Compromise, Don't Optimize: A Prior-Free Alternative to Perfect Bayesian Equilibrium
aa r X i v : . [ ec on . T H ] M a r Compromise, Don’t Optimize:A Prior-Free Alternative to Perfect Bayesian Equilibrium
Karl Schlag and Andriy Zapechelnyuk
Abstract.
Perfect Bayesian equilibrium is the classic solution concept forgames with incomplete information, where players optimize under given beliefsover states. We introduce a new concept called perfect compromise equilibrium,where players find compromise decisions that are good in all states. This so-lution concept is tractable even if states are high dimensional as it does notrely on priors, and it always exists. We demonstrate the power of our solu-tion concept in prominent economic examples, including Cournot and Bertrandmarkets, Spence’s signaling, and bilateral trade with common value.
JEL Classification:
D81, D83
Keywords: compromise, prior-free, loss, robustness, perfect Bayesian equilib-rium, perfect compromise equilibrium, solution concept
Date : March 6, 2020.
Schlag : Department of Economics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria.
E-mail: [email protected].
Zapechelnyuk : School of Economics and Finance, University of St Andrews, Castlecliffe, theScores, St Andrews KY16 9AR, UK.
E-mail: [email protected]. Introduction
The classic solution concept in games of incomplete information is perfectBayesian equilibrium (PBE). However, many aspects surrounding this conceptraise doubts as to whether this is the way we should solve such games. Players areuncertain about the primitives and yet they are willing to add even more detailsby assigning probabilities to possible events. PBE are often difficult to computeor are intractable, thus confining many of our insights to be limited to very simpleexamples with two types.We set out to introduce a new concept for solving games with informational un-certainty , where players do not know something about the environment or aboutwhat others know. On the one hand, the concept should avoid the above pitfalls.It should not be based on a specific prior over uncertain events. It should allowdifferent players to see the world differently. It should be tractable in salient ex-amples. On the other hand, the concept should not be too different from PBE.Preferences of players should be comparable to those used in PBE. In particular,known random events, such as tossing a coin, should be treated as in PBE. Thereshould be common knowledge of the equilibrium strategies. So there will be strate-gic certainty . Challenges are multifold, such as avoiding dynamic inconsistency,modeling learning in the game, and holding off the criticism of those who wouldlike us to choose a different model of optimization under uncertainty. Our solutionincludes a new rhetoric based on balancing losses and finding compromises. Ourexamples reveal the tractability of our concept. Our conviction is to provide analternative to PBE which is reasonable and insightful in applications.The key ingredient to our approach is that players do not optimize, they com-promise. Traditionally, players optimize with respect to some beliefs. However,this is not possible in our model as we do not assume a prior over uncertain events.Instead, players compromise. They look in each possible event and the loss of notplaying a best response in this event, and then make a choice that balances theselosses across all events.Our new solution concept for games with incomplete information is called perfectcompromise equilibrium (short, PCE). Uncertainty is captured by including a setof states (or events). At the outset, nature chooses one of these states, but noneof the players knows which state has been chosen. Each later information set isassigned a subset of states, those that are conceivable for the player who movesat that information set. Information about the state might be revealed to someof the players during the game, like a buyer learning her value. This is modeledby a move of player 0 who is in charge of all chance moves. The mixed strategyof player 0 is common knowledge. The rest of the game unfolds like a standardextensive form game. For each information set, there is a set of actions of the playerwho is making the choice, a set of conceivable states, and a belief, conditional ona conceivable state, over the decision nodes within this information set. Theseelements allow us to compute expected payoffs at each information set conditionalon each conceivable state using Bayes’ rule. The performance of an action at an
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 3 information set is evaluated for a conceivable state by comparing its payoff tothe best payoff that could have been attained if one had known the state andhad been allowed to choose a different action at this information set. We refer tothis difference as the loss at that information set. Actions are evaluated by their maximum loss across all conceivable states. An action is called a best compromise if no other action at this information set has a lower maximum loss.Analogous to sequential rationality in PBE, we impose a minimal consistencyrequirement on the conceivable states. A state that is conceivable at an informa-tion set must remain conceivable at all information sets that can be reached underthis state given the strategy profile and the beliefs.The elimination of priors over uncertain events comes with numerous advan-tages in comparison to PBE. Solutions are often easier to obtain. They are moreparsimonious as they do not change with a prior. Solutions are more intuitive asthey are simple and depend on concrete model primitives. Preferences conditionon observables and not on fictitious distributions. Solutions can be justified infront of others who have different priors. Uncertainty and ambiguity enter themodel distinctly.We do not introduce a new non-expected utility method for evaluating strate-gies. The evaluation of a strategy under a given state is as under PBE. Theterm “compromise” reflects the fact that the equilibrium strategy will be evalu-ated across all states, regardless of which state has been realized. The adjective“perfect” refers to the fact that optimization takes place at all information sets,not only at the outset.Learning about the uncertainty is modeled by updating conceivable sets. Dy-namic inconsistency that potentially arises by a player incorporating differentworst cases at different information sets is avoided by imposing deviations only atthe information set where a strategy is being evaluated. Future choices are notquestioned at that point as the player anticipates that she will reoptimize at eachfuture information set.Our concept has nice properties. It allows for addressing problems with richuncertainty that are intractable under PBE. It collapses to PBE is there is onlya single state. It always exists under standard assumptions. Players will makerational moves as strictly dominated strategies will not be chosen.Four examples are presented to show how the PCE concept applies. We firstconsider common uncertainty and investigate Cournot competition with unknowndemand. Demand is a state drawn by nature. Firms only know that it is boundedby two given linear functions. Next, we consider private information and analyzeBertrand competition where each firm only knows its own marginal cost. We thenmove on to signaling and look at Spence’s job market. Firms only know thatthe cost-productivity combinations of the worker are bounded by two given linearfunctions. Finally, we consider asymmetric information with common value andanalyze sequential bilateral trade where one side has private information.
SCHLAG AND ZAPECHELNYUK
We are proud to highlight some aspects of our findings in these examples interms of realism, good compromises, and new insights. All our examples are morerealistic than those found in the literature, as we do not have to confine ourselvesto parametric models of uncertainty and to models with two states (high and low).Our examples are concerned with strategic decision making under rich uncertaintywhere states are high dimensional and PBE analysis is no longer tractable. Com-promise values are negligible in the Cournot and Bertrand competition settings.In these contexts it makes little sense to think in more detail about which stateis really the true one, as payoffs would only be slightly higher in some states butcould be substantially lower in other states. New insights appear. We find thatadding uncertainty makes firms more competitive under Cournot competition andless competitive under Bertrand competition. In the separating equilibrium ofSpence’s job market signaling game, better educated workers are not necessarilymore productive, unlike in the classic model with two types. In the sequentialbilateral trade with common value, we find that trade is possible, as opposed tothe famous no-trade theorem for PBE in this context (Milgrom and Stokey, 1982).Under PCE the possibility that the trading partners have different valuations leadsto trade with positive probability, as ignoring this possibility generates losses thatthe traders want to minimize. Under PBE there is no trade as the trading partnersalways agree on the expected valuation of the good.
Related Literature.
Our paper contributes to the literature on robustness andambiguity in games of incomplete information.The most related paper to ours is Hanany, Klibanoff and Mukerji (2018). Theyconsider a game of incomplete information in which traditional players are replacedby players with smooth ambiguous preferences, as introduced in Klibanoff, Marinacci and Mukerji(2005). Hanany, Klibanoff and Mukerji (2018) show how to update informationin a dynamically consistent fashion, and this updating satisfies the one shot devi-ation principle. Hence, their approach is better founded in the axiomatic context.However, the intricacies that emerge from their mathematical formulation maketheir solution concept complex, and impede finding tractable solution in exampleswith rich state spaces, like the ones in this paper.An important ingredient of our solution concept is our use of compromise formaking choices when the true state is unknown. A popular alternative approach inthe literature on ambiguity is maximin preferences (Wald, 1950; Gilboa and Schmeidler,1989). These preferences have been brought to simultaneous-move games with in-complete information and multiple priors by Epstein and Wang (1996), Kajii and Morris(1997), Kajii and Ui (2005), and Azrieli and Teper (2011). While this approachcan be suitable in many applications, it leads to unintuitive results in our ex-amples. To obtain nontrivial results, additional structural assumptions need tobe added, such as assuming knowledge of the mean state, which reintroduces thepriors that we are trying to eliminate.Our idea of best compromise has origins in minimax regret (Savage, 1951) andconnects to ε optimality. Our optimization criterion differs from minimax regret PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 5 as evaluation occurs at each information set, while minimax regret traditionallyevaluates regret ex post. For an investigation of minimax regret under strategic un-certainty see Linhart and Radner (1989), and under partial strategic uncertaintysee Renou and Schlag (2010).PCE can be considered as a generalization of ex post Nash equilibrium (Cremer and McLean,1985). It can be thought of as an ε -ex post Nash equilibrium in which the small-est possible value of ε is chosen for each player. As a concept, ε -Nash equilibrium(Radner, 1980) is usually seen as a play under the restriction that deviations areonly undergone if payoff improvements are substantial. Our interpretation is dif-ferent. The value of ε does not measure the inertia that needs to be overcome, butinstead it measures the compromise needed to accommodate all possible states.In particular, the threshold ε is endogenous in a PCE.Stauber (2011) analyzes the local robustness of PBE to small degrees of am-biguity about player’s beliefs. In particular, it does investigate how the playersadjust their play to this ambiguity, unlike our paper.In fact, PCE can be interpreted as a globally robust version of PBE whererobustness (Huber, 1965) means to make choices that also perform well if the modelis slightly misspecified. Being a compromise, our suggested strategies perform wellin each state given how others make their choices, never doing too badly relativeto what could be achieved in that state. This stands in contrast to the maximinutility approach that focuses attention on the state where payoffs are lowest.We proceed as follows. In Section 2 we introduce our solution concept. InSection 3 we illustrate PCE in four self-contained examples. Section 4 concludes.All proofs are in Appendix A. Some additional examples are in Appendix B.2. Perfect Compromise Equilibrium
We introduce a solution concept called perfect compromise equilibrium (PCE) .The essential difference of PCE from perfect Bayesian equilibrium (PBE) is asfollows. In PBE, players choose strategies that are best under their beliefs aboutan uncertain state. Instead, in PCE players choose compromises that are good inall realizations of the state. When a player makes a move, she evaluates her actionin each state by her loss relative to the best payoff in that state, and then finds abest compromise action that achieves the lowest maximum loss.A formal definition of PCE is presented in Section 2.1 below. A reader whowishes to be spared with the formalities and seeks to understand the essenceof PCE and its applicability can jump to Section 3 that presents self-containedexamples.2.1.
Formal Setup.
Consider a finite extensive-form game described by ( N , G , u,s ), where N = { , , ..., n } is a set of players, G is a finite game tree, u is a profileof payoff functions, and s is a strategy of player 0 who is nonstrategic.Game tree G describes the order of players’ moves, players’ information sets, andactions that are available at each information set. It is defined by a set of linked SCHLAG AND ZAPECHELNYUK decision nodes and terminal nodes that form a tree. Each decision node is assignedthree elements: a player i , an information set φ that contains this decision node,possibly, together with other decision nodes that player i cannot distinguish, anda set of actions available to player i at that information set. Information sets andaction sets satisfy the standard assumptions of games with perfect recall. Let φ be the initial decision node of the game, let Φ be the set of all information setsexcept φ , and let T be the set of terminal nodes of the game. Let i ( φ ) be theplayer that makes a move at an information set φ , and let A ( φ ) be a finite set ofactions available at φ .The game starts with a move of nature. Nature moves only once, at the initialdecision node φ . An action of nature ω is called state and is chosen from a finiteset Ω of available states, so Ω = A ( φ ).The game terminates after finitely many moves at some terminal node, andplayers obtain payoffs. A payoff function of each player i ∈ N specifies the payoff u i ( τ ) of player i at each terminal node τ ∈ T . Player 0 is nonstrategic, so weassume that her payoff is always zero.A strategy of each player i ∈ N prescribes a mixed action s i ( φ ) for each infor-mation set φ ∈ Φ in which i makes a move, so i ( φ ) = i and s i ( φ ) ∈ ∆( A ( φ )). Player0 is non-strategic and follows an exogenously given strategy s . A strategy profile s describes the behavior of all players throughout the game, so s ( φ ) ∈ ∆( A ( φ ))for each φ ∈ Φ and each state ω ∈ Ω.Like in Bayesian games, we specify not only strategies, but also beliefs of theplayers. The crucial difference is that, in our setting, the players do not formpriors about the move of nature, that is, about states in Ω. Instead, ex ante theyconsider all states as possible. A player can rule out the possibility of some statesby being in an information set which cannot be reached under these states. Thus,the belief in each information set is decomposed into two elements: conceivable set and posterior . A conceivable set deals with the player’s ambiguity about the state.A posterior deals with the uncertainty that arises through probabilistic moves ofother players at earlier information sets.A state ω is called conceivable at an information set φ ∈ Φ if, upon reaching φ ,the possibility that the true state is ω is not ruled out. A conceivable set B ( φ )is a set of states that are conceivable at φ . Formally, for each information set φ ∈ { φ } ∪ Φ, B ( φ ) is a nonempty subset of Ω, with the convention that all statesare initially conceivable, so B ( φ ) = Ω.A posterior β ( φ | ω ) assigns to each information set φ ∈ Φ and each state ω ∈ B ( φ ) a probability distribution over decision nodes in φ conditional on the statebeing ω .Like in PBE, we will require consistency of beliefs. Definition 1.
A profile (
B, β ) of conceivable sets and posteriors is consistent witha strategy profile s if the following conditions hold for all ω ∈ Ω.(a) Let φ ′ ∈ Φ. If there does not exist a path in G from φ to φ ′ in whichnature’s move at φ is ω , then ω B ( φ ′ ). PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 7 (b) Let φ ∈ Φ and let α be a decision node in φ such that β ( φ | ω )( α ) > φ ′ ∈ Φ be an information set that is reached from node α in one moveunder s with a strictly positive probability. Then ω ∈ B ( φ ′ ) and β ( φ | ω )follows Bayes’ rule.Condition (a) stipulates which states must be excluded from the conceivableset. Every state under which the current information set cannot be reached nomatter what actions players choose must be ruled out. Condition (b) stipulateswhich states must be included in the conceivable set and how the posteriors arecomputed. Every state under which the current information set is reachable froman earlier information set under a given strategy profile is conceivable and can-not be ruled out. The posteriors follows Bayes’ rule whenever possible given theposteriors in the earlier information sets.We now define PCE. Consider a profile of strategies, conceivable sets, and pos-teriors ( s, B, β ). Let ¯ u i ( x i , s | ω, φ, β ) denote the expected payoff of player i fromchoosing a mixed action x i ∈ ∆( A ( φ )) in an information set φ where i makes hermove, so i ( φ ) = i , conditional on the state being ω ∈ B ( φ ) and assuming that theplay is given by s elsewhere in the game. The payoff differencesup a i ∈ A ( φ ) ¯ u i ( a i , s | ω, φ, β ) − ¯ u i ( x i , s | ω, φ, β )is called player i ’s loss from choosing mixed action x i at information set φ whenthe state is ω . It describes how much better off player i could have been atthis information set under state ω if, instead of choosing x i , she had chosen thebest action in this state, without changing actions prescribed by s in any otherinformation set. The maximum loss of player i from choosing mixed action x i ininformation set φ where i = i ( φ ) is given by l ( x i , s | φ, β ) = sup ω ∈ B ( φ ) sup a i ∈ A ( φ ) ¯ u i ( a i , s | φ, ω, β ) − ¯ u i ( x i , s | φ, ω, β ) ! . So the maximum (supremum) is sought over all states that are conceivable forplayer i at φ .Our equilibrium concept requires strategies to be chosen optimally in the senseof minimizing the players’ maximum losses given their beliefs, and the beliefs tobe consistent given the strategies. Definition 2.
A profile ( s, B, β ) is called a perfect compromise equilibrium if(a) each player chooses a best compromise in each of her information sets, so foreach φ ∈ Φ, strategy s i of player i = i ( φ ) minimizes her maximum loss at φ : s i ( φ ) ∈ arg min x i ∈ ∆( A ( φ )) l ( x i , s | φ, β ) , (1)(b) profile ( B, β ) of conceivable sets and posteriors is consistent with strategyprofile s . Remark 1.
In some applications, it is unrealistic to assume that players canchoose mixed actions. Our definition of PCE can be easily adjusted if players are
SCHLAG AND ZAPECHELNYUK only allowed to use pure actions. In this case, each player minimizes her maximalloss among her pure actions, so instead of (1) we require s i ( φ ) ∈ arg min a i ∈ A ( φ ) l ( a i , s | φ, β ) . (1 ′ )2.2. Properties of Perfect Compromise Equilibrium.
Let us mention a fewproperties of PCE.First, we establish existence of PCE.
Theorem 1.
A perfect compromise equilibrium exists.
The proof is in Appendix A.1.Second, PCE is equivalent to PBE when there is certainty about the state, sothe game is of complete information. In such a game, where the set of statesΩ is a singleton, conceivable sets are all singletons. So, an action minimizes themaximum loss of a player if and only if it is a best response.Third, an ex post Nash equilibrium (if it exists) is a PCE. Here, the term expost refers to the game in which the state (or move of nature) is observed byall. In an ex post Nash equilibrium, regardless of the realized state, each player’sstrategy in each information set is a best response. Thus, the maximum loss ineach information set is zero. No other strategy can further reduce this loss. So,this is a PCE in which all players have zero losses.Finally, the concept of PCE respects dominance and iterated dominance. Wesay that an action a i ∈ A ( φ ) at an information set φ is strictly dominated forplayer i = i ( φ ) if there exists a mixed action x i ∈ ∆( A ( φ )) such that player i ’spayoff from choosing a i is strictly worse than that from choosing x i , regardless ofthe state ω ∈ Ω and of the choices of other players at any of their informationsets. Iterated dominance is defined in a standard way: after having excludedactions that were strictly dominated in previous rounds, one checks the dominancecondition w.r.t. the remaining actions of each player. Observe that if an action a i is strictly dominated, then it cannot be a best compromise at the correspondentinformation set, and thus it cannot be a part of any PCE. This argument can beiterated, so any iterated strictly dominated action cannot be a part of any PCE.3. Examples
We illustrate our solution concept in a few applications that are prominent inthe literature. We consider Cournot and Bertrand duopoly, Spence’s job marketsignaling, and bilateral trade. Moreover, in Appendix B we analyze a forecastingproblem and a public good game.In these applications, actions traditionally belong to an interval or to the positivereals. The concept of PCE is easily extended to allow for infinite sets. Alterna-tively, one can discretize the sets of actions and states.Traditionally, uncertainty is incorporated in such models in a very simple fash-ion, often only considering two states, high and low. We consider richer (in some
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 9 cases, infinite dimensional) sets of uncertain events in order to capture more real-istic uncertainty.3.1.
Cournot Duopoly with Unknown Demand.
We investigate how twofirms compete in quantities when neither firm knows the demand.There are two firms that produce a homogeneous good. For clarity of exposition,we assume that there are no costs of production. Each firm i = 1 , q i ≥ P ( q + q ). Each firm i ’s profit is givenby u i ( q i , q − i ; P ) = P ( q i + q − i ) q i , i = 1 , . Neither firm knows the inverse demand P , they only know a set that contains P .Let¯ P ( q ) = ¯ a − ¯ bq and ¯ P ( q ) = ¯ a − ¯ bq, where ¯ a ≥ ¯ a > a/ ¯ b ≥ ¯ a/ ¯ b > . We assume that P belongs to the set P of inverse demand functions that satisfy P ( q ) is continuously differentiable,¯ P ( q ) ≤ P ( q ) ≤ ¯ P ( q ) and ¯ P ′ ( q ) ≤ P ′ ( q ) ≤ ¯ P ′ ( q ) . (2)A firm i ’s maximum loss of choosing quantity q i when the other firm chooses q − i is given by l i ( q i , q − i ) = sup P ∈P sup q ′ i ≥ u i ( q ′ i , q − i ; P ) − u i ( q i , q − i ; P ) ! . The maximum loss describes how much more profit firm i could have obtained if ithad known the demand P when anticipating the other firm to produce q − i . Firm i ’s best compromise is a quantity q ∗ i that achieves the lowest maximum loss for agiven choice q − i of the other firm: q ∗ i ∈ arg min q i ≥ l i ( q i , q − i ) . A strategy profile ( q ∗ , q ∗ ) is a perfect compromise equilibrium if each firm choosesa best compromise given the choice of the other firm.This application can be embedded in our formal setting as described in Section2. As the demand P is unknown, it is identified as the state. So, the set ofstates is P . In the formal game, first nature chooses the state, and then two firmssimultaneously choose their quantities without observing the state. Conceivablesets and posteriors are trivial, as this is a simultaneous move game with no privateinformation. Proposition 1.
There exists a unique perfect compromise equilibrium whose strat-egy profile ( q ∗ , q ∗ ) is given by q ∗ i = 13 (cid:16) √ ¯ b + √ ¯ b (cid:17) (cid:18) ¯ a √ ¯ b + ¯ a √ ¯ b (cid:19) , i = 1 , . (3) The associated maximum losses are l i ( q ∗ i , q ∗− i ) = (¯ a ¯ b − ¯ a ¯ b ) b ¯ b (cid:16) √ ¯ b + √ ¯ b (cid:17) , i = 1 , . (4)The proof is in Appendix A.2.Let us discuss the strategic concerns underlying the PCE in this game. Eachfirm i , when deciding about the quantity to produce and facing unknown demand,worries about two possibilities. It could be that the demand is actually very high,so the firm is losing profit by producing too little. The greatest such loss occurswhen the demand is the highest, so P = ¯ P . Alternatively, it could be that thedemand is actually very low, so the firm is losing profit by producing too much.The greatest such loss occurs when the demand is the lowest, so P = ¯ P . The firmthus chooses the best compromise q ∗ i that balances these two losses, assuming thatthe other firm follows its equilibrium strategy q ∗− i . Remark 2.
It is generally intractable to find a PBE in this game with such arich set of possible demand functions. It can only be done under very specific,degenerate priors about the demand. For example, PBE can be found if the priorhas support only on linear demand functions, P ( q ) = a − bq . Note that whenthe slope is known, so ¯ b = ¯ b = b , and the intercept a is uniformly distributedon [¯ a, ¯ a ], then the firms’ strategies in PBE and PCE are identical and given by q ∗ = q ∗ = (¯ a + ¯ a ) / (6 b ). Remark 3.
Our equilibrium analysis can shed light on how the firms’ behaviorchanges in response to increasing uncertainty. For comparative statics, let usconsider as a benchmark a linear demand function P ( q ) = a − b q . We normalizethe constants a and b such that the monopoly profit is equal to 1, that is,sup q ≥ ( a − b q ) q = a b = 1 . Suppose that there is a small uncertainty of the magnitude ε about the demandrelative to the benchmark. Specifically, for ε > P ( q ) satisfy (2) where¯ P ( q ) = (cid:16) − ε (cid:17) a − (cid:16) ε (cid:17) b q and ¯ P ( q ) = (cid:16) ε (cid:17) a − (cid:16) − ε (cid:17) b q. Denote by q ε = ( q ε , q ε ) the strategies of the PCE as given by Proposition 1. Wethen obtain d q εi d ε = 2 ε a + O ( ε ) > . So the firms optimally respond to the growing uncertainty about the demand byincreasing their output, and do so at an increasing rate as ε grows. Next, considerthe associated maximum losses l i ( q εi , q ε − i ) = ε + O ( ε ) , i = 1 , . Moreover, if ε = 0 .
1, then l i ( q εi , q ε − i ) ≈ .
01. So the firms lose no more than about1% of the maximum profit due to not knowing the demand.
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 11
Bertrand Duopoly with Private Costs.
We now consider how two firmscompete in prices when the cost of the rival firm is unknown.There are two firms that produce a homogeneous good. Each firm i = 1 , p i . Choices are made simultaneously. The consumers only buyfrom the firm that offers a lower price. So the quantity that firm i sells is givenby q i ( p i , p − i ) = Q ( p i ) , if p i < p − i ,Q ( p i ) / , if p i = p − i , , if p i > p − i , where Q ( p ) is the demand function. For clarity of exposition we assume that thedemand function is given by Q ( p ) = max (cid:26) a − pb , (cid:27) The cost producing q i units is c i q i . Each firm i ’s profit is given by u i ( p i , p − i ; c i ) = ( p i − c i ) q i ( p i , p − i ) , i = 1 , . Each firm i knows her own marginal cost c i but not that of the other, and it iscommon knowledge that c , c ∈ [¯ c, ¯ c ] , where 0 ≤ ¯ c ≤ ¯ c ≤ a/ i ’s pricing strategy p ∗ i ( c i ) describes its choice of the price given its marginalcost c i .For each marginal cost c i , firm i ’s maximum loss of choosing a price p i whenfacing pricing strategy p ∗− i of the other firm is given by l i ( p i , p ∗− i ; c i ) = sup c − i ∈ [¯ c, ¯ c ] sup p ′ i ≥ u i ( p ′ i , p ∗− i ( c − i ); c i ) − u i ( p i , p ∗− i ( c − i ); c i ) ! . The maximum loss describes how much more profit i could have obtained if ithad known the other firm’s marginal cost c − i , anticipating the other firm to followthe pricing strategy p ∗− i . Firm i ’s best compromise given c i is a price p ∗ i ( c i ) thatachieves the lowest maximum loss for a given strategy p ∗− i of the other firm: p ∗ i ( c i ) ∈ arg min p i ≥ l i ( p i , p ∗− i ; c i ) . A strategy profile ( p ∗ , p ∗ ) is a perfect compromise equilibrium if each firm i choosesa best compromise given its marginal cost c i and the strategy p ∗− i of the other firm.This application can be embedded in our formal setting as described in Section2. As the firms’ marginal costs are not common knowledge, the pair ( c , c ) isidentified as the state. So the set of states is C = [¯ c, ¯ c ] . In the formal game, firstnature chooses a state ( c , c ), then each firm i observes its own cost c i , and thenthe two firms simultaneously choose their prices. A firm’s conceivable set containsall cost pairs that include their own cost. The posteriors are trivial, as this is asimultaneous move game. Proposition 2.
There exists a unique perfect compromise equilibrium whose strat-egy profile p ∗ = ( p ∗ , p ∗ ) is given by p ∗ i ( c i ) = 12 (cid:16) a + c i − p ( a − ¯ c ) + (¯ c − c i ) (cid:17) , i = 1 , . (5) The associated maximum losses are l i ( p ∗ i ( c i ) , p ∗− i , c i ) = ( a − ¯ c )(¯ c − c i )2 ≤ ( a − ¯ c )(¯ c − ¯ c )2 , i = 1 , . (6)The proof is in Appendix A.3.Let us discuss the strategic concerns underlying the PCE in this game. Eachfirm i , when deciding about the price p i > c i and facing unknown cost of theother firm, worries about two possibilities. It could be that the other firm choosesa weakly lower price p − i ≤ p i . Thus, firm i could have obtained more profitby undercutting p − i . The greatest such loss occurs when the other firm’s pricemarginally undercuts p i . Alternatively, it could be that the other firm chooses ahigher price, p − i > p i . Thus, firm i is losing profit by charging too little. Thegreatest such loss occurs when the other firm’s cost is the highest possible, ¯ c .The firm thus chooses the best compromise p ∗ i ( c i ) that balances these two losses,assuming that the other firm follows its equilibrium strategy.We find that the PCE price p ∗ i ( c i ) is strictly increasing in c i and lies strictlyabove its marginal cost c i whenever c i < ¯ c . Moreover, p ∗ i (¯ c ) = ¯ c . So, any salewith the cost below ¯ c leads to a positive profit. The fact that the price does notlie above ¯ c is intuitive. It is common knowledge that the costs are at most ¯ c , soif the prices were above ¯ c , each firm would have incentive to undercut the other,regardless of what its cost is. Also, the largest price cannot lie below ¯ c , as a firmwith cost ¯ c will charge the price ¯ c in order to ensure a loss equal to zero.Note that the lowest price p ∗ i (¯ c ) is strictly positive, even if ¯ c = 0. This is becausewhen the price is very low, then the potential loss due to not undercutting theother firm is small, while the potential loss due to not setting a price much higheris large. This has an upward effect on prices. Remark 4.
It is generally intractable to find a PBE in this application under anyreasonable prior, even in this simplest setting with linear demand and constantmarginal costs. The PBE strategy profile for this simplest setting is implicitlydefined by a differential equation with no closed form solution (see Spulber, 1995).
Remark 5.
As in Section 3.1, our equilibrium analysis can shed light on how thefirms’ behavior changes in response to increasing uncertainty. For comparativestatics, let us consider as a benchmark marginal cost c = a/ ≤ c i ≤ a/
2, so c = a/ a and b of the demand function Q ( p ) = ( a − p ) /b such that a = 1 and the monopolyprofit is equal to 1, that is,sup p ≥ ( p − c ) a − pb = ( a − c ) b = 1 . PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 13
Suppose that there is a small uncertainty of the magnitude ε about the privatecost relative to the benchmark. Specifically, for 0 < ε < c i ∈ [¯ c, ¯ c ], i = 1 , c = (cid:16) − ε (cid:17) c and ¯ c = (cid:16) ε (cid:17) c . Denote by p ε = ( p ε , p ε ) the PCE strategy profile as given by Proposition 2. Wethen obtain d p εi ( c i )d ε = ( a + c i − c ) c p ( a − ¯ c ) + (¯ c − c i ) > , because, using our assumptions on the parameters, a + c i − c ≥ a − c = 1 − (cid:16) ε (cid:17) c = 14 (2 − ε ) > . So the firms optimally respond to the growing uncertainty about the demandby increasing their prices. They become less competitive. Next, consider theassociated maximum losses: l i ( p εi ( c i ) , p ε − i , c i ) ≤ ε − ε , i = 1 , . Moreover, if ε = 0 .
1, then the maximum losses are bounded by 0 .
01. So the firmslose no more than about 1% of the maximum profit due to not knowing the costof the other firm.3.3.
Job Market Signaling.
Here we investigate Spence’s job market signalingwhen the worker’s cost of education is unknown to the firms.There is a single worker and two firms. The worker has productivity θ ∈ [0 , e L ) or high ( e H ), tosignal her productivity to the firms. The cost of low education is zero. The costof high education depends on the worker’s productivity and is given by c ( θ ). Thefirms observe the worker’s education level e and simultaneously offer wages w and w . The worker chooses the better of the two wages. Her payoff is given by v ( w , w , e ; θ, c ) = max { w , w } − ( , if e = e L ,c ( θ ) , if e = e H . Each firm i ’s payoff is given by u i ( w i , w − i ; θ ) = θ − w i , if w i > w − i , ( θ − w i ) / , if w i = w − i , , if w i < w − i . The worker knows her productivity type θ and the cost of high education c ( θ ).The firms know neither. They only know that θ ∈ [0 ,
1] and that the cost function c is bounded by two functions, ¯ c and ¯ c . Specifically, let¯ c ( θ ) = 1 − bθ and ¯ c ( θ ) = 1 + δ − bθ, where 0 ≤ δ < b ≤ . The firms know that c ( θ ) is strictly decreasing,¯ c ( θ ) ≤ c ( θ ) ≤ ¯ c ( θ ) for all θ ∈ [0 , . (7)Define Ω to be the set of all pairs ( θ, c ) of productivities θ ∈ [0 ,
1] and cost functions c ( θ ) that satisfy (7).The worker’s strategy e ∗ ( θ, c ) describes her choice of the education level for eachpair ( θ, c ) ∈ Ω. Each firm i ’s strategy w ∗ i ( e ) describes its wage offer conditionalon each education level e ∈ { e L , e H } . In addition, each firm has a conceivable set B i ( e ). This is the set of all pairs ( θ, c ) that firm i considers possible after observingthe level of education e ∈ { e L , e H } .A conceivable set B i ( e ) is consistent with the worker’s strategy e ∗ if it includesall pairs ( θ, c ) under which the worker chooses e , so ( θ, c ) ∈ B i ( e ) if e ∗ ( θ, c ) = e .For each education level e , firm i ’s maximum loss of choosing wage w i when theother firm chooses the wage according to strategy its w ∗− i is given by l i ( w i , w ∗− i ; e ) = sup ( θ,c ) ∈ B i ( e ) sup w ′ i ≥ u i ( w ′ i , w ∗− i ( e ); θ ) − u i ( w i , w ∗− i ( e ); θ ) ! . The maximum loss describes how much more profit firm i could have obtained ifit had known the true productivity of the worker, anticipating that the other firmfollows its strategy w ∗− i . Firm i ’s best compromise given e is a wage w ∗ i ( e ) thatachieves the lowest maximum loss for a given strategy w ∗− i of the other firm: w ∗ i ( e ) ∈ arg min w i ≥ l i ( w i , w ∗− i ; e ) . (8)Observe that the worker has complete information. There is no need for a com-promise. So, the worker simply makes a best-response choice: e ∗ i ( θ, c ) ∈ arg max e ∈{ e L ,e H } v ( w ∗ ( e ) , w ∗ ( e ) , e ; θ, c ) . (9)A profile ( e ∗ , w ∗ , w ∗ , B , B ) of strategies and conceivable sets is a perfect com-promise equilibrium (PCE) if two conditions hold. First, the strategies satisfy (8)and (9), so each firm i chooses a best compromise, and the worker chooses a bestresponse to the strategies of the others. Second, the firms’ conceivable sets areconsistent with the worker’s strategy e ∗ .This application can be embedded in our formal setting as described in Section2. As the pair ( θ, c ) is unknown to the firms, it is identified as the state. So the setof states is Ω. In the formal game, first nature chooses a state ( θ, c ) ∈ Ω. Then theworker observes ( θ, c ) and chooses an education level e . Finally, the firms observethe worker’s choice e and simultaneously choose their wages. Conceivable sets foreach firm i are given by B i . Posteriors are trivial, because the worker plays a purestrategy, and there are no chance moves.A PCE is pooling if the worker chooses the same level of education for all ( θ, c ) ∈ Ω. A PCE is separating if the set Ω can be partitioned into two subsets such thata different level of education is chosen in each element of the partition.
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 15
Proposition 3. (i) There exists a pooling PCE in which the worker chooses loweducation, so e ∗ ( θ ) = e L for all ( θ, c ) ∈ Ω , and the firms’ wages are given by w ∗ i ( e H ) = w ∗ i ( e L ) = 12 , i = 1 , . After each observed education level e , each firm believes about the worker’s pro-ductivity that θ ∈ [0 , .(ii) If δ ≥ b − b , then a separating PCE does not exist.(iii) If δ < b − b , then there exists a separating PCE in which the worker chooseshigh education if and only if her cost is at most b ( b − δ ) , so for all ( θ, c ) ∈ Ω e ∗ ( θ, c ) = ( e H , if c ( θ ) ≤ b ( b − δ ) ,e L , if c ( θ ) > b ( b − δ ) , and the firms’ wages are given by w ∗ i ( e H ) = 12 + b + δ b and w ∗ i ( e L ) = δ b + b + δ b , i = 1 , . (10) After each observed education level e , each firm believes about the worker’s pro-ductivity that θ ∈ (cid:20) , b + δ b + δb (cid:21) if e = e L , and θ ∈ (cid:20) b + δ b , (cid:21) if e = e H . (11)The proof is in Appendix A.4.Let us discuss the strategic concerns underlying these two PCE. In either PCE,each firm i , when deciding about the wage offer w i and facing unknown productiv-ity of the worker, worries about two possibilities. It could be that the productivityis high, so offering a wage that is marginally greater than that of the competitorwould improve profit. The greatest such loss occurs when the productivity is thehighest possible. Alternatively, it could be that the productivity is low, so offeringa wage that is smaller than the competitor’s would cut the loss. The greatestsuch loss occurs when the productivity is the lowest possible. The firm thus offersthe best compromise wage that balances these two losses, assuming that the otherfirm follows its equilibrium strategy.In equilibrium, the firms offer the same wage and do not try to outbid eachother, because they can lose equal amounts by stealing the worker for themselvesif she is unproductive and by giving up the worker if she is highly productive.An essential detail in the above considerations is that the greatest and smallestproductivities are now endogenous and can depend on the level of education e thatthe worker chooses. In the pooling equilibrium, e = e L does not provide any usefulinformation, so all productivity types are possible. However, in the separatingequilibrium, the firms believe that the productivity belongs to a different intervalwhen observing a different level of education. For example, if b = 1 and δ = 1 / then the firms believe that θ ≤ / θ ≥ / θ ∈ [5 / , / θ can have different costsof education c ( θ ) that can fall below or above the threshold at which high educationis profitable. Clearly, this result cannot emerge in the traditional setting wherethere are only two types of workers.The parameter δ captures the firms’ uncertainty about the worker’s cost of ed-ucation conditional on knowing her productivity type. As δ goes up, the rangeof productivity types that cannot be identified with a specific cost of educationincreases. When δ is sufficiently large, education signaling is not very informa-tive. A costly signal does not allow to rule out low productivity types, and theseparating PCE does not exist.3.4. Bilateral Trade with Common Value.
We now examine bilateral tradewith common value and private information. In this example we illustrate the roleof the order of moves when traders are asymmetrically informed. A seller wants to sell an indivisible good to a buyer. The value v of the good isthe same for each of them. If the good is traded at some price p , then the buyerobtains v − p and the seller obtains p − v . If the good is not traded, then bothparties obtain zero.The value v is given by v = x + y , x ∈ [0 ,
1] and y ∈ [0 , , so v ∈ [0 , x , while neither trader observes y . Thus x represents the seller’s private information and y represents the common uncer-tainty among the two traders. Both traders know that x ∈ [0 ,
1] and y ∈ [0 , x . Then one trader (proposer) offers a price p ∈ [0 , α ∈ [0 ,
1] with which the offer is accepted, in which case the trade takes place. Ifthe offer is not accepted, then the trade does not take place.This application can be embedded in our formal setting as described in Section2. We identify the state with ( x, y ). So the set of states is Ω = [0 , . In theformal game, first nature chooses a state ( x, y ). Then the traders proceed asdescribed in the bargaining protocol. Each trader’s conceivable set contains allpairs of ( x, y ) that are possible given the trader’s information. The posteriors aretrivial, because the proposer plays a pure strategy, and there are no chance moves. The traditional bilateral trade model with private values and simultaneous offers is analyzed inAppendix B.1.
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 17
Proposer is Buyer (Uninformed Trader).
We first assume that the buyer isthe proposer and the seller is the responder. Because the buyer is uninformed,her strategy is a choice of a price p ∈ [0 , α ∗ ( x, p ) that depends on her private information x and the offeredprice p . Note that the offer of the buyer does not contain any information aboutthe value, so there is no signaling.The seller’s maximum loss of an accepting an offer p with probability α , givenprivate information x , is l s ( α ; x, p ) = sup y ∈ [0 , (cid:18) max (cid:26) p − x + y , (cid:27) − (cid:18) p − x + y (cid:19) α (cid:19) . It describes how much more the seller could have obtained if she knew the missinginformation y . The buyer’s maximum loss of offering price p , given the seller’sacceptance strategy α ∗ , is l b ( p, α ∗ ) = sup ( x,y ) ∈ [0 , sup p ′ ∈ [0 , (cid:18) x + y − p ′ (cid:19) α ∗ ( x, p ′ ) − (cid:18) x + y − p (cid:19) α ∗ ( x, p ) ! . It describes how much more the buyer could have obtained if she knew x and y , anticipating that the seller would follow her strategy α ∗ . Each trader’s bestcompromise is a choice that achieves the lowest maximum loss for a given strategyof the other trader. A strategy profile ( p ∗ , α ∗ ) is a perfect compromise equilibrium if each trader chooses a best compromise given the strategy of the other trader.Consider first how the seller finds an acceptance probability α as a best com-promise to a price offer in an abstract setting where the seller believes that thevalue is in an interval [ v , v ]. Lemma 1.
Let price be p ∈ [0 , . Suppose that the seller believes that v ∈ [ v , v ] .Then the seller’s best compromise acceptance probability is α = , if p ≤ v , ( p − v ) / ( v − v ) , if v < p < v , , if p ≥ v . The seller’s associated maximum loss is α max { v − p, } . The proof is in Appendix A.5.The intuition for this result is simple. Of course, if p ≥ v , then the payoff p − v is nonnegative for all v ∈ [ v , v ], so the optimal choice is to accept the offer.Similarly, if p ≤ v , then the optimal choice is to reject the offer. However, if v < p < v , then, for a given choice of the acceptance probability α , the sellerworries about two possibilities. First, the value could be smaller than the price,so p − v >
0. In that case, she could have obtained a better payoff by acceptingthe proposal with certainty. The greatest such loss occurs when v = v , and it isequal to ( p − v ) − α ( p − v ) = (1 − α )( p − v ) . Alternatively, the value could be greater than the price, so p − v <
0. In that case,she could have obtained a better payoff by rejecting the proposal with certainty,and thus getting zero. The greatest such loss occurs when v = v , and it is equalto 0 − α ( p − v ) = α ( v − p ) . The seller chooses α as a best compromise to balances these two losses, that is, α solves the equation (1 − α )( p − v ) = α ( v − p ).We now present a PCE of this game. Proposition 4.
There exists a perfect compromise equilibrium, in which the buyeroffers p ∗ = 1 / , and the seller accepts the offer with probability α ∗ ( x, p ) = , if x ≥ p , p − x, if p − < x < p , , if x ≤ p − . (12)The proof is in Appendix A.6.Under this PCE, trade can occurs with positive probability. The seller acceptsthe equilibrium offer of 1 / { / − x, } .Let us discuss how this PCE is computed. In the second stage, the seller,who has observed x and p and has no information about y ∈ [0 , v = ( x + y ) / x/ , (1 + x ) / α ∗ is given by Lemma 1 with [ v , v ] = [ x/ , (1 + x ) / y and of the seller’s private information x . As the price goes up,on the one hand the buyer’s gain from trade decreases, but on the other hand, theprobability that the seller accepts this price increases. Unlike in our earlier exam-ples, here the computation of the maximum loss involves more than just checkingthe extreme cases. However, notice that the price should not be too low, as thisoffer is likely to be rejected, so the buyer has a high loss when the common valueof the good is high. The price should not be too high either, as this offer is likelyto be accepted, so the buyer has a high loss when the common value of the goodis low. As best compromise, the buyer chooses the price that balances these twoconsiderations, anticipating the seller’s equilibrium behavior in the second stage.3.4.2. Proposer is Seller (Informed Trader).
Now we assume that the seller isthe proposer and the buyer is the responder. Because the seller observes x , herpricing strategy p ∗ ( x ) depends upon x . The buyer then responds by an acceptanceprobability α ∗ ( p ) that depends on the offered price p .Unlike when the buyer was the proposer, here the seller’s price can be informa-tive about the seller’s private information x . Let B ( p ) be the buyer’s conceivableset that describes what pairs of ( x, y ) the buyer considers possible after observingthe seller’s move. A conceivable set B ( p ) is consistent with the seller’s strategy p ∗ if it includes all pairs ( x, y ) under which the seller chooses p , so ( x, y ) ∈ B ( p ) if p ∗ ( x ) = p . PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 19
The buyer’s maximum loss from accepting p with probability α is l b ( α ; p ) = sup ( x,y ) ∈ B ( p ) (cid:18) max (cid:26) x + y − p, (cid:27) − (cid:18) x + y − p (cid:19) α (cid:19) . The seller’s maximum loss of offering price p , given her private information x andthe buyer’s acceptance strategy α ∗ , is l s ( p, α ∗ ; x ) = sup y ∈ [0 , sup p ′ ∈ [0 , (cid:18) p ′ − x + y (cid:19) α ∗ ( p ′ ) − (cid:18) p − x + y (cid:19) α ∗ ( p ) ! . Each trader’s best compromise is a choice that achieves the lowest maximum lossfor a given strategy of the trader. A strategy profile ( p ∗ , α ∗ ) is a perfect compromiseequilibrium if each trader chooses a best compromise given the strategy of the othertrader, and the buyer’s conceivable sets are consistent.We now present a PCE. In the proposition below, instead of expressing whatpairs ( x, y ) the buyer includes in the conceivable set, we simply state what valuesof v = ( x + y ) / Proposition 5.
There exists a perfect compromise equilibrium in which the seller’soffer is p ∗ ( x ) = 3 / for all x ∈ [0 , ,the buyer believes that v ∈ [0 , after observing p = 3 / and that v ∈ [0 , / afterobserving p = 3 / , and her acceptance probability is α ∗ ( p ) = ( / , if p = 3 / , max { − p, } if p = 3 / . The proof is in Appendix A.7.This PCE is pooling, in the sense that the equilibrium behavior is independent ofthe seller’s private information. In the second stage, the buyer, who has observed p , knows that in equilibrium the seller must choose p = 3 /
4. This price reveals noinformation about x . So the buyer cannot rule out any values, thus believing that v ∈ [0 , /
4. Alternatively,if the buyer observes p = 3 /
4, which cannot happen in equilibrium, then thereare no constraints on what values are conceivable. In this case the equilibriumconceivable set is specified to be such that x = 0 and y ∈ [0 , v ∈ [0 , /
2] and accepts the offer with probability max { − p, } .In the first stage, the seller observes x , but still faces different possible realiza-tions of the common uncertainty y . She anticipates the buyer’s acceptance of theprice of p = 3 / /
4, and the acceptance of a price p = 3 / { − p, } . The price of 3 / p = 3 / x .3.4.3. Discussion.
Propositions 4 and 5 stand in stark contrast to the no-tradetheorem under common values as predicted by PBE (Milgrom and Stokey, 1982).
We observe that trade can happen with a positive probability in a PCE. This istrue when the (uninformed) buyer is the proposer as well as when the (informed)seller is the proposer. When the buyer is the proposer, the probability of trade is α ∗ ( x, p ∗ ) = max { / − x, } > x < /
2. When the seller is the proposer,the probability of trade is 1 /
4, regardless of the seller’s private information. Thetrade is possible because the traders cannot rule out the possibility of two opposingcontingencies: winning and losing from trade. They do not want to miss a winningopportunity, but also they do not want to lose from trade. They compromise bytrading with a positive probability when the informed trader moves first, as wellas in many cases when the uninformed trader moves first.4.
Conclusion
We introduce the concept of perfect compromise equilibrium (PCE) as an alter-native to perfect Bayesian equilibrium (PBE) for solving sequential games whenthere is uncertainty about some of the primitives. These primitives can be specificcharacteristics of other players or the environment in which the game takes place.Traditionally, following Harsanyi (1967), such uncertainty is reduced to risk. Astate is introduced to capture the details underlying this uncertainty. At the out-set of the game a move of nature determines this state. This move is drawn froma distribution that is commonly known among the players in the game. We followthe same approach except for the last assumption, and assume instead that thedistribution determining the state is unknown.With our concept, dynamic strategic decision making can be illustrated in thetraditional fashion with a game tree. Beliefs are added whenever an informationset contains more than one node. This allows to evaluate choices at informationsets that cannot be reached. These beliefs are conditional on a state to separatewhat can happen in the different states. Bayes’ rule is still in place, applyingconditional on a state. A set of conceivable states is added to each information setin order to track any resolution of uncertainty. The concept of best compromisedetermines how to evaluate a strategy when looking at its performance acrossdifferent conceivable states. This stands in contrast to the classic PBE settingwhere strategies are evaluated using a prior over the different states.The description of the state governs the type of uncertainty that is modeled.States can describe values of market participants, thereby allowing for many differ-ent possible environments. They can specify distributions of such values in somesmall neighborhood, thereby modeling slight uncertainty around a given under-standing of the world. States can also include features of the choices of a player,thereby incorporating strategic uncertainty into the analysis. In particular, ourassumption of strategic certainty comes without loss of generality. Any uncer-tainty can be captured in the definition of a state. The common strategy profiledescribes all details that can be predicted if the state were known, provided thisstrategy profile is common knowledge.
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 21
Uncertainty seems to mean that details are hard to describe. And yet traditionalmodels focus on two types of workers, high and low, or assume linear demandfunctions. Uncertainty seems to preclude that players agree on likelihoods ofuncertain events and yet this is done in PBE. PCE opens the door to understandingmore realistic uncertainty.We demonstrate the usefulness of our solution concept in relevant economic ex-amples. The underlying game trees are simple while the uncertainty is rich. Thisrichness, such as allowing for any demand function in a neighborhood, precludesa tractable analysis of PBE. PCE yields tractable results with simple proofs asplayers focus on extreme situations, allowing them to ignore intermediate constel-lations. New insights come to light.The traditional PBE framework reveals a different solution for each prior. Suchflexibility can be useful to fit data. But flexibility in terms of a multitude ofdifferent answers gives little guidance to those who need to make choices. Oneeasily looses the big picture if there are many details that determine what happens.On a more abstract level, traditional PBE analysis reveals parsimonious resultsby limiting attention to only few types of players. In contrast, PCE generatesparsimonious results by forcing players to find a compromise in many differentsituations.Acceptance of the common knowledge assumption is dwindling. The literatureon decision making and game playing under uncertainty has now developed al-ternative concepts. We hope to add to this literature. Numerous paths to futureresearch open up in a search for new insights and for a clearer exposition of existingunderstanding of economic and strategic principles.
Appendix A. Proofs.
A.1.
Proof of Theorem 1.
Suppose that the strategy s of the nonstrategicplayer 0 is fully mixed. This is without loss of generality, as any zero-probabilityaction of player 0 can be removed from the game tree.We now introduce the notion of feasibility. A state ω is feasible at informationset φ ∈ Φ if there exists a path in the game tree G from the initial node φ to φ in which nature chooses ω at φ . Let ¯ B ( φ ) be the set of all feasible states at φ .Let S s be the set of strategy profiles s where the strategy of the nonstrategicplayer 0 is exogenously given by s . Let conceivable sets be equal to feasible sets, B ( φ ) = ¯ B ( φ ) for all φ ∈ Φ. This profile of conceivable sets is consistent withevery strategy profile s ∈ S s (see Definition 1). Let B be the set of belief systems β = ( β ω ) ω ∈ Ω .We now argue that there exists a PCE ( s, ¯ B, β ) where s ∈ S s and β ∈ B . Notethat we fix the profile of conceivable sets to be equal to ¯ B .Consider an arbitrary ( s, ¯ B, β ) that satisfies s ∈ S s and β ∈ B . Let ( s ′ φ , s − φ )denote the strategy profile where s ′ φ ∈ ∆( A ( φ )) is played at information set φ and s − φ is the profile of strategies at all information sets other than φ . For each information set φ ∈ Φ let U φ ( s ; β ) be the negative of the maximum loss at φ , so U φ ( s ; β ) = − sup ω ∈ ¯ B ( φ ) sup a φ ∈ A ( φ ) ¯ u i ( a φ , s − φ ; φ, ω, β ω ) − ¯ u i ( s φ , s − φ ; φ, ω, β ω ) ! . We now construct an augmented game (Φ , G , U, s , µ ) as follows. Let each in-formation set φ ∈ Φ be associated with a different player, so the set of players isthe set of information sets Φ. The game tree G remains unchanged. Nature movesfirst by choosing a state ω ∈ Ω = A ( φ ) in the initial node φ . We now model thechoice of nature by a given distribution µ over the states. We assume that µ hasfull support over nature’s actions in Ω. Each player φ ∈ Φ moves only once, ather information set φ , by choosing an action from the set A ( φ ).A strategy profile s describes a choice s φ ∈ ∆( A ( φ )) of each player φ . For eachinformation set φ that belongs to the nonstrategic player 0 in the original game, thestrategy at φ is exogenously given by s ( φ ). A posterior β ω ( φ ) is the probabilitydistribution over decision nodes in φ conditional on state ω . The interim payoff ofeach player φ ∈ Φ at the information set φ is given by U φ ( s ; β ), and U = ( U φ ) φ ∈ Φ .The augmented game (Φ , G , U, s , µ ) can be seen as a game of incomplete in-formation with a nonstandard specification of the players’ payoffs. While in astandard game the payoffs are ex post and specified at each terminal node, in thisaugmented game the payoff U φ of each player φ ∈ Φ is interim and specified at theinformation set where the player makes a move. Because each player moves onlyonce, the specification of the interim payoffs is sufficient to apply the concept ofPBE or sequential equilibrium to the augmented game.Another nonstandard feature of the augmented game is that each player’s in-terim payoff U φ ( s ; β ) is independent of nature’s choice ω . That is, for each state ω ∈ ¯ B ( φ ) the interim payoff U φ ( s ; β ) at φ is the same, and for each state ω ¯ B ( φ )the information set φ cannot be reached. So, nature’s distribution over states µ does not affect the best-response actions by the players, it only affects the likeli-hood of reaching different information sets in the game tree.Observe that maximizing U φ ( s ′ φ , s − φ ; β ) with respect to player φ ’s own decision s ′ φ ∈ ∆( A ( φ )) is the same as minimizing the maximum loss at φ in the originalgame. Consequently, if ( s ∗ , β ∗ ) is a sequential equilibrium of the augmented game,then ( s ∗ , ¯ B, β ∗ ) is a PCE of the original game. The existence of PCE follows fromthe existence of sequential equilibrium for finite games. We refer the reader toChakrabarti and Topolyan (2016) for the backward-induction proof of existenceof sequential equilibrium that uses interim payoffs at information sets to determineplayers’ best-response correspondences. (cid:3) A.2.
Proof of Proposition 1.
For derivations, we assume that the quantitiesand the price are always nonnegative, and then we verify that this is indeed thecase in equilibrium.Let x ∗ i ( q − i , P ) be a best response strategy of player i given the knowledge of q − i and the inverse demand function P . The loss of firm i from choosing quantity q i , PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 23 given q − i and P , is denoted by ∆ u i ( q i , q − i ; P ) and given by∆ u i ( q i , q − i ; P ) = P ( x ∗ i ( q − i , P ) + q − i ) x ∗ i ( q − i , P ) − P ( q i + q − i ) q i . By (2), the marginal revenue of firm i satisfies¯ P ( q i + q − i )+¯ P ′ ( q i + q − i ) q i ≤ P ( q i + q − i )+ P ′ ( q i + q − i ) q i ≤ ¯ P ( q i + q − i )+ ¯ P ′ ( q i + q − i ) q i . Therefore, for given q j and P , the best-response quantity x ∗ i ( q − i , P ) of firm i always lies between x ∗ i ( q − i , ¯ P ) and x ∗ i ( q − i , ¯ P ). While the profit function need notbe concave in general, it is concave when P = ¯ P or when P = ¯ P . So the highestloss will always be attained in one of these two extreme cases: l i ( q i , q − i ) = sup P ∆ u i ( q i , q − i ; P ) = max { ∆ u i ( q i , q − i ; ¯ P ) , ∆ u i ( q i , q − i ; ¯ P ) } . It is easy to see that the maximum loss is minimized by balancing the two expres-sions under the maximum:∆ u i ( q i , q − i ; ¯ P ) = ∆ u i ( q i , q − i ; ¯ P ) . Substituting ¯ P and ¯ P and simplifying the expressions yields the equation(¯ a − ¯ bq − i ) b − (¯ a − ¯ b ( q i + q − i )) q i = (¯ a − ¯ bq − i ) b − (¯ a − ¯ b ( q i + q − i )) q i . (13)Solving for q i yields q ∗ i = ¯ a √ ¯ b + ¯ a √ ¯ b b √ ¯ b + ¯ b √ ¯ b ) − q j , i = 1 , . Solving this pair of equations for ( q ∗ , q ∗ ), we find (3). It is easy to verify that underour assumptions, q ∗ i >
0, and moreover, P ( q ∗ + q ∗ ) ≥ ¯ P ( q ∗ + q ∗ ) >
0. Substitutingthe solution into (13) yields the maximum loss of each firm (4). (cid:3)
A.3.
Proof of Proposition 2.
For derivations, we assume that each firm pricesat or above marginal cost, and then we verify that this is indeed the case inequilibrium.Consider firm i with type c i ∈ [¯ c, ¯ c ]. Let p m ( c i ) be the monopoly price, so p m ( c i ) = ( a + c i ) /
2. Since we have assumed that ¯ c ≤ a/
2, this means that p m ( c i ) ≥ ¯ c for all c i . The monopoly profit is ( a − c i ) / (4 b ).Fix the other firm’s strategy p ∗− i ( c − i ) and let ¯ p be the maximum price of theother firm, so ¯ p = sup c − i ∈ [¯ c, ¯ c ] p ∗− i ( c − i ). Given the other firm’s cost c − i , and thusthe price p − i = p ∗− i ( c − i ), firm i ’s maximum profit is u ∗ i ( p − i ; c i ) = sup x i ≥ u i ( x i , p − i ; c i ) = , if p − i ≤ c i , ( p − i − c i ) a − p − i b , if c i < p − i ≤ p m ( c i ) , ( a − c i ) b , if p − i > p m ( c i )= max (cid:26) , ( p − i − c i ) a − p − i b , ( a − c i ) b (cid:27) . Let p i be a price of firm i . We now find the maximum loss of firm i from choosing p i , given its marginal cost c i and the strategy p ∗− i of the other firm. There arethree cases.First, suppose that p − i ≤ c i ≤ p i . Then firm i cannot make positive profit, so p i is a best response. Thus, firm i behaves optimally in this case, so the loss is zero.Second, suppose that c i < p − i ≤ p i . Then firm i could have been better off bymarginally undercutting p − i . Maximizing the loss over p − i ∈ ( c i , p i ], we obtainsup p − i ∈ ( c i ,p i ) ( u ∗ i ( p − i ; c i ) − u i ( p i , p − i ; c i )) = ( ( p i − c i ) a − p i b , if p i ≤ p m ( c i ), ( a − c i ) b , if p i > p m ( c i ) . (14)Third, suppose that p i < p − i . Then firm i could have made more profit byincreasing its price, so its maximum loss issup p − i ∈ ( p i , ¯ p ] ( u ∗ i ( p − i ; c i ) − u i ( p i , p − i ; c i )) = u ∗ i (¯ p ; c i ) − u i ( p i , ¯ p ; c i )= − ( p i − c i ) a − p i b + ( (¯ p − c i ) a − ¯ pb , if p i ≤ p m ( c i ), ( a − c i ) b , if p i > p m ( c i ) . (15)To minimize the maximum loss, we need to minimize the greater of the expressionsin (14) and (15). Observe that, by the definition of p m ( c i ), the right-hand sidein (14) is constant and the right-hand side in (15) is strictly increasing in p i for p i > p m ( c i ). So we only need to consider p i ≤ p m ( c i ). Under this assumption, thegreater of the expressions in (14) and (15) can be simplified to l i ( p i , p ∗− i ; c i ) = max (cid:26) ( p i − c i ) a − p i b , (¯ p − c i ) a − ¯ pb − ( p i − c i ) a − p i b (cid:27) . Because one expression is increasing and the other is decreasing in p i for p i ≤ p m ( c i ), the maximum loss is minimized at the solution of( p i − c i ) a − p i b = (¯ p − c i ) a − ¯ pb − ( p i − c i ) a − p i b . (16)Solving the above for p ∗ i ( c i ), we obtain (5).To see that p ∗ i ( c i ) ≥ c i , observe that p ∗ i ( c i ) − c i = 12 (cid:16) a − c i − p ( a − ¯ c ) + (¯ c − c i ) (cid:17) ≥ a > ¯ c ≥ c i . Moreover, p ∗ i ( c i ) > c i when c i < ¯ c , and p ∗ i (¯ c ) = ¯ c . Finally, substituting p ∗ i ( c i ) into the maximum loss expression in (16)yields (6). (cid:3) A.4.
Proof of Proposition 3.
First we find the equilibrium wages w H and w L after the worker’s level of education e H and e L . For each j = L, H , the firms havea conceivable set B ( e j ) ⊂ Ω, that is, the set of states ( θ, c ) ∈ Ω that the firmsconsider possible. Let ¯ θ j and ¯ θ j be the lowest and highest productivity levels given e j , so¯ θ j = inf { θ : ( θ, c ) ∈ B ( e j ) } and ¯ θ j = sup { θ : ( θ, c ) ∈ B ( e j ) } , j = L, H. (17)
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 25
Consider a firm i , some wages w i and w − i , and a state ( θ, c ). Firm i ’s maximumprofit u ∗ i ( w − i ; θ ) is obtained by marginally outbidding w − i when it is below θ , andby choosing the wage below w − i and thus giving up the worker if θ ≤ w − i , so u ∗ i ( w − i ; θ ) = sup w i ≥ u i ( w i , w − i ; θ ) = max { θ − w − i , } . Observe that we only need to consider w i and w − i in [¯ θ j , ¯ θ j ]. A wage above ¯ θ j is dominated and cannot be a best compromise; a wage below ¯ θ j will always beoverbid by the rival’s wage, as there is common knowledge that θ ≥ ¯ θ j .Suppose that w i < w − i , so u i ( w i , w − i ; θ ) = 0. Then the largest loss is obtainedwhen θ is the greatest:sup θ :( θ,c ) ∈ B ( e j ) ( u ∗ i ( w − i ; θ ) − u i ( w i , w − i ; θ )) ≤ max { ¯ θ j − w − i , } . Next, suppose that w i > w − i , so u i ( w i , w − i ; θ ) = θ − w i . Then the largest loss isobtained when θ is the smallest:sup θ :( θ,c ) ∈ B ( e j ) ( u ∗ i ( w − i ; θ ) − u i ( w i , w − i ; θ )) = max { θ − w − i , } − ( θ − w i ) ≤ w i − ¯ θ j . Finally, suppose that w i = w − i , so u i ( w i , w − i ; θ ) = ( θ − w i ) /
2. Thensup θ :( θ,c ) ∈ B ( e j ) ( u ∗ i ( w − i ; θ ) − u i ( w i , w − i ; θ )) = max { θ − w − i , } − θ − w i ≤ max { , ¯ θ j − w − i , ( w i − ¯ θ j ) / } . The maximum loss l i ( w i , w − i ) is given by the greatest of the three expressions, so l i ( w i , w − i ) = max { , ¯ θ j − w − i , w i − ¯ θ j . } . The wages w i that minimizes the maximum loss satisfies w i = ¯ θ j + ¯ θ j − w − i , i = 1 , . So, we have obtained two equations, one for each i = 1 ,
2. Solving this pair ofequations for w and w yields the best compromise w ∗ i ( e j ) for each firm i , where w ∗ i ( e j ) = ¯ θ j + ¯ θ j , i = 1 , . (18)The associated maximum losses are l i ( w ∗ i ( e j ) , w ∗− i ( e j )) = w ∗ i ( e j ) − ¯ θ j . (19)Next, observe that the worker operates under complete information. Given eachchoice of e j , she anticipates the wages w j = w ∗ ( e j ) = w ∗ ( e j ), j ∈ { L, H } . So,given a state ( θ, c ), the worker chooses e = e H if and only if w H − c ( θ ) ≥ w L . The tie breaking is arbitrary, because the set of types is a continuum.
Recall that c ( θ ) is strictly decreasing, and denote by c − its inverse. Then, theworker chooses e = e H if and only if her type θ satisfies θ ≥ c − ( w H − w L ) . Pooling PCE. If w H ≤ w L , then every type chooses low level of education e L , sothe equilibrium is pooling. After observing e = e L , the consistent conceivable set B L is thus the entire set of states, B L = Ω. By (7), the highest and lowest θ in B L are ¯ θ L = 1 and ¯ θ L = 0. By (18), we obtain the equilibrium wages w i ( e L ) = 1 / e = e H , the conceivable set B H must induce the wage w ∗ i ( e H ) ≤ w ∗ i ( e L ). In particular, we can assume B H = Ω,and thus w ∗ i ( e H ) = 1 / w ∗ i ( e ) = 1 / θ L = 0into (19), we obtain the maximum loss for each firm i , l i ( w ∗ i ( e j ) , w ∗− i ; e j ) = 12 , i = 1 , , j = L, H.
Separating PCE.
Consider now w H > w L , so that the worker with cost c ( θ ) ≤ w H − w L chooses high education. Let B L = { ( θ, c ) ∈ Ω : c > w H − w L } and B H = { ( θ, c ) ∈ Ω : c ( θ ) ≤ w H − w L } be the conceivable sets of each firm when the level of education is e L and e H ,respectively. So, B L and B H contain all pairs ( θ, c ) such that low and high edu-cation is chosen, respectively. These sets thus satisfy the consistency requirement(Definition 1).By (7) and (17), the highest and lowest θ in B H are given by¯ θ H = 1 and ¯ θ H = ¯ c − ( w H − w L ) = 1 − w H + w L b . (20)Similarly, the highest and lowest θ in B L are given by¯ θ L = ¯ c − ( w H − w L ) = 1 + δ − w H + w L b and ¯ θ L = 0 . (21)From (18), we have w H = ¯ θ H + ¯ θ H w L = ¯ θ L + ¯ θ L . (22)Solving the system of six equations in (20), (21), and (22), with six unknowns( w H , w L , ¯ θ H , ¯ θ H , ¯ θ L , and ¯ θ L ), we obtain the equilibrium wages and the boundson the productivity types as shown in (10) and (11).Observe that the lowest possible cost of high education is inf { c ( θ ) : ( θ, c ) ∈ Ω } = ¯ c − (1) = 1 − b . Therefore, there exist states ( θ, c ) where high education e H is chosen if and only if w H − w L > − b . Substituting our solution for w H and w L given by (10), we obtain that w H − w L > − b if and only if δ < b − b. This condition is thus necessary and sufficient for the existence of separating PCE.
PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 27
Finally, substituting the wage w H and the productivity lower bound ¯ θ H into(19), we obtain firm i ’s maximum loss when e = e H , l i ( w ∗ i ( e H ) , w ∗− i ( e H ); e H ) = w H − ¯ θ H = 12 − b + δ b . Substituting the wage w L and the productivity lower bound ¯ θ L into (19), we obtainthe maximum loss when e = e L , l i ( w ∗ i ( e L ) , w ∗− i ( e L ); e L ) = w L − ¯ θ L = δ b + b + δ b . (cid:3) A.5.
Proof of Lemma 1.
Let p be an offered price and let α be an acceptanceprobability. The seller’s payoff in case of trade at price p is p − v . If p ≤ v , thenthis payoff is nonpositive for all v ∈ [ v , v ]. The seller can achieve the best payoff(and thus zero maximum loss) by rejecting the proposal, so α = 0. Similarly, if p ≥ v , then the seller’s payoff from accepting p is nonnegative for all v ∈ [ v , v ].The seller can achieve the best payoff (and thus zero maximum loss) by acceptingthe proposal, so α = 1.Suppose that v < p < v . For each v ∈ [ v , p ], the best payoff is p − v , so theloss from choosing α is( p − v ) − α ( p − v ) = (1 − α )( p − v ) ≤ (1 − α )( p − v ) . For each v ∈ [ p, v ], the best payoff is 0, so the loss from choosing α is0 − α ( p − v ) = α ( v − p ) ≤ α ( v − p ) . Thus, the maximum loss is l ( p, α ; v ) = max { (1 − α )( p − v ) , α ( v − p ) } . The first term is decreasing and the second term is increasing in α . The maximumloss is thus minimized by the solution of (1 − α )( p − v ) = α ( v − p ), so α =( p − v ) / ( v − v ). (cid:3) A.6.
Proof of Proposition 4.
Let us find a conceivable set and a strategy ofthe seller in the second stage for our PCE. The seller, who has observed x and p and has no information about y ∈ [0 , B ( x, p ). Let B ( x, p )contain all states that cannot be ruled out given this information, so B ( x, p ) = { ( x ′ , y ′ ) ∈ [0 , : x ′ = x, y ′ ∈ [0 , } . So, B ( x, p ) is consistent (Definition 1). Values are thus contained in (cid:26) x + y x, y ) ∈ B ( x, p ) (cid:27) = [ x/ , (1 + x ) / . By Lemma 1, with [ v , v ] = [ x/ , (1 + x ) / α ∗ ( x, p ) is given by (12). We now find the buyer’s strategy in the first stage. The buyer’s maximum payoffgiven ( x, y ) is sup p ′ ∈ [0 , (cid:18) x + y − p ′ (cid:19) α ∗ ( x, p ′ ) = y . Let p be a price. The buyer’s loss from choosing p , given ( x, y ), is given by y − (cid:18) x + y − p (cid:19) α ∗ ( x, p ) ≤ max (cid:26) − (cid:18) x + 12 − p (cid:19) α ∗ ( x, p ) , − (cid:16) x − p (cid:17) α ∗ ( x, p ) (cid:27) = , if x ≥ p , (2 p − x ) + max (cid:8) − p + x , (cid:9) , if 2 p − < x < p,p − x , if x ≤ p − . (23)The inequality follows from convexity of the loss in y , so we can evaluate it at y = 1 and y = 0, and the equality is by the substitution of α ∗ ( x, p ) from (12).Observe that (23) is convex in x for x < p and constant for x ≥ p . So tomaximize it w.r.t. x , we only need to evaluate it at x = 0 and x = 1. It is thenstraightforward to see that p ∗ = 1 / / l b ( p ∗ , α ∗ ) = 18 and max x ∈ [0 , l s ( p ∗ , α ∗ ; x ) = 18 . (cid:3) A.7.
Proof of Proposition 5.
Let us first determine a conceivable set and thebest compromise acceptance strategy α ∗ ( p ) for the buyer in the second stage inour PCE. The derivation is analogous Lemma 1. Specifically, let v ∈ [ v , v ]. Thebuyer’s payoff in case of trade is v − p . If p ≥ v , then the buyer’s payoff isnonpositive for all v ∈ [ v , v ]. The buyer can achieve the best payoff (and thusthe maximum loss equal to zero) by rejecting the proposal, so α ∗ ( p ) = 0. Similarly,if p ≤ v , then the buyer’s payoff is nonnegative for all v ∈ [ v , v ]. The buyer canachieve the best payoff (and thus the maximum loss equal to zero) by acceptingthe proposal, so α ∗ ( p ) = 1.Suppose that v < p < v . For each p ≤ v , the best payoff is v − p , so the lossfrom accepting p with probability α is( v − p ) − α ( v − p ) = (1 − α )( v − p ) ≤ (1 − α )( v − p ) . For each p ≥ v , the best payoff is 0, so the loss from accepting p with probability α is 0 − α ( v − p ) = α ( p − v ) ≤ α ( p − v ) . Thus, the maximum loss is l ( p, α ; v ) = max { ( v − α )(1 − p ) , α ( p − v ) } . (24) PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 29
The first term is decreasing and the second term is increasing in α . The maximumloss is thus minimized by the solution of (1 − α )( v − p ) = α ( p − v ), so α ∗ ( p ) =(¯ v − p ) / (¯ v − ¯ v ).The buyer knows that the seller chooses p ∗ = 3 / x ∈ [0 , x . Thus, the buyer who has observed p ∗ = 3 / B ( p ∗ ) = Ω. The interval of values isthus [ v , v ] = [0 , α ∗ (3 /
4) = v − / v − v = 14 . Next, any price p = 3 / B ( p ) = { (0 , y ) : y ∈ [0 , } . The interval of values is thus[ v , v ] = [0 , / p ≥ v = 0. The buyer’s best compromise acceptanceprobability is α ∗ ( p ) = max (cid:26) v − pv − v , (cid:27) = max { − p, } , p = 3 / . We now find the seller’s best compromise strategy in the first stage. Anticipatingthe buyer to play α ∗ , the seller can obtain the following payoffs. Given v =( x + y ) /
2, the seller’s payoff is from p ∗ = 3 / u s ( v, p ∗ ) = ( p ∗ − v ) α ∗ ( p ∗ ) = (cid:18) − v (cid:19) . The seller’s payoff is from p = 3 / u s ( v, p ) = ( p − v ) α ∗ ( p ) = ( ( p − v ) (1 − p ) , if p < / , if p ≥ / p = 3 / . The maximum payoff among all prices p = 3 / p =3 / ˆ u s ( v, p ) ≥ ( (1 − v ) , if v < / , if v ≥ / p = 3 / v ∈ [ x , x ] (ˆ u s ( v, p ∗ ) − ˆ u s ( v, p )) ≥ sup v ∈ [ x , x ] (cid:18)
18 (1 − v ) − (cid:18) − v (cid:19) (cid:19) = 332 > . On the other hand, the maximum loss of choosing p ∗ = 3 / l s ( p ∗ , a ∗ ; x ) = sup v ∈ [ x , x ] sup p =3 / ˆ u s ( v, p ) − ˆ u s ( v, p ∗ ) ! = sup v ∈ [ x , x ] (cid:18)
18 (1 − v ) − (cid:18) − v (cid:19) (cid:19) . (25) It can be seen that the expression under the maximum in (25) is convex, so itneeds to be evaluated at y = 0 and y = 1. So, for all x ∈ [0 , l s ( p ∗ , a ∗ ; x ) = max (cid:26) x − x − , x − (cid:27) = 2 x − ≤ . We thus conclude that choosing p ∗ = 3 / p = 3 /
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Appendix B (
For Online Publication ) In this appendix we analyze three examples that complement those presentedin the main part of the paper.B.1.
Bilateral Trade with Simultaneous Offers.
We continue the theme ofbilateral trade in Section 3.4. This time, we consider the classic model of tradewith private values and simultaneous moves (Chatterjee and Samuelson, 1983). Aseller wants to sell an indivisible good to a buyer. The good has value v s for theseller and v b for the buyer. Each trader is privately informed about their ownvalue, and it is commonly known that both values belong to [0 , s and b , respectively, with s, b ∈ [0 , s ≤ b , then the goodis sold at the price p = ( s + b ) /
2, so the seller and the buyer obtain the payoffsof p − v s and v b − p , respectively. If s > b , then the good is not sold, and bothparties obtain zero payoffs.Let s ( v s ) and b ( v b ) denote the seller’s and the buyer’s strategies, respectively.We now find a perfect compromise equilibrium under the following regularity con-dition:Strategies s ( v s ) and b ( v b ) are continuous and strictly increasing. (A )We now present a PCE of this game. Proposition 6.
A pair of strategies ( s ∗ , b ∗ ) that satisfies (A ) is a PCE if andonly if for all v s , v b ∈ [0 , s ∗ ( v s ) = max (cid:26) v s ,
14 + 2 v s (cid:27) and b ∗ ( v b ) = min (cid:26) v b ,
112 + 2 v b (cid:27) . (26) The associated maximal losses are l s ( v s ) = max (cid:26) − v s − v s ) , (cid:27) ≤ and l b ( v b ) = max (cid:26) − − v b v b , (cid:27) ≤ . The PCE in Proposition 6 coincides with the PBE when v b and v s are uniformlydistributed on [0 ,
1] (Chatterjee and Samuelson, 1983).
Proof.
Denote by s ∗ ( v s ) and b ∗ ( v b ) the strategies of the seller and the buyer in abest compromise equilibrium. Also, denote¯ s ∗ = inf v s ∈ [0 , s ∗ ( v s ) and ¯ b ∗ = sup v b ∈ [0 , b ∗ ( v b ) . Suppose that the seller’s value is v s . Let us find the seller’s maximum loss forbidding s . If v s < ¯ b ∗ , so there is a potential gain from trade when the buyer’s bid b ∗ ( v b ) is high enough, then there are two outcomes that the seller worries about.First, it is s < b ∗ ( v b ), so the seller could have made a greater bid, s ′ = b ∗ ( v b ), andsold the good at a higher price, thus obtaining a payoff increment ofsup v b (cid:20)(cid:0) b ∗ ( v b ) − v s (cid:1) − (cid:18) s + b ∗ ( v b )2 − v s (cid:19)(cid:21) = ¯ b ∗ − s . PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 33
Second, it is s > b ∗ ( v b ) > v s , so the good is not sold, but the seller could have madea smaller bid, s ′ = b ∗ ( v b ), and sold the good, thus obtaining a payoff increment ofsup v b [ b ∗ ( v b ) − v s ] = s − v s . The seller’s maximum loss is thus c s ( v s , s, b ∗ ) = max (cid:26) ¯ b ∗ − s , s − v s (cid:27) if v s < ¯ b ∗ . The value of s that minimizes the above maximum loss is s ∗ ( v s ) = ¯ b ∗ v s , if v s < ¯ b ∗ . Alternatively, if there is no gain from trade, so v s ≥ ¯ b ∗ , then the seller obtainszero loss by choosing any bid that guarantees no trade, for example, s ∗ ( v s ) = v s .Symmetrically, we obtain that the buyer’s best compromise strategy satisfies b ∗ ( v b ) = ( ¯ s ∗ + v b , if v b > ¯ s ∗ ,v b , if v b ≤ ¯ s ∗ . Since v s , v b ∈ [0 , s ∗ = s ∗ (0) = ¯ b ∗ b ∗ = b ∗ (1) = 2 + ¯ s ∗ . Thus, ¯ s ∗ = 1 / b ∗ = 3 /
4. Therefore, the PCE ( s ∗ , b ∗ ) given by (26). (cid:3) B.2.
Public Good Provision.
Here we investigate the provision of a public goodwhen each beneficiary knows her own value but not that of the others.There are n agents, each has a private value v i ∈ [0 , ¯ v ] for the public good.Each agent i commits to contribute at most x i ∈ [0 , ¯ v ] in case the public good isprovided. Agents make their commitments simultaneously.The cost of providing the public good is c >
0. If the sum of committedcontributions does not cover the cost, so P ni =1 x i < c , then the public good is notprovided, and each agent i obtains zero payoff. Otherwise, if P ni =1 x i ≥ c , thenthe public good is provided, and each agent i obtains the payoff v i − t i ( x ) , where t i ( x ) is the final transfer of agent i that depends on the profile of committedcontributions x = ( x , ..., x n ). We assume that the transfer rule t i must satisfy:(a) t i ( x ) ≤ x i , so no agent pays more than her committed contribution,(b) P ni =1 t i ( x ) ≥ c ,(c) t = ( t , ..., t n ) is symmetric, so agents are treated ex ante equally.In addition, we assume that the cost of public good provision is relatively small,specifically, c ≤ ( n − v. (27)This assumption simplifies the exposition. The complementary case can also beeasily analysed. Let s i ( v i ) be a strategy of agent i , so x i = s i ( v i ) specifies the maximal contribu-tion of agent i . As in Section B.1, we restrict attention to strategies that satisfythe following assumption.Strategies s i are continuous, strictly increasing, and s i ( v ) = s j ( v ) for all i, j = 1 , ..., n and all v ∈ [0 , ¯ v ]. (A )We compare three simple transfer rules.(i) Pay-as-you-bid rule.
Each agent pays as much as she commits to contributewhenever the good is provided, so t i ( x ) = x i . (28)(ii) Proportional rule.
Each agent i pays proportionally to her commitment x i whenever the good is provided, so t i ( x ) = cx i P nj =1 x j . (29)(iii) Additive rule.
Each agent i pays the equal share c/n plus the differencebetween her commitment and the average commitment, so t i ( x ) = cn + x i − n n X j =1 x j . (30)The assumptions (a), (b), and (c) are easily verified for these transfer rules.We will measure the efficiency of a PCE profile s ∗ by the ratio of the maximumwelfare loss to the maximum possible surplus n ¯ v . Our measure is denoted by¯ C ( s ∗ ) and is given by¯ C ( s ∗ ) = sup ( v ,...,v n ) ∈ [0 , ¯ v ] n ( n ¯ v max { , P i v i − c } , if P i s ∗ i ( v i ) ≥ c ,0 , if P i s ∗ i ( v i ) < c .It turns out in the PCE presented below that the inefficiency emerges only whenthe public good is not provided when it is efficient to do so. Proposition 7.
A strategy profile s ∗ = ( s ∗ , ..., s ∗ n ) that satisfies (A ) is a PCE ifand only if for all i = 1 , ..., n and all v i ∈ [0 , ¯ v ] ,(i) if t i ( x ) is the pay-as-you-bid rule, then s ∗ i ( v i ) = v i and ¯ C ( s ∗ ) = 12 ; (ii) if t i ( x ) is the proportional rule, then s ∗ i ( v i ) = v i − c + 12 q v i + 4 c and ¯ C ( s ∗ ) = n n + 1 ; (iii) if t i ( x ) is the additive rule, then s ∗ i ( v i ) = n n − v i and ¯ C ( s ∗ ) = n − n − . Note that > n n +1 > n − n − . So, the additive rule is the most efficient amongthese three transfer rules. PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 35
Proof.
An agent who chooses x i worries about two contingencies. First, it couldbe that the total contribution is just below c , so P j x j = c − ε , so the good isnot provided, but had i contributed ε more it would have been provided. So, as ε →
0, agent i ’s loss is max { v i − x i , } .Second, it could be that all other agents contribute enough to cover c , so P j = i x j ≥ c , so the agent could have contributed nothing and still got the good.In this case the loss is the amount of contribution, t i ( x ). Under all three assump-tions, this loss is maximized when the other agents’ contributions exactly equalto the cost, so P j = i x j = c . If t i ( x ) is given by (28), then the loss is x i . If t i ( x ) isgiven by (29), then the loss is cx i / ( c + x i ). If t i ( x ) is given by (30), then the lossis ( n − x i /n .As the first type of loss is weakly decreasing and the second type of loss isstrictly increasing in x i , we equalize the two and solve for x ∗ i ( v i ) that minimizesthe maximum loss. For each of the assumptions about t i ( x ), the solution x ∗ i ( v i ) isgiven by (i), (ii), and (iii) in Claim 7.The maximum individual surplus is v i , so the maximal individual loss ε i ( v i ) issimply max { v i − x ∗ i ( v i ) , } /v i = 1 − x ∗ i ( v i ) /v i . Substituting the obtained solutions x ∗ i ( v i ) into this expression, we obtain ε i ( v i ) in (i), (ii), and (iii) in Claim 7.It remains to verify that there exist values v j for each j = i such that P j = i x ∗ ( v j ) ≥ c . If t i ( x ) is given by (28), thensup X j = i x ∗ ( v j ) = sup X j = i v j n − v , If t i ( x ) is given by (29), thensup X j = i x ∗ ( v j ) = sup X j = i (cid:18) v j − c + 12 q v j + 4 c (cid:19) = ( n − (cid:18) ¯ v − c + 12 √ ¯ v + 4 c (cid:19) ≥ ( n − v . Finally, if t i ( x ) is given by (30), thensup X j = i x ∗ ( v j ) = sup X j = i nv j n − n − n ¯ v n − ≥ ( n − v . Since c ≤ ( n − v by assumption (27), we obtain that there exist values v j for each j = i such that P j = i x ∗ ( v j ) ≥ c . (cid:3) B.3.
Forecasting.
Here we consider the problem of forecasting of a variablewhose underlying distribution is unknown. With this example we illustrate hownoise influences learning. To keep the focus on learning, we present a one-shotdecision problem of a single player.Consider a forecaster who has to predict θ ∈ [0 , θ is drawn froma distribution F on [0 , u ( a, θ ) = − ( a − θ ) . Before making a prediction, the forecaster observes a noisy signal z about θ .We analyze two variations of this model. In one variation, the forecaster knowshow the noisy signal z is generated but she is uncertain about the distributionof the fundamental variable θ . In the other variation, the forecaster knows thedistribution of θ but she is uncertain about the signal generating process.B.3.1. Unknown Distribution of Variable θ . Here we are interested in how to makea prediction without knowing how the variable of interest is distributed.Suppose that the forecaster does not know the distribution F . She only knowsthe expected value of of this distribution, denoted by θ . We allow for any suchdistribution F that admits a density f such that δ ≤ f ( θ ) ≤ /δ for some δ ∈ (0 , F δ ofsuch distributions is thus given by F δ = { F ∈ ∆([0 , E F [ θ ] = θ and δ ≤ f ( θ ) ≤ /δ for all θ ∈ [0 , } . The forecaster can condition her prediction on a noisy signal z about θ . Thesignal generating process is known and given by a conditional probability distri-bution G ( z | θ ). For a given ε >
0, signal z reveals the true value θ with probability1 − ε and a uniform draw from [0 ,
1] with probability ε , so G ε ( z | θ ) = ( εz, if z < θ ,1 − ε + εz, if z ≥ θ .Had the forecaster known the distribution F ∈ F δ , she could have formed aposterior about θ conditional on the signal z . Let E F,G ε [ ·| z ] denote the expectationunder this posterior.The maximum loss of a prediction a ∈ [0 ,
1] given z ∈ [0 ,
1] is l ( a ; z ) = sup F ∈F δ sup a ′ ∈ [0 , E F,G ε [ − ( a ′ − θ ) | z ] − E F,G ε [ − ( a − θ ) | z ] ! . A best compromise is a prediction a ∗ ( z ) that achieves the least maximum loss, so a ∗ ( z ) ∈ arg min a ∈ [0 , l ( a ; z ) . This problem can be embedded in our formal setting as described in Section 2.As the distribution F is unknown, it is identified as the state. So the set of statesis F δ . In the formal game, first nature chooses a state F . Then the nonstrategicplayer 0 observes F and generates a signal z . Finally, the forecaster observes z and makes a prediction. Proposition 8.
Let ε ∈ [0 , and δ ∈ (0 , . The forecaster’s best compromise is a ∗ ( z ) = (1 − λ ) z + λθ , where λ = ε (cid:18) δ − ε (1 − δ ) + 1 δ + ε (1 − δ ) (cid:19) . PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 37
We postpone the proof to the end of this subsection. To describe the intuition,let us start with a simple observation.
Lemma 2. l ( a ; z ) = sup F ∈F δ ( a − E F,G ε [ θ | z ]) . The intuition is as follows. The variance of θ conditional on a signal z entersthe payoffs additively, and thus cancels out when computing the loss. As a result,the maximum loss l ( a ; z ) is simply the maximum quadratic distance between a and the expected value of θ conditional on z . Proof of Lemma 2.
Fix G ε . Let ¯ a F ( z ) = E F,G ε [ θ | z ]. Observe that¯ a F ( z ) ∈ arg max a ′ ∈ [0 , E F,G ε [ − ( a ′ − θ ) | z ] . (31)So, we havesup a ′ ∈ [0 , E F,G ε [ − ( a ′ − θ ) | z ] − E F,G ε [ − ( a − θ ) | z ] = E F,G ε [ − (¯ a F ( z ) − θ ) + ( a − θ ) | z ]= E F,G ε [( a − ¯ a F ( z ))( a + ¯ a F ( z ) − θ ) | z ] = ( a − ¯ a F ( z )) , where the first equality is by (31) and the last equality is by E F,G ε [ θ | z ] = ¯ a F ( z ).Thus, l ( a ; z ) = sup F ∈F δ ( a − ¯ a F ( z )) = sup F ∈F δ ( a − E F,G ε [ θ | z ]) . (cid:3) So, different distributions F ∈ F δ induce different posterior means E F,G ε [ θ | z ].When making a prediction a , the forecaster worries about a possible loss of( a − E F,G ε [ θ | z ]) . The best compromise is thus the midpoint between the highestposterior mean H ( z ) and the lowest posterior mean L ( z ) conditional on z , where H ( z ) = sup F ∈F δ E F,G ε [ θ | z ] and L ( z ) = inf F ∈F δ E F,G ε [ θ | z ] . (32)Note that the posterior mean E F,G ε [ θ | z ] is always between the prior mean θ and the observed signal z . So, the best compromise a ∗ ( z ) can be expressed asa weighted average of z and θ .Note that the best compromise a ∗ is continuous in ε and satisfies a ∗ ( z ) → z as ε → a ∗ ( z ) → θ as ε → . As the noise vanishes, the signal becomes the best predictor. As the noise becomesdominant, so the signal becomes uninformative, the ex ante mean θ becomes thebest predictor.Now fix ε > a ∗ ( z ) → z + θ δ → . When the only assumption imposed on the set of distributions F is that E F [ θ ] = θ ,the best predictor is the midpoint between z and θ . In particular, even thoughthe noise ε may be very small, it is fixed and plays no role in this limit. Intuitively,depending on distribution F , the signal z might be the best predictor. Or it mightbe useless when the value z of this signal under F is very unlikely, in which case the signal contains hardly any information, and θ is the best predictor. The bestcompromise balances the losses in these two extreme cases.Note the discontinuity of a ∗ ( z ) at ε = 0 and δ = 0. This is because the limit of a ∗ depends on the order of the limits of ε and δ . Proof of Proposition 8.
Let H ( z ) and L ( z ) be the highest and lowest posteriormeans of θ conditional on z given by (32). We have l ( a ; z ) = sup F ∈F δ ( a − E F,G ε [ θ | z ]) = max (cid:8) ( a − H ( z )) , ( a − L ( z )) (cid:9) where the first equality is by Lemma 2, and the last equality is by the convexityof the expression. Thus, a ∗ ( z ) = inf a ∈ [0 , l ( a ; z ) = 12 ( H ( z ) + L ( z )) . It remains to find H ( z ) and L ( z ). Suppose that z ≥ θ . Observe that E F,G ε [ θ | z ] = (1 − ε ) f ( z ) z + ε R θf ( θ )d θ (1 − ε ) f ( z ) + ε R f ( θ )d θ = (1 − ε ) f ( z ) z + εθ (1 − ε ) f ( z ) + ε is increasing in f ( z ). Using the assumption that f ( z ) ≤ /δ , we have H ( z ) = sup F ∈F δ (1 − ε ) f ( z ) z + εθ (1 − ε ) f ( z ) + ε = (1 − ε ) f ( z ) z + εθ (1 − ε ) f ( z ) + ε (cid:12)(cid:12)(cid:12)(cid:12) f ( z )=1 /δ = (1 − ε ) z + εδθ − ε + εδ . Using the assumption that f ( z ) ≥ δ , we have L ( z ) = inf F ∈F δ (1 − ε ) f ( z ) z + εθ (1 − ε ) f ( z ) + ε = (1 − ε ) f ( z ) z + εθ (1 − ε ) f ( z ) + ε (cid:12)(cid:12)(cid:12)(cid:12) f ( z )= δ = (1 − ε ) δz + εθ (1 − ε ) δ + ε . Analogously, for z ≤ θ we obtain H ( z ) = (1 − ε ) δz + εθ (1 − ε ) δ + ε and L ( z ) = (1 − ε ) z + εδθ − ε + εδ . Thuswe obtain a ∗ ( z ) = 12 ( H ( z ) + L ( z )) = 12 (cid:18) (1 − ε ) z + εδθ − ε + εδ + (1 − ε ) δz + εθ (1 − ε ) δ + ε (cid:19) . (cid:3) B.3.2.
Unknown Distribution of Signal z . Here we are interested in how uncertainnoise influences prediction.Suppose that the forecaster knows the distribution F of θ , but is uncertainabout how the signal z is generated. The signal generating process is given asfollows. Let δ >
0. Let the signal z be given by the sum of the variable θ and anoise y , so z = θ + y, where y is drawn independently from the interval [ − δ, δ ]. Note that z = θ + y ∈ [ − δ, δ ], so z can be outside of [0 , ε ∈ [0 , − ε , the noise y is drawn from a givendistribution G on [ − δ, δ ]. With the complementary probability ε , the noise y isdrawn from a distribution G over [ − δ, δ ]. We assume that the forecaster knowsthis process except that she does not know G . We allow for all distributions G whose support is within [ − δ, δ ]. So the forecaster is fairly certain that the noise y is PRIOR-FREE ALTERNATIVE TO PERFECT BAYESIAN EQUILIBRIUM 39 drawn from G , but puts probability ε that it is drawn from another distribution.Thus, a state is identified with the distribution G , and the set of states G δ is theset of all distributions on [ − δ, δ ].Let E F,G,ε [ ·| z ] denote the expectation of θ conditional on z for a given G ∈ G δ this posterior. The maximum loss associated with a prediction a ∈ [0 ,
1] given asignal z ∈ [0 ,
1] is calculated as in Section B.3, except that now the set of statesis G δ , so l ( a ; z ) = sup G ∈G δ sup a ′ ∈ [0 , E F,G,ε [ − ( a ′ − θ ) | z ] − E F,G,ε [ − ( a − θ ) | z ] ! . In the next result, the distribution F of θ is defined on R and its density is zerooutside of [0 , Proposition 9.
Let ε ∈ [0 , and δ > . The forecaster’s best compromise is a ∗ ( z ) = 12 ( H ( z ) + L ( z )) , where H ( z ) = sup G ∈G δ E F,G,ε [ θ | z ] = sup x ∈ [ − δ,δ ] εf ( z − x )( z − x ) + (1 − ε ) R δ − δ ( z − y ) f ( z − y )d G ( y ) εf ( z − x ) + (1 − ε ) R δ − δ f ( z − y )d G ( y ) ,L ( z ) = inf G ∈G δ E F,G,ε [ θ | z ] = inf x ∈ [ − δ,δ ] εf ( z − x )( z − x ) + (1 − ε ) R δ − δ ( z − y ) f ( z − y )d G ( y ) εf ( z − x ) + (1 − ε ) R δ − δ f ( z − y )d G ( y ) . The proof is analogous to that of Proposition 8 and thus omitted.Just like in Section B.3, in this model the best compromise is the midpointbetween the highest posterior mean H ( z ) and the lowest posterior mean L ( z )conditional on z . The difference from Section B.3 is how these extreme posteriormeans are calculated. Observe that they are always in the δ -neighborhood of z .As the neighborhood size δ approaches to 0, the extreme posterior means approach z , so H ( z ) → z and L ( z ) → z as δ → . Consequently, a ∗ ( z ) → z as δ → δ > ε . As the noise vanishes, ε → G , so H ( z ) → E F,G , [ θ | z ] and L ( z ) → E F,G , [ θ | z ] as ε → . Consequently, a ∗ ( z ) → E F,G , [ θ | z ] as ε →
0. For instance, if G is the uniform dis-tribution, then the best predictor converges to the expected value of θ conditionalon being within δ of the signal.Finally, as ε →
1, so the role of the benchmark G disappears and any noisewithin [ − δ, δ ] becomes possible. We obtain H ( z ) → z + δ and L ( z ) → z − δ as ε → . Consequently, a ∗ ( z ) → z as ε →
1. So, as the prior G loses its value, the signal zz