Mechanism Design with Limited Commitment
aa r X i v : . [ ec on . T H ] A ug Mechanism Design with Limited Commitment *Laura Doval † Vasiliki Skreta ‡ August 27, 2020
Abstract
We develop a tool akin to the revelation principle for mechanism design withlimited commitment. We identify a canonical class of mechanisms rich enoughto replicate the outcomes of any equilibrium in a mechanism-selection gamebetween an uninformed designer and a privately informed agent. A corner-stone of our methodology is the idea that a mechanism should encode not onlythe rules that determine the allocation, but also the information the designerobtains from the interaction with the agent. Therefore, how much the designerlearns, which is the key tension in design with limited commitment, becomesan explicit part of the design. We show how this insight can be used to transformthe designer’s problem into a constrained optimization problem: To the usualtruthtelling and participation constraints, one must add the designer’s sequen-tial rationality constraint.K
EYWORDS : mechanism design, limited commitment, revelation principle, information de-signJEL
CLASSIFICATION : D84, D86 * We would like to thank Rahul Deb, Françoise Forges, David Miller, Dan Quigley, Luciano Pomatto,Pablo Schenone, Omer Tamuz and especially Michael Greinecker and Max Stinchcombe, as well asaudiences at Cowles, Gerzensee, and Stony Brook, for thought-provoking questions and illuminatingdiscussions. Alkis Georgiadis-Harris, Nathan Hancart, and Ignacio Núñez provided excellent researchassistance. Vasiliki Skreta is grateful for generous financial support through the ERC consolidatorgrant 682417 “Frontiers in design.” This research is supported by grants from the National ScienceFoundation (SES-1851744 and SES-1851729). † Columbia University, New York, NY 10027. E-mail: [email protected] ‡ University of Texas at Austin, University College London, and CEPR. E-mail: [email protected]
NTRODUCTION
The standard assumption in dynamic mechanism design is that the designer cancommit to long-term contracts. This assumption is useful: It allows us to charac-terize the best possible payoff for the designer in the presence of adverse selectionand/or moral hazard, and it is applicable in many settings. Often, however, this as-sumption is made for technical convenience. Indeed, when the designer can committo long-term contracts, the mechanism-selection problem can be reduced to a con-strained optimization problem thanks to the revelation principle . However, as theliterature starting with Laffont and Tirole (1987, 1988) shows, when the designer cancommit only to short-term contracts, the tractability afforded by the revelation prin-ciple is lost. Indeed, mechanism design problems with limited commitment are dif-ficult to analyze without imposing auxiliary assumptions either on the class of con-tracts the designer can choose from, as in Gerardi and Maestri (2020) and Strulovici(2017), or on the length of the horizon, as in Skreta (2006, 2015).This paper provides a “revelation principle” for dynamic mechanism-selection gamesin which the designer can only commit to short-term contracts. We study a gamebetween an uninformed designer and an informed agent with persistent private in-formation. Although the designer can commit within each period to the terms of theinteraction–the current mechanism–he cannot commit to the terms the agent faceslater on, namely, the mechanisms that are chosen in the continuation game. First, weshow there is a class of mechanisms that is sufficient to replicate all equilibrium out-comes of the mechanism-selection game. Second, we show how this insight can beused to transform the designer’s problem into a constrained optimization problem:To the usual truthtelling and participation constraints, one must add the designer’ssequential rationality constraint.The starting point of our analysis is the class of mechanisms we allow the designerto select from. Following Myerson (1982) and Bester and Strausz (2007), we considermechanisms defined by a general communication device as illustrated in Figure 1a: The “revelation principle” denotes a class of results in mechanism design; see Gibbard (1973),Myerson (1979), and Dasgupta et al. (1979). M S × A r (⋅∣ θ ) β (⋅∣ m ) Figure (a) General Mechanisms linebreak Θ M S A r (⋅∣ θ ) β (⋅∣ m ) α (⋅∣ s ) Figure (b) Canonical Mechanisms: M = Θ , S = ∆ ( Θ ) Figure 1: MechanismsHaving observed her private information (her type, θ ∈ Θ ), the agent privately reportsan input message, m ∈ M , to the mechanism; this then determines the distribution, β (⋅∣ m ) , from which an output message, s ∈ S , and an allocation, a ∈ A , are drawn.The output message and the allocation are publicly observable : They constitute thecontractible parts of the mechanism.When the designer has commitment, the revelation principle implies that, withoutloss of generality, we can restrict attention to mechanisms satisfying the followingthree properties: (i) M = Θ , (ii) ∣ M ∣ = ∣ S ∣ , and (iii) β is such that by observing theoutput message, the designer learns the input message, in this case the agent’s typereport. Moreover, the revelation principle implies that we can restrict attention toequilibria in which the agent truthfully reports her type, which means that the de-signer not only learns the agent’s type report upon observing the output messagebut also learns the agent’s true type.It is then clear why restricting attention to mechanisms that satisfy properties (i)-(iii) and truthtelling equilibria is with loss of generality under limited commitment:Upon observing the output message, the designer learns the agent’s type report andhence her type. Then the agent may have an incentive to misreport if the designercannot commit to not react to this information. This is precisely the intuition be-hind the main result in Bester and Strausz (2001), which is the first paper to providea general analysis of optimal mechanism design with limited commitment. Insteadof allowing the designer to choose any mechanism, the authors restrict attention tomechanisms such that the cardinality of the set of input and output messages is thesame and β is such that by observing the output message, the designer learns the in-put message. They show that to sustain payoffs in the Pareto frontier, mechanismsin which input messages are type reports are without loss of generality. However,focusing on truthtelling equilibria is with loss of generality. In a follow-up paper,Bester and Strausz (2007) lift the restrictions on the class of mechanisms (i.e., (ii) and(iii) above) and show in a one-period model that focusing on mechanisms in which The class of mechanisms considered in Bester and Strausz (2001) encompasses the mechanismsconsidered by most papers in the literature on limited commitment starting from Laffont and Tirole(1988). S = ∆ ( Θ ) . Theorem 1 shows that thefollowing two games between an uninformed designer and an informed agent im-plement the same set of equilibrium distributions over types and allocations. In thefirst, the mechanism-selection game , the principal can offer the agent mechanismsas in Figure 1a. In the second, the canonical game, the principal can offer the agentmechanisms in which input messages are type reports, output messages are beliefs,and, conditional on the output message, the allocation is drawn independently of theagent’s type report (see Figure 1b). Moreover, Theorem 1 shows that any equilibriumof the canonical game can be replicated by an equilibrium in which (a) the agentalways participates in the mechanism, and (b) input and output messages have a lit-eral meaning: The agent truthfully reports her type, and if the mechanism outputs agiven posterior, this posterior coincides with the belief that the designer holds aboutthe agent’s type given the agent’s strategy and the mechanism. Given that any equi-librium distribution over types and allocations can be replicated by mechanisms inwhich input messages are type reports and output messages are beliefs about theagent’s type, we call this class of mechanisms canonical. Theorem 1 implies that in mechanism design with limited commitment, the mech-anism serves a dual role within a period. On the one hand, it determines the allo-cation for that period. On the other hand, it determines the information about theagent that is carried forward in the interaction. An advantage of the language of pos-terior beliefs is that it avoids potential infinite-regress problems. Indeed, in a finitehorizon problem, an alternative set of output messages could be a recommendationfor an allocation today and a sequence of allocations from tomorrow on. In the fi-nal period, the revelation principle in Myerson (1982) pins down the implementableallocations. Therefore, the recommended allocations can be determined via back-ward induction. This idea cannot be carried to an infinite horizon setting: These setsof output messages would necessarily have to make reference to the continuationmechanisms, which are themselves defined by a set of output messages.Theorem 1 affords the analyst two main simplifications. First, it follows from its proof4hat it is without loss of generality to restrict attention to the analysis of the canon-ical game, since it implements the same set of distributions over types and alloca-tions, and thus payoffs, as the mechanism-selection game. Second, it provides theresearcher with a tractable way to analyze problems of mechanism design with lim-ited commitment by making how much the principal learns about the agent an ex-plicit part of the design. The three constraints that the mechanism must satisfy, theparticipation and truthtelling constraints for the agent, and the Bayes’ plausibilityconstraint, provide us with a tractable representation both of the agent’s behavior ina given period and of its impact on the mechanism offered in the next via the infor-mation that is generated about the agent’s type in the given period. A major chal-lenge in the received literature on limited commitment is how to keep track of howthe agent’s best response to the mechanism affects the information that the principalobtains from the interaction, which in turn affects the principal’s incentives to offerthe mechanism in the first place. Instead, our framework allows us to reduce theagent’s best response to the principal’s mechanism and its informational feedbackto a familiar set of constraints that the mechanism must satisfy. This avoids havingto consider complicated mixed strategies on the part of the agent (see, for instance,Laffont and Tirole (1988); Bester and Strausz (2001)) and transforms it instead into aprogram that combines elements of mechanism design and information design.While Theorem 1 assumes that the agent’s type is fully persistent, this is not neces-sary for its conclusion to hold. Theorem 2 extends Theorem 1 to a version of whatPavan et al. (2014) denote as
Markov environments . These are settings where (i) theagent’s private information follows a possibly nonhomogeneous Markov process,(ii) the principal and the agent’s payoffs are time-separable, and their flow payoffsdepend only on today’s allocation and the agent’s current type, and (iii) the transi-tion probability may depend both on today’s type and today’s allocation. Theorem 2shows that in Markov environments it is without loss of generality to restrict atten-tion to the characterization of equilibrium payoffs of the canonical game and tostrategy profiles where the agent participates and truthfully reports her current typeto characterize the set of payoffs the designer can implement in the mechanism-selection game.We illustrate how Theorem 1 and Theorem 2 can be used to shed new light on seem-ingly well-understood problems and open the door to the analysis of new problemswith two examples in Section 4 and Section 6. Both examples feature a continuum oftypes, but for simplicity, consider two-period settings. Theorem 1 also opens the door to the analysis of optimal mechanisms under limited commitment rationing can strictly dominate posted prices (Remark 2discusses how canonical mechanisms differ from the mechanisms in Skreta (2006),which explains the difference in the results.) This allows us to connect the mecha-nism design literature on the sale of a durable good with the work in theoretical in-dustrial organization on alternative strategies for a durable good monopolist, such asrationing (Denicolo and Garella (1999); Gilbert and Klemperer (2000); McAfee and Wiseman(2008)) and clearance sales (Nocke and Peitz (2007)). Indeed, Proposition 1 providesa microfoundation for a mechanism first suggested by Denicolo and Garella (1999)and a new rationale for the use of clearance sales.Section 6 considers the interaction between a seller and a buyer over two periodsand two different transactions. Contrary to the previous example, the buyer’s privateinformation is not fully persistent. Instead, we assume that the buyer’s type in pe-riod 1 is informative about her valuation in period 2. While stylized, this example isprototypical of many situations where a buyer interacts with a seller across differentpurchases, as in online shopping, or the purchase of a basic good, followed by anadd-on. We use this example to highlight the trade-off that arises in limited com-mitment between the benefits of the use of the information in period 1 to customizethe allocation to the buyer’s period 1 valuation and the costs of doing so via its im-pact on the mechanism offered in period 2. Proposition 2 shows that the trade-offbetween the benefits and the costs of the information generated by the mechanismin period 1 leads to a rich set of distortions in the period 1 allocation, above and be-yond those that follow from the rent extraction motive, which include both under-and overprovision in period 1. This rich pattern of distortions comes together with arich information disclosure policy, which features (sometimes multiple) pooling andseparation intervals.Our work brings forth a new application of information design by placing its toolsat the service of characterizing optimal mechanisms under limited commitment.By highlighting the canonical role of beliefs as the signals employed by the mech-anism, Theorem 1 and Theorem 2 underscore the importance of jointly determin- in infinite-horizon settings. We illustrate this in Doval and Skreta (2020a), where we solve an infinite-horizon binary-type version of the sale of a durable good.
Related Literature:
The paper contributes to the literature on mechanism designwith limited commitment with an informed agent with persistent private informa-tion, referenced throughout the introduction. Following the seminal contributionof Bester and Strausz (2001), a body of work studies optimal mechanisms under lim-ited commitment in settings with finitely many types and finite horizon (e.g., Bisin and Rampini(2006); Hiriart et al. (2011); Fiocco and Strausz (2015); Beccuti and Möller (2018)). Sincethe results in Bester and Strausz (2001) do not extend to settings with a continuumof types and/or infinite horizon, the characterization of optimal mechanisms underlimited commitment in these settings has proven elusive. On the one hand, Skreta(2006); Deb and Said (2015); Skreta (2015) study mechanism-selection games witha continuum of types and finite horizon. All three papers leverage the assumptionof finite horizon, which pins down the optimal mechanism in the final period, tocharacterize the implications of the principal’s sequential rationality constraints forthe set of incentive-feasible outcomes. On the other hand, the small set of papersthat study infinite-horizon problems of design under limited commitment do so un-der assumptions on either the set of mechanisms the designer is allowed to offer(e.g., Acharya and Ortner (2017); Strulovici (2017); Gerardi and Maestri (2020)) or thesolution concept (e.g., Acharya and Ortner (2017)). All these papers use the set ofmechanisms in Laffont and Tirole (1988); Bester and Strausz (2001).Due to the difficulties with the revelation principle, a large body of work in pub-lic finance, political economy and taxation considers optimal time-consistent poli-cies in settings where private information is fully nonpersistent (see, for instance,Sleet and Yeltekin (2008); Farhi et al. (2012); Golosov and Iovino (2016)). Moreover,a large literature studies the effect of limited commitment within a specific class of“mechanisms”: The papers in the durable-good monopolist literature (Bulow (1982);Gul et al. (1986); Stokey (1981)) study price dynamics and establish (under some con-ditions) Coase’s conjecture whereby a monopolist essentially loses all profits if itlacks commitment. In an analogous vein, Burguet and Sakovics (1996), McAfee and Vincent(1997), Caillaud and Mezzetti (2004), and Liu et al. (2019) study equilibrium reserve- A designer’s lack of commitment can take various forms that are not considered in this paperbut have been studied in others. See, for instance, McAdams and Schwarz (2007), Vartiainen (2013),and Akbarpour and Li (2020), in which the designer cannot commit even to obeying the rules of thecurrent mechanism.
Organization:
The rest of the paper is organized as follows. Section 2 describes themodel and notation. Section 3 introduces the main theorem and provides a sketchof the proof. Section 4 analyzes a two-period version of the model in Skreta (2006)to illustrate how one can apply Theorem 1 in a setting with a continuum of typesand shed new light on a classic problem. Section 5 presents Theorem 2, which ex-tends Theorem 1 to
Markov environments . Section 6 illustrates Theorem 2 with anapplication to nonlinear pricing. All proofs are in Appendix B and the supplemen-tary material, Doval and Skreta (2020c) (Appendices C-F).2 M
ODEL
Primitives:
Two players, a principal (he) and an agent (she), interact over T ≤ ∞ periods. Before the game starts, the agent observes her type, θ ∈ Θ , which is dis-tributed according to a full support distribution µ . Each period, as a result of theinteraction between the principal and the agent, an allocation a ∈ A is determined.Let A T + denote the set × Tt = A . For the principal, assume that there exists a function W ∶ A T + × Θ ↦ R such that his payoff from allocation a T + ∈ A T + when the agent’stype is θ is given by W ( a T + , θ ) . Similarly, for the agent, when her type is θ , her payofffrom allocation a T + ∈ A T + is given by U ( a T + , θ ) .For every t ≥ a t = ( a , a , . . . , a t − ) , the prin-cipal can only choose a t ∈ A( a t ) in period t . That is, there is a correspondence A ∶ ∪ Tt = A t ↦ A such that for t ∈ {
1, . . . , T } and a t ∈ A t , A( a t ) describes the set ofallocations that the principal can offer in period t given the allocations he has of-fered through period t −
1. On the one hand, the correspondence A encodes thatthe set of feasible allocations may be time dependent, so that A depends on a t onlythrough the time index t , as in the application in Section 6. On the other hand, itallows for the case in which the past allocations restrict what the principal can offerthe agent in the future, as in the application in Section 4. Assume that there exists anallocation a ∗ ∈ A such that a ∗ is always available. Below, allocation a ∗ plays the role8f the agent’s outside option. Given the general structure of payoffs, it is without lossof generality to take it to be time-independent.We impose some technical restrictions on our model. The sets Θ and A are Polish,that is, completely metrizable, separable, topological spaces. They are endowed withtheir Borel σ -algebra. We also assume that Θ is compact. Endowing product sets withtheir product σ -algebra, we assume that the principal and the agent’s utility func-tions, W and U , are bounded measurable functions. Similarly, the correspondence A is measurable. Mechanisms:
In each period, the principal offers the agent a mechanism, M t = ( M M t , S M t , β M t ) , where M M t and S M t are Polish, and β M t is a transition proba-bility from M M t to S M t × A . We endow the principal with a collection ( M i , S i ) i ∈I ofinput and output message sets in which ∣ Θ ∣ ≤ ∣ M i ∣ and ∣ ∆ ( Θ )∣ ≤ ∣ S i ∣ . Moreover, weassume that ( Θ , ∆ ( Θ )) is an element in that collection. Denote by M the set of allmechanisms with message sets ( M i , S i ) i ∈I , i.e., { β ∶ M i ↦ ∆ ( S j × A ) ∶ i , j ∈ I } .Three remarks are in order. First, the restriction that M i has at least as many mes-sages as types is without loss of generality. The principal can always replicate a mech-anism with a smaller set of input messages by using a larger set of input messages. Second, we restrict the principal to choosing input and output messages within theset ( M i , S i ) i ∈I . This allows us to have a well-defined set of deviations for the princi-pal, thereby avoiding set-theoretic issues related to self-referential sets. The analysisthat follows shows that the choice of the collection plays no further role in the analy-sis. Finally, we note that all aspects of the environment, except the agent’s type θ ∈ Θ ,are common knowledge between the principal and the agent. Timing:
In each period t , the game proceeds as follows. The principal offers theagent a mechanism, M t , with the property that for all m ∈ M M t , β M t ( S M t ×A( a t )∣ m ) =
1, where a t describes the allocations implemented through period t −
1. Observingthe mechanism, the agent decides whether to participate in the mechanism ( p = ) In what follows, we adopt the following notational conventions. First, all Polish spaces are en-dowed with their Borel σ -algebra. For a Polish space, X , B X denotes its Borel σ -algebra. Second,product spaces are endowed with their product σ -algebra. Third, for a Polish space, Y , we let ∆ ( Y ) denote the set of all Borel probability measures over Y , endowed with the weak ∗ topology. Thus, ∆ ( Y ) is also a Polish space (Aliprantis and Border (2013)). For any two measurable spaces X and Y , a map-ping ζ ∶ X ↦ ∆ ( Y ) is a transition probability from X to Y if for any measurable C ⊆ Y , ζ ( C ∣ x ) ≡ ζ ( x )( C ) is a measurable real valued function of x ∈ X . To see this, suppose that the principal would rather use a mechanism, M ′ t , with a message space M M ′ t with cardinality strictly less than ∣ Θ ∣ . Then he can choose a mechanism M t with M M t = Θ , choose β M t to coincide with β M ′ t on the first ∣ M M ′ t ∣ messages, and have β M t coincide with β M ′ t ( ⋅ ∣ m ′ ) for allremaining messages.
9r not ( p = ) . If she does not participate in the mechanism, a ∗ is implemented andthe game proceeds to t +
1. Instead, if she chooses to participate, she sends a message m ∈ M M t , which is unobserved by the principal. An output message and an allocation ( s t , a t ) are drawn according to β M t (⋅∣ m ) ; the output message and the allocation areobserved by both the principal and the agent.The above defines an extensive-form game, which we dub the mechanism-selectiongame and denote by G M . Public histories in this game are h t = ( M , p , s , a , . . . , M t − , p t − , s t − , a t − ) ,where p r ∈ {
0, 1 } denotes the agent’s participation with the restriction that p r = ⇒ s r = ∅ , a r = a ∗ . Given a mechanism M t , let z ∅ ( M t ) , denote the tuple M t , 0, ∅ , a ∗ andlet z ( s t , a t ) ( M t ) , denote the tuple M t , 1, s t , a t . Note that any public history at the endof period t can be written as ( h t , z ∅ ( M t )) or ( h t , z ( s t , a t ) ( M t )) .Public histories capture what the principal knows through period t . Let H t denotethe set of all period t public histories. A history for the agent consists of the public history of the game together with the agent’s inputs into the mechanism (henceforth,the agent history) and her private information. Formally, an agent history is an ele-ment h tA = ( M , p , m , s , a , . . . , M t − , p t − , m t − , s t − , a t − ) ,with p r = ⇒ m r = ∅ . Given a public history h t , let H tA ( h t ) denote the set of agenthistories consistent with h t . The agent also knows her type, and hence a historythrough period t is an element of { θ } × H tA when her type is θ . Strategies:
Since the principal’s action space, M , is an uncountable set of functions,we model the principal’s behavioral strategy, ( σ Pt ) Tt = , following Aumann (1964).That is, endow [
0, 1 ] with the Borel σ -algebra and the Lebesgue measure, λ . Then, σ Pt is defined as a jointly measurable function from H t × [
0, 1 ] to M . We denotethe collection ( σ Pt ) Tt = by σ P . The agent’s participation strategy is a transition prob-ability, π t , from Θ × H tA ×M to {
0, 1 } . Conditional on participating in the mechanism M t , her reporting strategy is a transition probability, r t , from Θ × H tA ( h t ) ×M× { } to While there is no output message when the agent does not participate in the mechanism, wedenote this by s = ∅ to keep the length of all the histories the same. To keep notation simple, we do not add the restriction that if M t is in the support of σ Pt ( h t ) , then β M t ( S M t × A( a t )∣ m ) =
1, where a t is the allocation up to period t according to h t . i ∈I M i such that r t ( θ , h tA , M t ) ∈ ∆ ( M M t ) . We denote the tuple ( π t , r t ) by σ At , andthe collection ( σ At ) Tt = by σ A .The definitions above assume that one can define a measurable structure on M suchthat we can define the principal and the agent’s strategies as measurable functions ofthe histories. As noted by Aumann (1961), this requires that M be a standard Borelspace. As we explain in Appendix C, this is the case when Θ is finite or countable.Instead, when Θ is a continuum, the set M is not a standard Borel space. When Θ is a continuum, there are two approaches one may follow. The first approach isto allow the principal to choose from a subset M ′ ⊂ M which is a standard Borelspace, in which case it is correct to write the principal and the agent’s strategies asconditioning on the past chosen mechanisms. The second approach relies on theidea that one can represent a very large set of mechanisms in terms of a standardBorel space, without ex-ante restricting the mechanisms the principal is allowed tochoose from. This approach, which we implicitly use in our applications, requiresdefining the principal and the agent’s strategies in a different way, so we relegate thisdiscussion to Appendix C. A belief for the principal at the beginning of time t , history h t , is a distribution µ t ( h t ) ∈ ∆ ( Θ × H tA ( h t )) . The principal is thus uncertain both about the agent’s payoff-relevant type, θ , and her payoff-irrelevant private history, h tA . The collection ( µ t ) Tt = denotes the belief system.An assessment is a tuple ( σ Pt , σ At , µ t ) Tt = ≡ ( σ P , σ A , µ ) . Our focus is on studyingthe equilibria of the mechanism-selection game. By equilibrium, we mean PerfectBayesian equilibrium (henceforth, PBE), defined as follows: Definition 1. A Perfect Bayesian Equilibrium is a tuple ( σ P , σ A , µ ) such that the fol-lowing holds:1. ( σ P , σ A , µ ) is sequentially rational (Definition A.1), and2. The belief system satisfies Bayes’ rule where possible (Definition A.2). The formal statement is in Appendix A. For now, we note that if the principal’s strat-egy space were finite, Θ is finite, and the mechanisms used by the principal have fi- While technically the agent’s reporting strategy should be written r t ( θ , h tA , M t ,1 ) to account forthe agent’s decision to participate, we omit the 1 to simplify notation. The issue of choosing mechanisms at random also shows up in the competing principals liter-ature, where it is typical to restrict attention to pure strategy equilibria of the mechanism-selectiongame. The prior µ together with a strategy profile ( σ P , σ A ) determine a distribution overthe terminal nodes Θ × H T + A . We are interested instead in the distribution they in-duce over the payoff-relevant outcomes, Θ × A T + . We say that γ ∈ ∆ ( Θ × A T + ) is aPBE outcome of the mechanism-selection game if there exists a PBE of the mechanism-selection game that induces γ . We denote by O ∗M the set of PBE outcomes of G M .Our main result establishes the equivalence between the set of PBE outcomes of themechanism-selection game and those of another game, which we dub the canonicalgame and introduce next. The canonical game:
The canonical game is essentially the same as the mechanism-selection game except for two features. First, in every period the principal can onlychoose canonical mechanisms, which we denote by M C . Canonical mechanismsdiffer from the mechanisms introduced above in two respects. First, the sets of inputand output messages are given by ( Θ , ∆ ( Θ )) . Second, if ( Θ , ∆ ( Θ ) , ˜ β ) ∈ M C , thenthere exist two transition probabilities, β from Θ to ∆ ( Θ ) and α from ∆ ( Θ ) to A suchthat, for all θ ∈ Θ , and for all measurable subsets ˜ U × ˜ A ⊆ ∆ ( Θ ) × A ˜ β ( ˜ U × ˜ A ∣ θ ) = ∫ ˜ U α ( ˜ A ∣ µ ) β ( d µ ∣ θ ) ,so that conditional on the output message µ ∈ ∆ ( Θ ) , the allocation is drawn indepen-dently of the type report. Third, at the beginning of each period both players observethe realization of a public randomization device ω ∼ U [
0, 1 ] . We denote the canonicalgame by G and its set of PBE outcomes by O ∗ . Remark 1 (An auxiliary game) . The proofs of our results require translating strategyprofiles from the mechanism-selection game to the canonical game, which is nota-tionally involved. To facilitate the presentation of our results we rely on a third aux-iliary game, G A M , which is exactly like the mechanism-selection game, except that atthe beginning of each period the principal and the agent observe a draw from a publicrandomization device ω ∼ U [
0, 1 ] . Note that one can trivially adapt any strategy pro-file σ of the mechanism-selection game to a strategy profile σ ′ of the auxiliary gamesimply by specifying that σ ′ follows σ for every realization of the public randomizationdevice. Similarly, any strategy profile σ ′ of the auxiliary game in which the principal The only difference between Bayes’ rule where possible and consistency in sequential equilibriumis the following. Under PBE, the principal can assign zero probability to a type and then, after theagent deviates, can assign positive probability to that same type. This distribution, which exists via the Ionescu Tulcea extension theorem (Tulcea (1949)), is for-mally defined in Appendix A. hooses a canonical mechanism at every history can trivially be adapted to a strategyprofile ˜ σ of the canonical game simply by specifying that ˜ σ coincides with σ ′ only athistories where the principal has offered canonical mechanisms throughout. AIN RESULT
Section 3 presents the main result of the paper. Theorem 1 shows that the mechanism-selection game and the canonical game have the same set of equilibrium outcomes.Moreover, any PBE assessment of the mechanism-selection game can be replicatedby a PBE assessment of the canonical game in which (a) the agent always participatesin the mechanism, and (b) input and output messages have a literal meaning: Theagent truthfully reports her type, and if the mechanism outputs µ ∈ ∆ ( Θ ) at the endof period t , then µ is indeed the belief the principal holds about the agent at the endof that period. Theorem 1.
The mechanism-selection game and the canonical game have the sameset of PBE outcomes, i.e., O ∗M = O ∗ for any collection of mechanisms, M , with whichwe endow the principal.Moreover, for any PBE outcome of G M , there exists a PBE assessment ( σ P , σ A , µ ) of Gthat implements the same outcome and satisfies the following properties:1. The agent’s strategy depends only on her private type and the public history.2. For all public histories h t , for all θ in the support of µ t ( h t ) , the agent participatesin the mechanism offered by the principal at that history and with probability onetruthfully reports her type,3. For all public histories h t , if the mechanism offered by the principal at h t outputsa posterior µ ′ , the principal’s updated equilibrium beliefs about the agent coincidewith µ ′ . That is, for all measurable subsets ˜ Θ , ˜ U , and ˜ A of Θ , ∆ ( Θ ) , and A, ∫ Θ ∫ ˜ U ∫ ˜ A µ t + ( ˜ Θ ∣ h t , z ( µ ′ , a t ) ) α M t ( d a t ∣ µ ′ ) β M t ( d µ ′ ∣ θ ) µ t ( d θ ∣ h t ) == ∫ ˜ Θ ∫ ˜ U α M t ( ˜ A ∣ µ ′ ) β M t ( d µ ′ ∣ θ ) µ t ( d θ ∣ h t ) = ∫ Θ ∫ ˜ U ∫ ˜ A µ ′ ( ˜ Θ ) α M t ( d a t ∣ µ ′ ) β M t ( d µ ′ ∣ θ ) µ t ( d θ ∣ h t ) . Theorem 1 plays the same role in mechanism design with limited commitment as therevelation principle does in the commitment case. First, it identifies a well-definedset of mechanisms, M C , to which we can restrict the principal’s choice set withoutloss of generality. Second, it simplifies the analysis of the behavior of the agent in the13ame induced by the mechanisms chosen by the principal: We can always restrictattention to assessments where the agent participates and truthfully reports her type.As is evidenced by the applications in Sections 4 and 6, this allows us to reduce theagent’s behavior to a set of constraints that the mechanism must satisfy, exactly as inthe case of commitment.The proof that any PBE outcome of the mechanism-selection game can be achievedas a PBE outcome of the canonical game that satisfies the properties listed in Theorem 1relies on four steps, which we review next. Input messages as type reports:
To fix ideas, consider the typical proof for the stan-dard revelation principle in static settings. The mechanism, M , together with theagent’s reporting strategy induces a transition probability from Θ to S M × A . This al-lows us to conclude that we can replace the set of input messages with the set of typereports, as illustrated in Figure 2a. Θ S M × AM M r β M β M ○ r Figure (a) Static revelationprinciple Θ × H tA ( h t ) S M t × AM M t r β M t β M t ○ r Figure (b) Mechanism selection game
Figure 2: Type reports as input messagesIn the dynamic setting, however, this argument would only allow us to conclude thatwe can rewrite the mechanism as a transition probability from Θ × H tA ( h t ) to S M t × A , as illustrated in Figure 2b. Indeed, to replicate the agent’s reporting strategy, themechanism needs to obtain all the information on which the agent conditions herstrategy, which potentially is ( θ , h tA ) .To circumvent this difficulty, we show that given a PBE in which the agent conditionsher strategy on the payoff-irrelevant part of her private history at some public history h t , there exists another outcome-equivalent PBE in which she does not and in whichthe principal obtains the same payoff after each continuation history consistent with h t and the equilibrium strategy (see Proposition B.1). Thus, conditional on the pub-lic history h t , the agent’s reporting strategy and the mechanism induce a transitionprobability from Θ to S M t × A , so we can always take the set of input messages to In Doval and Skreta (2020b), we provide a proof of these four steps for the case in which Θ is finiteand the principal can only offer mechanisms M such that, for all m ∈ M M , the support of β M ( ⋅ ∣ m ) isfinite. That proof mirrors the one in Appendix B, but is technically simpler and more accessible.
14e the set of type reports. This result relies on two observations. First, becauseinput messages are payoff irrelevant and unobserved by the principal, if the agentchooses different strategies at ( θ , h tA ) and ( θ , ˜ h tA ) with h tA , ˜ h tA ∈ H tA ( h t ) , then she isindifferent between these two strategies. However, the principal may not be indif-ferent between these two strategies. The second step is to show that we can buildan alternative strategy for the agent that conditions only on ( θ , h t ) and yields theprincipal the same payoff.This first step also gives us an important conceptual insight: The principal cannotpeek into his past mechanisms. To do so, he would have to ask the agent to reporther previous communication to him. Our result implies that this information cannotbe elicited in any payoff-relevant way.It follows from this first step that it is without loss of generality to focus on equilib-rium assessments of the mechanism-selection game such that the principal offersmechanisms where the set of input messages are type reports, i.e., M M t = Θ , and theagent truthfully reports her type. In what follows, when we refer to a PBE assessmentof G M , we mean one that satisfies these properties. Output messages as beliefs:
To understand the steps involved in showing that out-put messages can be taken without loss of generality to be the principal’s beliefsabout the agent’s type, it is useful to consider the other uses the principal may havefor the output messages beyond encoding information about the agent.First, the principal could use S M t to encode randomizations on the allocation, e.g.,two tuples, ( s t , a t ) and ( s ′ t , a ′ t ) , may be associated with the same posterior belief.This is not an issue, however, since a canonical mechanism allows the principal torandomize on the allocation conditional on the posterior belief.Second, the principal could use S M t to coordinate continuation play, e.g., two tuples ( s t , a t ) , ( s ′ t , a t ) may be associated with two different continuation equilibria, evenif they induce the same posterior belief. This is where canonical mechanisms andthe property that output messages must coincide with the principal’s updated beliefs This result is useful also in applications. It states that in our game the set of PBE payoffs coincideswith the set of Public PBE payoffs (Athey and Bagwell (2008)). In games with time-separable payoffs,Public PBE payoffs have a recursive structure and are amenable to self-generation techniques, as inAbreu et al. (1990). For an example, see Doval and Skreta (2020a). The same idea arises in the literature on Bayesian persuasion. Implicit in the result inKamenica and Gentzkow (2011) that any experiment can be written as a distribution over posteriorsis the assumption that the receiver breaks ties in favor of the sender. Unlike in Bayesian persuasion,it is not clear that the players may be indifferent between two continuation equilibria, so the publicrandomization device does not generally reduce to simple tie-breaking. M : Beliefs are not a rich enough lan-guage to encode both the principal’s updated beliefs and the suggested continuationplay. Nevertheless, the canonical game has a feature that the mechanism-selectiongame does not: the public randomization device. As we explain next, this allows usto subsume the second role of the output message.The potential challenge in using the public randomization device to subsume thesecond role of the output message is that, by definition, the use of the public ran-domization device in the canonical game can only depend on publicly available in-formation, while the output message in the mechanism-selection game is drawn asa function of the agent’s type, since the agent is reporting truthfully. We leveragehere that canonical mechanisms use beliefs as output messages. To see this, notethat beliefs are a sufficient statistic for the information about the agent’s type thatis encoded in the output messages of the mechanism-selection game. Thus, con-ditional on the induced belief and the allocation, the selection of continuation playcontains no further information about the agent’s type. This is how we are able todecompose the mechanism in the mechanism-selection game into a mechanism inthe canonical game that uses beliefs as output messages and a public randomizationdevice.The above argument also explains why in a canonical mechanism, conditional onthe output message, the allocation can be drawn independently of the agent’s type(report). Ultimately, conditional on the induced belief, the allocation contains nofurther information about the agent’s type.Formally, the proof of this result proceeds as follows (see Proposition B.3). Supposethat the principal offers M t in period t . The principal’s belief about the agent’s type,together with the agent’s reporting strategy and the mechanism β M t induces a jointdistribution P over Θ × S M t × A × ∆ ( Θ ) . Since conditional on the induced posterior, ( s t , a t ) ∈ S M t × A carries no further information about the agent’s type, this allows usto “split” the mechanism into a transition probability ˜ β from Θ to ∆ ( Θ ) , a transition Note that this is not a matter of cardinality, but a consequence of the restriction that suggested be-liefs must coincide with equilibrium beliefs. Ultimately, by Kuratowski’s theorem (see Parthasarathy(2005)), ∆ ( Θ ) is in bijection with ∆ ( Θ ) × [ ] so there are enough messages to encode both the prin-cipal’s updated beliefs and the suggested continuation play. Formally, assume that the mechanism M t is such that M M t = Θ and the agent truthfully reportsher type. Let µ ∈ ∆ ( Θ ) denote the principal’s prior belief at h t . Then, the posterior beliefs µ ′ satisfy ∫ Θ ∫ ˜ S × ˜ A µ ′ ( ˜ Θ ∣ s , a ) β M t ( d ( s , a )∣ θ ) µ ( d θ ) = ∫ ˜ Θ β M t ( ˜ S × ˜ A ∣ θ ) µ ( d θ ) for any measurable subsets ˜ Θ , ˜ S , ˜ A of Θ , S M t , and A . Note that the posterior beliefs define a transition probability from S M t × A to ∆ ( Θ ) .Denote it by T . Then, the joint distribution P is defined by P ( ˜ Θ × ˜ S × ˜ A × ˜ U ) = ∫ ˜ Θ ∫ ˜ S × ˜ A [ T ( s , a ) ∈ ˜ U ] β M t ( d ( s , a )∣ θ ) µ ( d θ ) , where ˜ U denotes a measurable subset of ∆ ( Θ ) . α from ∆ ( Θ ) to A , and a transition probability ω from ∆ ( Θ ) × A to S M t .The transition probability α plays the first role of the output message and highlightsthe importance of allowing the principal to offer randomized allocations. The tran-sition probability ω corresponds to the public randomization device: By Kuratowski’stheorem we can always embed S M t into [
0, 1 ] (see Parthasarathy (2005)). Three conceptual insights arise from this result. First, when the mechanism is canon-ical, the principal can separate the design of the information that the mechanism en-codes about the agent’s type from the design of the allocation. Second, the allocationhas to be measurable with respect to the information generated by the mechanism:The more the principal desires to tailor the allocation to the agent’s type, the more hehas to learn about the agent’s type through the mechanism (see Section 6). Third,it provides a microfoundation for the public randomization device in the canonicalgame: it represents the principal’s attempt to coordinate play in the mechanism-selection game.
Bayes’ rule and participation:
Underlying the previous step is the assumption thatthe beliefs associated with the output messages are determined via Bayes’ rule. Inparticular, the principal is never surprised by any output message he observes. Toachieve this we show in Proposition B.2 that we can “eliminate" from the mecha-nism all input messages that are used only by types to whom the principal assigns 0probability. This, of course, may change the participation decision for these types,which is why Theorem 1 only guarantees participation for those types in the supportof the principal’s beliefs. Finally, we show in Proposition B.4 that without loss of generality the agent partici-pates in the mechanism whenever her type is in the support of the principal’s beliefs.The logic is similar to that in the case of commitment: Whatever the agent obtainswhen she does not participate can be replicated by making her participate. However, Strausz (2003) also stresses the importance of allowing for randomized allocations for the standardrevelation principle to hold. One can then apply the integral transform theorem to make the distribution U [ ] . Contrast this with the case in which the principal has commitment, where we write a mechanismas a menu of options, one for each type of the agent. We do this even if the optimal mechanismoffers the same allocation to a set of agent types. When the principal has commitment, it is irrelevantwhether the allocation reveals more information beyond the set of types that receive that allocation,since additional information can always be ignored. Under limited commitment, however, this is notthe case and the principal in general trades off tailoring the allocation to the agent’s type and theinformation that is learned through this. If an agent’s type has zero probability at a specific public history, this means that she can onlyreach it through a deviation from σ A . This change to the mechanism actually makes the deviationless attractive, and hence this “disincentivizes" the agent from deviating in the first place. The public randomization device allows us to replicate the dis-tribution over continuations the agent faces in the PBE of the mechanism-selectiongame for those types that found it optimal to randomize between participating andnot participating in the PBE of the mechanism-selection game.The arguments so far describe why the mechanism-selection game implements nomore PBE outcomes than the canonical game. While the canonical game has fewerdeviations for the principal, it follows from our proof that the canonical game can-not sustain more PBE outcomes than the mechanism-selection game. The reasonfor this is that our construction applies to each history in the mechanism-selectiongame, not only those that are on the path of the equilibrium assessment under con-sideration. This essentially shows that whatever deviation the principal could en-tertain, he can also achieve it employing canonical mechanisms. Thus, the smallerset of deviations in the canonical game fails to provide the principal with “more com-mitment."4 E
XAMPLE : S
ALE OF A DURABLE GOOD
We now apply Theorem 1 to study the sale of a durable good in two periods with acontinuum of types. The advantage of this setting is that we know what the seller’soptimal mechanism is within the set of mechanisms considered previously in the lit-erature on limited commitment: Skreta (2006) shows that posted prices are optimal.The main result of this section, Proposition 1 shows that once we allow the sellerto select from a richer set of mechanisms, posted prices may no longer be optimal.In particular, we show that rationing may dominate posted prices, allowing us todraw connections with the literature in industrial organization that studies alterna-tive strategies to posted prices for durable good monopolists (e.g., Denicolo and Garella Starting from an equilibrium in which the mechanism is rejected with positive probability, thisbelief is also determined via Bayes’ rule This stands in contrast to the literature on the informed principal and on competing principals,where oftentimes revelation principle-style arguments apply on the path of play, but not off the path.For an eloquent discussion of this, see Peters (2001). θ ∈ Θ = [ θ , θ ] .Let F denote the seller’s prior belief over Θ . We assume that F has full support andis such that the virtual values , ϕ ( θ , F ) ≡ θ − ( − F ( θ ))/ f ( θ ) are nondecreasing. Anallocation is a pair ( q , x ) ∈ {
0, 1 } × R ≡ A , where q indicates whether the good is sold( q =
1) or not ( q = x is a payment from the buyer to the seller. If the good issold in period 1, the game ends. Moreover, if the buyer rejects the mechanism, thegood is not sold and no payments are made, that is, a ∗ = (
0, 0 ) . Payoffs are as follows.If in period t ∈ {
1, 2 } , the allocation is ( q , x ) , the flow payoffs are u ( q , x , θ ) = θ q − x and w ( q , x , θ ) = x . The buyer and the seller share a common discount factor δ ∈ (
0, 1 ) and maximize the expected discounted sum of payoffs.We proceed as follows. First, we show that we can characterize the seller-optimalPBE as the solution to a constrained optimization problem that only involves theseller (Equation 1). This is already in stark contrast to the existing work in mecha-nism design with limited commitment, which needs to keep track of how the buyer’sbest response to the seller’s mechanism determines the information that the sellerobtains from the interaction, which in turn affects the seller’s incentives to offer themechanism in the first place. Second, Proposition 1 characterizes the solution tothat program under the restriction that the mechanism induces at most one poste-rior, F D , such that the seller does not sell the good when the posterior is F D ( delay ).It only remains to verify that it is indeed optimal for the seller to choose such a distri-bution over posteriors. While we do not pursue this here, we conjecture based on ourprevious work, Doval and Skreta (2020a), that this is indeed the optimal informationstructure for the seller.To arrive at the program that characterizes the seller’s maximum revenue, we appealto Theorem 1. First, in what follows, we restrict attention to the canonical game.Second, it is without loss of generality to consider assessments where the buyer’s We use the standard cdf notation, F , instead of µ to denote the principal’s prior belief sinceunlike in Theorem 1, we are now assuming that Θ is a subset of R . Formally, A (( x )) = { } × R and A (( x )) = { } × R . θ . Let F denote the seller’s belief in period 2. Since the sellerhas commitment in period 2, Proposition 2 in Skreta (2006) implies that the optimalmechanism in period 2 is a posted price regardless of the properties of F . We denoteby ˆ θ ( F ) a solution to his maximization problem. Third, it is without loss of generality to consider assessments where (i) the buyer’sbest response to the seller’s optimal choice of mechanism in period 1 is to participateand truthfully report her type with probability 1, and (ii) when the output messageis F , the seller updates his belief to F . Moreover, the assumption of quasilinearityimplies that, without loss of generality, the seller does not randomize on the trans-fers: Below x ( F ) denotes the expected payment conditional on F and q ( F ) ∈ [
0, 1 ] denotes the probability with which the good is sold. Thus, we can write the seller’sproblem in period 1 as follows: max ( q , x , β ) ∫ Θ ∫ ∆ ( Θ ) ( x ( F ) + ( − q ( F )) δ ˆ θ ( F ) [ θ ≥ ˆ θ ( F )]) β ( d F ∣ θ ) F ( d θ ) (1)s.t. ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ( ∀ θ ∈ Θ ) U ( θ ) = ∫ ∆ ( Θ ) ( θ q ( F ) − x ( F ) + ( − q ( F )) δ u ∗ ( θ , F )) β ( d F ∣ θ ) ≥ ( ∀ θ , ˜ θ ∈ Θ ) U ( θ ) ≥ ∫ ∆ ( Θ ) ( θ q ( F ) − x ( F ) + ( − q ( F )) δ u ∗ ( θ , F )) β ( d F ∣ ˜ θ )( ∀ ˜ Θ × ˜ U ∈ B Θ ⊗ ∆ ( Θ ) ) ∫ Θ ∫ ˜ U F ( ˜ Θ ) β ( d F ∣ θ ) F ( d θ ) = ∫ ˜ Θ β ( ˜ U ∣ θ ) F ( d θ ) . That the seller’s belief about the buyer’s type updates to F when the output messageis F appears twice in the above expression: first, in the third constraint, which isthe Bayes plausibility constraint and, second, in the objective function, where theseller’s payoff in period 2 when the agent’s type is θ and his belief is F correspondsto whether θ buys the good at a price of ˆ θ ( F ) .The two remaining constraints are the buyer’s participation and incentive compati-bility constraints. The buyer’s payoff in the mechanism, U ( θ ) , is determined as fol-lows. For each F in the support of β ( ⋅ ∣ θ ) , she receives the good with probability q ( F ) and makes a payment of x ( F ) ; with the remaining probability, there is notrade, and she obtains a continuation payoff, u ∗ ( θ , F ) , which describes her optimaldecision of whether to buy the good at ˆ θ ( F ) . The participation constraint statesthat the buyer has to earn a payoff of at least 0 by participating. Indeed, since non-participation is a 0 probability event, we can specify that upon rejection of the mech-anism the seller believes that the buyer’s valuation is θ , so that in period 2 the sellerchooses a price of θ when the buyer chooses not to participate. The incentive com- It may be that the seller is indifferent among several prices. We determine the tie-breaking rule asa solution to the problem in period 1. θ the buyer cannot obtain a higherpayoff by reporting that her type is ˜ θ ≠ θ . When the buyer reports ˜ θ , she obtainsa different distribution over output messages β ( ⋅ ∣ ˜ θ ) ; however, in period 2, she stillchooses optimally whether to buy the good, which explains the term u ∗ ( θ , F ) .The three constraints in Equation 1 provide us with a tractable representation of boththe buyer’s behavior and its impact on the mechanism offered in period 2 via theinformation that is generated about the buyer’s type in period 1. This allows us tocharacterize the seller-optimal PBE by focusing only on the period 1’s seller choiceof mechanism, knowing that as long as the mechanism satisfies the constraints weare able to find a buyer’s strategy in the game to fully specify the PBE assessment thatimplements the seller’s maximum revenue. As we show in Appendix D, the incentive constraints deliver the envelope represen-tation of the buyer’s payoffs, so we can replace the transfers out of the seller’s payoffand reduce Equation 1 to the following program. The seller in period 1 chooses adistribution over posteriors, P ∆ ( Θ ) , and, for each posterior he induces, a probabilityof trade q ( F ) , to solvemax P ∆ ( Θ ) , q ∫ ∆ ( Θ ) [ q ( F ) ∫ θθ ϕ ( θ , F ) F ( d θ ) + ( − q ( F )) δ ∫ θ ˆ θ ( F ) ϕ ( θ , F ) F ( d θ )] P ∆ ( Θ ) ( d F ) ,(2)subject to (i) P ∆ ( Θ ) must be Bayes’ plausible given F and (ii) a monotonicity con-dition, which states that, in expectation, higher types must trade with higher prob-ability (see Equation D.4 in Appendix D). Equation 2 describes the seller’s payoff interms of the distribution over posteriors induced by the mechanism. If at posterior F the seller sells the good ( q ( F ) = F , but the virtual values are calculated using F .This reflects that the probability with which the seller pays rents to a buyer of type θ is measured by the probability F ( θ ) that buyer types below θ receive the good. In-stead, if at F the seller does not sell the good ( q ( F ) =
0) he obtains the (discounted)expected virtual surplus of selling the good at price ˆ θ ( F ) . While the posted pricein period 2 is optimal with respect to the posterior virtual values ϕ ( θ , F ) , it may notbe for the prior virtual values. This reflects the conflict between the period 1 and pe-riod 2 sellers: if they hold different beliefs about the buyer’s type, they pay rents withdifferent probabilities, and therefore may prefer different mechanisms. The same idea applies in our application to the infinite horizon analysis of the sale of the durablegood in Doval and Skreta (2020a). q ( F ) = F D . While we do not show that this is optimal, the analysis ofthis simpler class of mechanisms already expands on what is known about optimalmechanisms for this particular setting. In a slight abuse of notation, let β D ( θ ) de-note the probability that a buyer of type θ is delayed, i.e., β D ( θ ) = β ( F D ∣ θ ) . Then,the seller’s problem reduces to max β D ∫ θθ ϕ ( θ , F )( − β D ( θ )) F ( d θ ) + δ ∫ θ ˆ θ ( F D ) ϕ ( θ , F ) β D ( θ ) F ( d θ ) , (OPT β D )subject to the constraint that β D is nonincreasing. Proposition 1 shows that thesolutions to OPT β D are indexed by three parameters ˜ θ , ˜ θ , γ , with γ ≤ β D ( θ ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ θ ≥ ˜ θ γ if θ ∈ ( ˜ θ , ˜ θ ) γ >
0, the seller observes whether the good is sold in period 1, but notwhether the buyer’s type is above or below ˜ θ . Equation 3 encompasses three sellingstrategies for durable good monopolists considered in the literature. First, when γ = θ = ˜ θ , this corresponds to a posted price mechanism, with types above ˜ θ obtain-ing the good in period 1. Under a posted price mechanism we necessarily obtain adecreasing sequence of prices: Conditional on the good not being sold in period 1,the seller learns that the buyer’s type is below ˜ θ , so the period 2 demand, 1 − F D ( θ ) ,is lower than the period 1 demand, 1 − F ( θ ) . Second, when γ > θ = θ , thiscorresponds to what Denicolo and Garella (1999) denote as proportional rationing .Under proportional rationing, buyer types above ˜ θ obtain the good with probabilityless than 1 in period 1. This allows the seller to induce a stronger demand in period2 and hence avoid decreasing the period 2 price, at the cost of selling the good lessoften to high-valuation buyers in period 1. For discount factors close to 1, this costis small compared to the benefit of inducing higher prices in period 2. Finally, when γ > θ < θ , the mechanisms in Equation 3 correspond to what Nocke and Peitz(2007) denote as clearance sales : the seller sets two prices in period 1, one at which Because the objective function in Equation 2 is linear in F when q ( F ) =
1, it is without lossto group all the terms where q ( F ) = q ( F ) = Under the restriction that there is at most one belief at which the seller sets q ( F ) = β D nonin-creasing is equivalent to the monotonicity condition (see Equation D.4). θ < θ implies that the period 2 demand is lower than the period 1 demand. The proofof Proposition 1 suggests, however, why such a mechanism may be optimal: It allowsthe period 1 seller to satisfy the period 2 sequential rationality constraint while si-multaneously maximizing the probability with which high buyer types are served inperiod 1. Proposition 1.
Any solution to OPT β D is as in Equation 3. Moreover,1. If ˆ θ ( F D ) < ˜ θ , it is optimal to set γ = .2. Otherwise, the seller chooses ˜ θ , ˜ θ , γ subject to the constraint that ˜ θ ≤ ˆ θ ( F D ) . Proposition 1 shows that we can reduce the search for the solution to OPT β D to thecomparison between (i) the revenue-maximizing posted price mechanism, and (ii)the revenue-maximizing “rationing” mechanism such that the period 2 price ex-cludes at least as many buyer types as the period 1 mechanism.Proposition 1 clarifies and provides a foundation for the analysis in Denicolo and Garella(1999). The authors consider a two-period model of limited commitment wherein each period the seller can choose both a price at which to sell the good, anda probability, γ ( θ ) , with which a buyer of type θ , who wants to buy the good atthe posted price, receives the good. Implicit in their analysis is that the seller onlyobserves whether the good is sold but not whether the buyer is willing to buy thegood at the posted price. They focus on the case of proportional rationing and pro-vide conditions under which it dominates posted prices. While thought-provoking,their analysis neither shows how the seller can actually implement this alternativemechanism, nor that this is the optimal mechanism when the seller only observeswhether the good is sold. Our analysis allows us to interpret the seller’s strategy setin Denicolo and Garella (1999) as a special case of a mechanism that induces at mostone posterior at which the seller sets q ( F ) =
0. While Proposition 1 shows that pro-portional rationing is among the candidate mechanisms, it highlights that clearancesales may also be optimal. Nocke and Peitz (2007) show that clearance sales may Indeed, in footnote 3 Denicolo and Garella (1999) write the following:“One issue we do not analyze is the determination of the optimal rationing function γ ( v ) . To implement such a function, the monopolist would have to know the cus-tomer’s type-which presumably would allow him to engage in perfect static price dis-crimination. Alternatively, a self-selection constraint should be imposed if rationing isnot proportional." β D , let ˆ θ denote the highestbuyer type such that β D ( θ ) =
1. The first step shows that any incentive-compatible β D that induces a period 2 price below ˆ θ is dominated by a posted price mecha-nism. That is, if rationing is not conducive to more exclusion in period 2 than inperiod 1, the seller prefers to post a price in period 1 to reap the benefits of trad-ing with the high-valuation buyers. Thus, for a non-posted price mechanism to beoptimal it must be that ˆ θ ≤ ˆ θ ( F D ) .The second step is to show that for any incentive-compatible policy β D such thatˆ θ ( F D ) ≥ ˆ θ there exists an alternative incentive-compatible β ′ D of the form describedin Equation 3 which induces the same price in period 2, but leads to a higher revenuein period 1. This is where the assumption that virtual values, ϕ ( θ , F ) , are mono-tone matters the most. Whenever β D is not as in Equation 3, we show that the sellercan change the probability with which the buyer’s different types are delayed to in-crease the probability with which he sells to high types and reduce the probabilitywith which he sells to low types in period 1, while also implementing the same pe-riod 2 price. Thus, the candidate mechanisms in Equation 3 satisfy the period 2 se-quential rationality constraint at the lowest cost for the period 1 seller by allowingthe period 1 seller to maximize the probability with which he trades with high buyertypes.Figure 3 illustrates the seller’s revenue from the best posted price ( γ =
0) and the opti-mal mechanism as a function of the discount factor in the case in which F is U [
0, 1 ] .As anticipated above, proportional rationing and clearance sales can only be optimalwhen the discount factor is close to 1. Under this parametrization, proportional ra-tioning is never optimal. Note that when δ = δ =
1, posted prices achieve thecommitment payoff, which is 0.25 when F is U [
0, 1 ] .We conclude Section 4 by discussing the connection with previous analysis of this Denicolo and Garella (1999) make a similar observation in Proposition 1 of their paper. However,as we explain in Appendix D, they do not account for the incentive costs of implementing β D whenthey argue that a posted price dominates. Indeed, a posted price mechanism leaves more rents tothe buyer than other nonincreasing policies β D . Our proof shows that despite this, a posted pricedominates when ˆ θ ( F D ) < ˆ θ . Revenue 0 0.5 10.220.25 δ Revenue0.8 0.9 10.220.25Figure 3: Seller’s optimal revenue when F is U [
0, 1]: posted prices (solid), optimalmechanism (dashed), proportional rationing (right panel, dotted). The right panelreproduces the left for δ ∈ ( ) problem in the literature, particularly its difference with the main result in Skreta(2006). A reader interested in the case in which the agent’s information evolves overtime and the application to nonlinear pricing can proceed to Sections 5 and 6 with-out loss of continuity. Remark 2. [Comparison with Skreta (2006)] To understand the difference between theresult in Proposition 1 and that in Skreta (2006), it is instructive to compare the incen-tive constraints in Equation 1 to those implied by mechanisms where the seller ob-serves the buyer’s choice of input message as in, for instance, Hart and Tirole (1988);Laffont and Tirole (1988); Bester and Strausz (2001); Skreta (2006). While not expressedin the language of type reports or beliefs, the incentive constraints in Skreta (2006) re-quire that for each F in the support of β ( ⋅ ∣ θ ) , the buyer prefers the tuple ( q ( F ) , x ( F ) , u ∗ ( θ , F )) to any other tuple ( q ( F ′ ) , x ( F ′ ) , u ∗ ( θ , F ′ )) in the mechanism. In particular, the buyermust be indifferent between any two tuples that she chooses with positive probability.Contrast this with the incentive constraints in Equation 1, where the buyer does notnecessarily have to be indifferent between the tuples ( q ( F ) , x ( F ) , u ∗ ( θ , F )) in thesupport of β ( ⋅ ∣ θ ) , although in expectation, the lottery she faces over such tuples undertruthtelling must be better than the one she faces by lying. Indeed, when posted pricesfail to be optimal, the seller exploits the weaker incentive constraints in Equation 1:Buyer types in ( ˜ θ , ˜ θ ) are not indifferent between receiving the good in period at therationing price and receiving the good in period at price ˆ θ ( F D ) .Note, however, that the property that the seller attains a higher payoff when he choosescanonical mechanisms than that he obtains in the two-period version of the model n Skreta (2006) is an artifact of the two-period model. Indeed, for any belief thatthe seller may have in period , his payoff is the same in both models conditional onthe good not being sold in period . Proceeding via backward induction, the seller inour model chooses from a larger set of mechanisms in period , while facing the exactsame continuation values. For longer horizons, however, the comparison of the seller’spayoffs in the two models is not obvious since the larger set of canonical mechanismsalso implies that the seller has a larger set of deviations in our model than in the modelin Skreta (2006). The previous discussion highlights that under limited commitment the principalmay benefit from employing mechanisms where the output message (and hence,the allocation) does not reveal the input message that the agent submitted into themechanism. This is in contrast with the standard revelation principle for the caseof commitment when the principal faces a privately informed agent (adverse se-lection): As we explained in the introduction, it follows from the result in Myerson(1982) that it is without loss of generality in that case to consider mechanisms wherethe principal learns the input message from observing the realization of the outputmessage. Instead, Myerson (1982) shows that adding “noise" to the communicationmay be essential when the principal also faces an agent whose actions are not con-tractible (moral hazard). Indeed, it may be beneficial to pool in the same output mes-sage different types of the privately informed agent to incentivize the agent whoseaction is not contractible to follow the recommendation. Mechanism design withlimited commitment is closer to the hybrid model of adverse selection and moralhazard in Myerson (1982) than it is to the model of pure adverse selection. Indeed,note that in a given period the principal faces, in a sense, two agents whose incen-tives he needs to manage: the privately informed agent (adverse selection) and hisfuture self, whose choice of mechanism is not contractible (moral hazard). That is,today’s principal needs to elicit the agent’s information while simultaneously ensur-ing that his future behavior is sequentially rational. In the same way that outputmessages are key in the presence of moral hazard in Myerson (1982), they featureprominently in our framework.Nevertheless, employing mechanisms where the output message does not reveal theinput message the agent submitted into the mechanism might come at a cost: Aswe have argued, the allocation has to be measurable with respect to the informationthat the mechanism generates about the agent. Thus, the principal trades off theshort-term gains of tailoring the allocation to the agent’s type against the long-termcosts in terms of his future sequential rationality constraints of releasing too much26nformation about the agent (see the application in Section 6). Contrast this witha model where, while the principal is allowed to choose a mechanism in each pe-riod, he does not observe the outcome of the mechanism, but only observes whetherthe relationship with the agent is still ongoing. In this case, within each period theprincipal could perfectly tailor the allocation to the agent’s type without having tolearn this information. Thus, the principal can potentially implement the commit-ment solution, unless the event that the relationship with the agent is still ongoingreveals information about the agent’s type. An example of such a model is that inCorreia-da Silva (2020), who analyzes a version of the model in this section underthe assumption that the seller only observes whether the good is unsold in period 2and shows that in that case the mechanisms in Proposition 1 are optimal. The sameway that our two-period model provides an upper-bound on the seller’s payoff inthe model of Skreta (2006), the model of Correia-da Silva (2020) provides an upperbound on the seller’s payoff in our two-period model.5 M ARKOV ENVIRONMENTS
The case in which the agent’s private information is fully persistent is the cornerstoneof the literature on mechanism design with limited commitment for good reason.There is, in a sense, a fixed amount of information to be learned about the agent andthe principal needs to trade off the short-term gains and the future losses from theuse of this information. In contrast, when the agent’s type is less than fully persistent,nature renews the principal’s uncertainty about the agent’s type. Thus, as observedin Battaglini (2005), dynamic mechanisms are more often time-consistent when theagent’s type evolves over time.Nevertheless, the case in which the agent’s information evolves over time is relevantfor applications as evidenced in, amongst other contributions, the recent public fi-nance applications of Farhi and Werning (2013); Kapiˇcka (2013); Stantcheva (2015)and the burgeoning literature in dynamic mechanism design (see Pavan et al. (2014)for references). Thus, we show in this section that Theorem 1 extends to the case inwhich the agent’s private information evolves over time. We do so for a version ofwhat Pavan et al. (2014) denote as Markov environments, which we define next. The environment is
Markov if the following holds. First, the agent’s private informa-tion is described by a nonhomogenous Markov process with states in Θ and tran- This would be the case, for instance, in our example in Section 6. We comment at the end of this section on how our results extend outside Markov environments. F t ∶ Θ × A ↦ ∆ ( Θ ) so that F t ( ˜ Θ ∣ θ t − , a t − ) describes the probability that theagent’s type in period t is in ˜ Θ when her type in t − θ t − and the allocation is a t − . Second, the principal and the agent’s payoffs are time separable and theirperiod- t flow payoff only depends on the current allocation and the agent’s period t type. That is, if ( a t , θ t ) ≤ t ≤ T describes the allocations and agent’s private informationthrough period T , then W (( a t , θ t ) ≤ t ≤ T ) = T ∑ t = δ t w t ( a t , θ t ) , U (( a t , θ t ) ≤ t ≤ T ) = T ∑ t = δ t u t ( a t , θ t ) ,denote the payoffs to the principal and the agent, respectively. Everything else is asin the model in Section 2. However, to keep track of the agent’s time-varying privateinformation, we index the agent’s private history differently. Namely, the agent’s pri-vate history through period t , h tA corresponds to a sequence ( θ , M , p , m , s , a ,. . . , θ t − , M t − , p t − , m t − , s t − , a t − ) . Since the agent knows her type at the beginningof period t , we index her information sets by ( h tA , θ t ) . To simplify notation, we writethe agent’s behavioral strategy σ At ( h tA , θ t , M t ) = ( π t ( h tA , θ t , M t ) , r t ( h tA , θ t , M t )) ∈ [
0, 1 ] × ∆ ( M M t ) .In Markov environments, it is important to keep track of two beliefs for the princi-pal. The first is the belief he holds about the agent’s type at the end of period t ; thesecond is the belief he holds at the beginning of period t +
1, after applying F t + . Wedenote the former by µ t + , anticipating that, as in Theorem 1, this is the belief thatwill be used as an output message. We denote the latter by ν t + : This is the beliefthat is used to determine which types have positive probability in period t +
1. Thatis, µ t + ( h tA , θ t , m , z ∣ h t , z ) is the probability that the principal assigns to the agent be-ing at information set ( h tA , θ t ) and sending message m at the end of period t when z is the outcome of the interaction in period t , while ν t + ( h tA , θ t , m , z , θ t + ∣ h t , z ) = µ t + ( h tA , θ t , m , z ∣ h t , z ) F t + ( θ t + ∣ θ t , a t ) is the probability that the principal assigns tothe agent being at information set ( h tA , θ t ) in period t and her type being θ t + inperiod t + t is z , where a t is theallocation in period t consistent with z .To introduce Theorem 2, we need one last piece of notation. Let E ∗M and E ∗ denotethe set of equilibrium payoffs of the mechanism-selection game and the canonicalgame, respectively. Note that assuming that the set of states is time-invariant is without loss of generality. See Ely (2017) for another model where the same choice is made. heorem 2. The mechanism-selection game and the canonical game have the sameset of equilibrium payoffs, i.e., E ∗M = E ∗ for any collection M with which we endow theprincipal.Moreover, for any equilibrium payoff in E ∗M , there exists a PBE assessment ( σ P , σ A , µ ) of G that achieves the same payoff and satisfies the following properties:1. The agent’s strategy depends on her current payoff-relevant type and the public his-tory. That is, for all periods t , all public histories h t , all h tA , h tA ∈ H tA ( h t ) , and alltypes θ t ∈ Θ , σ At ( h tA , θ t , M t ) = σ At ( h tA , θ t , M t )
2. For all public histories h t , if θ t is in the support of ν t ( ⋅ ∣ h t ) , then the agent partici-pates in the mechanism offered by the principal at that history and with probabilityone truthfully reports her current type,3. For all public histories h t , if the mechanism offered by the principal at h t outputsa posterior µ ′ , the principal’s updated equilibrium beliefs about the agent coincidewith µ ′ . That is, for all measurable subsets ˜ Θ , ˜ U , ˜ A of Θ , ∆ ( Θ ) , and A, ∫ Θ ∫ ˜ U ∫ ˜ A µ t + ( ˜ Θ ∣ h t , z ( µ ′ , a t ) ) α M t ( d a t ∣ µ ′ ) β M t ( d µ ′ ∣ θ ) ν t ( d θ ∣ h t ) == ∫ ˜ Θ ∫ ˜ U α M t ( ˜ A ∣ µ ′ ) β M t ( d µ ′ ∣ θ ) ν t ( d θ ∣ h t ) = ∫ Θ ∫ ˜ U ∫ ˜ A µ ′ ( ˜ Θ ) α M t ( d a t ∣ µ ′ ) β M t ( d µ ′ ∣ θ ) ν t ( d θ ∣ h t ) . Two remarks are in order. First, the history independence in Theorem 2 is, in a sense,stronger than that of Theorem 1. The agent does not condition her strategy on eitherher past communication or her past payoff-relevant types. This is where we moreprominently employ the restriction to Markov environments. This affords an impor-tant simplification: In each period t , the principal only needs to elicit the agent’scurrent payoff-relevant type θ t , not the past realizations. We believe this simplifi-cation is important for applications. In more general environments, a similar resultwould obtain, but the principal may need to elicit the whole realization ( θ , . . . , θ t ) .Note that the stronger form of history independence also implies that contrary toTheorem 1 the mechanism-selection and canonical games implement the same setof payoffs but not necessarily the same equilibrium outcomes. Second, the sepa-ration between the beliefs that the output message represents ( µ t + ) and the beliefsthat the principal uses in the next period to select mechanisms ( ν t + ) highlights thatthe principal in period t attempts to design his prior for period t + The revelation principle-style arguments in Peters (2001); Hart et al. (2017); Ben-Porath et al.(2019) are also in terms of payoff equivalence. F t + .)The proof of Theorem 2 is in Appendix E. Except for the proof of the stronger formof history independence, the remaining steps closely follow the proof of Theorem 1,and thus, we omit them. Moreover, since the extension to the case in which the set Θ is uncountable or the support of β M t is uncountable should be immediate fromthe proof of Theorem 1, we only present the proof under the assumption that Θ isfinite and the principal can only choose mechanisms, M , such that for all m ∈ M M ,the support of β M ( ⋅ ∣ m ) is finite.6 E XAMPLE : P
RODUCT LINE DESIGN UNDER LIMITED COMMITMENT
We apply Theorem 2 to characterize the revenue-maximizing mechanism for a sellerwho interacts with a buyer over two periods and two different transactions. We as-sume that the buyer’s period 1 type is informative of her period 2 valuation. Whilestylized, this example is prototypical of many situations where a buyer interacts witha seller across different purchases, as in online shopping, or the purchase of a basicgood, followed by an add-on. This example highlights the trade-off that arises underlimited commitment between the benefits of the use of information in period 1 tobetter cater to the buyer’s period 1 valuation and the costs of doing so via its impacton the mechanism offered in period 2. As we show in Proposition 2 below, this leadsto allocative distortions above and beyond those due to informational rents, whichinclude both under- and overprovision in period 1. This rich pattern of distortionscomes together with a rich information disclosure policy, which features (multiple)pooling and separation intervals. Contrary to the example in Section 4, however, theoptimal mechanism can be implemented with the seller observing the buyer choos-ing from a set of menus.Formally, we consider the following special case of the model in Section 5. A seller(the principal) and a buyer (the agent) interact over two periods. In period 1, theseller produces a good of variable quality, so that allocations are described by ( q , x ) ∈ [ Q ] × R , where q is the good’s quality and x is a payment from the buyer to theseller. The seller incurs cost c ( q ) = c q / q . In period 2, theseller produces an indivisible good at 0 marginal cost, so that allocations are de-scribed by ( q , x ) ∈ {
0, 1 } × R . The seller’s objective is to maximize his revenue. Thebuyer’s valuation for each of the goods is her private information. In period 1, if shepurchases a good of quality q and pays x , her flow payoff is u ( q , x , θ ) = θ q − x ,where θ ∼ U [
0, 1 ] . In period 2, if she purchases the good and pays x , her flow payoff30s v − x , where v ∈ { v L , v H } , 0 < v L < v H . In period 1, the buyer does not know hervaluation of the good in period 2. Conditional on the buyer’s type in period 1 being θ , her valuation in period 2 is v H with probability p ( θ ) = θ . Both the buyer and theseller are fully patient.Theorem 2 implies that from the perspective of the seller in period 1 what matters isthe belief that he holds about θ in period 2. Denote this belief by F . As in Section 4,the optimal mechanism in period 2 is a posted price regardless of F . It equals v L ,whenever E F θ < v L / v H , and v H whenever E F θ > v L / v H . We specify the price when E F θ = v L / v H as part of the solution to the seller’s problem in period 1. In what fol-lows, we denote the ratio v L / v H by µ .Let q ( v L , F ) = [ E F θ < µ ] denote the probability that the seller trades with the lowvaluation buyer in period 2 when his belief is F . Then, the agent’s payoff in period 1when her type is θ and she truthfully reports θ is given by: U ( θ ) = ∫ ∆ ( Θ ) [ θ q ( F ) − x ( F ) + θ ∆ v q ( v L , F )] β ( d F ∣ θ ) . (4)To see this, note that in period 2, the buyer makes a positive payoff only when hervaluation is v H and the seller sells the good at a price of v L , in which case she earns v H − v L = ∆ v . Theorem 2 implies that the seller can focus without loss of generalityon mechanisms where the buyer truthfully reports her type so that U ( θ ) ≥ U ( ˜ θ , θ ) = ∫ ∆ ( Θ ) [ θ q ( F ) − x ( F ) + θ ∆ v q ( v L , F )] β ( d F ∣ ˜ θ ) , (5)must hold for all θ , ˜ θ ∈ Θ . After misrepresenting her information in period 1 by re-porting ˜ θ , it is optimal for the buyer to truthfully report her valuation in period 2,which explains the right-hand side of the above equation.The seller’s profits in period 1 are given by: ∫ Θ ∫ ∆ ( Θ ) [ x ( F ) − c ( q ( F )) + q ( v L , F ) v L + ( − q ( v L , F )) p ( θ ) v H ] β ( d F ∣ θ ) F ( d θ ) .(6) Theorem 2 implies that the revenue-maximizing mechanism can be obtained by max-imizing the objective in Equation 6 subject to the incentive compatibility constraintsin Equation 5, the participation constraints that state that U ( θ ) ≥ θ ∈ Θ , andthe Bayes plausibility constraint. The right-hand side of the participation constraint is justified as follows. Since rejection of the
31s in Section 4, we can obtain an envelope representation of the buyer’s utility U ( θ ) which we can use to replace the transfers out of the seller’s payoff in Equation 6. This,in turn, allows us to reduce the seller’s mechanism design problem in period 1 to theproblem of choosing a distribution over posteriors, P ∆ ( Θ ) , and for each posterior aquality level, q ( F ) , to maximize: ∫ ∆ ( Θ ) ∫ Θ ⎡ ⎢⎢⎢⎢⎣ ϕ ( θ , F ) q ( F ) − c ( q ( F )) + ( − q ( v L , F )) p ( θ ) v H + q ( v L , F ) ( p ( θ ) v H + ( − p ( θ )) ( v L − p ′ ( θ ) − p ( θ ) − F ( θ ) f ( θ ) ∆ v )) ⎤⎥⎥⎥⎥⎦ F ( d θ ) P ∆ ( Θ ) ( d F ) ,(7) subject to the constraints that (i) P ∆ ( Θ ) is Bayes plausible given F and (ii) an av-erage monotonicity constraint (see Appendix F). As in Section 4, ϕ ( θ , F ) = θ − ( − F ( θ ))/ f ( θ ) denotes the buyer’s period 1 virtual value.It is easier to develop intuition about the virtual surplus without replacing the expres-sions for p ( θ ) , and F ( θ ) . The first two terms correspond to the static virtual surplusin the model of Mussa and Rosen (1978) but are expressed using canonical mecha-nisms. The second part of the virtual surplus corresponds to an adjusted version ofthe surplus in period 2. In period 2, with probability 1 − q ( v L , F ) , the seller tradesonly with the high-valuation buyer, in which case there is nonzero surplus only whenthe buyer’s valuation is v H , which occurs with probability p ( θ ) . With the remainingprobability, he trades with the buyer regardless of her valuation, in which case sur-plus is p ( θ ) v H + ( − p ( θ )) v L . However, because whenever the seller trades with thelow-valuation buyer, the high-valuation buyer makes rents, v L is modified to reflectthis. Like in Section 4, the adjustment by rents is accounted for using the prior. Unlike in Section 4, the adjustment reflects the dynamic nature of the buyer’s infor-mation. From the perspective of period 1, the seller only leaves rents for the second-period transaction because of the impact that the buyer’s current type has on her fu-ture valuation. This is why the inverse hazard rate is multiplied by p ′ ( θ )/( − p ( θ )) .Instead of fully solving the maximization problem implied by Equation 7, we focusfor the rest of this section on the relaxed problem , where we drop the monotonic-ity constraint (ii). In addition to describing the solution to the relaxed problem,Proposition 2 below shows that under a wide range of parameter configurations, thesolution to the relaxed problem satisfies the monotonicity constraint. In the relaxedproblem, the seller chooses q ( F ) = ( E F θ − )/ c when E F θ ≥ / q ( F ) = mechanism is an off-the-path event, the seller assigns probability 1 to the buyer’s type being θ = v H in period 2. From the perspective of the seller in period 2, the virtual value of the low valuation buyer is v L − E F θ − E F θ ∆ v . c ( ⋅ ) , p ( ⋅ ) , F ( ⋅ ) , and the optimal choice of q ( F ) reveals that the seller’s payoff is only a function of E F [ θ ] : ∫ ∆ ( Θ ) ( max { E F θ −
1, 0 }) c P ∆ ( Θ ) ( d F ) + (8) ∫ ∆ ( Θ ) [( − q ( v L , F )) v H E F θ + q ( v L , F ) ( v H E F θ + ( v L − ∆ v )( − E F θ ))] P ∆ ( Θ ) ( d F ) . This allows us to solve the problem by exploiting the techniques in information de-sign for continuum type spaces, which deal exclusively with the case in which thereceiver’s action and the seller’s payoff are a function of the posterior mean (e.g.,Gentzkow and Kamenica (2016); Kolotilin (2018); Dworczak and Martini (2019)).To understand the trade-off introduced by the seller’s limited commitment, it is in-structive to first consider an artificial problem where the seller can separately solvethe problems represented on each line of Equation 8. That is, suppose that the sellercould choose P ∆ ( Θ ) to maximize the first line and P ′ ∆ ( Θ ) to maximize the second line.Figure 4 below illustrates the seller’s payoff as a function of the posterior mean, m , ineach of these problems in the case in which µ > / m Figure (a) Period 1 payoff m µ Figure (b) Period 2 payoff
Figure 4: The seller’s payoff as a function of the posterior mean( ( v H , v L , c ) = (
1, 0.75, 2 ) )Figure 4a plots the integrand in the first line of Equation 8. Figure 4b plots the inte-grand in the second line of Equation 8. In period 2, the seller sets a price of v H when m > µ and v L otherwise. In period 1, the seller prefers that the price in period 2 is v L at m = µ , so we break ties accordingly.When µ > /
2, the seller in period 1 prefers that the price in period 2 is v L regardless of the induced posterior mean. It turns out that this is feasible: Since the prior mean33f θ is less than µ , by not disclosing any information about θ , the seller in period 1can guarantee that the period 2 price is v L with probability 1.In contrast, for the purposes of maximizing his revenue in period 1, the seller prefersan information policy that perfectly reveals the types above 1 /
2. This is intuitive: Theperiod 1 problem coincides with the linear-quadratic version of Mussa and Rosen(1978), which features full separation. It is optimal for the seller to perfectly tailor thequality provided to the buyer’s type.When the seller has commitment, he can obtain his maximum value in both prob-lems: He can implement the Mussa-Rosen solution in period 1, and ignoring theinformation about θ revealed by the allocation, he can set a period 2 price equal to v L . Indeed, this is the commitment solution: If using the information revealed bythe allocation is detrimental to revenue in period 2, the seller can commit to ignoreit.However, when the seller has limited commitment, the allocation has to be measur-able with respect to the information released by the mechanism. Thus, to achieve hismaximum payoff in period 1, the seller must bear the cost of pricing all types above1 / v H . To achieve his maximum payoff in period 2, the seller needs toreveal no information in period 1, which in this example implies that all buyer typesare excluded in period 1.Not surprisingly, the optimal solution turns out to be a compromise between thesetwo forces. Except for the case when µ < /
4, in which the seller can actually obtainthe commitment payoff, the optimal mechanism distorts quality provision in period1 to discipline the revelation of information about θ across periods. This is stated inProposition 2 and illustrated in Figure 5 below. Proposition 2.
The following is the solution to the relaxed problem:1. If µ < / , it is optimal to reveal whether θ is above or below / and to perfectlyreveal the types above / . Buyer types below / are excluded in period , whilebuyer types above / receive quality ( θ − )/ c. The price in period is v H .Otherwise, let m ∗ , m ∗ be such that E F [ θ ∣ θ ∈ [ m ∗ , m ∗ ]] = µ and l ( c ) = ( + c )/ ( + c ) .Then2. If / < µ < / , then m ∗ = and m ∗ > / . It is optimal to reveal whether θ is aboveor below m ∗ and to perfectly reveal the types above m ∗ . Buyer types below m ∗ are The proof is available from the authors upon request. xcluded in period , while buyer types above m ∗ receive quality ( θ − )/ c. Theprice in period is v H .3. If / < µ < l ( c ) , the seller reveals whether θ is below m ∗ or between m ∗ and m ∗ and perfectly reveals all types above m ∗ . Buyer types below m ∗ are excluded inperiod and receive a price of v L in period . Buyer types θ ∈ [ m ∗ , m ∗ ] receivequality ( µ − )/ c and a price of v L in period . Buyer types above m ∗ receive quality ( θ − )/ c and face a period price of v H .4. If l ( c ) < µ < , then the seller perfectly reveals θ ∈ [ / m ∗ ) ∪ ( m ∗ , 1 ] and pools θ ∈ [ m ∗ , m ∗ ] . Buyer types below / are excluded in period , buyer types in [ / m ∗ ) ∪ ( m ∗ , 1 ] receive quality ( θ − )/ c, and buyer types in [ m ∗ , m ∗ ] receive quality ( µ − )/ c. In period , buyer types below m ∗ receive a price of v L , and otherwise receivea price of v H .Moreover, the solution to the relaxed problem satisfies the monotonicity constraints incases 1 and 2, and, if µ ≥ ( + c )/( + c ) , also in case 4. In each of these cases, thereexists a menu of contracts such that the optimal mechanism can be implemented withthe seller observing the buyer’s choice from this menu. θ ( θ − c , v H ) separation ( v H ) pooling Figure (a) µ < / θ m ∗ ( θ − c , v H ) separation ( v H ) pooling Figure (b) 1 / < µ < / θ m ∗ m ∗ ( θ − c , v H ) separation ( µ − c , v L ) pooling ( v L ) Figure (c) 1 / < µ < l ( c ) θ m ∗ m ∗ ( θ − c , v H ) separation ( µ − c , v L ) pooling ( θ − c , v L ) separation ( v L ) Figure (d) l ( c ) < µ < Figure 5: Solution to the relaxed problem. In brackets, we display q , p for eachtype θ , and below the information disclosure policy.Figure 5 illustrates the period 1 quality and period 2 price faced by the buyer as a35unction of her period 1 type. In the figure, the black dashed lines depict the typesfor which the seller distorts the allocation to discipline how much he learns aboutthe buyer’s type. For instance, in Figure 5b, buyer types in ( / m ∗ ) are excluded,even though they would be served in the commitment solution. By pooling themwith types below 1 /
2, the seller is able to keep the period 2 price at v H for those typesas well. In contrast, in Figure 5c, the seller provides a positive level of quality to typesin [ m ∗ , m ∗ ] , even though m ∗ < /
2, so that he would have preferred to exclude typesin [ m ∗ , 1 / ] . However, by pooling them with types in ( / m ∗ ] , he guarantees thatthe latter also face a period 2 price of v L . Finally, note that although the commitmentsolution can be implemented when µ < /
4, limited commitment shapes how infor-mation is transmitted across periods. While in the commitment solution the sellerwould perfectly learn the buyer’s type for θ below 1/2 and then choose to ignore itin period 2, under limited commitment the seller only learns that the buyer received q = v H in period 2.It is interesting to contrast this example with that in Section 4. In Section 4, we con-sidered mechanisms that induce simple information structures and in that case, theseller exploited the shape of the incentive constraints to offer the buyer options amongstwhich she is not indifferent. In contrast, the example in this section features a richinformation policy with a simple implementation. Indeed, the seller can offer thebuyer a menu of qualities and payments in period 1, so that what the seller learnsfrom observing the buyer’s choices from the menu coincides with the informationthat is induced by the optimal mechanism.The above point illustrates yet another advantage of the framework we propose inthis paper. A major challenge in the received literature on limited commitment ishow to keep track of how the agent’s best response to the mechanism affects theinformation that the seller obtains from the interaction, which in turn affects theseller’s incentives to offer the mechanism in the first place. Instead, our frameworkallows us to reduce the agent’s best response to the principal’s mechanism and itsinformational feedback to a set of constraints that the mechanism must satisfy. Thisavoids having to consider complicated mixed strategies on the part of the agent (see,for instance, Laffont and Tirole (1988)) and transforms it instead into an informationdesign problem. Often, as in this example, the optimal information structure in themechanism turns out to be a more parsimonious way of describing the agent’s be-havior in the mechanism. The same property holds in Doval and Skreta (2020a). EFERENCES A BREU , D., D. P
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OLLECTED DEFINITIONS AND NOTATION
Histories and strategies:
We review in Appendix C how to endow the set M with ameasurable structure so that it is a standard Borel space (Aumann (1961)). In whatfollows, we take this as given.Denote by M t the set ∪ i ∈I t M ti ∪ { ∅ } and similarly let S t denote the set ∪ i ∈I t S ti ∪ { ∅ } .Let Z t = M × {
0, 1 } × S t × A and Z A , t = M × {
0, 1 } × M t × S t × A . By construction, Z t and Z A , t are standard Borel spaces.Public histories through period t are H t = × t − τ = Z τ and private histories are Θ × H tA = Θ × × t − τ = Z A , τ . Note that the information sets of the principal can be described bya measurable function ζ Pt ∶ Θ × H tA ↦ H t where ζ Pt is the projection of Θ × H tA onto × t − τ = Z τ . Similarly, the information sets of the agent can be described by a measurablefunction ζ At ∶ Θ × H tA ↦ Θ × H tA where ζ At is simply the identity. Induced distributions and payoffs:
Fix a mechanism M t . The strategy profile to-gether with the mechanism define a transition probability from Θ × H tA × M to M t × S t × A as follows: ρ σ A ( ˜ M × ˜ S × ˜ A ∣ θ , h tA , M t ) = ∫ ˜ M β M t ( ˜ S × ˜ A ∣ m ) r t ( θ , h tA , M t )( d m ) ,for all measurable subsets ˜ M , ˜ S , ˜ A of M t , S t , A .Given σ = ( σ P , σ A ) and ( θ , h tA ) we can define transition probabilities from Θ × H tA to Z A , t as follows: κ σ t ( ˜ M × { } × { ∅ } × { ∅ } × { a ∗ }∣ θ , h tA ) = ∫ { x ∶ σ Pt ( ζ Pt ( θ , h tA ) , x )∈ ˜ M} ( − π t ( θ , h tA , σ P ( ζ Pt ( θ , h tA ) , x ))) λ ( d x ) for any measurable subset ˜ M of M , and κ σ t ( ˜ M × { } × ˜ M × ˜ S × ˜ A ∣ θ , h tA )= ∫ { x ∶ σ Pt ( ζ Pt ( θ , h tA ) , x )∈ ˜ M} ρ σ A ( ˜ M × ˜ S × ˜ A ∣ θ , h tA , σ Pt ( ζ Pt ( θ , h tA ) , x )) π t ( θ , h tA , σ Pt ( ζ Pt ( θ , h tA ) , x )) λ ( d x ) on the measurable rectangles ˜ M × ˜ M × ˜ S × ˜ A ∈ B M⊗ M t ⊗ S t ⊗ A .41et µ denote the initial distribution on Θ . The Ionescu-Tulcea extension theorem(Tulcea (1949)) guarantees the existence of a sequence of probability measures P σ t = µ ⊗ ⊗ t − τ = κ στ defined on the product spaces ( Θ × H tA ) Tt = and a probability measure P σ on ( Θ × H T + A , B Θ ⊗⊗ T τ = B Z A , τ ) so that for any measurable ˜ Θ × ˜ H tA ⊆ Θ × H tA , P σ t ( ˜ Θ × ˜ H tA ) = P σ ( ˜ Θ × ˜ H tA × ∏ T τ = t + Z A , τ ) .Then, the principal’s payoff, W ( P σ ) , is given by ∫ Θ × H T + A W ( proj Θ × A T + ( θ , h T + A )) P σ ( d ( θ , h T + A )) = ∫ Θ × A T + W ( a T + , θ )( P σ ○ proj − Θ × A T + )( d ( θ , a T + )) , while the agent’s payoff when her type is θ , U ( P σ ) , is given by ∫ Θ × H T + A U ( proj Θ × A T + ( θ , h T + A )) P σ ∣ θ ( d ( θ , h T + A )) = ∫ Θ × H T + A U ( a T + , θ )( P σ ∣ θ ○ proj − Θ × A T + )( d ( θ , a T + )) , where P σ ∣ θ is the induced probability over Θ × H T + A determined by drawing θ = θ with probability one and drawing the terminal history using P σ .Fix a measurable strategy σ . Fix a period t , a belief p t ∈ ∆ ( Θ × H tA ( h t )) and mech-anism M t . Define the one-step ahead prediction equations on the measurable rect-angles as follows: f t ( p t , M t )( ˜ Θ × ˜ H tA × z ∅ ( M t )) = ∫ ˜ Θ × ˜ H tA ( − π t ( θ , h tA , M t )) p t ( d ( θ , h tA )) , (A.1) f t ( p t , M t )( ˜ Θ × ˜ H tA × M t × { } × ˜ M × ˜ S × ˜ A ) = ∫ ˜ Θ × ˜ H tA ρ σ A ( ˜ M × ˜ S × ˜ A ∣ θ , h tA , M t ) π t ( θ , h tA , M t ) p t ( d ( θ , h tA )) . The mapping f t ∶ ∆ ( Θ × H tA ) ×M ↦ ∆ ( Θ × H tA × Z A , t ) ≡ ∆ ( Θ × H t + A ) is Borel measurable.Let p t + ∈ ∆ ( Θ × H t + A ) , with marginal p t + H t + over H t + , we can construct ∫ ˜ H t + q ( ˜ Θ × ˜ H t + A ∣ h t + , p ) p t + H t + ( d h t + ) = ∫ ˜ Θ × ˜ H t + A [( θ , h t + A ) ∈ H ′ ] p t + ( d ( θ , h t + A )) ,on the measurable rectangles ˜ Θ × ˜ H t + A ⊂ Θ × H t + A , ˜ H t + ⊂ H t + . Note that as a func-tion of ( h t + , p ) , q is measurable.Fix a public history h t , a mechanism M t , and a belief µ t ( h t ) ∈ ∆ ( Θ × H tA ( h t )) . Definetransition probabilities from H τ × M to Θ × H τ + A for τ ≥ t , recursively as follows: ν t ( h t , M t ) = f t ( µ t ( h t ) , M t ) , ν τ ( h τ , M τ ) = f τ ( q ( h τ , ν τ − ( h τ − , ⋅ )) , M τ ) .Fix a belief p ∈ ∆ ( Θ × H t + A ) . This, together with the kernels ( κ στ ) τ ≥ t + , induces a se-42uence of distributions P σ ∣ p τ = p ⊗⊗ τ n = t + κ σ n on the product spaces ( Θ × H τ A ) τ ≥ t + anda measure P σ ∣ p on ( Θ × H T + A , B Θ × H TA ) , by the Ionescu-Tulcea Theorem. Proposition7.26, Lemma 7.28, and Corollary 7.29.1 in Bertsekas and Shreve (1978) imply that themappings p ↦ P σ ∣ p τ and p ↦ P σ ∣ p are Borel measurable. The following lemma is aconsequence of Lemma 10.4 in Bertsekas and Shreve (1978): Lemma A.1.
For every Borel measurable strategy profile σ and Borel measurable sub-set ˜ Θ × ˜ H τ A ⊆ Θ × H τ A P σ ∣ µ t ( h t ) τ ( ˜ Θ × ˜ H τ A ∣ h τ , M τ ) = ν τ ( h τ , M τ )( ˜ Θ × ˜ H τ A ) P σ ∣ µ t ( h t ) − almost everywhere .Note that ν t ( h t , M t ) = f t ( µ t ( h t ) , M t ) defines a transition probability from H t ×M to Θ × H t + A . Then, P σ ∣ ⋅ ○ ν t defines a transition probability from H t ×M to ∆ ( Θ × H T + A ) .It follows from this that we can define a bounded measurable function from H t × M to R as follows: W ( σ , ν ∣ h t , M t ) = ∫ Θ × H t + A E P σ ∣( θ , ht + A ) [ W ( a t , ⋅ , θ )] ν t ( d ( θ , h t + A )∣ h t , M t ) .Lemma A.1 implies that W ( σ , ν ∣ h t , M t ) represents the principal’s payoff conditionalon information set h t and having chosen M t when his beliefs are given by µ t ( h t ) . U ( σ ∣ θ , h tA , M t ) can be defined analogously.With this we can formally define Perfect Bayesian equilibrium: Definition A.1.
An assessment ( σ P , σ A , µ ) is sequentially rational if for all t and pub-lic histories h t ,1. If M t ∈ supp σ Pt ( h t ) , W ( σ , ν ∣ h t , M t ) ≥ W ( σ , ν ∣ h t , M ′ t ) for all M ′ t ∈ M ,2. For all M t ∈ M , U ( σ ∣ θ , h tA , M t ) ≥ U ( σ P , σ ′ A ∣ θ , h tA , M t ) for all θ ∈ Θ , h tA ∈ H tA ( h t ) , σ ′ A . Definition A.2.
A system of beliefs satisfies Bayes’ rule where possible if for all t , allpublic histories h t , all measurable subsets ˜ Θ , ˜ H tA of Θ , H tA ( h t ) , µ t + ( ˜ Θ × ˜ H tA × z ∅ ( M t )∣ h t , z ∅ ( M t )) ∫ Θ × H tA ( − π t ( θ , h tA , M t )) µ t ( d ( θ , h tA )∣ h t )= ∫ ˜ Θ × ˜ H tA ( − π t ( θ , h tA , M t )) µ t ( d ( θ , h tA )∣ h t ) , nd for all measurable subsets ˜ Θ , ˜ H tA , ˜ M , ˜ S , ˜ A of Θ , H tA ( h t ) , M M t , S M t , A, ∫ Θ × H tA ∫ ˜ S × ˜ A µ t + ( ˜ Θ × ˜ H tA × ˜ M × z ( s t , a t ) ( M t )∣ h t , z ( s t , a t ) ( M t )) ρ σ A S M t × A ( d ( s t , a t )∣ θ , h tA ) π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t )= ∫ ˜ Θ × ˜ H tA ( ∫ ˜ M β M t ( ˜ S × ˜ A ∣ m ) r t ( θ , h tA , M t )( dm )) π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) . Definition A.3.
An assessment ( σ P , σ A , µ ) is a Perfect Bayesian equilibrium if it issequentially rational and satisfies Bayes’ rule where possible.
If the system of beliefs satisfies Bayes’ rule where possible, then letting ν t ( h t , M t ) = f t ( µ t ( ⋅ ∣ h t ) , M t ) and ν t H t + denote the marginal of ν t on H t + , we have that ∫ ˜ H t + µ t + ( ˜ Θ × ˜ H t + A ∣ h t + ) ν t H t + ( d h t + ) = ν t ( h t , M t )( ˜ Θ × ˜ H t + A ) ,on the measurable rectangles ˜ Θ × ˜ H t + A ∈ B Θ ⊗ H t + A and ˜ H t + ∈ B H t + . Working forwardthrough the one-step ahead prediction equations we have that q ( h τ , ν τ − ( h τ − , ⋅ )) = µ τ ( h τ ) for those histories in the support of P σ ∣ µ t ( h t ) . Note the following: W ( σ , ν ∣ h t , M t ) = ∫ Θ × H t + A E P σ ∣( θ , h t + A ) [ W ( a t , ⋅ , θ )] ν t ( d ( θ , h t + A )∣ h t , M t ) (A.2) = ∫ Θ × H tA ⎛⎜⎝ ( − π t ( θ , h tA , M t )) E P σ ∣( θ , htA , z ∅( M t )) [ W ( a t , a ∗ , ⋅ , θ )] + π t ( θ , h tA , M t )∫ M M t × S M t × A E P σ ∣( θ , htA , m , z ( st , at )( M t )) [ W ( a t , a t , ⋅ , θ )] ρ σ A ( d ( m , s t , a t )∣ θ , h tA , M t )) ⎞⎟⎠ µ t ( d ( θ , h tA )∣ h t )= ∫ { h t + ∈ H t + ∶ h t ≺ h t + } ∫ Θ × H t + A E P σ ∣( θ , ht + A ) [ W ( a t , ⋅ , θ )] µ t + ( d ( θ , h t + A )∣ ν t , h t + ) ν t H t + ( d h t + ∣ h t , M t ) , where h t ≺ h t + denotes that public history h t precedes h t + . Disintegration:
The proofs in this appendix frequently make use of the notion of adisintegration. For any two measurable spaces, X and Y , and a Borel measure ν on X × Y , ν X and ν Y denote the marginals of ν on Y and X , respectively. Given a productspace X × Y , proj Y denotes the projection of X × Y onto Y .Given two Polish spaces, X and Y , and a joint measure ν on X × Y , the ( ν X , proj Y ) -disintegration of ν is the collection of measures on B X × Y , { η x ∶ x ∈ X } , where(i) η x is concentrated on X = x , i.e. η x ({ X ≠ x }) = ν X − almost everywhere,and for each non negative measurable function f on B X × Y :(ii) x ↦ ∫ Y f ( x , y ) η x ( d y ) is measurable,(iii) ∫ X × Y f ( x , y ) ν ( d ( x , y )) = ∫ X ∫ Y f ( x , y ) η x ( d y ) ν X ( d x ) .Proposition 3.6 in Crauel (2002) ensures that { η x ∶ x ∈ X } exists and is unique ν X -44lmost everywhere.In what follows, we denote the principal’s beliefs conditional on h t and the agent par-ticipating in mechanism M t by µ + t ( ⋅ ∣ h t ) ∈ ∆ ( Θ × H tA ( h t )) . That is, for any measurablesubset ˜ Θ × ˜ H tA ⊂ Θ × H tA ( h t ) , µ + t ( ˜ Θ × ˜ H tA ∣ h t , M t ) = ∫ ˜ Θ × ˜ H tA π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) ,Note that when µ + t ( Θ × H tA ( h t )∣ h t , M t ) ≠
0, we can actually take it to be a probabilitymeasure by normalizing it appropriately.B P
ROOF OF T HEOREM
Proposition B.1.
Fix a PBE assessment ( σ P , σ A , µ ) of G M and a public history h t , anda mechanism M t ∈ M . Then, there exists a continuation strategy σ ′ A such that:1. For all public histories h τ that succeed h t , σ ′ A τ ( θ , h τ A , M τ ) = σ ′ A τ ( θ , h τ A , M τ ) forall h τ A , h τ A ∈ H τ A ( h τ ) , M τ ∈ M , for all θ ∈ Θ ,2. For all public histories h τ that succeed h t , U ( M τ , σ P , σ ′ A ∣ θ , h τ A ) = U ( M τ , σ P , σ A ∣ θ , h τ A ) for all h τ A ∈ H τ A ( h τ ) , M τ ∈ M , θ ∈ Θ ,3. There exists a belief system ( µ ′ ) Tt = such that ( σ P , σ ′ A , µ ′ ) is also a PBE, and4. For all public histories h τ on the equilibrium path of ( σ P , σ A ) starting at h t ,W ( M τ , σ P , σ ′ A , µ ′ ∣ h τ ) = W ( M τ , σ P , σ A , µ ∣ h τ ) .Proof of Proposition B.1. Fix a public history h t , a mechanism M t and suppose thatthe set of agent types for which σ At ( θ , h tA , M t ) is not B Θ ⊗ H t -measurable has positivemeasure under µ t ( ⋅ ) . Note that the continuation strategy of ( θ , h tA ) is feasible for ( θ , h tA ) and vice versa. Thus, conditional on participating of M t , the agent at ( θ , h tA ) is not only indifferent between all the messages in the support of r t ( θ , h tA , M t )( ⋅ ) ,but is also indifferent between all messages in the support of r t ( θ , h tA , M t )( ⋅ ) . There-fore, the agent at ( θ , h tA ) is indifferent between r t ( θ , h tA , M t ) and any randomizationbetween r t ( θ , h tA , M t ) and r t ( θ , h tA , M t ) . This indifference also holds taking into ac-count the decision to participate. Moreover, this is true for any continuation publichistory that is reached from h t for the same reasons. That is, for any τ ≥ t and h τ that succeeds ( h t , M t ) and for any h τ A , h τ A that succeed h tA and h tA , respectively, the45gent of type θ is indifferent between her continuation strategy at ( h τ A , M τ ) and thatat ( h τ A , M τ ) . We now construct a new strategy for the agent which, by the above ar-guments, is payoff equivalent to σ A .The principal’s belief together with the agent’s participation strategy induce a mea-surable map from H t × M to ∆ ( Θ × {
0, 1 }) given by P π ( ˜ Θ × { }∣ h t , M t ) = ∫ ˜ Θ × H tA π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) ,with marginal µ t Θ ( ⋅ ∣ h t ) over the set of types. The disintegration theorem implies thatwe can write: P π ( ˜ Θ × { }∣ h t , M t ) = ∫ ˜ Θ π ′ t ( θ , h t , M t )({ }) µ t Θ ( d θ ∣ h t ) ,where, in a slight abuse of notation, { π ′ t ( θ , ⋅ )({ }) ∶ θ ∈ Θ } is the ( µ t Θ , proj { } ) -disintegration of P π ( ⋅ ∣ h t , M t ) .Similarly, if ∫ Θ × H tA ( h t ) π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) >
0, the principal’s belief togetherwith the agent’s reporting strategy induce a measurable map from H t × M to ∆ ( Θ × M M t ) given by P r ( ˜ Θ × ˜ M ∣ h t , M t ) = ∫ ˜ Θ × H tA r t ( θ , h tA , M t )( ˜ M ) µ + t ( d ( θ , h tA )∣ h t , M t ) ,with marginal µ + t Θ ( ⋅ ∣ h t , M t ) over the set of types. The disintegration theorem impliesthat we can write P r ( ˜ Θ × ˜ M ∣ h t , M t ) = ∫ ˜ Θ r ′ t ( θ , h t , M t )( ˜ M ) µ + t Θ ( d θ ∣ h t , M t ) ,where { r ′ t ( θ , h t , M t ) ∶ θ ∈ Θ } is the ( µ + t Θ , proj M M t ) -disintegration of P r ( ⋅ ∣ h t , M t ) . Since π ′ t , r ′ t are ( Θ × H t × M ) - measurable, they are a fortiori ( Θ × H tA × M ) -measurable.We can similarly redefine π ′ τ , r ′ τ for τ ≥ t +
1. Indeed, we have P π ( ˜ Θ × { }∣ h τ , M τ ) = ∫ ˜ Θ × H τ A π τ ( θ , h τ A , M τ ) µ τ ( d ( θ , h tA )∣ h τ ) = ∫ ˜ Θ π ′ τ ( θ , h τ , M τ )({ }) µ τ Θ ( d θ ∣ h τ ) , and, whenever ∫ Θ × H τ A π τ ( θ , h τ A , M τ ) µ τ ( d ( θ , h τ A )∣ h τ ) >
0, define: P r ( ˜ Θ × ˜ M ∣ h τ , M τ ) = ∫ ˜ Θ × H τ A r τ ( θ , h τ A , M τ )( ˜ M ) µ + τ ( d ( θ , h τ A )∣ h τ , M τ ) = ∫ ˜ Θ r ′ τ ( θ , h τ , M τ )( ˜ M ) µ + τ Θ ( d θ ∣ h τ , M τ ) . The definition of ( π τ , r τ ) for θ not in the support of the principal’s beliefs is irrelevant. f ′ τ ( µ ′ τ , M τ )( ˜ Θ × ˜ H τ A × z ∅ ( M τ )) = ∫ ˜ Θ × ˜ H τ A ( − π ′ τ ( θ , h τ A , M τ )) µ ′ τ ( d ( θ , h τ A )∣ h τ ) , f ′ τ ( µ ′ τ , M τ )( ˜ Θ × H ′ A × M τ × { } × ˜ M × ˜ S × ˜ A ) = ∫ ˜ Θ × ˜ H τ A ρ σ ′ A ( ˜ M × ˜ S × ˜ A ∣ θ , h τ A , M τ ) π ′ τ ( θ , h τ A , M τ ) µ ′ τ ( d ( θ , h τ A )∣ h τ ) . We now use these equations inductively to show that under the new strategies thedistribution over Θ × H T + induced by P σ ∣ ⋅ ○ µ t ( ⋅ ∣ h t ) is preserved. Suppose that wehave shown that for t ≤ τ ′ ≤ τ , µ ′ τ ′ ( ⋅× H τ ′ A ( h τ ′ )∣ h τ ′ ) coincides with µ τ ′ ( ⋅× H τ ′ A ( h τ ′ )∣ h τ ′ ) for P σ ∣ ⋅ ○ µ t ( ⋅ ∣ h t ) almost all h τ ′ . We now show that this holds for τ +
1. To see this, firstnote that for all measurable subset ˜ Θ of Θ , we have µ ′+ τ ( ˜ Θ × H τ A ( h τ )∣ h τ , M t ) = ∫ ˜ Θ × H τ A ( h τ ) π ′ τ ( θ , h τ A , M τ ) µ ′ τ ( d ( θ , h τ A )∣ h τ ) = ∫ ˜ Θ π ′ τ ( θ , h τ A , M τ ) µ ′ τ Θ ( d θ ∣ h τ ) = ∫ ˜ Θ π ′ τ ( θ , h τ A , M τ ) µ τ Θ ( d θ ∣ h τ ) = ∫ ˜ Θ × H τ A ( h τ ) π τ ( θ , h τ A , M τ ) µ τ ( d ( θ , h τ A )∣ h τ ) = µ + τ ( ˜ Θ × H τ A ( h τ )∣ h τ , M τ ) ,(B.1) where the second equality follows from Fubini’s theorem and π ′ t being B H t -measurableand the third from the inductive hypothesis. This automatically implies that ν ′ τ ( µ τ ( ⋅ ∣ h τ ) , M τ )( ˜ Θ × H τ A ( h τ ) × z ∅ ( M τ )) = ν τ ( µ τ ( ⋅ ∣ h τ ) , M τ )( ˜ Θ × H τ A ( h τ ) × z ∅ ( M τ )) ,and hence the updated belief µ τ + ( ⋅ × H τ A ( h τ )∣ h τ , M τ , z ∅ ( M τ )) remains unchangedif the event that the mechanism was rejected has positive probability in the originalstrategy profile. Moreover, for any measurable subset ˜ Θ of Θ , and for any measurablesubset ˜ M of M M τ we have ∫ ˜ Θ × H τ A ( h τ ) r ′ τ ( θ , h τ A , M τ )( ˜ M ) µ ′+ τ ( d ( θ , h τ A )∣ h τ , M τ ) = ∫ ˜ Θ r ′ τ ( θ , h τ A , M τ )( ˜ M ) µ ′+ τ Θ ( d θ ∣ h τ , M τ ) = ∫ ˜ Θ r ′ τ ( θ , h τ A , M τ )( ˜ M ) µ + τ Θ ( d θ ∣ h τ , M τ ) = ∫ ˜ Θ × H τ A ( h τ ) r τ ( θ , h τ A , M τ )( ˜ M ) µ + τ ( d ( θ , h τ A )∣ h τ , M τ ) , where the first equality uses Fubini’s theorem, the second equality uses Equation B.1, While we change the agent’s strategy at all histories which succeed ( h t , M t ) , Bayes’ rule wherepossible ties the beliefs at h t and the beliefs at h τ only at those histories h τ that are on the path of the(agent’s) strategy. This is why when we check that the principal’s beliefs over Θ have not changed wedo so along the path of the strategy profile starting at h t . This is enough to check that we have notchanged the principal’s (continuation) payoffs. r ′ t ( ⋅ ) . This implies that f ′ τ ( µ ′ τ (⋅∣ h τ ) , M τ )( ˜ Θ × H τ A ( h τ ) × M τ × { } × M M τ × ˜ S × ˜ A ) = f τ ( µ τ (⋅∣ h τ ) , M τ )( ˜ Θ × H τ A ( h τ ) × M τ × { } × M M τ × ˜ S × ˜ A ) , on the measurable rectangles ˜ Θ × M M τ × ˜ S × ˜ A ∈ B Θ ⊗ M τ × S τ × A . Thus, the (marginal) up-dated beliefs { µ ′ τ + ( ⋅× H τ A ( h τ ) × M τ × M M τ ×⋅ ∣ h τ , M t , z ) ∶ z ∈ Z τ } coincide with { µ t + ( ⋅× H tA ( h t ) × M t × M M t × ⋅ ∣ h t , M t , z ) ∶ z ∈ Z τ } , P σ ∣ µ t ( h t ) - almost surely. It follows that theprincipal’s payoff remains the same (see Equation A.2). Remark B.1.
Note that the PBE assessment one obtains from Proposition B.1 satisfiesthat P σ ′ ( proj Θ × A T + Θ × H T + A ) = P σ ( proj Θ × A T + Θ × H T + A ) . It follows that the set of PBEoutcomes of the mechanism-selection game is the same as the set of PBE outcomes ofthe mechanism-selection game when the agent’s strategy only depends on her payoffrelevant type and the public history. The outcome-equivalent PBE assessment one obtains from Proposition B.1 satis-fies the following property. On the equilibrium path, the principal’s beliefs over theagent’s payoff-relevant type, θ ∈ Θ , do not depend on her payoff-irrelevant history, h tA . However, at a public history h t reached after a deviation by the agent, the re-quirements of PBE do not rule out that the principal’s updated beliefs depend non-trivially on both θ and h tA . It follows from Proposition B.1 that without loss of gener-ality, we can assume that when the principal observes a deviation by the agent, hisupdated beliefs do not depend on h tA . The proof is available upon request.Given a mechanism M t , let ( S M t × A ) + = ⋃ m ∈ M M t supp β M t ( ⋅ ∣ m ) . (B.2)The set ( S M t × A ) ∖ ( S M t × A ) + has zero probability regardless of the agent’s strategy.Hence, if we remove from the tree those paths that are consistent with mechanism M t and ( s , a ) ∉ ( S M t × A ) + , this does not change the set of equilibrium outcomes.Hereafter, these histories are removed from the tree.Fix a PBE assessment of G M and a mechanism M t and define a measure on M M t × S M t × A as follows: R h t , M t ( ˜ M × ˜ S × ˜ A ) = ∫ Θ × H tA ( h t ) ∫ ˜ M β ( ˜ S × ˜ A ∣ m ) r t ( θ , h tA , M t )( d m ) π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) .(B.3) Proposition B.2.
Fix a PBE assessment ( σ P , σ A , µ ) of G M that satisfies Proposition B.1. hen, there exists an outcome-equivalent PBE assessment ( σ ′ P , σ ′ A , µ ′ ) such that for allpublic histories h t , for all mechanisms M t on the path of the equilibrium strategy ath t such that the agent participates with positive probability, M M t = Θ , ( S M t × A ) + isthe support of R h t , M t ( Θ × ⋅ ) . Moreover, the agent truthfully reports her type.Proof of Proposition B.2. Fix a PBE assessment ( σ P , σ A , µ ) of G M that satisfies Proposition B.1,a history h t , and a mechanism M t on the support of the principal’s strategy at h t ,such that the agent participates with positive probability. Let Θ + denote the supportof µ + t ( ⋅ ∣ h t , M t ) . By definition, Θ + is closed.Define a new mechanism M t as follows. Let ( M M t , S M t ) = ( Θ , S M t ) (Recall that Θ is afeasible set of input messages.) Define the transition probability from Θ + to S M t × A by specifying its values on the measurable rectangles, ˜ S × ˜ A ∈ B S M t ⊗ A , β M t ( ˜ S × ˜ A ∣ θ ) = ∫ M M t β M t ( ˜ S × ˜ A ∣ m ) r t ( θ , h tA , M t )( d m ) .By composition of measurable functions and since r t does not depend on h tA , thisdefines a measurable mapping from Θ + to ∆ ( S M t × A ) . Note that this modificationdoes not alter the principal’s beliefs about the agent’s type conditional on observing ( s t , a t ) ∈ ( S M t × A ) + .For θ ∉ Θ + , let Q ∗ ( θ ) = arg max q ∈ ∆ ( Θ + ) ∫ Θ + [ ∫ S M t × A E P σ ∣( θ , htA , ⋅ , z ( st , at )( M t )) [ U ( a t , a t , ⋅ , θ )] β M t ( d ( s t , a t )∣ ˜ θ )] q ( d ˜ θ ) ,where the payoff on the right-hand side of the above expression corresponds to thepayoff from reporting (possibly at random) a type in Θ + , and then conditional on ( s t , a t ) , play proceeding as in the original strategy profile. The objective is continu-ous in q and ∆ ( Θ + ) is compact since Θ + is compact (Theorem 15.11 in Aliprantis and Border(2013)). Then, the maximization is well-defined. Theorem 18.19 in Aliprantis and Border(2013) implies that there exists a measurable selector q ∗ ( θ ) ∈ Q ∗ ( θ ) . Use this to de-fine β M t ( ⋅ ∣ θ ) for θ ∉ Θ + on the measurable rectangles ˜ S × ˜ A of S M t × A as follows: β M t ( ˜ S × ˜ A ∣ θ ) = ∫ Θ + β M t ( ˜ S × ˜ A ∣ ˜ θ ) q ∗ ( θ )( d ˜ θ ) .Note that this defines β M t as a transition probability from Θ to S M t × A . The term in brackets is bounded above by the payoff the agent of type θ obtains in equilibrium.The results in Serfozo (1982) imply then continuity of the objective in q . To see this, fix a measurable subset C of ∆ ( S M t × A ) and let B denote a measurable subset of [ ] . M t instead of M t . Modify the con-tinuation strategies so that for any public history that succeeds ( h t , M t ) play followswhat would have transpired if instead M t had been played. Set r ′ t ( θ , h tA , M t ) = δ θ .For θ ∈ Θ + , it is a best response to set π ′ t ( θ , h tA , M t ) = π t ( θ , h tA , M t ) . Types not in Θ + may not find it optimal to participate in the mechanism; recompute their participa-tion strategies accordingly. Finally, use Equation A.1 equations to modify the beliefsystem. It is immediate to check that the new assessment is also a PBE.To finalize the proof, we need to show that the distribution R h t , M t ∈ ∆ ( M M t × S M t × A ) ∈ ∆ ( Θ × S M t × A ) satisfies that the support of R h t , M t ( Θ × ⋅ ) is ( S M t × A ) + . Clearly,supp R h t , M t ( Θ + × ⋅ ) ⊆ ( S M t × A ) + . Suppose the inclusion is strict and let ( s , a ) ∈ ( S M t × A ) + ∖ supp R h t , M t ( Θ + ×⋅ ) . Then, there exists an open neighborhood of ( s , a ) , N such that R h t , M t ( Θ × N ) =
0. We claim that supp R h t , M t ( ⋅ × S M t × A ) ≠ Θ + . To-wards a contradiction, assume that supp R h t , M t ( ⋅ × S M t × A ) = Θ + . Then, for all θ ∈ Θ + there exists an open neighborhood θ ∈ V θ such that R h t , M t ( V θ × S M t × A ) > ∫ Θ + β M t ( N ∣ θ ) µ + t ( d θ ∣ h t , M t ) = β M t ( N ∣ θ ) = θ ∈ Θ + . Thus, ( s , a ) ∉ ( S M t × A ) + , a contradiction.Two corollaries follow from Proposition B.2. First, from now on, we can focus on PBEassessments ( σ P , σ A , µ ) that satisfy Proposition B.1 and where the principal offersmechanisms with input messages equal to the set of types and the agent truthfullyreports her type conditional on participating. Second, if the agent participates in themechanism offered by the principal, the principal is never surprised by the tuples ( s t , a t ) that come out of the mechanism. This, instead, means that if the agent par-ticipates in the mechanism with positive probability, then beliefs, µ t + ( ⋅ , z ( s t , a t ) ( M t )∣ h t , z ( s t , a t ) ( M t )) , are determined via Bayes’ rule where possible.The following two propositions require lifting a PBE assessment, ( σ P , σ A , µ ) , from G M to one in the auxiliary game. Each proposition finds an outcome-equivalentPBE assessment, ( σ ′ P , σ ′ A , µ ′ ) , of the auxiliary game with certain properties. Proposition B.3.
Fix a PBE assessment ( σ P , σ A , µ ) of G M that satisfies the propertieslisted in Propositions B.1-B.2. Then, there exists an outcome-equivalent PBE assess-ment of the auxiliary game , ( σ ′ P , σ ′ A , µ ′ ) such that for all h t , and M t in the supportof σ Pt ( h t ) such that the agent participates with positive probability, the followingholds. First, M t is a canonical mechanism. Second, the agent truthfully reports her Then, the set { θ ∈ Θ ∶ β M t ( C ∣ θ ) ∈ B } = { θ ∈ Θ + ∶ β M t ( C ∣ θ ) ∈ B } ∪ { θ ∈ Θ ∖ Θ + ∶ β M t ( C ∣ θ ) ∈ B } . Each set isin B Θ by construction and therefore their union is in B Θ . ype. Third, the principal’s updated beliefs coincide with the output message.Proof of Proposition B.3. Let ( σ P , σ A , µ ) be as in the statement of Proposition B.3. Ina slight abuse of notation, lift ( σ P , σ A , µ ) so that it is a PBE assessment in the auxiliarygame. Let h t be a public history and let M t denote a mechanism on the path of theequilibrium strategy starting at h t . Let Θ + denote the support of µ + t ( ⋅ ∣ h t , M t ) .The kernel ν t ( h t , M t ) defines a joint probability on Θ × S M t × A via ν t ( h t , M t )( ˜ Θ × ˜ S × ˜ A ) = ∫ ˜ Θ β M t ( ˜ S × ˜ A ∣ θ ) µ + t Θ ( d θ ∣ h t , M t ) ,while the updated beliefs satisfy ν t ( h t , M t )( ˜ Θ × ˜ S × ˜ A ) = ∫ ˜ S × ˜ A µ t + ( ˜ Θ × H t + A ( h t , z ( s t , a t ) ( M t ))∣ h t , z ( s t , a t ) ( M t )) ν tS M t × A ( h t , M t )( d ( s t , a t )) . Recall that we can write the principal’s payoff as follows (Equation A.2): ∫ Θ × S M t × A E P σ ∣( θ , htA , z ( st , at )( M t )) [ W ( a t , a t , ⋅ , θ )] β M t ( d ( s t , a t )∣ θ ) µ + t Θ ( d θ ∣ h t , M t ) = ∫ S M t × A ∫ Θ E P σ ∣( θ , htA , z ( st , at )( M t )) [ W ( a t , a t , ⋅ , θ )] µ t + Θ ( d θ ∣ h t , z ( s t , a t ) ( M t )) ν tS M t × A ( d ( s t , a t )∣ h t , M t ) .By Kuratowski’s theorem, there exists a bijection ω ∶ S M t ↦ [
0, 1 ] (see Parthasarathy(2005)). Define the measurable function W ∶ A × ∆ ( Θ ) × [
0, 1 ] ↦ R as follows: W ( µ , ω ( s t ) , a t ) = ∫ Θ E P σ ∣( θ , htA , z ( st , at )( M t )) [ W ( a t , a t , ⋅ , θ )] µ ( d θ ) .We allow W to explicitly depend on s t since continuation payoffs may depend on s t beyond its impact on beliefs. Define the measurable map T ∶ S M t × A ↦ ∆ ( Θ ) × [
0, 1 ] × A , so that T ( s t , a t ) = ( µ t + ( ⋅ ∣ h t , z ( s t , a t ) ( M t )) , ω ( s t ) , a t ) . Define a measure over Θ × ∆ ( Θ ) × [
0, 1 ] × A by specifying it on the measurable rectangles: P ( ˜ Θ × ˜ U × ˜ Ω × ˜ A ) = ν t ( h t , M t )( ˜ Θ × T − ( ˜ U × ˜ Ω × ˜ A )) , (B.4)where we denote by ˜ Ω an element of B [ ] anticipating that this part of the output51essage will become the public randomization device. Note that we can write: ∫ S M t × A W ( µ ( h t , z ( s t , a t ) ( M t )) , ω ( s t ) , a t ) ν tS M t × A ( d ( s t , a t )∣ h t , M t ) (B.5) = ∫ S M t × A W ( T ( s t , a t )) ν tS M t × A ( d ( s t , a t )∣ h t , M t ) = ∫ T ( S M t × A ) W ( µ , ω , a t ) ν tS M t × A ○ T − ( d ( µ , ω , a t )) = ∫ ∆ ( Θ ) × [ ] × A W ( µ , ω , a t ) P ∆ ( Θ ) × [ ] × A ( d ( µ , ω , a t )) .Let { η ( µ , ω , a ) ∶ ( µ , ω , a ) ∈ ∆ ( Θ ) × [
0, 1 ] × A } denote the ( P ∆ ( Θ ) × [ ] × A , proj Θ ) disinte-gration of P . We have that on the measurable rectangles, ∫ ˜ U × ˜ Ω × ˜ A η ( µ , ω , a ) ( ˜ Θ ) P ∆ ( Θ ) × [ ] × A ( d ( µ , ω , a )) = P ( ˜ Θ × ˜ U × ˜ Ω × ˜ A ) = ν t ( h t , M t )( ˜ Θ × T − ( ˜ U × ˜ Ω × ˜ A )) = ∫ T − ( ˜ U × ˜ Ω × ˜ A ) T ∆ ( Θ ) ( s t , a t )( ˜ Θ ) ν tS M t × A ( d ( s t , a t )∣ h t , M t ) = ∫ ˜ U × ˜ Ω × ˜ A µ ( ˜ Θ ) P ∆ ( Θ ) × [ ] × A ( d ( µ , ω , a )) ,where T ∆ ( Θ ) is the first coordinate of T . It follows from Proposition 3.6 in Crauel(2002) that η ( µ , ω , a ) = µ P ∆ ( Θ ) × [ ] × A -almost everywhere. Intuitively this is just sayingthat “when the output message is ( µ , ω ) ", the principal updates his beliefs to µ .Now, let { η µ ∶ µ ∈ ∆ ( Θ )} denote the ( P ∆ ( Θ ) , proj Θ × [ ] × A ) -disintegration of P . For anymeasurable subset ˜ U of ∆ ( Θ ) , we have that on the measurable rectangles ˜ Θ × ˜ Ω × ˜ A of Θ × [
0, 1 ] × A , ∫ ˜ U η µ ( ˜ Θ × ˜ A × ˜ Ω ) d P ∆ ( Θ ) = P ( ˜ Θ × ˜ U × ˜ A × ˜ Ω ) = ∫ ˜ U × ˜ A × ˜ Ω µ ( ˜ Θ ) d P ∆ ( Θ )× A × Ω = (B.6) ∫ ˜ U × A × Ω [( a , ω ) ∈ ˜ A × ˜ Ω ] µ ( ˜ Θ ) d P ∆ ( Θ )× A × Ω = ∫ ˜ U ∫ A × Ω µ ( ˜ Θ ) [( a , ω ) ∈ ˜ A × ˜ Ω ] η µ ( d ( a , ω )) d P ∆ ( Θ ) = ∫ ˜ U µ ( ˜ Θ ) ∫ A × Ω [( a , ω ) ∈ ˜ A × ˜ Ω ] η µ ( d ( a , ω )) d P ∆ ( Θ ) = ∫ ˜ U µ ( ˜ Θ ) η µ ( ˜ A × ˜ Ω ) d P ∆ ( Θ ) , where the first and second equalities follow from the disintegration property, thethird equality is a rewriting of the integral, the fourth uses the ( P ∆ ( Θ ) , π A × Ω ) -disintegrationof P ∆ ( Θ ) × A × Ω , { η µ ∶ µ ∈ ∆ ( Θ )} , and the fifth equality uses that “conditional on µ ", µ ( ˜ Θ ) is constant. It follows from this that Θ á ( A , Ω )∣ ∆ ( Θ ) .Now, let { β θ ∶ θ ∈ Θ } denote the ( P Θ , proj ∆ ( Θ ) × Ω × A ) -disintegration of P . Theorem 1.25in Kallenberg (2017) implies that there exist two transition probabilities p ∶ Θ ↦ ∆ ( Θ ) , q ∶ Θ × ∆ ( Θ ) ↦ [
0, 1 ] × A such that β = p ⊗ q and hence, ∫ ˜ Θ β θ ( ˜ U × ˜ Ω × ˜ A ) P Θ ( d θ ) = ∫ ˜ Θ ( ∫ ˜ U q ( θ , µ ) ( ˜ Ω × ˜ A ) p θ ( d µ )) P Θ ( d θ ) .Equation B.6 and Theorem 1.27 in Kallenberg (2017) imply that q ( θ , µ ) = η µ P ∆ ( Θ ) -52lmost everywhere, so that we can write ∫ ˜ Θ β θ ( ˜ U × ˜ Ω × ˜ A ) P Θ ( d θ ) = ∫ ˜ Θ ( ∫ ˜ U η µ ( ˜ Ω × ˜ A ) p θ ( d µ )) P Θ ( d θ ) .Theorem 1.25 in Kallenberg (2017) implies that we can write η µ as the compositionof two transition probabilities, α from ∆ ( Θ ) to A and γ from ∆ ( Θ ) × A to [
0, 1 ] , so that ∫ ˜ Θ β θ ( ˜ U × ˜ Ω × ˜ A ) P Θ ( d θ ) = ∫ ˜ Θ ( ∫ ˜ U ( ∫ ˜ A γ ( µ , a ) ( ˜ Ω ) α µ ( d a )) p θ ( d µ )) P Θ ( d θ ) .Finally, note that ∫ ˜ Θ β θ ( ˜ U × ˜ Ω × ˜ A ) P Θ ( d θ ) = ν t ( h t , M t )( ˜ Θ × T − ( ˜ U × ˜ Ω × ˜ A )) = ∫ ˜ Θ β M t ( T − ( ˜ U × ˜ Ω × ˜ A )∣ θ ) P Θ ( d θ ) .(B.7) Let M ′ t be such that ( M M ′ t , S M ′ t ) = ( Θ , ∆ ( Θ )) and for θ ∈ Θ + , define β M ′ t ( ⋅ ∣ θ ) as thecomposition of the transition probabilities p from Θ to ∆ ( Θ ) and α from ∆ ( Θ ) to A .Continuation strategies are modified so that when the outcome of the mechanismis ( µ , a t ) , we draw ω ∈ [
0, 1 ] according to γ ( µ , a t ) and we play the continuation corre-sponding to ( h t , z ( s t , a t ) ( M t )) where T ( s t , a t ) = ( µ , a t , ω ) . That is, ( σ ′ P , σ ′ A )∣ ( h t , z ( µ , at ) ( M ′ t ) , ω , ⋅ ) = ( σ P , σ A )∣ ( h t , z T − ( µ , at , ω ) ( M t ) , ⋅ ) .With the continuation strategies at hand, for types not in Θ + , use the same argumentas in the proof of Proposition B.2 to extend β M ′ t to all of Θ . This completes the spec-ification of M ′ t . Equation B.7 implies that the agent receives the same payoff whenher type is in Θ + by truthfully reporting. If θ ∉ Θ + , the agent’s payoff from participat-ing in the mechanism may be lower, so that we recompute the participation strategyaccordingly. Equation B.5 implies that the principal receives the same payoff under M t and under M ′ t . It follows that the new profile is also a PBE. To facilitate checking the application of Theorem 1.27 in Kallenberg (2017) to our setting,we now use his notation. For any sets Y , X , and Z , and joint measure ν on Y × X × Z , let η Y X ∣ Z denote the ( ν Z ,proj Y × X ) − disintegration of ν . Then Equation B.6 shows that η Θ A [ ]∣ ∆ ( Θ ) = η Θ ∣ ∆ ( Θ ) ⊗ η A [ ]∣ ∆ ( Θ ) , P ∆ ( Θ ) -almost everywhere. By Theorem 1.25 in Kallenberg (2017), η Θ A [ ]∣ ∆ ( Θ ) = η Θ ∣ ∆ ( Θ ) ⊗ η A [ ] ∣ Θ ∣ ∆ ( Θ ) , which means that η A [ ] ∣ Θ ∣ ∆ ( Θ ) = η A [ ]∣ ∆ ( Θ ) P ∆ ( Θ ) -almost everywhere. The-orem 1.27 in Kallenberg (2017) shows that η A [ ] ∣ ∆ ( Θ )∣ Θ = η A [ ] ∣ Θ ∣ ∆ ( Θ ) P Θ∆ ( Θ ) - almost everywhere.Together with the observation that η ∆ ( Θ ) A [ ]∣ Θ = η ∆ ( Θ )∣ Θ η A [ ]∣ ∆ ( Θ )∣ Θ , completes the claim. At the risk of introducing more notation, one could use the probability integral transform andmake the distribution on [ ] be the uniform distribution. Now, the probability integral transformrequires that the distribution of ω be continuous. This can always be guaranteed by applying the resultin Lehmann et al. (1988), which shows that for any (real-valued) random variable X one can alwaysconstruct an information-equivalent random variable X ∗ the distribution of which is continuous. roposition B.4. Fix a PBE assessment ( σ P , σ A , µ ) in G M that satisfies B.1-B.2. Then,there is an outcome-equivalent PBE assessment ( σ ′ P , σ ′ A , µ ′ ) of the auxiliary game thatsatisfies Proposition B.3 and such that the following holds. For every t ≥ , for everypublic history h t and mechanism M t in the support of σ Pt ( h t ) , π t ( θ , h tA , M t ) = forall types in the support of µ t ( ⋅ × H tA ∣ h t ) .Proof of Proposition B.4. Let ( σ P , σ A , µ ) denote a PBE assessment of G M that satis-fies Propositions B.1- B.2. In a slight abuse of notation, let ( σ P , σ A , µ ) denote theoutcome-equivalent PBE of the auxiliary game as in Proposition B.3. Fix a publichistory h t and a mechanism M t on the path of σ Pt at h t such that ∫ Θ × H tA ( h t ) π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) < µ t + ( ⋅ ∣ h t , z ∅ ( M t )) ≡ µ ∅ is defined via Bayes’ rule usingthe equilibrium strategy profile. Let Θ + denote the support of µ t ( ⋅ × H tA ∣ h t ) .If M t is rejected with probability one, then modify the principal’s strategy so thatinstead of offering M t , he offers M ′ t such that for all θ ∈ Θ β M ′ t ( ⋅ ∣ θ ) = δ ( µ ∅ , a ∗ ) , where δ ⋅ denotes the Dirac measure. Modify the continuation strategies so that ( σ ′ P , σ ′ A )∣ ( h t , z ( µ ∅ , a ∗) ( M ′ t )) = ( σ P , σ A )∣ ( h t , z ∅ ( M t )) .Modify the agent’s strategy so that π ′ t ( θ , h tA , M ′ t ) = r ′ t ( θ , h tA , M ′ t ) = δ θ whenever θ isin Θ + ; otherwise, leave the agent’s strategy unchanged.Suppose now that the mechanism is accepted with positive probability, so that both µ ∅ and µ + t ( ⋅ ∣ h t , M t ) are determined via Bayes’ rule from the equilibrium strategy pro-file. By Proposition B.3, it is without loss of generality to assume that ( M M t , S M t ) = ( Θ , ∆ ( Θ )) . Define a new mechanism M ′ t as follows. Let ( M M ′ t , S M ′ t ) = ( M M t , S M t ) anddefine the transition probability β M ′ t from Θ + to ∆ ( Θ ) × A on the measurable rectan-gles as follows β M ′ t ( ˜ U × ˜ A ∣ θ ) = π t ( θ , h tA , M t ) β M t ( ˜ S × ˜ A ∣ θ ) + ( − π t ( θ , h tA , M t )) [ µ ∅ ∈ ˜ U , a ∗ ∈ ˜ A ] . Define a joint probability over Θ × {
0, 1 } × ∆ ( Θ ) × A as follows: P ( ˜ Θ × { } × ˜ U × ˜ A ) = [ µ ∅ ∈ ˜ U , a ∗ ∈ ˜ A ] ∫ ˜ Θ ( − π t ( θ , h tA , M t )) µ t ( d ( θ , h tA )∣ h t ) , P ( ˜ Θ × { } × ˜ U × ˜ A ) = ∫ ˜ Θ β M t ( ˜ U × ˜ A ∣ θ ) π t ( θ , h tA , M t ) µ t ( d ( θ , h tA )∣ h t ) .54he disintegration theorem implies that we can write for n ∈ {
0, 1 } P ( ˜ Θ × { n } × ˜ U × ˜ A ) = ∫ ˜ U × ˜ A η ( µ , a t ) ( ˜ Θ × { n }) P ∆ ( Θ ) × A ( d ( µ , a t )) .Let q = η ( µ ∅ , a ∗ ) ( Θ × { }) . Modify the continuation strategy so that for ω ∈ [ q ) ( σ ′ P , σ ′ A )∣ ( h t , z µ ∅ , a ∗ ( M ′ t ) , ω ) = ( σ P , σ A )∣ ( h t , z ∅ ( M t ) , ω q ) ,while for ω ∈ ( q , 1 ] , ( σ ′ P , σ ′ A )∣ ( h t , z ( µ ∅ , a ∗) ( M ′ t ) , ω ) = ( σ P , σ A )∣ ( h t , z ( µ ∅ , a ∗) ( M t ) , ω − q − q ) .Modify the agent’s strategy so that for types in Θ + , r ′ t ( θ , h tA , M ′ t ) = δ θ and π ′ t ( θ , h tA , M ′ t ) =
1. For types not in Θ + , extend β M ′ t to all of Θ as we did in the proof of Proposition B.2.Set r ′ t ( θ , h tA , M ′ t ) = δ θ . Their payoff from participating may be lower, so recomputetheir participation strategy accordingly. It is straightforward to check that the newassessment is also a Perfect Bayesian equilibrium.Propositions B.1-B.4 imply that for any equilibrium assessment, ( σ P , σ A , µ ) , of themechanism-selection game G M , there exists an equilibrium assessment, ( σ ′ P , σ ′ A , µ ′ ) ,of the auxiliary game that satisfies that for all periods t and public histories h t ,(i) theprincipal offers canonical mechanisms, (ii) the agent’s strategy satisfies the proper-ties listed in Theorem 1, and (iii) the beliefs employed by the mechanism coincidewith the principal’s equilibrium beliefs. Moreover, ( σ ′ P , σ ′ A , µ ′ ) implements the samedistribution over outcomes Θ × A T + as ( σ P , σ A , µ ) does.Remark 1 implies that we can lift ( σ ′ P , σ ′ A , µ ′ ) to an outcome-equivalent assessmentof the canonical game, ( ˜ σ P , ˜ σ A , ˜ µ ) . Furthermore, the construction highlights thatwhatever the principal can achieve starting at any public history h t with mecha-nisms in M , he can also alternatively achieve with canonical mechanisms. Thus, itfollows that for any PBE outcome γ ∈ ∆ ( Θ × A T + ) of the canonical game, there is anoutcome-equivalent assessment ( σ P , σ A , µ ) of the mechanism-selection game. Wedo not include the proof of this construction since it readily follows from the above.We do note that in the mechanism-selection game the public randomization devicein the canonical game must be subsumed in the mechanism. That this is feasiblefollows from Kuratowski’s theorem since S M t and ∆ ( Θ ) × [
0, 1 ] have the same cardi-nality. The notation assumes that the public randomization device is drawn U [ ] . This is without lossbecause of the probability integral transform (see the proof of Proposition B.3).. This is without lossbecause of the probability integral transform (see the proof of Proposition B.3).