) ( κ z − b ∪ κ z − s ) \ { s } (cid:1) , (cid:0) κ z − b , κ z − s (cid:1) otherwise,where, with some abuse of notation, we write (cid:0) κ z − b , κ z − s (cid:1) for (cid:0) κ z − b ( K , ξ ) , κ z − s ( K , ξ ) (cid:1) .Define ˜ V (cid:0) K b , K s (cid:12)(cid:12) ξ (cid:1) to be the equilibrium continuation payoff of seller s when the initial s -state is K = ( K b , K s ) and the sequence of link recognitions is ξ . The change in seller s ’scontinuation payoff when one more buyer j deems her guilty at the outset, for any j ∈ N B ,˜ V (cid:0) K b , K s (cid:12)(cid:12) ξ (cid:1) − ˜ V (cid:0) K b ∪ { j } , K s (cid:12)(cid:12) ξ (cid:1) = ∞ X z =1 e − rτ z X b ∈N B q ∗ BF I (cid:0) ℓ z = { s, b } and b ∈ κ z − b (( K b ∪ { j } , K s ) , ξ ) \ κ z − b (( K b , K s ) , ξ ) (cid:1) ≤ ∞ X z =1 e − rτ z X b ∈N B q ∗ BF I (cid:0) ℓ z = { s, b } and b ∈ κ z − b (( { j } , { s } ) , ξ ) \ κ z − b (( ∅ , { s } ) , ξ ) (cid:1) = ˜ V (cid:0) ∅ , { s } (cid:12)(cid:12) ξ (cid:1) − ˜ V (cid:0) { j } , { s } (cid:12)(cid:12) ξ (cid:1) , (6)where I is the indicator function. The weak inequality follows from K zb (cid:0) ( K b ∪ { j } , K s ) , ξ (cid:1) \ K zb ( K , ξ ) ⊆ K zb (cid:0) ( { j } , { s } ) , ξ (cid:1) \ K zb (cid:0) ( ∅ , { s } ) , ξ (cid:1) ,since, for fixed ξ , the set of players who learn about seller s ’s deviation via a path through12uyer j is decreasing in the number of other players who initially know of her deviation.Observe that V (cid:0) | K b | , | K s | (cid:1) = E ξ ˜ V (cid:0) K b , K s (cid:12)(cid:12) ξ (cid:1) . Therefore, since (6) holds for almostevery ξ , taking the expectation over ξ yields (5). (cid:3) In markets where traders cannot contractually commit to their terms of trade, word-of-mouthcommunication is viewed to be a powerful incentive: traders may cut ties with those revealedto be defectors, while continuing business with non-defectors. We begin with the premisethat traders may not truthfully communicate who is guilty unless they have an incentive todo so. Based on this premise, we find that markets in which traders on only one side havea myopic incentive to shirk can support significantly higher volumes of trade than those inwhich traders on both sides face moral hazard. The rationale is that traders who lack amyopic incentive to shirk become “guardians” who communicate truthfully to others. Theirtruthful communication deters traders on the other side of the market from defecting.While our model is stylized, these results may help us better understand when ostracismsucceeds or fails in practice. Certain situations naturally take the form of a sequential-movegame. For example, in financial lending, a lender first decides how much to lend, and a bor-rower then decides whether to repay. Our results speak to why ostracism, with informationabout borrowers being shared by lenders, is credible and ubiquitous. Analogously, in thelong-distance trade model proposed by Greif (1993), merchants first decide whether to trustagents, and agents later decide to reward or exploit that trust. Thus, even though Greifdoes not model players’ incentives to report or withhold information, our results imply thatmerchants would have no incentive to withhold information.More recently, Bernstein (2015) documents a network of relationships among originalequipment manufacturers (OEMS) and their suppliers. Within each OEM-supplier relation-ship, supplier behavior is contractually specified in great detail but OEM behavior is not.Thus, a supplier has no legal recourse if the OEM steals its innovation and then puts pro-duction out for bid. Recognizing this problem of one-sided moral hazard, one OEM formeda “Supplier Council” to promote communication among suppliers. Through the lens of ourmodel, we interpret this setting as a seller-first protocol (where OEMs are buyers), and theCouncil as a communication device that increases the rate of communication among sellers.In other contexts, one may envision markets where enforcement intermediaries mitigateincentive issues on one side. For instance, in supply contracts where quality is not legallyenforceable, buyers are often given the right to withhold payment if they deem the qual-13ty delivered to be “non-conforming.” Similarly, franchising arrangements impose detailed,legally enforceable requirements on the details of franchisees’ business operations, but im-pose few requirements on franchisors (Blair and Lafontaine 2011). Such arrangements enablemultilateral enforcement because parties who no longer have a myopic incentive to deviateare truthful conduits of information. By contrast, Bolton, Greiner and Ockenfels (2013) discuss how before eBay paymentswere made through Paypal, both buyers and sellers could deviate, but a two-sided feedbacksystem failed to produce reliable reviews and discipline players. Once it was feasible tostructure payments through Paypal, so that buyers no longer needed to be rated, a one-sided feedback system has remained, and such feedback influences sellers’ payoffs. Otherplatforms continue to face issues of two-sided moral hazard. As discussed by Tadelis (2016),Airbnb owners can misrepresent their unit, leave it dirty, etc., and renters too can cheat.In such cases, our theory highlights why players may have strategic reasons not to reportdeviations they have observed.Our stylized model omits several considerations. We study only one motive to concealinformation, ignoring costs of communication and the possibility for retribution. Moreover,one may view permanent ostracism to take the principle of “ostracizing the guilty, cooper-ating with the innocent” to a logical extreme. Perhaps after several individuals have beenostracized, it need not be the case that innocent players continue to trade with other innocentplayers. Our results suggest that combining permanent ostracism with other schemes (e.g.,contagion) could be fruitful in settings with two-sided moral hazard, but are unnecessary inthose with one-sided moral hazard.
Appendix A Definition of Permanent Ostracism
In a permanent ostracism equilibrium, each player i has a personal state variable, ω i ⊂ N that liststhe players that i deems guilty. Player i ’s behavior in each interaction depends on the history in away that is measurable with respect to ω i . For brevity, we define permanent ostracism only forsimultaneous and buyer-first protocols.At the start of the game, ω i = ∅ . Under a simultaneous protocol, any player can becomeguilty. However, under a buyer-first protocol, buyers cannot become guilty (they have “immunity”),because only sellers are subject to moral hazard. We write the set of players with immunity as I = ∅ Our point complements Acemoglu and Wolitzky (2020), who show how community enforcement cansubtly improve enforcement intermediation whereas we focus on the reverse channel. Although ω i is a function of player i ’s private history, we suppress the history argument except whereneeded for clarity. or a simultaneous protocol, and I = N B for a buyer-first protocol. When a buyer b and a seller s meet, and their personal states at the start of the trading stage are ω b and ω s respectively, thenunder a simultaneous protocol the buyer should pay p ∗ bs ( ω b ) and the seller should deliver quality q ∗ bs ( ω s ). Under a buyer-first protocol, the buyer should pay p ∗ bs ( ω b ), and then seller should deliverquality q ∗ bs ( ω s ) if the buyer paid correctly, but deliver quality zero otherwise. When player i meetsplayer j at time t , his personal state updates at the end of each stage of the interaction. We write ω i − for his state at the start of the stage, and ω i + for his state at the end of the stage. At the end ofthe communication stage, after the partners exchange messages m i and m j , i ’s state updates from ω i − to ω + i = (cid:0) ω − i ∪ m i ∪ m j (cid:1) \ I . Then, again at the end of the trading stage, the state updatesfrom ω i − to ω i + as follows, for each ℓ ∈ { i, j } : • For a simultaneous protocol: If ℓ / ∈ I and player ℓ plays any action other than p ∗ ( ω i − ) (ifplayer ℓ is the buyer) or q ∗ ( ω i − ) (if player ℓ is the seller), then ℓ ∈ ω i + ; • For a buyer-first protocol:
If player ℓ is the seller and either (1) the buyer paid p ∗ ( ω i − ) and ℓ delivers quality not equal to q ∗ ( ω i − ), or (2) the buyer paid any amount other than p ∗ ( ω i − )and ℓ delivers quality not equal to zero, then ℓ ∈ ω i + ; • Otherwise ℓ ∈ ω i − ⇐⇒ ℓ ∈ ω i + . Definition 1.
An assessment (a strategy profile and a system of beliefs) is a permanent os-tracism assessment if there exists a price function p ∗ bs : 2 N → R + and quality function q ∗ sb : 2 N → [0 , ¯ q ] for each buyer-seller pair sb ; each player i ’s personal state ω i evolves according to the rulegiven above; and for every player i and every partner j = i , if i meets j at time t , the following aresatisfied:1. In the communication stage, i sends the message ω i \ { i } .2. In the trading stage, if the protocol is simultaneous,(a) if { i, j } ∩ ω i = ∅ then i pays p ∗ ij ( m i ∪ m j ) (if i is the buyer) or delivers q ∗ ij ( m i ∪ m j ) (if i is the seller);(b) if j ∈ ω i , then i pays 0 (if i is the buyer) or delivers 0 (if i is the seller).3. In the trading stage, if the protocol is buyer-first:(a) if i is the buyer: i pays p ∗ ij ( m i ∪ m j ) if j / ∈ ω i , but pays zero otherwise;(b) if i is the seller: i delivers q ∗ ij ( m i ∪ m j ) if i / ∈ ω i and j paid p ∗ ij ( m i ∪ m j ) , but deliverszero otherwise;4. When player i ’s state is ω i at the start of any communication or trading stage when interactingwith player j , player i assigns probability 1 to the event that ω j ⊆ ω i . The requirement on beliefs (Item 4) embodies ostracism: as long as player i has seen noindication—either directly or via messages from other players—that player k may have deviated, i should not believe that k has deviated and caused other players to deem k guilty. eferences Acemoglu, Daron and Alexander Wolitzky , “Sustaining Cooperation: Community Enforce-ment versus Specialized Enforcement,”
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Proof of Proposition 1.
Suppose towards a contradiction that there is an interaction between play-ers i and j at which their private histories at the start of the communication stage are ( h i , h j ),their personal states are ( ω i ( h i ) , ω j ( h j )), and they exchange messages m i and m j , such that at thestart of the trading stage both i and j deem both i and j innocent, and q ∗ ( m i ∪ m j ) > q .Consider another private history ˆ h i that coincides with h i except that every player other than i and j has transitioned to being deemed guilty by player i (so ω i (ˆ h i ) = N \ { i, j } ) after thelast interaction in h i . Suppose player j communicates first and sends message m j = ω j ( h j ). Ina permanent ostracism equilibrium, player i deems player j innocent, and so should report m i = N \{ i, j } truthfully. Then they should trade at quality ˆ q = q ∗ ( N \{ i, j } ) and price ˆ p = p ∗ ( N \{ i, j } ).Note that ˆ q ≤ q and ˆ p ≤ p , since they must employ bilateral enforcement in their relationship whilepermanently ostracizing all other players. However, if player i is the seller then a deviation in whichhe reports m i = ω i ( h i ) rather than ω i (ˆ h i ) and shirks yields a payoff of p ∗ (cid:0) ω i ( h i ) ∪ ω j ( h j ) (cid:1) > p = p − c ( q ) + Z ∞ e − rt λ ( p − c ( q )) dt ,where the first inequality is by our supposition, the equality is by definition of p and q . Similarly,if player i is the buyer, falsely reporting ω i ( h i ) and then reneging on payment yields a payoff of q ∗ (cid:0) ω i ( h i ) ∪ ω j ( h j ) (cid:1) > q = q − p + Z ∞ e − rt λ ( q − p ) dt .Adding these inequalities yields p ∗ (cid:0) ω i ( h i ) ∪ ω j ( h j ) (cid:1) + q ∗ (cid:0) ω i ( h i ) ∪ ω j ( h j ) (cid:1) > p + q = q − c ( q ) + Z ∞ e − rt λ ( q − c ( q )) dt ≥ ˆ q − c (ˆ q ) + Z ∞ e − rt λ (ˆ q − c (ˆ q )) dt ,where the last inequality follows from q ≥ ˆ q and q − c ( q ) being strictly increasing. Therefore, atleast one of these deviations is strictly profitable, and so we have reached a contradiction. (cid:3)(cid:3)