Mechanism Design for Cumulative Prospect Theoretic Agents: A General Framework and the Revelation Principle
MMechanism Design for Cumulative Prospect TheoreticAgents: A General Framework and the RevelationPrinciple
Soham R. Phade ∗ and Venkat Anantharam †,‡ Abstract
This paper initiates a discussion of mechanism design when the participating agentsexhibit preferences that deviate from expected utility theory (EUT). In particular, weconsider mechanism design for systems where the agents are modeled as having cumu-lative prospect theory (CPT) preferences, which is a generalization of EUT preferences.We point out some of the key modifications needed in the theory of mechanism designthat arise from agents having CPT preferences and some of the shortcomings of the clas-sical mechanism design framework. In particular, we show that the revelation principle,which has traditionally played a fundamental role in mechanism design, does not con-tinue to hold under CPT. We develop an appropriate framework that we call mediatedmechanism design which allows us to recover the revelation principle for CPT agents.We conclude with some interesting directions for future work.
Keywords— mechanism design; game theory; revelation principle; cumulative prospecttheory
In nearly every application of mechanism design, the decision-making entities are predom-inantly human beings faced with uncertainties. These uncertainties, for example, couldarise from a combination of one or more factors from the following: (i) lack of informationabout the outcomes (e.g. oil lease auctions, kidney-exchange, insurance markets), (ii) eachplayer having uncertainty about other players’ behavior (e.g. voting behavior in elections,inclination to getting vaccinated in immunization programs), (iii) strategic interactions be-tween the players (e.g. players could employ randomized strategies to hedge their marketreturns), (iv) randomness introduced by design (e.g. Tullock contests, where the probabil-ity of winning a prize depends on the amount of effort an agent puts into it). Naturally,to realize the mechanism designer’s objectives, it is beneficial to consider as accurate andgeneral models for human preference behavior under uncertainty as possible. Cumulative ∗ Corresponding author † The authors are with the Department of Electrical Engineering and Computer Sciences, University of Cali-fornia, Berkeley, Berkeley, CA 94720. [email protected], [email protected] ‡ Research supported by the NSF Science and Technology Center grant CCF- 0939370: "Science of Informa-tion", the NSF grants ECCS-1343398, CNS-1527846, CIF-1618145, CCF-1901004 and CIF-2007965, and theWilliam and Flora Hewlett Foundation supported Center for Long Term Cybersecurity at Berkeley. a r X i v : . [ c s . G T ] J a n rospect theory (CPT), proposed by Tversky and Kahneman [1992], is one of the leadingtheories for decision making under uncertainty. Our goal here is to study mechanism designwhen players exhibit CPT preferences.We are interested in situations where the agents participating in the system have private types (comprised of private information and preferences). Player and agent are used as in-terchangeable terms. The system operator is in a position to set the rules of communicationand can control the implementation in the system. It aims to achieve certain goals, suchas social welfare or revenue generation, without getting to directly observe the types of theplayers. Studying these systems when agents have CPT preferences requires modificationsto the formal structures commonly encountered in classical mechanism design [Harsanyi,1967, Myerson, 1979, 1982, 2004, Mas-Colell et al., 1995]. But before engaging in a sys-tematic discussion of these issues, let us briefly describe our key result.This starts with the observation that if the players are assumed to have CPT preferencesinstead of expected utility theory (EUT) preferences, then the revelation principle [Myerson,1981], one of the fundamental principles in mechanism design, does not hold anymore. Arelated observation was made by Karni and Safra [1989], where the authors show that ina second-price sealed-bid auction the revelation principle holds in general if and only if theplayers have EUT preferences. Chew [1985] provides an example to show that the revelationprinciple fails in a second-price sealed-bid auction when the players have preferences givenby implicit weighted utility theory [Dekel, 1986, Chew, 1989].The classical mechanism design framework is comprised of a fixed number of players, anallocation set, a set of types for each player, and a signal set for each player. (In this paper,we will be concerned with the setting where all these sets are assumed to be finite.) Thesystem operator commits to an allocation function, i.e. a function from the signal profile ofthe players to an allocation (see (2.14) for the formal definition).The mechanism operates as follows:1. Each player sends a signal strategically to the system operator based on its type (whichis private knowledge to the player).2. The system operator implements the allocation based on the signals from all the play-ers in accordance to the allocation function that it committed to.If we assume a prior over the types of the players which is common knowledge to allthe agents and the system operator, and we assume that the signal sets of all the players,the allocation set and the allocation mapping are also common knowledge, then this consti-tutes a Bayesian game and one studies the outcome of such a game through its Bayes-Nashequilibria (see (2.18) for the formal definition). The revelation principle states that forthe question of implementability of social choice functions (see (2.5) and (2.19) for formaldefinitions of social choice functions and their implementability), it is enough to assumethe signal set to be the same as the type set for each player and confine attention to theequilibrium in which each player reports her type truthfully.We propose a modification to the above framework that we call a mediated mechanism .We introduce a new stage where the system operator acts like a mediator and sends eachplayer a private message sampled from a certain joint distribution on the set of messageprofiles. The allocation chosen by the system operator can now depend on both the messageprofile and the signal profile. Further, we explicitly allow the choice of the allocation to2e randomized, which turns out to have no advantage in the classical mechanism designframework but can lead to benefits with CPT agents.A mediated mechanism is therefore comprised of a fixed number of players, an allocationset, a set of types for each player, a message set for each player, and a signal set for eachplayer, all of which are generally assumed to be finite sets. The system operator commits toa mediator distribution, which is a probability distribution on the set of message profiles.It also commits to a mediated allocation function, which maps each pair of signal profileand message profile to a probability distribution on allocations (see (3.9) for the formaldefinition).The mechanism operates as follows:1. The system operator samples a message profile from the declared mediator distribu-tion and sends the individual messages to each player privately.2. Each player receives her mediator message and, based on this message and her pri-vately known type, sends a signal strategically to the system operator.3. Based on the signals collected from all the players and the sampled message profile,the system operator samples the allocation in accordance to the probability distribu-tion on allocations resulting from the mediated allocation function that it committedto.Similarly to the previous setting we assume a prior over the types of the players thatis common knowledge to all the agents and the system operator. We also assume that themessage sets and the signal sets of all the players, the mediator distribution, the alloca-tion set, and the allocation-outcome mapping are common knowledge. This along with themechanism operation stated above constitutes a Bayesian game and we study the outcomeof such a game through its Bayes-Nash equilibria (see (3.14) for the formal definition). Withthis modified framework, we recover a form of the revelation principle which states that itis enough to assume the signal set to be the same as the type set for each player and confineour attention to the equilibrium in which each player reports her true type irrespective ofthe private message she receives from the mediator. (See statement (i) of Theorem 3.4.)As the mediator message sets could be arbitrary, it might seem that the problem ofdesigning the signal sets has been transformed into the problem of designing the messagesets. Although this is true, notice that the revelation principle allows us to restrict ourattention to truthful strategies for each player, which have a simple form, thus resolving thedifficult task of finding all the Bayes-Nash equilibria of the resulting game. Further, the factthat truthful reporting does not depend on the private message received by a player makesit a practical and natural strategy for the players.We now resume our discussion of the different aspects involved in the study of mech-anism design when agents have CPT preferences. The majority of the mechanism designliterature has been restricted to EUT modeling of individual decision-making under uncer-tainty. Indeed, EUT has a nice normative interpretation and provides a useful and insightfulfirst-order approximation (see, for example, Schoemaker [1982]). However, systematic de-viations from the predictions of EUT have been observed in several empirical studies involv-ing human decision-makers [Allais, 1953, Fishburn and Kochenberger, 1979, Kahnemanand Tversky, 2013] (see Starmer [2000] for an excellent survey). With the advent of e-commerce activities and the ever-growing online marketplaces such as Amazon, eBay, and3ber, where the participating agents are largely human beings, who exhibit behavior thatis highly susceptible to these deviations from EUT, it has become crucial to account for suchbehavioral deviations in the modeling of these systems. (For example, Pavlou and Dimoka[2006] discuss the phenomenon of premium prices showing up in online marketplaces suchas eBay to differentiate among sellers based on their reputation and buyers’ perceived risks.)A typical environment in the traditional mechanism design setup consists of a set ofplayers that have private information about their types and an allocation set listing thepossible alternatives from which the system operator chooses one that is best suited giventhe players’ types. As mentioned earlier, we assume that the system operator controls theimplementation and the players do not have separate decision domains. (Recall that byprivate decision domains we mean possible actions for the player that directly affect theoutcomes.) This is typical in several online marketplaces. For example, in online advertisingplatforms such as Google Ads, the platform has complete control over where to place whichads. Note that although the agents can affect the implementation of the system throughtheir bids, these signals fall under the communication protocol set by the system, leaving theultimate implementation in the hands of the system operator. In online matching marketssuch as eBay and Uber, the platform matches the buyers to sellers as in eBay, or riders todrivers as in Uber. Even if the system operator has complete control over the implementation, it wants theimplementation to depend on the types of the agents. However, it does not have access tothese types, and hence needs to design a mechanism to achieve this goal. Thus the systemimplementation indirectly depends on the choices of the participating agents. Note that thee-commerce applications mentioned above—Amazon, eBay, and Uber—fit well in this setup.Indeed, these are instances of a delivery system, an auction house, and a clearinghouse,which have been topics of interest for several years in mechanism design. However, thenature of these applications, and the presence of vast data corresponding to several repeatedshort-lived interactions of the system with any given user, makes it feasible to incorporatethe behavioral features displayed by the users.It has been a convention to assume that the outcome set for each player is identical tothe allocation set, and hence the type for each player is assumed to capture her preferencesover the allocation set (see, for example, Vohra [2011]). However, in principle, the outcomeset for any player need not be the same as the allocation set. Indeed the allocation set is alist of the alternatives available to the system operator to implement, whereas the outcomeset consists of the outcomes realized by the players, and these can be quite different. Forexample, in the case of Amazon, the allocation set consists of alternative resource allocationsto fulfill the delivery of purchased products, whereas the outcome set of a buyer consistsof features such as time of delivery, place of delivery, etc. It makes sense to consider thepreferences of a player over her outcome set, and any consideration of her preferences overthe allocation set should be thought of as a pullback or a precomposition of her preferencesover the outcome set with respect to the (possibly random) function that maps allocationsto outcomes for this player.We allow the above mapping from allocations to outcomes for any player to be random-ized. Indeed, more often than not, the system operator does not have complete control overthe outcomes of the players due to intrinsic uncertainties present in the system. For ex-ample, fixed resource allocations by Amazon can lead to uncertainty in the delivery times,possibly due to factors not part of the system model. In the case of Uber, upon matching the4iders with the drivers in a certain way and choosing their corresponding routes, the arrivaltimes and the riding experience of the users remain uncertain. In an auction setting suchas eBay, if we consider the outcome set for any player to indicate if she receives the item ornot, then the mapping from allocation to outcomes is deterministic. However, if we modelthe outcome set to indicate whether the player is satisfied with the item she receives, thenwe have to allow the mapping to be randomized.Furthermore, the system operator might not be able to observe the outcome realization,for example the ride experience of a passenger. It can only try to learn this in hindsightthrough customer feedback. Besides, the outcome set for any single player is typically smallas compared to the allocation set and the product of the outcome spaces of all the partici-pating agents. Thus, treating each player’s outcome set separately would enable us to focuson the preference behavior of an individual player and have better models for this player’spreferences.The (random) mapping from allocations to outcomes for any player induces a lottery L on the outcome set of this player for each allocation. EUT satisfies the linearity propertywhich states that U ( αL +(1 − α ) L ) = αU ( L )+(1 − α ) U ( L ) , where ≤ α ≤ , L , L aretwo lotteries, and U ( · ) denotes the expected utility of the lottery within the parentheses.This property of EUT allows us to model the type of a player by considering her utilityvalues for each allocation. For any lottery L over her outcomes that is induced by a lotteryover the allocations µ , we can evaluate her utility U ( L ) by taking the expectation over herutility values of the allocations with respect to the distribution µ . CPT on the other handdoes not satisfy this linearity property (see, for example, Tversky and Kahneman [1992]),and hence it is important that we consider the general model with separate outcome sets.We formalize this general setup in subsection 2.2 and provide preliminary backgroundon CPT preferences in subsection 2.3. Then, in subsection 2.4, we consider the traditionalmechanism design framework where each player knows her (private) type and strategi-cally sends a signal to the system operator. The system operator collects these signals andimplements a lottery over the allocation set.We define a social choice function as a function mapping each type profile into a lotteryover the product of the outcome sets for each player (see (2.5) for the formal definition).As an intermediate step, we consider an allocation choice function (i.e. a function thatmaps type profiles into lotteries over the allocation set, see (2.7)). Each allocation choicefunction uniquely defines a social choice function through the allocation-outcome mapping(see (2.8)), which we think of as a mapping from allocations to probability distributions onthe product of the outcome sets of the agents. Note that there can be multiple allocationchoice functions that give rise to the same social choice function. We define the notion ofimplementability for an allocation choice function in Bayes-Nash equilibrium (see (2.19)).We say that a social choice function is implementable in Bayes-Nash equilibrium if thereexists an allocation choice function that is implementable in Bayes-Nash equilibrium andinduces this social choice function.We similarly define the notions of implementability in dominant equilibrium. Here,we identify an additional notion of implementability that we call implementable in belief-dominant equilibrium. Roughly speaking, a dominant strategy is a best response to all thestrategy profiles of the opponents (see (2.21)), and a belief-dominant strategy is a bestresponse to all the beliefs over the strategy profiles of the opponents (see (2.23)). UnderEUT, the notion of a dominant strategy is equivalent to that of a belief-dominant strategy.5owever, this is not true in general when the agents have CPT preferences, thus making itnecessary to distinguish between these two notions of equilibrium.In section 3, we define the notions of direct mechanism (see (3.1)) and truthful imple-mentation (see (3.2)). We then give an example that highlights the shortcoming of restrict-ing oneself to direct mechanisms when the players have CPT preferences, as opposed toEUT preferences. In particular, we consider a -player setting where the players have CPTpreferences that are not EUT preferences. Example 3.1 gives an allocation choice functionfor which the revelation principle does not hold for implementation in Bayes-Nash equilib-rium. We then introduce the framework of mediated mechanism design in subsection 3.1.We define the corresponding notions of Bayes-Nash equilibrium (see (3.14)), dominantequilibrium (see (3.18)), and belief-dominant equilibrium (see (3.19)) for mediated mech-anisms. In Theorem 3.4, we recover the revelation principle under certain settings (seetable 1).We conclude in section 4 with some remarks and directions for future work. Let {·} denote the indicator function that is equal to one if the predicate inside the bracketsis true and is zero otherwise. Let ∆( · ) denote the set of all probability distributions overthe (finite) set within the parentheses. Let supp( · ) denote the support of the probabilitydistribution within the parentheses. Let co( · ) denote the convex hull of the set within theparentheses. For a function f : X → ∆( Y ) , let f ( y | x ) = f ( x )( y ) denote the probability of y under the probability distribution f ( x ) . Let L = { ( p , z ); . . . ; ( p t , z t ) } denote a lottery with outcomes z j , ≤ j ≤ t , with their corresponding probabilities givenby p j . We assume the lottery to be exhaustive (i.e. (cid:80) tj =1 p j = 1 ). Note that we are allowedto have p j = 0 for some values of j and we can have z k = z l even when k (cid:54) = l . If a lottery L consists of a unique outcome z that occurs with probability , then with an abuse ofnotation we will denote the lottery L = { (1 , z ) } simply by L = z . Similarly, if a probabilitydistribution f ( x ) assigns probability to y , then again with an abuse of notation we willwrite f ( x ) = y . If, for each x , f ( x ) has a singleton support, then with an abuse of notationwe will treat f as a function from X to Y . Let [ n ] := { , , . . . , n } be the set of players participating in the system. Let A denote the setof allocations for this system. We assume unless stated otherwise that the set of allocationsis finite, say A := { α , . . . , α l } . For example, in the sale of a single item (or multiple items),it could represent the allocation of the item(s) to the different individuals. In a routingsystem, such as traffic routing or internet packet routing, it could represent the differentrouting alternatives. More generally, in a resource allocation setting it could represent theassignment of resources to the participating agents (with their corresponding payments)that respect the system (and budget) constraints. In a voting scenario, it could represent the6inning candidate. Thus, we imagine the allocations α ∈ A as being the various alternativesavailable to the system operator to implement.Traditionally, each player is assumed to have a value for each of the allocations, and thisdefines the type of this player. It describes the preferences of a player over the allocations,and further, by assuming EUT behavior, we get her preferences over the lotteries over theseallocations. Here, instead, we assume that for each player i ∈ [ n ] , we have a finite set of outcomes Γ i := { γ i , . . . , γ k i i } , and player i ’s type is defined by her CPT preferences overthe lotteries on this set Γ i . We imagine the set Γ i to capture the outcome features that arerelevant to player i . Thus the outcome set Γ i allows us to separate out the features that affectplayer i from the underlying allocations that give rise to these outcomes. We capture thisrelation between the allocation set and the outcome sets through a mapping ζ : A → ∆(Γ) that we call the allocation-outcome mapping , where Γ := (cid:81) i Γ i . Let ζ i : A → ∆(Γ i ) denoteallocation-outcome mapping for player i given by the marginal of ζ on the set Γ i .From a behavioral point of view it is natural to model a player’s preferences on theoutcome set Γ i rather than the allocation set A . Then why is it that the sets Γ i and themapping ζ are usually missing from the mechanism design framework prevalent in theliterature? At the end of subsection 2.4, after setting up the relevant notation, we will showthat under EUT, from the point of view of the typical goals of the mechanism designer,it is enough to consider a transformation of the system where Γ i = A , for all i , and theallocation-outcome mappings are trivial , namely, ζ i ( α ) = α , for all α ∈ A, i ∈ [ n ] (this isshown formally in Appendix C). We will also show that this does not hold in general whenthe players do not have EUT preferences, and in particular when they have CPT preferences.We model the preference behavior of the players using cumulative prospect theory (CPT)that we describe now. (For more details, see [Wakker, 2010].) Suppose Γ i is the outcome set for player i , who is associated with a value function v i : Γ i → R and two probability weighting functions w ± i : [0 , → [0 , . The value function v i partitionsthe set of outcomes Γ i into two parts: gains and losses ; an outcome γ i ∈ Γ i is said to bea gain if v i ( γ i ) ≥ , and a loss otherwise. The probability weighting functions w + i and w − i will be used for gains and losses, respectively. The probability weighting functions w ± i are assumed to satisfy the following: (i) they are strictly increasing, (ii) w ± i (0) = 0 and w ± i (1) = 1 . We say that ( v i , w ± i ) are the CPT features of player i .Suppose player i faces a lottery L i ∈ ∆(Γ i ) given by { ( p ji , γ ji ) } ≤ j ≤ k i . Let p i := ( p i , . . . , p k i i ) .Let β i := ( β i , . . . , β k i i ) be a permutation of (1 , . . . , k i ) such that v i ( γ β i i ) ≥ v i ( γ β i i ) ≥ · · · ≥ v i ( γ β kii i ) . (2.1)Let ≤ j r ≤ k i be such that v i ( γ β ji i ) ≥ , for ≤ j ≤ j r , and v i ( γ β ji i ) < , for j r < j ≤ k i .(Here j r = 0 when v i ( γ β ji i ) < for all ≤ j ≤ k i .) The CPT value V i ( L i ) of the lottery L i is evaluated using the value function v i and the probability weighting functions w ± i asfollows: V i ( L i ) := j r (cid:88) j =1 π + i ( j, p i , β i ) v i ( γ β ji i ) + k i (cid:88) j = j r +1 π − i ( j, p i , β i ) v i ( γ β ji i ) , (2.2)7here π + i ( j, p i , β i ) , ≤ j ≤ j r and π − i ( j, p i , β i ) , j r < j ≤ k i , are decision weights definedvia: π + i (1 , p i , β i ) := w + i ( p β i i ) ,π + i ( j, p i , β i ) := w + i ( p β i i + · · · + p β ji i ) − w + i ( p β i i + · · · + p β j − i i ) , for < j ≤ k i ,π − i ( j, p i , β i ) := w − i ( p β kii i + · · · + p β ji i ) − w − i ( p β kii i + · · · + p β j +1 i i ) , for ≤ j < k i ,π − i ( k i , p i , β i ) := w − i ( p β kii i ) . Although the expression on the right in equation (2.2) depends on the permutation β i , onecan check that the formula evaluates to the same value V i ( L i ) as long as the permutation β i satisfies (2.1). The CPT value in equation (2.2) can equivalently be written as: V i ( L i ) = j r − (cid:88) j =1 w + i (cid:32) j (cid:88) i =1 p β ji i (cid:33) (cid:20) v i ( γ β ji i ) − v i ( γ β j +1 i i ) (cid:21) + w + i (cid:32) j r (cid:88) i =1 p β ji i (cid:33) v i (cid:18) γ β jri i (cid:19) + w − i k i (cid:88) i = j r +1 p β ji i v i ( γ β jr +1 i i )+ k i − (cid:88) j = j r +1 w − i k i (cid:88) i = j +1 p β ji i (cid:20) v i ( γ β j +1 i i ) − v i ( γ β ji i ) (cid:21) . (2.3)A person is said to have CPT preferences if, given a choice between lottery L i and lottery L (cid:48) i , she chooses the one with higher CPT value.The expected utility of player i is completely characterized by her value function v i :Γ i → R . For a lottery L i = { ( p ji , γ ji ) } ≤ j ≤ k i , the expected utility is given by U i ( L i ) := k i (cid:88) j =1 p ji v i ( γ ji ) . A person is said to have EUT preferences if, given a choice between lottery L i and lottery L (cid:48) i ,she chooses the one with higher expected utility. Observe that, if the probability weightingfunctions are linear, i.e. w ± i ( p ) = p , for p ∈ [0 , , then V i ( L i ) = U i ( L i ) for all lotteries L i ∈ ∆(Γ i ) . Thus EUT preferences are a special case of CPT preferences. For each i , let Θ i denote the set from which the permissible types for player i are drawn.Corresponding to any type θ i for player i , let v i : Γ i → R be her value function , and w ± i :[0 , → [0 , be her probability weighting functions . Let V θ i i ( L i ) denote the CPT value ofthe lottery L i ∈ ∆(Γ i ) for player i having type θ i . Thus, the type θ i completely determinesthe preferences of player i over lotteries on her outcome set Γ i . We will assume that thesets Θ i are finite for all i .Let θ := ( θ , . . . , θ n ) denote the profile of types of the players, and let Θ := (cid:81) i Θ i . Weassume that each player knows her type but cannot observe the types of her opponents.8et the set of players [ n ] , their corresponding type sets Θ i , i ∈ [ n ] , the allocation set A , and the outcome spaces Γ i , i ∈ [ n ] , together with the mapping ζ form an environment ,denoted by E := (cid:0) [ n ] , (Θ i ) i ∈ [ n ] , A, (Γ i ) i ∈ [ n ] , ζ (cid:1) . (2.4)A social choice function g : Θ → ∆(Γ) (2.5)determines a lottery over the product of the outcome sets of the individual players given thetype profile θ of all the players. The outcome choice function for player i corresponding tothe social choice function g is g i : Θ → ∆(Γ i ) , (2.6)given by the restriction of g to the set Γ i , and represents the lottery faced by player i giventhe type profile θ of all the players. We will treat the social choice function g as the goal ofthe mechanism designer, i.e, the goal is to design a mechanism to implement a social choicefunction g without having knowledge of the true types of the players.Let an allocation choice function f : Θ → ∆( A ) (2.7)represent the choice of the allocation to be implemented by the system operator given atype profile θ ∈ Θ . Note that f ( θ ) is a probability distribution over the allocations A .Thus we allow the system operator to implement a randomized allocation. A deterministicallocation choice function maps each type profile to a unique allocation. Since the mapping ζ is fixed and a part of the environment description, the allocation choice function f effectivelycaptures the goal of a mechanism designer. More precisely, let F ( g ) denote the set of allallocation choice functions that induce the social choice function g , i.e. for all θ ∈ Θ , g ( θ ) is the mixture probability distribution of the probability distributions ( ζ ( α ) , α ∈ A ) withweights f ( α | θ ) . We note that the set F ( g ) is non-empty if and only if g ( θ ) ∈ co { ζ ( α ) : α ∈ A } , for all θ ∈ Θ . We wish to design a mechanism that would implement an allocation choicefunction in F ( g ) . Thus a social choice function is implementable if and only if we canimplement an allocation choice function f that satisfies g ( γ | θ ) = (cid:88) α ∈ A f ( α | θ ) ζ ( γ | α ) , (2.8)for all γ ∈ Γ , θ ∈ Θ . This raises the main question in mechanism design, namely whetherwe can design a game that results in the implementation of some given allocation choicefunction f under certain rationality conditions on the players even when the system operatorcannot observe the players’ types.First, let us look at the the relationship between lotteries on allocations and lotteries onthe outcome set of a given player. Any lottery µ ∈ ∆( A ) induces a lottery L i ( µ ) ∈ ∆(Γ i ) given by L i ( γ i | µ ) := (cid:88) α ∈ A µ ( α ) ζ i ( γ i | α ) . (2.9)9iven that player i has type θ i , we know that the CPT value of lottery L i ( µ ) is V θ i i ( L i ( µ )) .This induces a value for player i with type θ i on the lottery µ denoted by W θ i i ( µ ) := V θ i i ( L i ( µ )) . (2.10)This defines a utility function W θ i i : ∆( A ) → R that gives the preference relation over thelotteries µ ∈ ∆( A ) for a player i having type θ i . Let u θ i i ( α ) := V θ i i ( ζ i ( α )) = W θ i i ( α ) (2.11)be the CPT value of the lottery for player i corresponding to allocation α . If player i hasEUT preferences, then we have that W θ i i ( µ ) = (cid:88) α ∈ A µ ( α ) u θ i i ( α ) . (2.12)We now consider a mechanism M := ((Ψ i ) i ∈ [ n ] , h ) , (2.13)consisting of a collection of finite signal sets Ψ i , one for each player i , and an allocationfunction h : Ψ → ∆( A ) , (2.14)where Ψ := (cid:81) i ∈ [ n ] Ψ i . Note that the allocation function is allowed to be randomized. Let ψ i denote a typical element of Ψ i , and ψ := ( ψ i ) i ∈ [ n ] denote a typical element of Ψ , calleda signal profile .It is straightforward to incorporate the feature that the outcome sets Γ i might be dif-ferent from the allocation set A , and the corresponding allocation-outcome mapping ζ , soas to extend the definition of a Bayes-Nash equilibrium strategy profile for the mechanism M and the implementability of an allocation choice function f in Bayes-Nash equilibrium.To do this, assume that the types of the individual players are drawn according to a priordistribution F ∈ ∆(Θ) and that this distribution is common knowledge among the agentsand the system operator. Let F i ∈ ∆(Θ i ) denote the marginal of F on Θ i . Suppose player i has type θ i . Then the belief of player i about the types of other players is given by theconditional distribution F − i ( θ − i | θ i ) := F ( θ i , θ − i ) F i ( θ i ) , for all θ − i ∈ Θ − i , θ i ∈ supp F i , where θ − i := ( θ j ) j (cid:54) = i is the profile of types of all players other than player i , Θ − i := (cid:81) j (cid:54) = i Θ j .Recall that ψ i denotes a typical element of Ψ i , and ψ := ( ψ i ) i ∈ [ n ] denotes a typicalelement of Ψ . Let Ψ − i := (cid:81) j (cid:54) = i Ψ j with a typical element denoted by ψ − i . Let σ i : Θ i → ∆(Ψ i ) (2.15)be a strategy for player i , and let σ := ( σ , σ , . . . , σ n ) denote a strategy profile. Let σ − i :=( σ j ) j (cid:54) = i denote the strategy profile of all players other than player i . For any type θ i (such that F i ( θ i ) > ) and signal ψ i , consider the probability distribution µ i ( θ i , ψ i ; M , F, σ − i ) ∈ ∆( A ) given by µ i ( α | θ i , ψ i ; M , F, σ − i ) := (cid:88) θ − i ∈ Θ − i F − i ( θ − i | θ i ) (cid:88) ψ − i ∈ Ψ − i (cid:89) j (cid:54) = i σ j ( ψ j | θ j ) h ( α | ψ ) , (2.16)10or all α ∈ A . Suppose player i has type θ i (such that F i ( θ i ) > ), and she chooses to signal ψ i . Then, by Bayes’ rule, the lottery faced by player i is given by L i ( µ i ( θ i , ψ i ; M , F, σ − i )) . This comes from the assumption that player i knows her type θ i , the common prior F , thestrategies σ j , j (cid:54) = i of her opponents, and the mapping ζ i . Given that player i has type θ i ,her CPT value for the lottery L i ( µ i ( θ i , ψ i ; M , F, σ − i ) is given by W θ i i ( µ i ( θ i , ψ i ; M , F, σ − i )) = V θ i i ( L i ( µ i ( θ i , ψ i ; M , F, σ − i )) , where we recall that W θ i i ( µ ) is the CPT value of player i with type θ for the lottery L i ( µ ) ∈ ∆(Γ i ) induced by the distribution µ ∈ ∆( A ) . Let the best response strategy set BR i ( σ − i ) forplayer i to a strategy profile σ − i of her opponents consist of all strategies σ ∗ i : Θ i → ∆(Ψ i ) such that W θ i i ( µ i ( θ i , ψ i ; M , F, σ − i )) ≥ W θ i i ( µ i ( θ i , ψ (cid:48) i ; M , F, σ − i )) , (2.17)for all θ i ∈ supp F i , ψ i ∈ supp σ ∗ i ( θ i ) , ψ (cid:48) i ∈ Ψ i . The rationale behind this definition is that aplayer’s best response strategy σ ∗ should not assign positive probability to any suboptimalsignal ψ i given her type θ i .A strategy profile σ ∗ is said to be an F -Bayes-Nash equilibrium for the environment E and common prior F with respect to the mechanism M if, for each player i , we have σ ∗ i ∈ BR i ( σ ∗− i ) . (2.18)We will refer to σ ∗ simply as a Bayes-Nash equilibrium when the respective environment E ,the common prior F , and mechanism M are clear from the context.We say that the allocation choice function f is implementable in F -Bayes-Nash equilibrium by a mechanism if there exists a mechanism M and an F -Bayes-Nash equilibrium σ suchthat f is the induced distribution, i.e. for all θ i ∈ supp F i , α ∈ A , we have f ( α | θ ) = (cid:88) ψ ∈ Ψ (cid:89) i ∈ [ n ] σ i ( ψ i | θ i ) h ( α | ψ ) . (2.19)An alternative notion is that of an allocation choice function f being implementable indominant equilibrium . The traditional notion states that a strategy σ i is a dominant strategy for player i if the signals in the support of σ i ( θ i ) are optimal given player i ’s type θ i and anysignal profile ψ − i of the opponents. More precisely, if we let µ i ( θ i , ψ i ; M , ψ − i ) := h ( ψ i , ψ − i ) , (2.20)then σ ∗ i is a dominant strategy if, for all θ i ∈ θ i , ψ i ∈ supp σ ∗ i ( θ i ) , ψ (cid:48) i ∈ Ψ i , and ψ − i ∈ Ψ − i ,we have W θ i i ( µ i ( θ i , ψ i ; M , ψ − i )) ≥ W θ i i ( µ i ( θ i , ψ (cid:48) i ; M , ψ − i )) . (2.21)Thus, if player i employs a dominant strategy, then regardless of the signal profile of theopponents she always signals a best response given her type. A dominant equilibrium is onein which each player plays a dominant strategy. We say that an allocation choice function11 is implementable in dominant equilibrium if there exists a mechanism M and a strat-egy profile σ ∗ = ( σ ∗ , . . . , σ ∗ n ) where each σ ∗ i is a dominant strategy (equivalently, σ ∗ is adominant equilibrium) such that (2.19) holds for all θ i ∈ Θ , α ∈ A .Under EUT, if a signal ψ i is a best response of player i for all of the opponents’ signalprofiles, then it is also a best response for any belief G − i ∈ ∆(Ψ − i ) of player i over heropponents’ signal profiles. However, under CPT, this need not hold. (See example 2.1.)This observation leads us to the following stricter notion of dominant strategies under CPT.We call a strategy σ i a belief-dominant strategy for player i if the signals in the support of σ i ( θ i ) are optimal given player i ’s type θ i and any belief G − i ∈ ∆(Ψ − i ) she has on the signalprofile of her opponents. If we let µ i ( θ i , ψ i ; M , G − i ) := (cid:88) ψ − i G − i ( ψ − i ) h ( ψ i , ψ − i ) , (2.22)then ψ ∗ i is a belief-dominant strategy for player i if, for all θ i ∈ θ i , ψ i ∈ supp σ ∗ i ( θ i ) , ψ (cid:48) i ∈ Ψ i ,and G − i ∈ ∆(Ψ − i ) , we have W θ i i ( µ i ( θ i , ψ i ; M , G − i )) ≥ W θ i i ( µ i ( θ i , ψ (cid:48) i ; M , G − i )) . (2.23)It is straightforward to check that under EUT a strategy is dominant if and only if it isbelief-dominant. A belief-dominant equilibrium is one in which every player plays a belief-dominant strategy. We say that an allocation choice function f is implementable in belief-dominant equilibrium if there exists a mechanism M and a strategy profile σ ∗ = ( σ ∗ , . . . , σ ∗ n ) where each σ ∗ i is a belief-dominant strategy (equivalently, σ ∗ is a belief-dominant equilib-rium) such that (2.19) holds for all θ i ∈ Θ , α ∈ A .Note that if σ ∗ is a belief-dominant strategy profile, and thus a belief-dominant equi-librium, then it is a dominant strategy profile, i.e. a dominant equilibrium, and also an F -Bayes-Nash equilibrium with respect to any prior distribution F on type profiles. Example . Let n = 2 . Let Θ = Θ = { UP , DN } . Let A = { a, b, c } , Γ = { I , II , III , IV , V } ,and Γ = A . Let the allocation-outcome mapping be given by the product distribution of themarginals ζ and ζ , given by ζ ( a ) = { (1 / , I); (1 / , V) } , ζ ( b ) = { (1 / , II); (1 / , IV) } , ζ ( c ) = { (1 , (III)) } , and ζ ( α ) = α, ∀ α ∈ A. Let the probability weighting functions for gains for thetwo players be given by w +1 ( p ) = exp {− ( − ln p ) . } , w +2 ( p ) = p, for p ∈ [0 , . In this example, we consider only lotteries with outcomes in the gains do-main, and hence an arbitrary probability weighting function for the losses can be assumed.Here, for player ’s probability weighting function, we employ the form suggested by Prelec[1998] (see figure 1). Note that player has EUT preferences. Let the value functions v and v be given by v I II III IV V
UP 2 x x + 1 1 .
99 1 0DN 0 0 1 0 0 v a b c UP 1 0 2DN 0 1 2 where x := 1 /w +1 (0 .
5) = 2 . . Note that x = 4 . and x + 1 = 3 . . We have, V UP1 ( L ( a )) = 2 xw +1 (0 .
5) = 2 ,V UP1 ( L ( b )) = 1 + xw +1 (0 .
5) = 2 , .
25 0 . .
75 10 . . . p w + ( p ) w +1 ( p ) w +2 ( p ) Figure 1: The solid curve shows the probability weighting function w +1 for player fromexample 2.1 and example 3.5. The dotted curve shows the probability weighting function w +2 for player in example 2.1 and example 3.5, which is the linear function correspondingto EUT preferences.and, V UP1 (0 . L ( a ) + 0 . L ( b )) = w +1 (0 .
75) + xw +1 (0 .
5) + ( x − w +1 (0 . . . (Here, we have w +1 (0 .
25) = 0 . , w +1 (0 .
5) = 0 . , and w +1 (0 .
75) = 0 . .) Considerthe mechanism M = ((Ψ i ) i ∈ [ n ] , h ) , where Ψ = Ψ = { UP , DN } , and h is given by h (UP , UP) = a, h (UP , DN) = b, h (DN , UP) = c, h (DN , DN) = c. Consider the strategies σ i (UP) = UP , and σ i (DN) = DN , for both the players i . It is easy to see that these strategies are dominant for both theplayers. However, if player has type UP and believes that there is an equal chance ofplayer reporting her strategy to be UP and DN , then player ’s best response is to report DN . Thus, σ is not a belief-dominant strategy for player .We will now look at the remark made earlier about the absence of the distinction be-tween the allocation set and the outcome sets in classical mechanism design, and why itis important to consider this distinction under CPT. In Appendix C, we show that underEUT it suffices to consider the scenario where the outcome set of each player is the same asthe allocation set by the simple expedient of interpreting each type θ i ∈ Θ i in terms of theutility function on allocations that it defines via (2.11).While equation (2.12) holds under EUT, under CPT in general it does not hold, and ingeneral the utility function W θ i i is not completely determined by the values u θ i i ( α ) , ∀ α ∈ A .Thus, we can either characterize the type of a player by her utility function W θ i i or by her13PT features which, combined with the mapping ζ i , together define the utility function W θ i i .In any given setting, it is more natural to put behavioral assumptions on the CPT features ( v i , w ± i ) than on the utility function W θ i i . Hence, we include the sets Γ i and the mappings ζ i , for all i , in our system model under CPT. A mechanism M = ((Ψ i ) i ∈ [ n ] , h ) is called a direct mechanism if Ψ i = Θ i , for all i . Let M d := ((Θ i ) i ∈ [ n ] , h d ) denote a direct mechanism, where h d : Θ → ∆( A ) (3.1)is the direct allocation function . Corresponding to a direct mechanism, let σ di : Θ i → Θ i denote the truthful strategy for player i , given by σ di ( θ i ) = θ i , (3.2)for all θ i ∈ Θ i . An allocation choice function f is said to be truthfully implementable in F -Bayes-Nash equilibrium (resp. dominant equilibrium or belief-dominant equilibrium)if there exists a direct mechanism M d such that the truthful strategy profile σ d is an F -Bayes-Nash equilibrium (resp. dominant equilibrium or belief-dominant equilibrium), andit induces f .The revelation principle says that if an allocation choice function is implementable inBayes-Nash equilibrium (resp. dominant equilibrium or belief-dominant equilibrium) by amechanism, then it is also truthfully implementable in Bayes-Nash equilibrium (resp. domi-nant equilibrium or belief-dominant equilibrium) by a direct mechanism. When the playersare restricted to have EUT preferences and the outcome set of each player is assumed to bethe same as the allocation set with the trivial allocation-outcome mapping, Myerson [1982]proved that the revelation principle holds for both the versions - Bayes-Nash equilibriumand dominant equilibrium (and hence also for belief-dominant equilibrium, since dominantstrategies are equivalent to belief-dominant strategies under EUT). It is easy to extend thisresult to the general setting where some of the individual outcome sets might differ fromthe allocation set, provided the players are restricted to have EUT preferences. Indeed, inAppendix C it is proved that, under EUT, an allocation choice function f is implementablein F -Bayes-Nash (resp. dominant or belief-dominant) equilibrium by a mechanism M forthe environment E with the equilibrium strategy σ , if and only if, for the correspondingenvironment E (cid:48) (defined in Appendix C), the corresponding allocation choice function f (cid:48) isimplementable in F (cid:48) -Bayes-Nash (resp. dominant or belief-dominant) by the same mech-anism M with the corresponding equilibrium strategy σ (cid:48) . We now observe that M is adirect mechanism for environment E if and only if it is a direct mechanism for environment E (cid:48) . Also, σ i is the truthful strategy with respect to the environment E and a direct mecha-nism M , if and only if, the corresponding strategy σ (cid:48) i is the truthful strategy with respectto the environment E (cid:48) and the same direct mechanism M . These observations togethergive us the required revelation principle under EUT for the setting where the outcome setsof some of the players can differ from the allocation set.The following example shows that the revelation principle need not hold when playershave CPT preferences. We will consider implementability in Bayes-Nash equilibrium in thisexample. 14 . . . . . . . . p w ± ( p ) GainsLossesFigure 2: Probability weighting functions for the players in example 3.1.
Example . Let there be two players, i.e. n = 2 . Let each player belong to one of the threetypes: Mildly Favorable ( MF ), Unfavorable ( UF ), and Super Favorable ( SF ), i.e. Θ = Θ = { MF , UF , SF } . Let the outcome sets for both the players be Γ = Γ = { I , II , III , IV , V } . Letthe value functions v and v for both the players be as shown below.I II III IV V MF 13 .
616 8 .
616 5 .
816 3 . − − − −
50 0SF 0 0 1M 0 0
Observe that a player with type MF has mild gains for all the outcomes, a player with type UF has medium losses for all outcomes except outcome III , where she has a big loss, and aplayer of type SF has a huge gain for outcome III and zero gains otherwise.Let the probability weighting functions for both the players, for all of their types, begiven by the following piecewise linear functions: w + ( p ) = (8 / p, for ≤ p < (7 / , (1 /
4) + (2 / p − / , for (7 / ≤ p < / , (5 /
8) + (12 / p − / , for (25 / ≤ p < , for gains, and w − ( p ) = (3 / p, for ≤ p < (1 / , (3 /
16) + (1 / p − / , for (1 / ≤ p < / , (1 /
2) + 2( p − / , for (3 / ≤ p < , for losses. See figure 2.Let the prior distribution F be such that the types of the players are independentlysampled with probabilities, P (MF) = 1 / , P (UF) = 3 / , P (SF) = 1 / . (3.3)15et A = { a, b, c } be the allocation set, and let the allocation-outcome mapping be givenby ζ ( a ) = { (1 / , (I , I)); (1 / , (V , V)) } ,ζ ( b ) = { (1 / , (II , II)); (1 / , (IV , IV)) } ,ζ ( c ) = (III , III) . Consider the allocation choice function f ∗ given by f ∗ (SF , θ ) = f ∗ ( θ , SF) = c, ∀ θ ∈ Θ , θ ∈ Θ ,f ∗ (UF , θ ) = f ∗ ( θ , UF) = { (1 / , a ); (1 / , b ) } , ∀ θ ∈ { MF , UF } , θ ∈ { MF , UF } ,f ∗ (MF , MF) = { (1 / , a ); (1 / , b ) } . We will now show that f ∗ is not truthfully implementable in F -Bayes-Nash equilibrium bya direct mechanism. However, if we do not restrict ourselves to direct mechanisms, thenwe will show that it is possible to implement f ∗ in F -Bayes-Nash equilibrium. We will thenconclude that the revelation principle does not hold for Bayes-Nash implementability whenthe players have CPT preferences.We observe that if either of the players is of type SF then under the allocation c the play-ers with type SF get the maximum possible reward, i.e. . This motivates implementingallocation c if either of the players is of type SF . Now suppose none of the players has type SF . If player is of type UF , then in claim 3.2, we show that player ’s CPT value for thelottery L i ( µ ) corresponding to a distribution µ ∈ ∆( A ) is maximized when µ = { (1 / , a ); (1 / , b ); (0 , c ) } . (3.4)Thus, if at least one of the players has type UF and none of the players have type SF , then thedistribution in (3.4) gives the best CPT value for the players with type UF . This motivatesthe following definition: we will call an allocation choice function f special if it satisfies f (SF , θ ) = f ( θ , SF) = { (1 , c ) } , ∀ θ ∈ Θ , θ ∈ Θ , (3.5)and f (UF , θ ) = f ( θ , UF) = { (1 / , a ); (1 / , b ) } , ∀ θ , θ ∈ { MF , UF } . (3.6)Note that f ∗ is special.After proving claim 3.2, we will show that it is impossible to truthfully implement anyspecial allocation choice function in F -Bayes-Nash equilibrium by a direct mechanism. Inparticular, this would imply that f ∗ is not truthfully implementable by a direct mechanism.We will then give a mechanism M that implements f ∗ in F -Bayes-Nash equilibrium. Claim . The CPT value V UF1 ( L ( µ )) is maximized when µ is given by (3.4) .Proof of Claim 3.2. Consider a lottery µ = { ( x, a ); ( y, b ); ( z, c ) } , where x, y, z are nonnegative numbers with x + y + z = 1 . Then the outcome lottery forplayer is L ( µ ) = { ( x/ , I); ( y/ , II); ( z, III); ( y/ , IV); ( x/ , V) } . . . . . − − − − − x E ( x ) Figure 3: Plot of expression E ( x ) in Claim 3.2.CPT satisfies strict stochastic dominance [Chateauneuf and Wakker, 1999], i.e. shifting posi-tive probability mass from an outcome to a strictly preferred outcome leads to a strictly pre-ferred lottery. This implies that z = 0 in the optimal solution. Taking z = 0 and y = 1 − x ,from (2.3), we have E ( x ) := V UF1 ( { ( x/ , I); (1 / − x/ , II); (0 , III); (1 / − x/ , IV); ( x/ , V) } )= − w − ( x/ − w − (1 / − w − (1 − x/ . We can verify that this function is maximized at x = 1 / . See figure 3.Suppose we have a direct mechanism M d = h d that truthfully implements a specialallocation choice function f . Then the allocation function h d must be equal to the allocationchoice function f . Since f satisfies (3.5) and (3.6), the only freedom left is in the choice of f (MF , MF) . Let h d (MF , MF) = f (MF , MF) = { ( x (cid:48) , a ); ( y (cid:48) , b ); ( z (cid:48) , c ) } , where x (cid:48) , y (cid:48) , z (cid:48) are nonnegative numbers with x (cid:48) + y (cid:48) + z (cid:48) = 1 . The lottery faced by a playerof type MF signaling truthfully would then be L ( µ (MF , MF; M d , F, σ d − i )) = { (3 / x (cid:48) / , I); (3 /
32 + y (cid:48) / , II);(1 / z (cid:48) / , III); (3 /
32 + y (cid:48) / , IV); (3 /
32 + x (cid:48) / , V) } . We obtain this by using the belief F − ( ·| MF) of player on the type of player given by(3.3), the truthful strategy σ d for player , and the allocation function h d in (2.16). Claim . For any nonnegative x (cid:48) , y (cid:48) , z (cid:48) such that x (cid:48) + y (cid:48) + z (cid:48) = 1 , we have V MF1 ( L ( µ (MF , MF; M d , F, σ d − i ))) < . . . . . . . . . z (cid:48) E ( z (cid:48) ) Figure 4: Plot of expression E ( z (cid:48) ) in Claim 3.3. Proof of Claim 3.3.
We have V MF1 L ( µ (MF , MF; M d , F, σ d − i )) =3 . w + (29 / − x (cid:48) /
4) + 2 . w + (18 /
32 + z (cid:48) / . w + (14 / − z (cid:48) /
4) + 5 w + (3 /
32 + x (cid:48) / . We observe that the expression, E ( z (cid:48) ) := 2 . w + (18 /
32 + z (cid:48) /
4) + 2 . w + (14 / − z (cid:48) / , is maximized at z (cid:48) = 0 with value E (0) = 2 . . See figure 4.We can therefore set z (cid:48) = 0 , since this choice would also lead to the least constrainedproblem of maximizing the expression E ( x (cid:48) ) := 3 . w + (29 / − x (cid:48) /
4) + 5 w + (3 /
32 + x (cid:48) /
4) + 2 . , which we can see is maximized at x (cid:48) = 0 and x (cid:48) = 1 . At z (cid:48) = 0 , and either x (cid:48) = 0 or x (cid:48) = 1 ,we have V MF1 L ( µ (MF , MF)) = 5 . . See figure 5.This establishes the claim.Thus, player will defect from the truthful strategy and report SF when her true typeis MF , because if she does so the allocation c will be implemented by the system oper-ator, which results in her outcome being III , hence giving her a value of . . Hence,truthful strategies do not form a Bayes-Nash equilibrium under the allocation function h d .And hence, any allocation choice function f that satisfies (3.5) and (3.6) is not truthfullyimplementable by a direct mechanism.We will now show that the allocation choice function f ∗ is implementable in Bayes-Nashequilibrium. Consider the mechanism M = ((Ψ i ) i , h ) with the signal sets for the playersbeing Ψ = Ψ = { MF a , MF b , UF , SF } , and the allocation function h given by h (SF , ψ ) = h ( ψ , SF) = c, ∀ ψ ∈ Ψ , ψ ∈ Ψ ,h (UF , UF) = { (1 / , a ); (1 / , b ) } , . . . . . . . . . x (cid:48) E ( x (cid:48) ) Figure 5: Plot of expression E ( x (cid:48) ) in Claim 3.3. h (UF , MF a ) = a,h (UF , MF b ) = b,h (MF a , UF) = a,h (MF b , UF) = b,h (MF a , MF a ) = a,h (MF b , MF b ) = b,h (MF a , MF b ) = h (MF b , MF a ) = { (1 / , a ); (1 / , b ) } . Now consider the strategies σ ∗ and σ ∗ given by σ ∗ i (SF) = SF ,σ ∗ i (UF) = UF ,σ ∗ i (MF) = { (1 / , MF a ); (1 / , MF b ) } , (3.7)for i = 1 , .One can check that the allocation function h and the strategy profile σ ∗ induce theallocation choice function f ∗ defined above. We will now verify that σ ∗ is a Bayes-Nashequilibrium and thus conclude that f ∗ is implementable in Bayes-Nash equilibrium.If player i has type SF then clearly SF is a best response signal for her. To see this,observe that amongst all the lotteries L i ∈ ∆(Γ i ) , V SF i ( L i ) is maximized when L i = III (this follows from the first order stochastic dominance property of CPT preferences). Sincesignaling SF produces the lottery III for player i , we get that it is her best response. If player i has type UF , then signaling UF dominates signaling SF . To see this, note that amongstall the lotteries L i ∈ ∆(Γ i ) , V UF i ( L i ) is minimized when L i = III (this follows from thefirst order stochastic dominance property of CPT preferences). Since signaling SF producesthe lottery III for player i , we get that it is dominated by all other strategies, in particular,19ignaling UF . As for comparing with signaling MF a or MF b , if she signals UF then she willface the lottery L i ( µ i (UF , UF; M , F, σ ∗− i )) = { (7 / , I); (7 / , II); (1 / , III); (7 / , IV); (7 / , V) } . If she signals MF a , then she will face the lottery L i ( µ i (UF , MF a ; M , F, σ ∗− i )) = { (3 / , I); (1 / , II); (1 / , III); (1 / , IV); (3 / , V) } . If she signals MF b , then she will face the lottery L i ( µ i (UF , MF b ; M , F, σ ∗− i )) = { (1 / , I); (3 / , II); (1 / , III); (3 / , IV); (1 / , V) } . The CPT values in each of these scenarios are as follows: V UF i ( L i ( µ i (UF , UF; M , F, σ ∗− i )))= − w − (25 / − w − (18 / − w − (11 / − w − (4 / − . ,V UF i ( L i ( µ i (UF , MF a ; M , F, σ ∗− i )))= − w − (20 / − w − (18 / − w − (16 / − w − (4 / − . , and, V UF i ( L i ( µ i (UF , MF b ; M , F, σ ∗− i )))= − w − (30 / − w − (18 / − w − (6 / − w − (4 / − . . Thus, signaling UF is the best response of a player with type UF .Finally, let player i have type MF . Depending on what she signals, we have the followinglotteries: L i ( µ i (MF , MF a ; M , F, σ ∗− i )) = { (3 / , I); (1 / , II); (1 / , III); (1 / , IV); (3 / , V) } ,L i ( µ i (MF , MF b ; M , F, σ ∗− i )) = { (1 / , I); (3 / , II); (1 / , III); (3 / , IV); (1 / , V) } ,L i ( µ i (MF , UF; M , F, σ ∗− i )) = { (7 / , I); (7 / , II); (1 / , III); (7 / , IV); (7 / , V) } ,L i ( µ i (MF , SF; M , F, σ ∗− i )) = III . The corresponding CPT values are as follows: V MF i ( L i ( µ i (MF , MF a ; M , F, σ ∗− i )))= 3 . w + (20 /
32) + 2 . w + (18 /
32) + 2 . w + (14 /
32) + 5 w + (12 / . ,V MF i ( L i ( µ i (MF , MF b ; M , F, σ ∗− i )))= 3 . w + (30 /
32) + 2 . w + (18 /
32) + 2 . w + (14 /
32) + 5 w + (2 / . , MF i ( L i ( µ i (MF , UF; M , F, σ ∗− i )))= 3 . w + (25 /
32) + 2 . w + (18 /
32) + 2 . w + (14 /
32) + 5 w + (7 / . , and, V MF i ( L i ( µ i (MF , SF; M , F, σ ∗− i ))) = 5 . . Thus σ ∗ i (MF) has support on optimal signals, and hence is a best response. This completesthe verification that σ ∗ is a Bayes-Nash equilibrium. With this, we end our example.In the previous example, let us focus on the behavior of player i when she has type MF .For any mechanism with the signal sets for the players being Ψ = Ψ = { MF a , MF b , UF , SF } as above (the mechanism M = ((Ψ i ) i , h ) considered above is an instance of such a mech-anism), the signals MF a and MF b allow this player to play so that the lotteries faced byher are L (cid:48) i := L i ( µ i (MF , MF a ); M , F, σ − i ) and L (cid:48)(cid:48) i := L i ( µ i (MF , MF b ); M , F, σ − i ) respec-tively, where F denotes the prior distribution on types (i.e. the product distribution withmarginals given as in (3.3) above) and σ − i denotes the strategy of the other player. Thelotteries L (cid:48) i and L (cid:48)(cid:48) i are equally preferred by player i when she has type MF , and they arepreferred over the lotteries corresponding to signaling UF or SF , when the mechanism is M = ((Ψ i ) i , h ) as considered in example 3.1, and the other player plays according to thestrategy prescribed in (3.7). Under the equilibrium strategy σ ∗ i , as defined in (3.7), whenplayer i has type MF she signals MF a or MF b each with probability half.We can think of player as tossing a fair coin to decide whether to signal MF a or MF b when her type is MF , and similarly for player . The outcome of the coin toss is privateknowledge to the player tossing the coin. The equilibrium strategies in (3.7) correspond toeach player signaling UF or SF truthfully if that is her type, while if her type is MF thenshe signals MF a or MF b depending on the outcome of her coin toss. From our analysis inthe above example, we have that these strategies form an F -Bayes-Nash equilibrium for thisgame and induce the allocation choice function f ∗ .An alternate viewpoint is to think of the coins being tossed at the beginning as before,but now let us assume that the system operator observes the outcomes of both the coins. Wecontinue to assume that each player does not observe the result of the coin toss of the otherplayer. Suppose each player only has the option to signal from { MF , UF , SF } . The systemoperator collects these signals and implements a lottery on the allocation set according tothe following rule: If player i signals UF or SF then let ψ (cid:48) i = UF or ψ (cid:48) i = SF respectively. Ifplayer i signals MF then, depending on the outcome of coin toss i , let ψ (cid:48) i = MF a or MF b .The system operator implements h ( ψ (cid:48) , ψ (cid:48) ) . Now consider the strategy where each playerreports her type truthfully. We observe that this strategy is an F -Bayes-Nash equilibriumfor this game and induces f ∗ .Thus the issue with the revelation principle is superficial in the sense that the reasonthat it does not hold is not that player i does not wish to reveal her type, but rather that shewould like to have an asymmetry in the information of the players. In the above example,this asymmetry comes from the coin tosses and, as seen in the latter viewpoint, these cointosses can be thought of as shared between each individual player and the system operator,so one could even think of the coins as being tossed by system operator, with the result ofeach individual coin toss being shared with the respective player. To capture this intuition,we propose a framework where there is a mediator who sends messages to each individual21layer before collecting their signals. As we will see now, this way we can recover a form ofthe revelation principle. We now lay out the framework for a mechanism with messages from the mediator, along thelines of the augmented framework for mechanism design motivated by the example above.Let Φ i be a finite message set for each player i , with a typical element denoted by φ i , and let Φ := (cid:81) i Φ i . Let D ∈ ∆(Φ) denote a mediator distribution from which the mediator draws aprofile of messages φ := ( φ , . . . , φ n ) . Let D i ∈ ∆(Φ i ) denote the marginal of D on Φ i . Forany φ i ∈ supp D i , let the conditional distribution be given by D − i ( φ − i | φ i ) := D ( φ i , φ − i ) D i ( φ i ) , for all φ − i ∈ Φ − i , (3.8)where φ − i := ( φ j ) j (cid:54) = i and Φ − i := (cid:81) j (cid:54) = i Φ j . Let Ψ i be a finite set of signals as before. Let h : Φ × Ψ → ∆( A ) (3.9)be a mediated allocation function . The message sets Φ i , i ∈ [ n ] , a mediator distribution D ∈ ∆(Φ) , and a mediated allocation function h together constitute a mediated mechanism ,denoted by M := ((Φ i ) i ∈ [ n ] , D, (Ψ i ) i ∈ [ n ] , h ) . (3.10)The mediator first draws a profile of messages φ from the distribution D . Each player i observes her message φ i , and then sends a signal ψ i to the mediator. The mediator collectsthe signals from all the players and then chooses an allocation according to the probabilitydistribution h ( φ, ψ ) . A strategy for any player i is thus given by τ i : Φ i × Θ i → ∆(Ψ i ) . (3.11)Let τ i ( ψ i | φ i , θ i ) denote the probability of signal ψ i under the distribution τ i ( φ i , θ i ) . Let τ := ( τ , . . . , τ n ) denote the profile of strategies. Suppose player i has received message φ i and has type θ i (thus, φ i ∈ supp D i , and θ i ∈ supp F i ), and she chooses to signal ψ i (so ψ i ∈ supp τ i ( φ i , θ i ) ); then consider the probability distribution µ i ( φ i , θ i , ψ i ; M , F, τ − i ) ∈ ∆( A ) given by µ i ( α | φ i , θ i , ψ i ; M , F, τ − i ) := (cid:88) φ − i D − i ( φ − i | φ i ) (cid:88) θ − i F − i ( θ − i | θ i ) × (cid:88) ψ − i (cid:89) j (cid:54) = i τ j ( ψ j | φ j , θ j ) h ( α | φ, ψ ) . (3.12)The best response strategy set BR i ( τ − i ) of player i to a strategy profile τ − i of her oppo-nents consists of all strategies τ ∗ i : Φ i × Θ i → ∆(Ψ i ) such that W θ i i ( µ i ( φ i , θ i , ψ i ; M , F, τ − i )) ≥ W θ i i ( µ i ( φ i , θ i , ψ (cid:48) i ; M , F, τ − i )) , (3.13)for all φ i ∈ supp D i , θ i ∈ supp F i , ψ i ∈ supp τ ∗ i ( φ i , θ i ) , ψ (cid:48) i ∈ Ψ i .22 strategy profile τ ∗ is said to be an F -Bayes-Nash equilibrium for the environment E with respect to the mediated mechanism M if for each player i we have τ ∗ i ∈ BR i ( τ ∗− i ) . (3.14)We will say that an allocation choice function f : Θ → ∆( A ) is implementable in F -Bayes-Nash equilibrium by a mediated mechanism if there exists a mediated mechanism M and an F -Bayes-Nash equilibrium τ with respect to this mediated mechanism such that f is the induced allocation choice function, i.e. for all θ ∈ supp F, α ∈ A , we have f ( α | θ ) = (cid:88) φ D ( φ ) (cid:88) ψ (cid:32)(cid:89) i τ i ( ψ i | φ i , θ i ) (cid:33) h ( α | φ, ψ ) . (3.15)A mediated mechanism M = ((Φ i ) i ∈ [ n ] , D, (Ψ i ) i ∈ [ n ] , h ) is called a direct mediated mech-anism if Ψ i = Θ i for all i , and we write it as M d = ((Φ i ) i ∈ [ n ] , D, (Θ i ) i ∈ [ n ] , h d ) , where h d : Φ × Θ → ∆( A ) is the corresponding direct mediated allocation function .For a direct mediated mechanism, the truthful strategy τ di for player i should satisfy τ di ( φ i , θ i ) = θ i , for all φ i ∈ Φ i , and θ i ∈ Θ i . Thus, if player i receives a message φ i andhas type θ i , she reports her true type θ i irrespective of her received message. In a way,the messages are present only to align the beliefs of the players appropriately so that truth-telling is an equilibrium strategy (depending on the type of equilibrium under consideration,i.e. Bayes-Nash, dominant, or belief-dominant equilibrium). Note that in the definition ofthe truthful strategy τ di for player i we require τ di ( φ i , θ i ) = θ i , for all θ i ∈ Θ i and φ i ∈ Φ i , andnot just for θ i ∈ supp F i (when discussing an F -Bayes-Nash equilibrium) and φ i ∈ supp D i .This is done to make the notion of a truthful strategy uniquely defined.An allocation choice function f is said to be truthfully implementable in mediated F -Bayes-Nash equilibrium if there exists a direct mediated mechanism M d such that the truth-ful strategy profile τ d is a mediated F -Bayes-Nash equilibrium and it implements f .Let µ i ( φ i , θ i , ψ i ; M , ψ − i ) := (cid:88) φ − i D − i ( φ − i | φ i ) h ( φ, ψ ) , (3.16)denote the lottery faced by player i with type θ i , who has received message φ i (thus, φ i ∈ supp D i ) and believes that her opponents are going to report ψ − i . Similarly, let µ i ( φ i , θ i , ψ i ; M , G − i ) := (cid:88) φ − i D − i ( φ − i | φ i ) (cid:88) ψ − i G − i ( ψ − i ) h ( φ, ψ ) , (3.17)denote the lottery faced by player i with type θ i , who has received message φ i ∈ supp D i and has belief G − i ∈ ∆(Ψ − i ) over her opponents’ signal profiles. We define strategy τ ∗ i tobe dominant if, for all φ i ∈ supp D i , θ i ∈ Θ i , ψ i ∈ supp τ ∗ i ( φ i , θ i ) , ψ (cid:48) i ∈ Ψ i , and ψ − i ∈ Ψ − i ,we have W θ i i ( µ i ( φ i , θ i , ψ i ; M , ψ − i )) ≥ W θ i i ( µ i ( φ i , θ i , ψ (cid:48) i ; M , ψ − i )) . (3.18)Similarly, we define strategy τ ∗ i to be belief-dominant if, for all φ i ∈ supp D i , θ i ∈ Θ i , ψ i ∈ supp τ ∗ i ( φ i , θ i ) , ψ (cid:48) i ∈ Ψ i , and G − i ∈ ∆(Ψ − i ) , we have W θ i i ( µ i ( φ i , θ i , ψ i ; M , G − i )) ≥ W θ i i ( µ i ( φ i , θ i , ψ (cid:48) i ; M , G − i )) . (3.19)23n allocation choice function f is said to be implementable in dominant equilibrium bya mediated mechanism if there is a mediated mechanism M and a dominant equilibrium τ (i.e. a strategy profile comprised of dominant strategies for the individual players) such that f is the allocation choice function induced by τ under M , i.e. (3.15) holds for all θ ∈ Θ and α ∈ A . f is said to be truthfully implementable in dominant equilibrium by a direct mediatedmechanism if there is a directed mediated mechanism M d such that the truthful strategyprofile is a dominant equilibrium and induces f under M d . The notions of implementabilityby a mediated mechanism and truthful implementability by a direct mediated mechanismof an allocation choice function in belief-dominant equilibrium can be similarly defined.If the message set Φ i is a singleton for each player i , then we get back the originalmechanism design framework. Thus, the mediated mechanism design framework definedabove is a generalization of the mechanism design framework. This generalization allowsus to establish a form of the revelation principle even when players have CPT preferences.A special case of the mediated mechanism design framework is when the mediator mes-sage profile φ is publicly known. That is, each player knows the entire message profileinstead of privately knowing only her own message. This would happen if Φ i = Φ ∗ , for all i ∈ [ n ] , and D is a diagonal distribution, i.e. D ( φ ) = 0 for all message profiles φ = ( φ i ) i ∈ [ n ] such that φ i (cid:54) = φ j for some pair i, j ∈ [ n ] . Let Φ ∗ denote the common message set and let D ∗ ∈ ∆(Φ ∗ ) denote the mediator distribution on this set. Let M ∗ := (Φ ∗ , D ∗ , (Ψ i ) i ∈ [ n ] , h ∗ ) denote such a mediated mechanism with common messages, where now h ∗ : Φ ∗ × Ψ → ∆( A ) . We will call M ∗ a publicly mediated mechanism . The notions of an allocation choice func-tion being implementable in publicly mediated Bayes-Nash equilibrium, publicly mediateddominant equilibrium, or publicly mediated belief-dominant equilibrium can be definedsimilarly to the corresponding earlier definitions that were made for general message sets.The notions of an allocation choice function being truthfully implementable in direct pub-licly mediated Bayes-Nash equilibrium, direct publicly mediated dominant equilibrium, ordirect publicly mediated belief-dominant equilibrium can also be defined similarly to thecorresponding earlier definitions that were made for general message sets.We are now in a position to state one of our main results. Theorem 3.4 (Revelation Principle) . We have the following three versions of the revelationprinciple:(i) If an allocation choice function is implementable in Bayes-Nash equilibrium by a mediatedmechanism, then it is also truthfully implementable in Bayes-Nash equilibrium by a directmediated mechanism.(ii) If an allocation choice function is implementable in dominant equilibrium by a publiclymediated mechanism, then it is also truthfully implementable in dominant equilibriumby a direct publicly mediated mechanism.(iii) If an allocation choice function is implementable in belief-dominant equilibrium by a me-diated (resp. publicly mediated) mechanism, then it is also truthfully implementable inbelief-dominant equilibrium by a direct mediated (resp. direct publicly mediated) mech-anism. (cid:56) (cid:56) (cid:52)
Dominant Strategies (cid:56) (cid:52) ? Belief-dominant Strategies (cid:56) (cid:52) (cid:52)
Table 1: Settings in which the revelation principle holds.We prove this theorem in Appendix A. Theorem 3.4, in particular, implies that if anallocation choice function is implementable in Bayes-Nash equilibrium by a non-mediatedmechanism then it is truthfully implementable in Bayes-Nash equilibrium by a direct me-diated mechanism. Similarly, if an allocation choice function is implementable in domi-nant strategies (resp. belief-dominant strategies) by a non-mediated mechanism, then itis truthfully implementable in dominant strategies (resp. belief-dominant strategies) by adirect publicly mediated mechanism. Table 1 summarizes the different implementabilitysettings under which the revelation principle does and does not hold. Example 3.1 showsthat the revelation principle does not hold for the setting with Bayes-Nash equilibrium andnon-mediated mechanism. In example 3.5, we show that the revelation principle does nothold for the settings with dominant equilibrium or belief-dominant equilibrium and non-mediated mechanism. In example 3.6, we show that the revelation principle does not holdfor the settings with Bayes-Nash equilibrium and publicly mediated mechanism. The ques-tion of whether the revelation principle holds in the setting with dominant equilibrium andmediated mechanism remains open for future investigation.
Example . Consider the setting from example 2.1 with two players. Recall that Θ =Θ = { UP , DN } , A = { a, b, c } , Γ = { I , II , III , IV , V } , Γ = A . The allocation-outcomemapping is given by the product distribution of the marginals ζ and ζ , given by ζ ( a ) = { (1 / , I); (1 / , V) } , ζ ( b ) = { (1 / , II); (1 / , IV) } , ζ ( c ) = { (1 , (III)) } , and ζ ( α ) = α, ∀ α ∈ A . The probability weighting functions for gains for the two players are w +1 ( p ) = exp {− ( − ln p ) . } , w +2 ( p ) = p, for p ∈ [0 , (see figure 1). Let the value functions v and v be given by v I II III IV V
UP 2 x x + 1 1 .
99 1 0DN 0 0 1 0 0 v a b c UP 1 0 2DN 0 1 2 where x := 1 /w +1 (0 .
5) = 2 . .Consider a mechanism M = { (Ψ , Ψ ) , h } , where Ψ = { a, b, c } , Ψ = { UP , DN } ,and h ( a, ψ ) = a,h ( b, ψ ) = b,h ( c, ψ ) = c, for all ψ ∈ Ψ . The CPT values for player having type UP for the lotteries over her25utcomes corresponding to the different allocations are given by V UP1 ( L ( a )) = 2 xw +1 (0 .
5) = 2 ,V UP1 ( L ( b )) = w +1 (1) + xw +1 (0 .
5) = 2 ,V UP1 ( L ( c )) = 1 . . Further, the CPT values for player having type DN for the above lotteries are given by V DN1 ( L ( a )) = 0 , V DN1 ( L ( b )) = 0 , V DN1 ( L ( c )) = 1 . Since the allocation choice function h does not depend on the signal of player , from theabove the above calculations, we observe that the strategy σ given by σ ( ·|
0) = { (0 . , a ); (0 . , b ) } ,σ ( ·|
1) = c, is a dominant strategy. and a belief-dominant strategy. Let σ be the truthful strategy forplayer . Again, since the allocation choice function h does not depend on the signal ofplayer , σ is trivially a dominant strategy and a belief-dominant strategy. Thus σ = ( σ , σ ) is a dominant equilibrium and a belief-dominant equilibrium. The corresponding socialchoice function f is given by f (UP , θ ) = { (0 . , a ); (0 . , b ) } ,f (DN , θ ) = c. Thus, the allocation choice function f is implementable in dominant (resp. belief-dominant)equilibrium. Suppose there were a direct mechanism M d = h d that truthfully implementsthe allocation choice function f in dominant (resp. belief-dominant) equilibrium. Then, h d = f . As observed in example 2.1, the CPT value for player having type UP for thelottery corresponding to { (0 . , a ); (0 . b ) } is V UP1 ( L ( { (0 . , a ); (0 . b ) } )) = 1 . . If player has type UP and believes that player ’s type report is UP (or equivalently, anyother distribution over player ’s type report), then player would deviate from her truthfulstrategy and report DN instead, because it gives her a higher CPT value. Hence the truthfulstrategy σ d is not a dominant (resp. belief-dominant) equilibrium for the direct mechanism M d . Thus f is not truthfully implementable in dominant (resp. belief-dominant) equilib-rium by a direct mechanism.We will now show that the revelation principle does not hold for the setting with Bayes-Nash equilibrium and publicly mediated mechanism. Let us first make an observation re-garding the allocation choice functions that are truthfully implementable in F -Bayes-Nashequilibrium by a direct publicly mediated mechanism. Let f be an allocation choice functionthat is truthfully implementable in F -Bayes-Nash equilibrium by a direct publicly mediatedmechanism M d ∗ = (Φ ∗ , D ∗ , (Θ i ) i ∈ [ n ] , h d ∗ ) , (3.20)where h d ∗ : Φ ∗ × Θ → ∆( A ) , (3.21)26s the direct mediated allocation function for this direct publicly mediated mechanism. Sincetruthful strategies τ d are an F -Bayes-Nash equilibrium, for each φ ∗ ∈ supp D ∗ , we have W θ i i ( µ i ( φ ∗ , θ i , θ i ; M d ∗ , F, τ d − i )) ≥ W θ i i ( µ i ( φ ∗ , θ i , ˜ θ i ; M d ∗ , F, τ d − i )) , (3.22)for all θ i ∈ supp F i , ˜ θ i ∈ Θ i , i ∈ [ n ] , where µ i ( φ ∗ , θ i , ˜ θ i ; M d ∗ , F, τ d − i ) = (cid:88) θ − i F − i ( θ − i | θ i ) h d ∗ ( φ ∗ , ˜ θ i , θ − i ) , is the lottery induced on the allocations for player i receiving message φ ∗ , having type θ i ,and deciding to report type ˜ θ i . Now, fix φ ∗ ∈ Φ ∗ with D ∗ ( φ ∗ ) > , and consider a non-mediated direct mechanism M d := ((Θ i ) i ∈ [ n ] , h d ) , with its direct allocation function being h d ( · ) := h d ∗ ( φ ∗ , · ) : Θ → ∆( A ) . It follows from (3.22) that truthful strategies correspondingto mechanism M d form an F -Bayes-Nash equilibrium. Thus, we note that h d ∗ ( φ ∗ , · ) is the al-location function truthfully implemented by the non-mediated direct mechanism M d . Sincemechanism M d ∗ truthfully implements the allocation function f in F -Bayes-Nash equilib-rium, we have that f ( θ ) = (cid:88) φ ∗ D ∗ ( φ ∗ ) h d ∗ ( φ ∗ , θ ) , for all θ ∈ supp F . From these two observations, we conclude that if f is an allocation choicefunction that is truthfully implementable in F -Bayes-Nash equilibrium by a direct publiclymediated mechanism, then f is a convex combination of allocation choice functions each ofwhich is truthfully implementable in F -Bayes-Nash equilibrium by a non-mediated directmechanism. It is easy to see that the converse of this statement is also true.In the following example, we will use this observation to establish that the revelationprinciple does not hold for the setting with Bayes-Nash equilibrium and publicly mediatedmechanism. Example . Let there be two players, i.e. n = 2 . Let Θ = Θ = { UP , DN } . Let Γ = Γ = { I , II , III , IV , V } . Let the value function v for player be as shown below v I II III IV V
UP 80 57 34 17 0DN 0 0 100 0 0 and let the value function v for player be as shown below v I II III IV V UP − − − −
17 0DN 0 0 100 0 0
Let the probability weighting functions for both the players, for both types, for gains andlosses, be given by the following piecewise linear function: w ± ( p ) = w ± ( p ) = w ( p ) = (8 / p, for ≤ p < (7 / , (1 /
4) + (2 / p − / , for (7 / ≤ p < / , (5 /
8) + (12 / p − / , for (25 / ≤ p < , F be such that the types of the players are independently sampled with probabilities, P (UP) = 3 / , P (DN) = 1 / . (3.23)Let A = { a, b, c } . Let ζ ( a ) = { (1 / , (I , I)); (1 / , (V , V)) } ,ζ ( b ) = { (1 / , (II , II)); (1 / , (IV , IV)) } ,ζ ( c ) = (III , III) . Consider the allocation choice function f ∗ given by f ∗ (DN , θ ) = f ∗ ( θ , DN) = c, ∀ θ ∈ Θ , θ ∈ Θ f ∗ (UP , UP) = { (1 / , a ); (1 / , b ) } . We will now show that f ∗ is implementable in F -Bayes-Nash equilibrium by a publiclymediated mechanism. In fact, we will show that f ∗ is implementable in F -Bayes-Nashequilibrium by a non-mediated mechanism. We will then show that f ∗ cannot be a convexcombination of allocation choice functions each of which is truthfully implementable by anon-mediated direct mechanism. This will give us that f ∗ is not truthfully implementable in F -Bayes-Nash equilibrium by a direct publicly mediated mechanism. We will then concludethat the revelation principle does not hold for the setting with Bayes-Nash equilibrium andpublicly mediated mechanism.Consider the mechanism M = ((Ψ i ) i ∈ [ n ] , h ) , where Ψ = { UP a , UP b , DN } , Ψ = { UP , DN } , and the allocation function h is given by h (DN , ψ ) = h ( ψ , DN) = c, ∀ ψ ∈ Ψ , ψ ∈ Ψ ,h (UP a , UP) = a,h (UP a , UP) = b. Consider the strategy σ for player given by σ (UP) = { (1 / , UP a ); (1 / , UP b ) } ,σ (DN) = DN , and the strategy σ for player given by σ (UP) = UP ,σ (DN) = DN . It is easy to see that this induces the allocation choice function f ∗ .We will now verify that σ is an F -Bayes-Nash equilibrium for M . If player has type UP , then the CPT values of the lotteries faced by her corresponding to her signals are asfollows: W UP1 ( µ (UP , UP a ; M , F, σ − )) = V UP1 ( { (3 / , I); (0 , II); (1 / , III); (0 , IV); (3 / , V) } )= 46 w (3 /
8) + 34 w (5 / . UP1 ( µ (UP , UP b ; M , F, σ − )) = V UP1 ( { (0 , I); (3 / , II); (1 / , III); (3 / , IV); (0 , V) } )= 23 w (3 /
8) + 17 w (5 /
8) + 17= 34 .W UP1 ( µ (UP , DN; M , F, σ − )) = V UP1 (III) = 34 . Thus player is indifferent between all signals when she has type UP and so the strategyof signaling σ (UP) = { (1 / , UP a ); (1 / , UP b ) } is optimal for her.If player has type DN , then III is the best outcome and she receives this lottery if shesignals DN . Thus DN dominates any other strategy, in particular, signaling UP a or UP b .If player has type UP , then the CPT values of the lotteries faced by her correspondingto her signals are as follows: W UP2 ( µ (UP , UP; M , F, σ − )) = V UP1 ( { (3 / , I); (3 / , II); (1 / , III); (3 / , IV); (3 / , V) } )= − w (3 / − w (3 / − w (5 / − w (13 / − . .W UP2 ( µ (UP , DN; M , F, σ − )) = V UP1 (III) = − . Hence the strategy of signaling σ (UP) = UP is optimal for player when she has type UP .If player has type DN , then III is the best outcome and she receives this lottery if shesignals DN . Thus DN dominates any other strategy, in particular, signaling UP .This shows that σ is an F -Bayes-Nash equilibrium for M , and hence establishes that f ∗ is implementable in F -Bayes-Nash equilibrium by a non-mediated mechanism.Suppose f ∗ were a convex combination of allocation choice functions each of which istruthfully implementable by a non-mediated direct mechanism. Let f be one of the alloca-tion choice functions in this convex combination. Since f ∗ (DN , θ ) = f ∗ ( θ , DN) = c for all θ , θ , and since { c } is an extreme point of the simplex ∆( A ) , we get that f (DN , θ ) = f ( θ , DN) = c, ∀ θ ∈ Θ , θ ∈ Θ . (3.24)Similarly, since f ∗ (UP , UP) lies on the line joining the vertices { a } and { b } of the simplex ∆( A ) , we get that f (UP , UP) = { ( x, a ); (1 − x, b ) } , (3.25)where ≤ x ≤ .Let f be truthfully implementable in F -Bayes-Nash equilibrium by the non-mediateddirect mechanism M d = h d . Then h d = f . If player has type UP , then the lottery facedby her if she reports UP is given by L ( µ (UP , UP; M d , F, σ d − )) = { (3 x/ , I); (3(1 − x ) / , II); (1 / , III); (3(1 − x ) / , IV); (3 x/ , V) } , where σ d − = σ d is the truthful strategy of player . Let E ( x ) := 23 w (cid:18) x (cid:19) + 23 w (cid:18) (cid:19) + 17 w (cid:18) (cid:19) + 17 w (cid:18) − x (cid:19) , .
58 13433 . x E ( x ) Figure 6: Plot of expression E ( x ) in example 3.6.for x ∈ [0 , . We observe that E ( x ) is maximum at x = 0 and x = 1 , and for all x ∈ (0 , , E ( x ) < . (See figure 6.)Now, unless x = 0 or x = 1 , player will defect from the truthful strategy and report DN when her true type is UP , because if she does so the allocation c will be implementedby the system operator, which results in her outcome III , hence giving her a value of .Thus, x = 0 or x = 1 .If player has type UP , then the lottery faced by her if she reports UP is given by L ( µ (UP , UP; M d , F, σ d − )) = { (3 x/ , I); (3(1 − x ) / , II); (1 / , III); (3(1 − x ) / , IV); (3 x/ , V) } , where σ d − = σ d is the truthful strategy of player . If x = 0 , then the CPT value for player is given by V UP2 ( { (0 , I); (3 / , II); (1 / , III); (3 / , IV); (0 , V) } = − w (3 / − w (5 / − − . . If x = 1 , then the CPT value for player is given by V UP2 ( { (3 / , I); (0 , II); (1 / , III); (0 , IV); (3 / , V) } = − w (3 / − w (5 / − . . Now, if x = 0 or x = 1 , player will defect from the truthful strategy and report DN whenher true type is UP , because if she does so the allocation c will be implemented by thesystem operator, which results in her outcome III , hence giving her a value of − . Thus x cannot be or , leading to a contradiction. Thus, f ∗ cannot be a convex combinationof allocation choice functions each of which is truthfully implementable by a non-mediateddirect mechanism. This completes the argument.30 Remarks and future directions
Generally in the settings where agents exhibit deviations from expected utility behavior,one would expect that the participating agents do not possess large computational power.Hence, truthful strategies are especially suitable for such settings in contrast to the morecomplicated strategies that are permitted by the concept of Bayes-Nash equilibrium. On theother hand, if our participating agents do not possess large computational power, then itis natural to question if they have the ability to exhibit strategic behavior, in particular therequirement that the strategies form a Bayes-Nash equilibrium (or dominant equilibriumor belief-dominant equilibrium). However, there can also be agents in the system who dopossess large computational power. Indeed, most of the systems such as online auctions andmarketplaces or networked-systems such as transportation networks, Internet routing net-works, etc. are comprised of players having varying degrees of computational and strategicabilities. For example, a firm participating in an online marketplace has the resources toestimate the common prior and other players’ strategies through extensive data collection,and thus can develop optimal strategies. On the other hand, individual agents participatingin the same system often lack such resources. When truthful strategies are in equilibrium,we get the best of both the worlds – it is easy for the players with limited resources to im-plement optimal strategies and at the same time there is no incentive for the players withlarge resources to deviate from these strategies.Consider the setting when players have independent types, i.e. the common prior F onthe type profiles has a product distribution F = (cid:81) i F i . Let M d = ((Φ i ) i ∈ [ n ] , D, (Θ i ) i ∈ [ n ] , h d ) be a direct mediated mechanism in such a setting. We note that the lottery induced on theoutcome set of player i when she receives a message φ i , has type θ i , and decides to report ˜ θ , is independent of her own type θ i . This is because her belief F − i ( ·| θ i ) on the type profilesof her opponents is independent of her type θ i . With an abuse of notation, let us denote thisbelief by F − i ∈ ∆(Θ − i ) . Then the lottery induced on the outcome set of player i when shereceives a message φ i ∈ supp D i , and decides to report ˜ θ , is given by L φ i , ˜ θ i i ( γ i ) := (cid:88) φ − i D − i ( φ − i | φ i ) (cid:88) θ − i F − i ( θ − i ) (cid:88) α h d ( α | φ, ˜ θ, θ − i ) ζ i ( γ i | α ) , γ i ∈ Γ i . We will now interpret the message profile as determining the menu of options to be pre-sented to each player. For example, if the message profile φ ∈ Φ is drawn from the distri-bution D , then player i would be presented with the menu comprised of lotteries, one foreach type ˜ θ i ∈ Θ i of the player. Let L i ( φ i ) := { L φ i , ˜ θ i i } ˜ θ i ∈ Θ i , denote the list of lotteries presented to player i when her message is φ i ∈ Φ i . Dependingon the player’s type, she chooses the lottery that gives her maximum CPT value. If truthfulstrategies form an F -Bayes-Nash equilibrium, then the lottery L φ i ,θ i i is indeed the best optionfor a player with type θ i .In several practical situations, the players are unaware of the type sets of other players Θ j , j (cid:54) = i , the allocation set A , the allocation-outcome mapping ζ , and the common prior F .It might also be preferable to relieve the players from the burden of knowing the messagesets and the mediator distribution D . Note that the system operator has enough knowledgeto construct the list of lotteries L i ( φ i ) for each player i based on her sampled message φ i .31ow, using the knowledge of her own type θ i , namely her preferences on the lotteries overher outcome set, player i can select the lottery that is optimal for her from the list L i ( φ i ) .This provides a way to operate the mechanism M d under reasonable assumptions on theplayers’ information.Further, it is beneficial to limit the complexity of the list L i ( φ i ) presented to the players.A way to do this would be to limit the size of the list and the complexity of each individuallottery in the list. The complexity of each individual lottery can be restricted, for example,by limiting the size of the outcome set Γ i and by restricting the probabilities of each outcometo belong to a grid { k/K : 0 ≤ k ≤ K } , where K > determines the granularity of the grid.Our framework with separate allocation and outcome sets is helpful in imposing restrictionson the size of the outcome set Γ i . Subsequently, for any lottery L i ∈ ∆(Γ i ) , we can find anapproximate lottery ˜ L i = { ( p i ( γ i ) , γ i ) } γ i ∈ Γ i such that p i ( γ i ) ∈ { k/K : 0 ≤ k ≤ K } for all γ i ∈ Γ i .On the other hand, the size of the list L i ( φ i ) is same as the size of the type set Θ i inthe worst case. This could make things practically infeasible. For example, when consid-ering type spaces comprised of general CPT preferences, it might be impossible in practiceto elicit the probability weighting functions from the agents. Restricting the type spacecan lead to inefficient social choice functions. The mediated mechanism design frameworkcould allow us to limit the size of menu options and at the same time have diversity inthe social choice function across different types of the players, facilitated by the messag-ing stage. Such multiple communication rounds have been studied under EUT and there isan extensive literature concerning the communication requirements in mechanism design.(See Mookherjee and Tsumagari [2014] and the references therein. See also the litera-ture on computational mechanism design [Conitzer and Sandholm, 2004].) Given that thenon-EUT preferences can reliably be applied only to non-dynamic decision-making, we areespecially interested in mechanisms that have a single stage of mediator messages to whichthe participating agents respond optimally by choosing their best option. It would be in-teresting to study the design of mechanisms that optimally elicit CPT preferences undercommunication restrictions such as limiting the size of the menu options. For example, wecould consider mechanism designs where the mediated allocation function h d for a directmediated mechanism has to satisfy |{L i ( φ i }| ≤ B , for all messages φ i , for some bound B .In this paper, we focused on the mechanism design framework and the revelation prin-ciple for agents having CPT preferences. It is just the first step towards mechanism designfor non-EUT players, with several interesting directions for future work. Notes Myerson [1982] refers to the mechanism design framework as a generalized principal-agent problem. Incontrast to our framework, Myerson is interested in problems where the agents have private decision domainsin addition to private information. Here, by private decision domains, we mean possible actions for the playerthat directly affect the outcomes. For example, in employment contracts the actions of the employee directlyaffect the outcome. These actions should not be confused with the signals of the player in the communicationprotocol set up by the system operator. We prefer to call the entity in control as the system operator insteadof the principal to emphasize that the system operator alone controls the system implementation. We restrictourselves to situations where agents do not have private decision domains because such situations involvedynamic decision-making, and non-EUT models face several issues in such situations (see section 4 for more onthis). Thus our model cannot account for moral hazard. Here, strictly speaking, given a matching by the platform, the users can refuse to go through with the atching. Although these decisions fall under the separate decision domains of the agents, they are rare andcan be accounted for separately. Since we have assumed that the type of a player completely determines her CPT features, we are implic-itly assuming private preferences , i.e. the preference over lotteries on the outcome set for each player is herprivate information and does not depend on other players’ information or types, also known as informationalexternalities (see Williams [2008]). Even if u θ i i = u ˜ θ i i for some θ i (cid:54) = ˜ θ i it is sometimes convenient to retain the connection to the underlyingtype. Notice that we have allowed different types of player i to have the same CPT features. Later, when wediscuss mechanism design with a common prior, which is a distribution on the types of all the players, it willlet us differentiate between the types of players that have identical CPT features but distinct beliefs on theopponents’ types. Mechanism design often focuses on “naive type sets”, that is, the type set Θ i for each player i is assumed to be comprised of exactly one element for each “preference type” of the player. Here, by preferencetype of a player we mean the preferences of the player on her outcome set. We borrow the expression “naivetype sets” from Börgers and Oh [2011]. In this paper, we do not assume the type sets to be naive. Such anassumption would entail a bijective correspondence between the types θ i and the CPT features ( v i , w ± i ) for eachplayer i . This distinction is relevant because besides having a preference type, a player can also have a “belieftype”. For example, the prior F could be such that F − i ( θ i ) (cid:54) = F − i (˜ θ i ) even when the value function and theprobability weighting functions corresponding to the types θ i and ˜ θ i coincide. (For more on this, see Bergemannand Morris [2005], Liu [2009], and Börgers and Krahmer [2015, Chapter 10].) Note that, in general, the preferences defined by the utility function W θi over the lotteries over the allocationset may not be given by CPT preferences directly, i.e. there need not exist any probability weighting functions ˜ w ± i such that, for all µ ∈ ∆( A ) , W θ i i ( µ ) is equal to the CPT value corresponding to the value function u θ i i on A and the probability weighting functions ˜ w ± i . To see this, consider a type θ i for player i such that V θ i i ( L (cid:48) i ) = V θ i i ( L (cid:48)(cid:48) i ) > V θ i i (0 . L (cid:48) i + 0 . L (cid:48)(cid:48) i ) , for lotteries L (cid:48) i , L (cid:48)(cid:48) i ∈ ∆(Γ i ) . See Phade and Anantharam [2018] for anexample of CPT preferences and lotteries (over outcomes) that satisfy the above condition. Let there be twoallocations α (cid:48) and α (cid:48)(cid:48) such that ζ i ( α (cid:48) ) = L (cid:48) i and ζ i ( α (cid:48)(cid:48) ) = L (cid:48)(cid:48) i . If W θ i i were to correspond to any CPT preferencedirectly on the allocation set then, by the first order stochastic dominance property of CPT, we would get W θ i i (0 . α (cid:48) + 0 . α (cid:48)(cid:48) ) = W θ i i ( α (cid:48) ) = W θ i i ( α (cid:48)(cid:48) ) . But, since this is not true for the setting under consideration, weget that W θ i i cannot correspond to any CPT preference directly over A . This is the version of the revelation principle commonly referred to in the mechanism design context.Another version of the revelation principle appears in the context of correlated equilibrium [Aumann, 1974,1987]. This is concerned with an n -player non-cooperative game in normal form. A mediator draws a messageprofile, comprised of a message for each player, from a fixed joint probability distribution on the set of messageprofiles, and sends each player her corresponding message. The joint distribution over message profiles usedis assumed to be common knowledge between the mediator and all the players. Based on her received signal,each player chooses her action (possibly from a probability distribution over her action set). When the messageset for each player is the same as her action set and the probability distribution on the set of message profiles(or equivalently action profiles) is such that truthful strategy, i.e. the strategy of choosing the action that isreceived as a message from the mediator, is a Nash equilibrium, then such a probability distribution is said to bea correlated equilibrium. Under EUT, the set of all correlated equilibria of a game is characterized as the unionover all possible message sets and mediator distributions, of the sets of joint distributions on the action profilesof all players, arising from all the Nash equilibria for the resulting game. See Phade and Anantharam [2018] fora discussion on the revelation principle for correlated equilibrium when players have CPT preferences. Myerson[1986] has considered a further generalization to games with incomplete information in which each player firstreports her type. Analyzing such settings under CPT would entail dynamic decision making and is beyond thescope of this paper. Appendices
A Proof of the Revelation Principle
We will first consider the revelation principle in the setting of mediated mechanisms. Thiscorresponds to statement (i) and a part of statement (ii) of theorem 3.4. In this setting33e will show that if an allocation choice function f is implementable in Bayes-Nash equi-librium (resp. belief-dominant equilibrium) by a mediated mechanism then it is truthfullyimplementable in Bayes-Nash equilibrium (resp. belief-dominant equilibrium) by a directmediated mechanism. We will then consider the setting of publicly mediated mechanismsand show that if an allocation choice function f is implementable in dominant equilibrium(resp. belief-dominant equilibrium) by a publicly mediated mechanism then it is truthfullyimplementable in dominant equilibrium (resp. belief-dominant equilibrium) by a directpublicly mediated mechanism. This will complete the proof of statement (ii) and the re-maining part of statement (iii) of theorem 3.4.For the first setting, let M = ((Φ i ) i ∈ [ n ] , D, (Ψ i ) i ∈ [ n ] , h ) , be a mediated mechanism and let τ be a strategy profile that induces f for this mechanism.Consider now the direct mediated mechanism M d = ((Φ (cid:48) i ) i ∈ [ n ] , D (cid:48) , (Θ i ) i ∈ [ n ] , h d ) , where the message set is given by Φ (cid:48) i := Φ i × (Ψ i ) Θ i , (A.1)with a typical element denoted by φ (cid:48) i := ( φ i , ( ψ θ (cid:48) i i ) θ (cid:48) i ∈ Θ i ) , (A.2)and the mediator distribution D (cid:48) is given by D (cid:48) ( φ (cid:48) ) := D ( φ ) (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) for all φ (cid:48) ∈ Φ (cid:48) . (A.3)The modified mediator messages and the mediator distribution can be interpreted as en-capsulating the randomness in the strategies of the players for each of their types into theirprivate messages.We now observe that D (cid:48) i ( φ (cid:48) i ) = D i ( φ i ) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) , (A.4)and (cid:88) φ (cid:48) i ∈ Φ (cid:48) i D (cid:48) i ( φ (cid:48) i ) = (cid:88) φ (cid:48) ∈ Φ (cid:48) D (cid:48) ( φ (cid:48) ) = 1 . Thus, D (cid:48) ∈ ∆(Φ (cid:48) ) is indeed a valid distribution. Equation (A.4) can be formally proved asfollows: D (cid:48) i ( φ (cid:48) i ) = (cid:88) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48) ( φ (cid:48) i , φ (cid:48)− i ) (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j (cid:88) ψ θ (cid:48) jj ∈ Ψ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j = D i ( φ i ) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) . Let the direct mediated allocation function be given by h d ( φ (cid:48) , θ (cid:48) ) := h (cid:18) φ, (cid:16) ψ θ (cid:48) i i (cid:17) i ∈ [ n ] (cid:19) for all φ (cid:48) ∈ Φ (cid:48) , θ (cid:48) ∈ Θ . (A.5)Note that the construction of the direct mediated mechanism is independent of the priordistribution F .The modified mediator messages and the direct mediated allocation function h d essen-tially transfer the randomness in the strategies of the players to the mediator messages, thusallowing each player to simply report her type. We observe that the truthful strategies τ di (˜ θ i | φ (cid:48) i , θ i ) = { ˜ θ i = θ i } , for all players i , implement the allocation choice function f for the direct mediated mecha-nism M d . Here is a formal proof.Let us compute the distribution on the allocation set induced by the truthful strategy forthe direct mediated mechanism. For any fixed θ ∈ Θ and α ∈ A , we have (cid:88) φ (cid:48) ∈ Φ (cid:48) D (cid:48) ( φ (cid:48) ) (cid:88) ˜ θ ∈ Θ (cid:89) i ∈ [ n ] τ di (˜ θ i | φ (cid:48) i , θ i ) h d ( α | φ (cid:48) , ˜ θ )= (cid:88) φ (cid:48) ∈ Φ (cid:48) D (cid:48) ( φ (cid:48) ) (cid:88) ˜ θ ∈ Θ (cid:89) i ∈ [ n ] { ˜ θ i = θ i } h d ( α | φ (cid:48) , ˜ θ ) ... because τ d is a truthful strategy = (cid:88) φ (cid:48) ∈ Φ (cid:48) D (cid:48) ( φ (cid:48) ) h d ( α | φ (cid:48) , θ )= (cid:88) φ (cid:48) ∈ Φ (cid:48) D ( φ ) (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) h d ( α | φ (cid:48) , θ ) ... from (A.3)35 (cid:88) φ ∈ Φ D ( φ ) (cid:88) ( ψ θ (cid:48) ii ) θ (cid:48) i ∈ Θ i,i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] (Ψ i ) Θ i (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) h d ( α | φ (cid:48) , θ ) ... from (A.2) = (cid:88) φ ∈ Φ D ( φ ) (cid:88) ( ψ θ (cid:48) ii ) θ (cid:48) i ∈ Θ i,i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] (Ψ i ) Θ i (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) h (cid:18) α (cid:12)(cid:12) φ, (cid:16) ψ θ i i (cid:17) i ∈ [ n ] (cid:19) ... from (A.5) = (cid:88) φ ∈ Φ D ( φ ) (cid:88) ( ψ θii ) i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] Ψ i (cid:88) ( ψ θ (cid:48) ii ) θ (cid:48) i (cid:54) = θi,i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] (Ψ i ) Θ i \ θi (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i (cid:54) = θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) × (cid:89) i ∈ [ n ] τ i (cid:16) ψ θ i i | φ i , θ i (cid:17) h (cid:18) α (cid:12)(cid:12) φ, (cid:16) ψ θ i i (cid:17) i ∈ [ n ] (cid:19) = (cid:88) φ ∈ Φ D ( φ ) (cid:88) ( ψ θii ) i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] Ψ i (cid:89) i ∈ [ n ] τ i (cid:16) ψ θ i i | φ i , θ i (cid:17) h (cid:18) α (cid:12)(cid:12) φ, (cid:16) ψ θ i i (cid:17) i ∈ [ n ] (cid:19) × (cid:88) ( ψ θ (cid:48) ii ) θ (cid:48) i (cid:54) = θi,i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] (Ψ i ) Θ i \ θi (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i (cid:54) = θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) = (cid:88) φ ∈ Φ D ( φ ) (cid:88) ( ψ θii ) i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] Ψ i (cid:89) i ∈ [ n ] τ i (cid:16) ψ θ i i | φ i , θ i (cid:17) h (cid:18) α (cid:12)(cid:12) φ, (cid:16) ψ θ i i (cid:17) i ∈ [ n ] (cid:19) × (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i (cid:54) = θ i (cid:88) ψ θ (cid:48) ii ∈ Ψ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) = (cid:88) φ ∈ Φ D ( φ ) (cid:88) ( ψ θii ) i ∈ [ n ] ∈ (cid:81) i ∈ [ n ] Ψ i (cid:89) i ∈ [ n ] τ i (cid:16) ψ θ i i | φ i , θ i (cid:17) h (cid:18) α (cid:12)(cid:12) φ, (cid:16) ψ θ i i (cid:17) i ∈ [ n ] (cid:19) (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i (cid:54) = θ i ... because τ i ( ·| φ i , θ (cid:48) i ) ∈ ∆(Ψ i )= (cid:88) φ ∈ Φ D ( φ ) (cid:88) ψ ∈ Ψ (cid:89) i ∈ [ n ] τ i ( ψ i | φ i , θ i ) h (cid:0) α (cid:12)(cid:12) φ, ψ (cid:1) f ( α | θ ) if θ ∈ supp F ... from (3.15).This confirms that the truthful strategy profile implements the social choice function for thedirect mediated mechanism M d .We will now show that if τ is an F -Bayes-Nash equilibrium for M , then τ d is an F -Bayes-Nash equilibrium for M d . We will then show that if τ is a belief-dominant equilibrium for M , then τ d is a belief-dominant equilibrium for M d . To prove these two statements, wefirst make the following observation concerning the lottery induced over the allocationsfor player i in the setting of the direct mediated mechanism M d , when she receives themessage φ (cid:48) i := ( φ i , ( ψ θ (cid:48) i i ) θ (cid:48) i ∈ Θ i ) ∈ supp D (cid:48) i , has type θ i ∈ Θ i , has a belief G (cid:48)− i ∈ ∆(Θ − i ) onthe opponents’ type reports (which are the signals of the opponents in this direct mediatedmechanism), and decides to report ˜ θ i . The lottery induced over the allocations for player i satisfies µ (cid:48) i ( φ (cid:48) i , θ i , ˜ θ i ; M d , G (cid:48)− i ) := (cid:88) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48)− i ( φ (cid:48)− i | φ (cid:48) i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h d ( φ (cid:48) , ˜ θ i , θ − i )= (cid:88) φ − i ∈ Φ − i D − i ( φ − i | φ i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ψ − i (cid:89) j (cid:54) = i τ j ( ψ j | φ j , θ j ) h (cid:16) φ, ψ ˜ θ i i , ψ − i (cid:17) . (A.6)We give a formal proof of this in Appendix B. Let us see how this observation helps us provethe two statements above, namely, τ d is an equilibrium ( F -Bayes-Nash or belief-dominantresp.) of M d given that τ is an equilibrium ( F -Bayes-Nash or belief-dominant resp.) of M .Suppose F is the common prior and τ is an F -Bayes-Nash equilibrium for the mediatedmechanism M . Let φ (cid:48) i ∈ supp D (cid:48) i and θ i ∈ supp F i . From (A.4), we know that D (cid:48) i ( φ (cid:48) i ) > implies D i ( φ i ) > and τ i ( ψ θ (cid:48) i i | φ i , θ (cid:48) i ) > , for all θ (cid:48) i ∈ Θ i , (and in particular, we have τ i ( ψ θ i i | φ i , θ i ) > ). Since τ is a Bayes-Nash equilibrium for M , we have W θ i i (cid:16) µ i ( φ i , θ i , ψ θ i i ; M , F, τ − i ) (cid:17) ≥ W θ i i (cid:16) µ i ( φ i , θ i , ˜ ψ i ; M , F, τ − i ) (cid:17) , for all ˜ ψ i ∈ Ψ i . (Note that ψ θ i i ∈ supp τ i ( ·| φ i , θ i ) , φ i ∈ supp D i , θ i ∈ supp F i .) Taking G (cid:48)− i = F − i ( ·| θ i ) in (A.6), we get that µ (cid:48) i ( φ (cid:48) i , θ i , ˜ θ i ; M d , F, τ d − i ) = µ i ( φ i , θ i , ψ ˜ θ i i ; M , F, τ − i ) , (A.7)for all ˜ θ i ∈ Θ i , and thus, W θ i i (cid:16) µ (cid:48) i ( φ (cid:48) i , θ i , θ i ; M d , F, τ d − i ) (cid:17) = W θ i i (cid:16) µ i ( φ i , θ i , ψ θ i i ; M , F, τ − i ) (cid:17) ≥ W θ i i (cid:16) µ i ( φ i , θ i , ψ ˜ θ i i ; M , F, τ − i ) (cid:17) = W θ i i (cid:16) µ (cid:48) i ( φ (cid:48) i , θ i , ˜ θ i ; M d , F, τ d − i ) (cid:17) , (A.8)for all ˜ θ i ∈ Θ i . This establishes that the truthful strategy τ d is an F -Bayes-Nash equilibriumfor M . 37ow suppose τ is a belief-dominant strategy for M . Let φ (cid:48) i ∈ supp D (cid:48) i and θ i ∈ Θ i . Again,this implies D i ( φ i ) > and ψ θ i i ∈ supp τ i ( φ i , θ i ) . Corresponding to a belief G (cid:48)− i ∈ ∆(Θ − i ) ,consider the belief G − i ∈ ∆(Ψ − i ) given by G − i ( ψ − i ) := (cid:88) φ − i ∈ Φ − i D − i ( φ − i | φ i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) (cid:89) j (cid:54) = i τ j ( ψ j | φ j , θ j ) (A.9)Then, from (A.6), we have that µ (cid:48) i ( φ (cid:48) i , θ i , ˜ θ i ; M d , G (cid:48)− i ) = µ i ( φ i , θ i , ψ ˜ θ i i ; M , G − i ) (A.10)Noting that ψ ˜ θ i i ∈ supp τ i ( φ i , ˜ θ i ) for all ˜ θ i ∈ Θ i and φ i ∈ supp D i , and τ i being a belief-dominant strategy, we get that W θ i i (cid:16) µ (cid:48) i ( φ (cid:48) i , θ i , θ i ; M d , G (cid:48)− i ) (cid:17) ≥ W θ i i (cid:16) µ (cid:48) i ( φ (cid:48) i , θ i , ˜ θ i ; M d , G (cid:48)− i ) (cid:17) (A.11)for all ˜ θ i ∈ Θ i . Thus, the truthful strategy τ d is a belief-dominant strategy for M d .This completes the proof of statement (i) in theorem 3.4 and part of statement (iii)corresponding to mediated mechanisms. We now consider the setting of publicly mediatedmechanisms and establish the rest of the theorem.Let M ∗ = (Φ ∗ , D ∗ , (Ψ i ) i ∈ [ n ] , h ∗ ) be a publicly mediated mechanism and for each player i let τ i : Φ ∗ × Θ i → ∆(Ψ i ) be herstrategy such that the strategy profile τ induces the allocation choice function f for thismechanism. We now consider the direct publicly mediated mechanism M d ∗ := (Φ (cid:48)∗ , D (cid:48)∗ , (Θ i ) i ∈ [ n ] , h d ∗ ) , where the message set is given by Φ (cid:48)∗ := Φ ∗ × n (cid:89) i =1 (Ψ i ) Θ i , with a typical element denoted by φ (cid:48)∗ := ( φ ∗ , ( ψ θ (cid:48) i i ) θ (cid:48) i ∈ Θ i ,i ∈ [ n ] ) , (A.12)and the mediator distribution D (cid:48)∗ is given by D (cid:48)∗ ( φ (cid:48)∗ ) := D ∗ ( φ ∗ ) (cid:89) i ∈ [ n ] (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) for all φ (cid:48) ∈ Φ (cid:48) . (A.13)Similar to the previous setting, here the modified mediator messages and the mediatordistribution can be interpreted as encapsulating the randomness in the strategies of theplayers for each of their types into the public messages. We can similarly verify that D (cid:48)∗ isindeed a probability distribution on Φ (cid:48)∗ . The direct mediated allocation function h d ∗ in thedirect publicly mediated mechanism M d is given by h d ∗ ( φ (cid:48)∗ , θ (cid:48) ) := h ∗ (cid:18) φ ∗ , (cid:16) ψ θ (cid:48) i i (cid:17) i ∈ [ n ] (cid:19) for all φ (cid:48)∗ ∈ Φ (cid:48)∗ , θ (cid:48) ∈ Θ . (A.14)38e can similarly verify that the truthful strategies τ d ( φ (cid:48)∗ , θ i ) = θ i implement the allocation choice function f for M d ∗ .Fix φ (cid:48)∗ ∈ supp D (cid:48)∗ . Note that h d ∗ ( φ (cid:48)∗ , ˜ θ i , θ − i ) = h ∗ ( φ ∗ , ψ ˜ θ i i , ( ψ θ j j ) j (cid:54) = i ) , (A.15)for all ˜ θ i ∈ Θ i . From (A.13), we have φ ∗ ∈ supp D ∗ and ψ θ i i ∈ supp τ i ( φ ∗ , θ i ) for all θ i ∈ Θ i .Now suppose τ is a dominant equilibrium for M ∗ . The lottery induced over the alloca-tions for player i when she receives a publicly mediated message φ (cid:48)∗ , has type θ i , believesthat the opponents are reporting θ − i , and decides to report ˜ θ i is given by µ (cid:48) i ( φ (cid:48)∗ , θ i , ˜ θ i ; M d ∗ , θ − i ) = h d ∗ ( φ (cid:48)∗ , ˜ θ i , θ − i ) . (A.16)We get this from (3.16) by considering the special case of publicly mediated mechanisms.From (A.15), we get that this is equal to the lottery induced over the allocations for player i when she receives a publicly mediated message φ ∗ , has type θ i , believes that the opponentsare reporting ψ θ j j , j (cid:54) = i , and decides to report ψ ˜ θi , namely, µ i ( φ ∗ , θ i , ψ ˜ θi ; M ∗ , ( ψ θ j j ) j (cid:54) = i ) = h ∗ ( φ ∗ , ψ ˜ θi , ( ψ θ j j ) j (cid:54) = i ) . Since τ i is a dominant strategy, φ ∗ ∈ supp D ∗ , and ψ θ i i ∈ supp τ i ( φ ∗ , θ i ) , we have W θ i i ( µ i ( φ ∗ , θ i , ψ θ i i ; M ∗ , ( ψ θ j j ) j (cid:54) = i )) ≥ W θ i i ( µ i ( φ ∗ , θ i , ˜ ψ i ; M ∗ , ( ψ θ j j ) j (cid:54) = i )) , for all ˜ ψ i ∈ Ψ i . Hence, we have W θ i i ( µ (cid:48) i ( φ (cid:48)∗ , θ i , θ i ; M d ∗ , θ − i )) ≥ W θ i i ( µ (cid:48) i ( φ (cid:48)∗ , θ i , ˜ θ i ; M d ∗ , θ − i )) , for all ˜ θ i ∈ Θ i . Thus, τ d is a dominant equilibrium of M d ∗ .Now suppose that τ is a belief-dominant equilibrium for M ∗ . Consider the fixed message φ (cid:48)∗ = ( φ ∗ , ( ψ θ (cid:48) i i ) θ (cid:48) i ∈ Θ i ,i ∈ [ n ] ) ∈ supp D (cid:48)∗ , as before. Corresponding to a belief G (cid:48)− i ∈ ∆(Θ − i ) , consider G − i, ∗ ∈ ∆(Ψ − i ) given by G − i, ∗ ( ˜ ψ − i ) := (cid:88) θ − i ∈ Θ − i s.t. ψ θjj = ˜ ψ − i , ∀ j (cid:54) = i G (cid:48)− i ( θ − i ) , (A.17)for all ˜ ψ − i ∈ Ψ − i , where ψ θ j j are the signals corresponding to the types as defined by themessage φ (cid:48)∗ .As observed in equation (A.16), the lottery induced over the allocations for player i ,when she receives message φ (cid:48)∗ , has type θ i , believes that the opponents’ are reporting θ − i ,and decides to report ˜ θ i is given by h d ∗ ( φ (cid:48)∗ , ˜ θ i , θ − i ) . Now suppose that she has belief G (cid:48)− i on39er opponents’ type report instead. Then, the induced lottery over the allocations for player i is given by µ (cid:48) i ( φ (cid:48)∗ , θ i , ˜ θ i ; M d ∗ , G (cid:48)− i ) = (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h d ∗ ( φ (cid:48)∗ , ˜ θ i , θ − i )= (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h ∗ ( φ ∗ , ψ ˜ θ i i , ( ψ θ j j ) j (cid:54) = i ) ... from (A.15) = (cid:88) ˜ ψ − i ∈ Ψ − i h ∗ ( φ ∗ , ψ ˜ θ i i , ˜ ψ − i ) (cid:88) θ − i ∈ Θ − i s.t. ψ θjj = ˜ ψ − i , ∀ j (cid:54) = i G (cid:48)− i ( θ − i )= (cid:88) ˜ ψ − i ∈ Ψ − i h ∗ ( φ ∗ , ψ ˜ θ i i , ˜ ψ − i ) G − i, ∗ ( ˜ ψ − i ) ... from (A.17) = µ i ( φ ∗ , θ i , ψ ˜ θ i i ; M ∗ , G − i, ∗ ) . Since τ is a belief-dominant equilibrium, φ ∗ ∈ supp D ∗ , and ψ θ i i ∈ supp τ i ( φ ∗ , θ i ) for all θ i ∈ Θ i , we have W θ i i ( µ i ( φ ∗ , θ i , ψ θ i i ; M ∗ , G − i, ∗ ) ≥ W θ i i ( µ i ( φ ∗ , θ i , ˜ ψ i ; M ∗ , G − i, ∗ ) , for all ˜ ψ i ∈ Ψ i . Hence, we have W θ i i ( µ (cid:48) i ( φ (cid:48)∗ , θ i , θ i ; M d ∗ , G (cid:48)− i )) ≥ W θ i i ( µ (cid:48) i ( φ (cid:48)∗ , θ i , ˜ θ i ; M d ∗ , G (cid:48)− i )) , for all ˜ θ i ∈ Θ i . Thus, τ d is a belief-dominant equilibrium of M d ∗ .This completes the proof of the theorem. B Proof of (A.6)
Let us recall the first setting considered in Appendix A. We have a mediated mechanism M = ((Φ i ) i ∈ [ n ] , D, (Ψ i ) i ∈ [ n ] , h ) , and a corresponding strategy profile τ . We had constructed a direct mediated mechanism M d = ((Φ (cid:48) i ) i ∈ [ n ] , D (cid:48) , (Θ i ) i ∈ [ n ] , h d ) , given by (A.1), (A.2), (A.3), and (A.5). We are interested in the situation when player i receives message φ (cid:48) i := ( φ i , ( ψ θ (cid:48) i i ) θ (cid:48) i ∈ Θ i ) ∈ supp D (cid:48) i , has type θ i ∈ Θ i , and belief G (cid:48)− i ∈ ∆(Θ − i ) on the opponents’ type reports, and decides to report ˜ θ i . Since D (cid:48) ( φ (cid:48) i ) > byassumption, we have (cid:88) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48)− i ( φ (cid:48)− i | φ (cid:48) i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h d ( φ (cid:48) , ˜ θ i , θ − i ) (cid:80) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48) ( φ (cid:48) i , φ (cid:48)− i ) (cid:80) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h d ( φ (cid:48) , ˜ θ i , θ − i ) (cid:80) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48) ( φ (cid:48) i , φ (cid:48)− i ) . Let the denominator be denoted by C := (cid:88) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48) ( φ (cid:48) i , φ (cid:48)− i ) = D (cid:48) i ( φ (cid:48) i ) . We now focus on the numerator, to get (cid:88) φ (cid:48)− i ∈ Φ (cid:48)− i D (cid:48) ( φ (cid:48) i , φ (cid:48)− i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h d ( φ (cid:48) , ˜ θ i , θ − i )= (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) × (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h d ( φ (cid:48) , ˜ θ i , θ − i ) ... from (A.3) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) × (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) ... from (A.5) = (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) × (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) Let C := (cid:89) θ (cid:48) i ∈ Θ i τ i (cid:16) ψ θ (cid:48) i i | φ i , θ (cid:48) i (cid:17) . We have, (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) × h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ( ψ θjj ) j (cid:54) = i ∈ (Ψ j ) j (cid:54) = i (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j \ θj,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j \ θj (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j \ θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) × (cid:89) j (cid:54) = i τ j (cid:16) ψ θ j j | φ j , θ j (cid:17) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ( ψ θjj ) j (cid:54) = i ∈ (Ψ j ) j (cid:54) = i (cid:89) j (cid:54) = i τ j (cid:16) ψ θ j j | φ j , θ j (cid:17) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) × (cid:88) ( ψ θ (cid:48) jj ) θ (cid:48) j ∈ Θ j \ θj,j (cid:54) = i ∈ (cid:81) j (cid:54) = i (Ψ j ) Θ j \ θj (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j \ θ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ( ψ θjj ) j (cid:54) = i ∈ (Ψ j ) j (cid:54) = i (cid:89) j (cid:54) = i τ j (cid:16) ψ θ j j | φ j , θ j (cid:17) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) × (cid:89) j (cid:54) = i (cid:89) θ (cid:48) j ∈ Θ j \ θ j (cid:88) ψ θ (cid:48) jj ∈ Ψ j τ j (cid:18) ψ θ (cid:48) j j | φ j , θ (cid:48) j (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) (cid:88) ( ψ θjj ) j (cid:54) = i ∈ (Ψ j ) j (cid:54) = i (cid:89) j (cid:54) = i τ j (cid:16) ψ θ j j | φ j , θ j (cid:17) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) × (cid:89) j (cid:54) = i (cid:89) ˜ θ (cid:48) j ∈ Θ j \ θ j = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ( ψ θjj ) j (cid:54) = i ∈ (Ψ j ) j (cid:54) = i (cid:89) j (cid:54) = i τ j (cid:16) ψ θ j j | φ j , θ j (cid:17) h (cid:18) φ i , φ − i , ψ ˜ θ i i , (cid:16) ψ θ j j (cid:17) j (cid:54) = i (cid:19) = (cid:88) φ − i ∈ Φ − i D ( φ i , φ − i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ψ − i (cid:89) j (cid:54) = i τ j ( ψ j | φ j , θ j ) h (cid:16) φ i , φ − i , ψ ˜ θ i i , ψ − i (cid:17) = D i ( φ i ) (cid:88) φ − i ∈ Φ − i D − i ( φ − i | φ i ) (cid:88) θ − i ∈ Θ − i G (cid:48)− i ( θ − i ) × (cid:88) ψ − i (cid:89) j (cid:54) = i τ j ( ψ j | φ j , θ j ) h (cid:16) φ i , φ − i , ψ ˜ θ i i , ψ − i (cid:17) . We recall that D i ( φ i ) C /C = 1 from (A.4), and hence, we get (A.6). C Outcome sets can be identified with the allocation set underEUT
Consider a setting in which all the players have EUT preferences for all their types. For thisrestricted setting, we will now construct an environment E (cid:48) := (cid:0) [ n ] , (Θ (cid:48) i ) i ∈ [ n ] , A, (Γ (cid:48) i ) i ∈ [ n ] , ζ (cid:48) (cid:1) , that we call the reduced environment corresponding to the environment (as defined in (2.4)) E := (cid:0) [ n ] , (Θ i ) i ∈ [ n ] , A, (Γ i ) i ∈ [ n ] , ζ (cid:1) . From (2.12), we observe that, since we are dealing with EUT preferences, the utilityfunction W θ i i is completely determined by the values u θ i i ( α ) , ∀ α ∈ A . Suppose the mecha-nism designer models the outcome set of each player i by Γ (cid:48) i = A instead of the true outcomeset Γ i , with the trivial allocation-outcome mapping ζ (cid:48) i instead of the original allocation-outcome mapping ζ i . Let ζ (cid:48) denote the product of the trivial allocation-outcome mappings ζ (cid:48) i , i ∈ [ n ] . Corresponding to a type θ i ∈ Θ i for player i , the mechanism designer modelsher type by θ (cid:48) i , which is characterized by the utility function u θ i i : Γ (cid:48) i → R as defined in432.11). Since the players are assumed to have EUT preferences, the probability weightingfunctions under each type θ (cid:48) i are modeled to be w ± i ( p ) = p, ∀ p ∈ [0 , . Let Θ (cid:48) i denote the setcomprised of all types θ (cid:48) i corresponding to the types θ i ∈ Θ i . Let T i : Θ i → Θ (cid:48) i denote thefunction for this correspondence. Suppose the mechanism designer treats the environmentas if given by E (cid:48) := (cid:0) [ n ] , (Θ (cid:48) i ) i ∈ [ n ] , A, (Γ (cid:48) i ) i ∈ [ n ] , ζ (cid:48) (cid:1) . Let Θ (cid:48) := (cid:81) i Θ (cid:48) i . Let T : Θ → Θ (cid:48) denote the product transformation defined by thefunctions T i , i ∈ [ n ] . Notice that the function T i is a bijection since, as pointed out earlier,even if u θ i i = u ˜ θ i i for some θ i (cid:54) = ˜ θ i , we will treat T i ( θ i ) and T i (˜ θ i ) as different elements of Θ (cid:48) i .For any prior F ∈ ∆(Θ) , let F (cid:48) ∈ ∆(Θ (cid:48) ) be the corresponding prior induced by the bijection T . Note that, for any player i , having any type θ i , and any lottery µ ∈ ∆( A ) , we have W θ i i ( µ ) = W T i ( θ i ) i ( µ ) . (C.1)(Here, W T i ( θ i ) i should be interpreted as the utility function for player i with type θ (cid:48) i = T i ( θ i ) corresponding to the reduced environment E (cid:48) .) Let f (cid:48) : Θ (cid:48) → A be an allocation choicefunction that is implementable in F (cid:48) -Bayes-Nash equilibrium σ (cid:48) := ( σ (cid:48) i ) i ∈ [ n ] (where σ (cid:48) i :Θ (cid:48) i → ∆(Ψ (cid:48) i ) ) for the mechanism M = ((Ψ i ) i ∈ [ n ] , h ) . Now suppose the system operator uses the same mechanism M in environment E .Consider the allocation choice function f : Θ → ∆( A ) given by f ( θ ) = f (cid:48) ( T ( θ )) . For each player i , consider the strategy σ i : Θ i → ∆(Ψ i ) given by σ i ( θ i ) = σ (cid:48) i ( T i ( θ i )) . Similar to (2.16), for any θ (cid:48) i ∈ supp F (cid:48) i and signal ψ i , let µ (cid:48) i ( θ (cid:48) i , ψ i ; M , F (cid:48) , σ (cid:48)− i ) := (cid:88) θ − i ∈ Θ − i F (cid:48)− i ( θ − i | θ i ) (cid:88) ψ − i ∈ Ψ − i (cid:89) j (cid:54) = i σ j ( ψ j | θ (cid:48) j ) h ( ψ ) , (C.2)be the belief of player i on the allocation set corresponding to the reduced environment E (cid:48) .Note that µ i ( θ i , ψ i ; M , F, σ − i ) = µ (cid:48) i ( T i ( θ i ) , ψ i ; M , F (cid:48) , σ (cid:48)− i ) . From observation (C.1) and the definition of F -Bayes-Nash equilibrium in (2.17) and (2.18),we get that the allocation choice function f is implementable in F -Bayes-Nash equilibriumby the mechanism M with the equilibrium strategy σ .On the other hand, suppose we have an allocation choice function f : Θ → ∆( A ) .Consider the corresponding allocation choice function f (cid:48) : Θ (cid:48) → ∆( A ) given by f (cid:48) ( θ (cid:48) ) = f ( T − ( θ )) .
44e now observe that if f is implementable in F -Bayes-Nash equilibrium by a mechanism M and an F -Bayes-Nash equilibrium σ , then so is f (cid:48) by the same mechanism M and the F (cid:48) -Bayes-Nash equilibrium σ (cid:48) comprised of σ (cid:48) i ( θ (cid:48) i ) = σ i ( T − i ( θ (cid:48) i )) , for all i ∈ [ n ] , θ (cid:48) i ∈ Θ (cid:48) i .We can similarly show that if f (cid:48) is implementable in dominant (resp. belief-dominant)equilibrium by a mechanism M with the equilibrium strategy profile σ (cid:48) for the reduced en-vironment E (cid:48) , then so is f by the same mechanism M with the corresponding equilibriumstrategy profile σ for the environment E , and vice versa.Hence, under EUT, from the mechanism designer’s point of view, it is enough to modelthe types of player i by setting the outcome set Γ (cid:48) i = A , assuming the trivial allocation-outcome mapping ζ (cid:48) i , and the types θ (cid:48) i ∈ Θ (cid:48) i . References
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