Optimal mechanism for the sale of a durable good
aa r X i v : . [ ec on . T H ] J a n Optimal mechanism for the sale of a durablegood ∗ Laura Doval † Vasiliki Skreta ‡ January 22, 2020
Abstract
We show that posted prices are the optimal mechanism to sell a durablegood to a privately informed buyer when the seller has limited commitmentin an infinite horizon setting. We provide a methodology for mechanism de-sign with limited commitment and transferable utility. Whereas in the case ofcommitment, subject to the buyer’s truthtelling and participation constraints,the seller’s problem is a decision problem, in the case of limited commitment,the seller’s problem corresponds to an intrapersonal game, where different“incarnations” of the seller represent the different beliefs he may have aboutthe buyer’s valuation. K EYWORDS : mechanism design, limited commitment, intrapersonal equilibrium,information design, self-generation, posted priceJEL
CLASSIFICATION : D84, D86 ∗ We would like to thank Rahul Deb, Frederic Koessler, Pablo Schenone, and especially MaxStinchcombe, as well as audiences at Cowles, SITE, and Stony Brook, for thought-provokingquestions and illuminating discussions. Vasiliki Skreta is grateful for generous financial supportthrough the ERC consolidator grant 682417 “Frontiers in design.” This research was supported bygrants from the National Science Foundation (SES-1851744 and SES-1851729). † California Institute of Technology, Pasadena, CA 91125. E-mail: [email protected] ‡ University of Texas at Austin, University College London, and CEPR. E-mail: [email protected] Introduction
A classic problem in mechanism design is how to sell a good to a buyer so as tomaximize the seller’s revenue. The answer is surprisingly simple: out of the manyways in which a seller could achieve this feat, a posted price is optimal. Thisresult does not depend on the length of the interaction between the buyer andthe seller: even if they would interact for infinitely many periods, the revenue-maximizing mechanism is to post the same price in each period. An importantassumption behind this result is that the seller can commit to the whole sequenceof mechanisms that the buyer faces. The commitment assumption is importantbecause the optimal mechanism is, in general, time-inconsistent . When the buyerdoes not buy the good at the posted price, the seller has to be able to not lower theprice even if there is common knowledge of unrealized gains from trade.However, much less is known about how to sell a durable good when the sellercan commit to today’s mechanism, but not to the mechanism he will offer if no saleoccurs. In an infinite horizon game in which the seller may only post a price ineach period, Ausubel and Deneckere (1989) show the seller can achieve monopolyprofits provided that (i) the buyer’s valuation is drawn from a continuum, withthe lowest valuation being smaller than the seller’s marginal cost (i.e., the no gap case), and (ii) we consider the limit case in which the seller is fully patient. Ina finite horizon game in which the seller can choose mechanisms in each periodand the buyer’s valuation is drawn from a continuum, Skreta (2006) shows theseller’s optimal mechanism under limited commitment is a sequence of postedprices. These results rely, however, on the assumption that the seller directly ob-serves the buyer’s choices – the rejection of the price in Ausubel and Deneckere(1989) and the choice of messages in Skreta (2006) . Indeed, in a two-period model,Denicolo and Garella (1999) show that when the seller only observes whether tradehappens or not, the seller may instead prefer to ration high valuation buyers in thefirst period, in order to induce a strong demand in the second period.In this paper, we study an infinite-horizon mechanism-selection game between anuninformed seller, who owns one unit of a durable good, and a privately informedbuyer. The buyer’s valuation is binary and fully persistent. At the beginning ofeach period, as long as the good has not been sold, the seller offers the buyer a mechanism , the rules of which determine the allocation for that period. FollowingBester and Strausz (2007) and Doval and Skreta (2018), the set of mechanisms theseller may choose from allows the seller to design how much he observes aboutthe buyer’s choices, and hence, design his beliefs about the buyer’s type. Since2he sellers beliefs define the demand schedule he faces, it is feasible for him toimplement, amongst others, the demand schedule that obtains from rationing inDenicolo and Garella (1999).The main result of this paper is that, among all mechanisms, posted pricesare optimal when the seller has limited commitment. Theorem 1 characterizesthe revenue-maximizing Perfect Bayesian equilibrium and a strategy profile thatachieves it. The latter is such that, as long as a sale has not occurred, the seller willchoose a mechanism that can be implemented as a posted price. Contrary to thecase in which the seller has commitment to the entire sequence of mechanisms, thesequence of posted prices is not constant, reflecting the fact that the seller makesinferences about the buyer’s valuation in the event that no trade happens.The optimality of posted prices echoes the result for the case in which the sellerhas commitment, showing that the shape of the mechanism does not depend onwhether there is commitment to short- or long-term mechanisms. Moreover, thisresult provides a microfoundation for the strategy space used by the literaturethat studies the sale of a durable good by a monopolist. Inasmuch as we areinterested in understanding the best outcome for the monopolist, assuming hecan post prices is without loss of generality.The key conceptual innovation of the analysis is to translate the problem of find-ing the revenue-maximizing PBE in the mechanism-selection game between theseller and the buyer into an auxiliary problem, which only involves the seller. Thisauxiliary problem is an intrapersonal game , where the seller is the only player, andeach of his “incarnations” represents the different beliefs he may have about thebuyer’s valuation. This step establishes a formal connection between the liter-atures on mechanism design with limited commitment and that on time incon-sistency. Time inconsistency of optimal mechanisms under full commitment liesat the heart of the difficulties in mechanism design with limited commitment(Laffont and Tirole (1990)). This paper is the first to show that these two prob-lems, mechanism design with limited commitment and intrapersonal games, areformally related and to use the solution of one to solve the other.To understand how we arrive to this auxiliary problem, a review of the fourmain steps involved in the proof of Theorem 1 is useful (We contrast our ap-proach to previous work on mechanism design with limited commitment, espe-cially Skreta (2006) in the related literature). First, we rely on the results in ourprevious work, Doval and Skreta (2018), to simplify the class of mechanisms theseller will offer in any equilibrium of the game and, more importantly, the buyer’s3quilibrium behavior. This step reduces the search of the optimal sequence ofmechanisms to those that satisfy, loosely speaking, a sequence of participation and truthtelling constraints. Second, we derive necessary conditions that a revenue-maximizing PBE assessment must satisfy. These necessary conditions are the dy-namic analogue of establishing which participation and incentive compatibilityconstraints are binding in the case of commitment. We use these conditions to ob-tain a recursive program which provides an upper bound on the seller’s payoff inthe payoff-maximizing PBE, analogous to the maximization of the virtual surplus in the case of commitment. Indeed, as in the case of commitment, in this recursiveprogram, the seller’s payoff is written solely as a function of the allocation: thetransfers have been removed by using the binding constraints.The third step shows this recursive program has a unique solution. Whereasin the case of commitment the maximization of the virtual surplus is a decisionproblem, it corresponds to an intrapersonal game in the case of limited commitment.Indeed, the best response condition of the intrapersonal equilibrium captures therestrictions imposed by sequential rationality on the seller’s behavior. Wheneverthe seller chooses an allocation that involves no trade, he internalizes how hisbeliefs will change and how that change will determine the choice of mechanismsin the continuation.The output of this third step is a mapping that assigns a choice of a mecha-nism to each belief the seller may hold about the buyer’s valuation. The final stepis to return to our dynamic incomplete information game and show how to usethis mapping to construct the revenue-maximizing PBE assessment. Crucial tothis step is a result in our previous work that allows us to apply self-generationtechniques as in Abreu et al. (1990); Athey and Bagwell (2008) to characterize thewhole set of equilibrium payoffs.Notwithstanding the conceptual contributions of the analysis, our work alsomakes a methodological contribution in that it provides a recipe for analyzingmechanism design problems with limited commitment with transferable utility.Indeed, the four steps we described above follow very closely the prototypicalsteps in classical mechanism design: (i) invoke the revelation principle to simplifythe class of mechanisms and the agent’s behavior, (ii) find the binding constraints,(iii) maximize the virtual surplus, and (iv) show the solution satisfies any con-straints that may have been ignored. As will become clear from the analysis thatfollows, these four steps will also be necessary in other problems of mechanismdesign with limited commitment and transferable utility beyond the one we con-sider here. 4ur goal is to highlight the conceptual and technical nuances that arise in aninfinite-horizon mechanism selection game when the designer has limited commit-ment, and thus our focus on the case of binary valuations. This allows us to bringto the forefront the complexities that arise because of the designer’s rich actionspace. The extension to the case in which the buyer’s valuation is drawn from acontinuum is of interest, and we plan to address it in future work. When possible,we highlight in footnotes how the results presented extend to the continuum.
Related Literature:
The paper contributes mainly to four strands of literature.The first strand, similar to this paper, derives optimal mechanisms when the de-signer has limited commitment, but unlike this paper, these studies consider finitehorizon settings (see Laffont and Tirole (1988); Skreta (2006, 2015); Deb and Said(2015); Fiocco and Strausz (2015); Beccuti and M ¨oller (2018)). In these papers, theobservability of the agent’s choices in the mechanism makes it hard to disentanglethe principal’s choice of mechanism from the agent’s best response, which deter-mines, in turn, the principal’s continuation beliefs, and hence, his continuationpayoffs. In their seminal paper, to get around this difficulty, Bester and Strausz(2001) allow the agent to misrepresent her type with positive probability. Instead,Skreta (2006) circumvents the same problem by leveraging the properties of theset of incentive feasible outcomes and the finite horizon, which pins down theoptimal mechanism in the final period, to study the implications of the seller’ssequential rationality constraints for the set of incentive feasible outcomes.Our approach allows us to reduce the seller’s search for the optimum to a setof mechanisms that satisfy the truthtelling and participation constraints of thebuyer, without having to impose any restrictions on the horizon of the game. Thissimplifies the analysis by allowing us, for the most part, to ignore the buyer asa player. This approach relies on endowing the seller with a larger set of mech-anisms, which comes “at the cost” of having to rule out as optimal a richer setof possibilities. As we discuss Section 3.2, the richer set of mechanisms and noexogenous deadline on the interaction between the seller and the buyer, make theanalysis differ from the case in which the horizon is finite.The second strand, similar to this paper, studies infinite horizon problems inwhich the designer has limited commitment and faces an agent with one of finitelymany types. Given the difficulties with the revelation principle when the de- Beccuti and M ¨oller (2018) lie somewhat in between these two strands because they take limitsof their finite horizon results to draw conclusions about the infinite horizon setting. However, and establish (under some conditions) Coase’s conjecture. Related to this literature is the problem of dynamic bargaining with one-sided in-complete information, where an uninformed proposer each period makes a take-it-or-leave-it offer to a privately informed receiver. Sobel and Takahashi (1983),Fudenberg et al. (1985), and Ausubel and Deneckere (1989) characterize the equi-libria of this game for the case of finite horizon, infinite horizon and gap, infinitehorizon and no gap, respectively. In an analogous vein, Burguet and Sakovics(1996), McAfee and Vincent (1997), Caillaud and Mezzetti (2004), and Liu et al.(2019) study equilibrium reserve-price dynamics without commitment in differ-ent settings. The common thread in all these papers is that the seller’s inabilityto commit reduces monopoly profits.The last strand is the literature on games with time-inconsistent preferences,which builds on the seminal works of Strotz (1955), Peleg and Yaari (1973), and they do not show that this limit corresponds to the seller’s revenue-maximizing equilibrium in theinfinite horizon game. Even when the seller can commit not to revise prices, price dynamics can arise if there aredemand or cost changes. See, for example, Stokey (1979), Conlisk et al. (1984), and Garrett (2016). There is also a literature that analyzes different variations on the sale of a durable good modelto understand whether Coase’s conjecture still obtains. See, for instance, McAfee and Wiseman(2008); Board and Pycia (2014). More recently, Peski (2019) studies alternating bargaining games where players can offermenus. Dilm´e and Li (Forthcoming) study revenue management when the seller cannot commit torevise prices.
Organization
The rest of the paper is organized as follows. Section 2 describesthe model; Section 2.1 summarizes the results in Doval and Skreta (2018) used tosimplify the analysis that follows. Section 3 formally states the main result and de-scribes the main steps in the proof of Theorem 1. Section 3.1 derives the recursiveformulation that is the basis of the intrapersonal game; Section 3.2 characterizesthe unique equilibrium of the intrapersonal game; Section 3.3 uses the intraper-sonal equilibrium to build a PBE assessment that delivers the revenue-maximizingPBE. Section 4 concludes.
Primitives:
Two players, a seller and a buyer, interact over infinitely many pe-riods. The seller owns one unit of a durable good to which he attaches value 0.The buyer has private information: before her interaction with the seller starts,she observes her valuation v ∈ { v L , v H } ≡ V , with 0 ≤ v L < v H . Let µ denotethe probability that the buyer’s valuation is v H at the beginning of the game. Inwhat follows, we denote by ∆ ( V ) the set of distributions on V .An allocation in period t is a pair ( q , x ) ∈ {
0, 1 } × R , where q indicates whetherthe good is traded ( q =
1) or not ( q = x is a payment from the buyer to theseller. The game ends the first time the good is traded.Payoffs are as follows. If in period t , the allocation is ( q , x ) , the flow payoffs are u B ( q , x , v ) = vq − x and u S ( q , x ) = x for the buyer and the seller, respectively. Theseller and the buyer maximize the expected discounted sum of flow payoffs. Theyshare a common discount factor δ ∈ (
0, 1 ) . Mechanisms:
To introduce the timing of the game, we first define the actionspace of the seller in each period. In each period, the seller offers the buyer a mech-7nism. Following Myerson (1982); Bester and Strausz (2007); Doval and Skreta(2018), we define a mechanism as follows. A mechanism, M = h ( M M , β M , S M ) , ( q M , x M ) i ,consists of a communication device β M , which maps an input message m ∈ M M to a distribution with finite support on S M , and an allocation rule ( q M , x M ) , whichmaps each element s in S M to a probability of trade, q M ( µ ′ ) , and a payment fromthe buyer to the seller, x M ( µ ′ ) . We endow the seller with a collection ( M i , S i ) i ∈I of input and output messages in which each M i is finite, contains at least two ele-ments, and each S i contains ∆ ( M i ) . Denote by M . the set of all mechanisms withmessage sets ( M i , S i ) i ∈I . Timing:
If in period t , the good is yet to be traded, then t . 0 Both players observe a draw from a public randomization device ω ∈ [
0, 1 ] . t . 1 The seller offers the buyer a mechanism M = ( h M M , β M , S M , i , ( q M , x M )) . t . 2 The buyer observes the mechanism and decides to participate ( p =
1) or not( p = t . 2.1 If the buyer does not participate in the mechanism, the good is not soldand no payments are made; that is, the allocation is ( q , x ) = (
0, 0 ) . t . 2.2 If the buyer participates in the mechanism,i. She chooses a report m ∈ M M , which is unobserved by the seller,ii. An output message s ∈ S M is drawn from β M ( ·| m ) , which, in turn,determines the probability of trade q M ( s ) , and the payment, x M ( s ) .The output message and the allocation are observed by both the sellerand the buyer. If the good is not traded, the game proceeds to period t + A priori, we could have allowed a mechanism to offer a randomization over allocations, i.e. arandomization over {
0, 1 } × R . However, because both players have quasilinear payoffs, the sellerdoes not benefit from randomizing over transfers (see Lemma II.1 in Doval and Skreta (2019) for aproof). He may, however, benefit from randomizing over whether the good is traded. We restrict the seller to choose mechanisms with input and output messages in ( M i , S i ) i ∈I tohave a well-defined action space for the seller. This allows us to have a well-defined set of devia-tions, avoiding set-theoretic issues related to self-referential sets. The analysis in Doval and Skreta(2018) shows that the choice of the collection plays no further role in the analysis. Public randomization devices are standard in repeated games (see, e.g.,Fudenberg and Maskin (1986)). G ∞ M ( µ ) to remind the reader that the seller’s beliefs at the beginningof the game are given by µ . Public histories in this game are h t = ( ω , M , p , s , ( q , x ) , . . . , ω t − , M t − , p t − , s t − , ( q t − , x t − ) , ω t ) ,where p k ∈ {
0, 1 } denotes the buyer’s participation decision, with the restrictionthat if p k =
0, then s k = ∅ , i.e., no signal is generated, and ( q k , x k ) = (
0, 0 ) . Let H t denote the set of all period t public histories; they capture what the seller knowsthrough period t . A strategy for the seller is then Γ : ∪ ∞ t = H t ∆ ( M C ) .A history for the buyer consists of the public history of the game together withthe buyer’s reports into the mechanism (henceforth, the buyer history) and herprivate information. Formally, a buyer history is an element h tB = ( ω , M , p , m , s , ( q , x ) , . . . , ω t − , M t − , p t − , m t − , s t − , ( q t − , x t − ) , ω t ) .Let H tB denote the set of buyer histories through period t . The buyer also knowsher valuation and hence, when her valuation is v , a history through period t is anelement of { v } × H tB . The buyer’s participation strategy is π v : ∪ ∞ t = ( H tB × M ) [
0, 1 ] . Conditional on participating in the mechanism M , her reporting strategy isa distribution r v ( h tB , M , 1 ) ∈ ∆ ( M M ) for each of her types v and each h tB ∈ H tB .A belief for the seller at the beginning of time t , history h t , is a distribution µ ( h t ) ∈ ∆ ( V × H tB ( h t )) , where H tB ( h t ) is the set of buyer histories consistent withthe public history, h t . Solution concept:
We are interested in studying the Perfect Bayesian equilib-rium (henceforth, PBE) payoffs of this game, where PBE is defined as follows. Anassessment, h Γ , ( π v , r v ) v ∈ V , µ i , is a PBE if the following hold:1. Given µ ( h t ) , Γ ( h t ) is sequentially rational given ( π v , r v ) v ∈ V ,2. Given Γ ( h t ) , π v ( h tB , · ) , r v ( h tB , · , 1 ) are sequentially rational for all h tB ∈ H tB and v ∈ V ,3. µ ( h t ) is derived via Bayes’ rule where possible (see Definition 4 in Section A.1). While there is no output message when the buyer does not participate in the mechanism, wedenote this by s = ∅ to keep the length of all histories the same. Section A.1 contains the formal statement of Bayes’ rule where possible. To wit, we impose the
9e denote by E ∗M ( µ ) ⊆ R the set of PBE payoffs of G ∞ M ( µ ) and we use ( u S , u H , u L ) to denote a generic element of E ∗M ( µ ) , where u S is the seller’s payoffand u H , u L denote the buyer’s payoff when her type is v H , v L , respectively.The seller’s highest equilibrium payoff in G ∞ M ( µ ) is of particular interest to us: u ∗ S ( µ ) = max { u S : ( u S , u H , u L ) ∈ E ∗M ( µ ) } . (1)Theorem 1 characterizes u ∗ S ( µ ) and an assessment that achieves it for all µ ∈ ∆ ( V ) , which we denote by h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i . There are at least two reasons that the game G ∞ M ( µ ) is not a simple to analyze.First, the seller’s action space is large and, a priori, it is not clear which mecha-nisms could be ruled out from consideration. Second, fixed a seller’s strategy, andhence a sequence of mechanisms faced by the buyer, we still need to understandthe buyer’s best response in the game induced by the sequence of mechanisms.The main result of Doval and Skreta (2018) allows us to deal with these two com-plications by (i) pinning down a well-defined set of mechanisms that we can re-strict attention to, and (ii) pinning down the buyer’s behavior as a best responseto the sequence of mechanisms offered by the seller.Let M C denote the set of all mechanisms where ( M , S ) = ( V , ∆ ( V )) . Foreshad-owing the result in Lemma 1, we label these mechanisms canonical . Let G ∞ ( µ ) de-note the same game in the previous section, except that in each period the seller’saction space is M C and let E ∗ ( µ ) denote the set of equilibrium payoffs of G ∞ ( µ ) .In Doval and Skreta (2018), we show that to characterize the set of equilibriumpayoffs of G ∞ M ( µ ) , it is enough to characterize the set of equilibrium payoffs of G ∞ ( µ ) . Moreover, we also show that it is without loss of generality to focus ona particular class of assessments to characterize all equilibrium payoffs in E ∗ ( µ ) .Lemma 1 below summarizes these results for future reference: following requirement. Let h and h ′ be two consecutive information sets for the seller. The beliefsat h ′ are obtained via Bayes’ rule from the beliefs at h if one of the following holds: (i) conditionalon reaching h , h ′ is reached with positive probability under the equilibrium strategy profile, or (ii)conditional on reaching h , h ′ is reached from h through a deviation by the seller and the buyerplaying the equilibrium strategy profile. emma 1 (Doval and Skreta (2018)) . G ∞ M ( µ ) and G ∞ ( µ ) have the same set of equi-librium payoffs, i.e., E ∗M ( µ ) = E ∗ ( µ ) .Moreover, let ( u S , u H , u L ) ∈ E ∗ ( µ ) . Then, there exists a PBE assessment h Γ , ( π v , r v ) v ∈ V , µ i of G ∞ ( µ ) that achieves payoff ( u S , u H , u L ) and satisfies the following properties:1. For all histories h t , the buyer participates in the mechanism offered by the seller atthat history and truthfully reports her type, with probability ,2. For all histories h t , if the mechanism offered by the seller at h t outputs posterior µ ′ ,the seller’s updated equilibrium beliefs about the buyer coincide with µ ′ ,3. The strategy of the buyer depends only on her private valuation and the public his-tory. Lemma 1 has two implications. First, it is without loss of generality to restrictthe seller to offering mechansims where the set of input messages are the set oftype reports and the set of output messages is the set of beliefs the seller has aboutthe buyer’s type. Second, it is without loss of generality to restrict attention toassessments that satisfy certain properties.Part 1 of Lemma 1 implies the mechanisms chosen by the seller in equilibriummust satisfy a participation constraint and an incentive compatibility constraint foreach buyer type and each public history. For completeness, these constraintsare stated in Section A.2; see Equation PC v , h t and Equation IC v , h t . As in the caseof commitment to long-term mechanisms, part 1 simplifies the analysis of thebuyer’s behavior, by reducing it to a series of constraints. Part 2 implies we caninterpret the seller’s choice of a communication device as a choice of a distributionover posteriors that satisfies a Bayes’ plausibility constraint (see Equation BC µ ( h t ) in Section A.2). As a consequence, our analysis involves aspects of informationdesign . Part 3 implies the set of PBE payoffs of G ∞ ( µ ) coincides with the set of Public PBE payoffs of G ∞ ( µ ) (Athey and Bagwell (2008)), allowing us to invokeself-generation techniques as in Abreu et al. (1990); Athey and Bagwell (2008) toargue the assessment we construct in Section 3.3 is indeed a PBE assessment.A corollary of Lemma 1 is that u ∗ S ( µ ) is also the highest equilibrium payoffthe seller can obtain in G ∞ ( µ ) . The rest of the paper focuses on studying theequilibrium payoffs of G ∞ ( µ ) and when we refer to a PBE assessment, we meanone that satisfies the conditions of Lemma 1.In what follows, we abuse notation in the following way. Because valuationsare binary, we can think of an element in ∆ ( V ) (a distribution over v L and v H )11s an element of the interval [
0, 1 ] (the probability assigned to v H ). We use thelatter formulation in what follows. That is, whereas the mechanism outputs adistribution over v L and v H , we index this distribution by the probability of v H . Section 3 contains the main result of the paper: the optimal mechanism for a sellerof a durable good is a sequence of posted prices. To state the result, we proceed asfollows. First, we define what it means for the seller to attain a certain equilibriumpayoff via a sequence of posted prices, when the seller’s action space consists ofmechanisms (Definition 1). This is enough to informally state our main result. Sec-ond, we describe a PBE assessment, h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i , that achieves u ∗ S ( µ ) andargue that, indeed, u ∗ S ( µ ) can be implemented via a sequence of posted prices.The rest of this section describes the main steps to prove h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i in-deed achieves u ∗ S ( µ ) . Definition 1.
Let u S denote a seller’s payoff in E ∗ ( µ ) . The payoff u S can be im-plemented via a sequence of posted prices if a PBE assessment h Γ , ( π v , r v ) v ∈ V , µ i existssuch that in each history h t , the seller’s mechanism M satisfies the following: thereexist µ ′ = µ ′ and a price p ( h t ) such that1. β M ( µ ′ | v L ) = β M ( µ ′ | v H ) = ( q M ( µ ′ ) , x M ( µ ′ )) = (
0, 0 ) and ( q M ( µ ′ ) , x M ( µ ′ )) = ( p ( h t )) .Thus, a mechanism is a posted price whenever it can be implemented as if thebuyer could choose one of two options: either she buys the good at price p ( h t ) , orshe does not buy the good, in which case she pays nothing. Note that when theseller posts a price in each period, he does not use noisy communication devices:by observing the posterior, the seller knows which report the buyer submitted intothe mechanism. However, the buyer’s report need not be truthful in this implemen-tation. Theorem 1.
Let u ∗ S ( µ ) denote the seller’s maximum revenue in a PBE. Then, u ∗ S ( µ ) can be implemented via a sequence of posted prices. Theorem 1 shows that, amongst all trading protocols, posted prices are optimal.This provides a microfoundation for the seller’s strategy space in the literaturethat studies the sale of a durable good. 12heorem 1 echoes the result for the case in which the seller has commitmentand faces one buyer, where it is also well known that a posted price is optimal.However, in the latter case, the seller would post the same price at all histories,which would fail to be sequentially rational in G ∞ ( µ ) . As we show in the analysisthat follows, we find that the seller’s mechanism determines a price path that isdecreasing in the seller’s belief that the buyer’s valuation is v H , as in the literaturethat analyzes the Coase conjecture. PBE assessment that achieves u ∗ S ( µ ) : The proof of Theorem 1 singles out a par-ticular PBE assessment, h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i , that achieves u ∗ S ( µ ) , which we nowdefine informally (the formal statement is in Section E.2). After defining it, weuse it to argue that u ∗ s ( µ ) can be implemented via a sequence of posted prices.In what follows, the ratio v L / v H is important and we denote it by µ . To see why µ is important, recall the following fact from second-degree price discrimination:when the seller sells at a price of v L , he leaves rents, ∆ v ≡ v H − v L , to v H , so thatwhen v H has probability µ , the seller’s revenue, v L , can be written as v L = µ ( v H − ∆ v ) + ( − µ ) v L = µ v H + ( − µ )( v L − µ − µ ∆ v ) = µ v H + ( − µ ) ˆ v L ( µ ) .The first equality represents revenue as the surplus extracted from each type. Thesecond equality represents revenue as the virtual surplus , where the value of allo-cating the good to v L is adjusted to capture the fact that when v L is served, so is v H , which leaves rents to v H . µ is the belief at which ˆ v L ( µ ) =
0. At that belief,the seller is indifferent between serving the buyer for both of her valuations andexcluding the low-valuation buyer.We focus here and in the rest of this section on the case in which v L >
0; wereturn to the (less interesting) case of v L = v L >
0, we show a sequence 0 = µ < µ = v L / v H < · · · < µ n < . . . exists suchthat the following holds: Chapter 10 in Fudenberg and Tirole (1991) solves for the equilibrium of the price posting gamefor the binary type case. Hart and Tirole (1988) compare price paths and the seller’s revenue acrossdifferent sale modes (sale vs. leasing) in a finite-horizon setting with limited commitment andbinary valuations. As in their model with sale, the optimal price path in our model is determinedby the beliefs at which the seller is indifferent between trading with the low-valuation buyer in n versus n +
13. Along the equilibrium path:(a) If at history h t , the seller’s beliefs, µ ∗ ( h t ) , are in [ µ , µ ) , he chooses amechanism such that ( q M ∗ t ( µ ∗ ( h t )) , x M ∗ t ( µ ∗ ( h t ))) = ( v L ) and the com-munication device satisfies that β M ∗ t ( µ ∗ ( h t ) | v ) = v ∈ { v L , v H } .(b) If at history h t , the seller’s beliefs, µ ∗ ( h t ) , are in [ µ n , µ n + ) for n ≥
1, theseller’s mechanism satisfies the following. First, it induces two posteriors, µ n − and 1. Second, the allocation rule satisfies that ( q M ∗ t ( ) , x M ∗ t ( )) =( v L + ( − δ n ) ∆ v ) , whereas ( q M ∗ t ( µ n − ) , x M ∗ t ( µ n − )) = (
0, 0 ) . Finally, thecommunication device maps v L to µ n − , whereas it maps v H to both µ n − and 1 with positive probability. The probabilities β M ∗ t ( µ n − | v H ) , β M ∗ t ( | v H ) are chosen so that when the seller observes µ n − , his updated belief coin-cides with µ n − .2. Off the equilibrium path, the seller’s strategy coincides with the above, exceptthat when µ ∗ ( h t ) = µ n for some n ≥
1, the seller may randomize between themechanism he offers on the path of play when his belief is µ n and the one heoffers on the path of play when his belief is µ n − .
3. At each history h t , the buyer’s best response to the seller’s equilibrium offer at h t is to participate in the mechanism and truthfully report her valuation. Implementation via posted prices:
We now argue that if the assessment de-scribed above achieves u ∗ S ( µ ) , then u ∗ S ( µ ) can be achieved via a sequence ofposted prices. Clearly, when the seller’s beliefs are below µ , the seller’s mecha-nism corresponds to selling the good at a price of v L . Consider then the case inwhich the seller’s beliefs are in [ µ , µ ) . Note that when the buyer’s valuation is v H and the realized allocation is trade, then her payoff is v H − v L − ( − δ ) ∆ v = δ ∆ v .On the other hand, when the buyer’s valuation is v H and the realized allocationis no trade, then the seller’s beliefs next period are µ =
0, so that the buyer’scontinuation payoff is δ ∆ v . That is, the buyer with valuation v H is indifferentbetween obtaining the good at price v L + ( − δ ) ∆ v and not obtaining the good,and paying a price of v L in the next period. Since the buyer with valuation is v H The need for mixing arises for technical reasons: it ensures that the buyer’s continuationpayoffs when her valuation is v H are upper-semicontinuous and, thus, guarantees that a best re-sponse exists after any deviation by the seller (see Section E.1). Indeed, we appeal to the resultsin Simon and Zame (1990) to simultaneously determine the buyer’s best response and the seller’smixing.
14s indifferent between these two options, she is willing to mix between buyingat price v L + ( − δ ) ∆ v and not obtaining the good. She does so in a way thatthe seller’s belief is µ when the allocation is (
0, 0 ) . Since µ =
0, it implies thatwhen the seller’s belief is in [ µ , µ ) , the buyer buys with probability 1 at a priceof v L + ( − δ ) ∆ v . Working recursively through the equations, one can show thatwhen the seller’s prior is in [ µ n , µ n + ) , the mechanism is equivalent to posting aprice of v L + ( − δ n ) ∆ v . In this case, the low-valuation buyer chooses the (
0, 0 ) allocation, whereas the high-valuation buyer mixes so that the seller’s belief is µ n − when the allocation is (
0, 0 ) .It is interesting to contrast the implementation under the PBE assessment h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i described above with the implementation via posted prices.In the former, the buyer is truthful and the seller rations the high valuation buyeras in Denicolo and Garella (1999), which slows down the rate at which the seller’sbeliefs fall conditional on the good not being sold. Instead, in the implementa-tion via posted prices, the high valuation buyer misreports her type with positiveprobability, which, like rationing, prevents the seller from becoming pessimistictoo quickly about the buyer’s valuation. Since both implementations are payoffequivalent, the seller cannot do better with rationing than with posted prices.The rest of this section describes the steps to show the assessment, h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i ,achieves u ∗ S ( µ ) . Section 3.1 derives necessary conditions that the PBE assessment, h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i , must satisfy. They are the basis for the formulation of a re-cursive problem, the intrapersonal game, studied in Section 3.2. The main outputof Section 3.2 is a mapping γ ∗ : ∆ ( V )
7→ M C , which describes for each prior µ ′ the seller may have about the buyer, his optimal choice of mechanism. Section3.3 discusses then how we use γ ∗ to construct h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i . Finally, weappeal to self-generation techniques to show h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i is indeed a PBEassessment. The main output of Section 3.1 is Problem 1 at the end of this section, which relieson Proposition 1 below. To introduce Proposition 1, we introduce the definition ofincentive efficiency (see Bester and Strausz (2001)): Incentive efficiency as defined in Bester and Strausz (2001) is related, but different from,incentive efficiency as defined in Holmstr ¨om and Myerson (1983). While the definition inHolmstr ¨om and Myerson (1983) would allow for Pareto improvements for both the buyer and efinition 2 (Incentive efficiency) . Let h Γ , ( π v , r v ) v ∈ V , µ i be a PBE assessment and ( u S , u H , u L ) its associated payoff. h Γ , ( π v , r v ) v ∈ V , µ i is incentive efficient if no u ′ S > u S exists such that ( u ′ S , u H , u L ) ∈ E ∗ ( µ ) .In an incentive efficient PBE, the seller at the beginning of the game is earninghis best payoff consistent with the buyer’s equilibrium payoff. Proposition 1. If h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i is the PBE assessment that delivers the seller’sbest payoff in E ∗ ( µ ) , u ∗ S ( µ ) , the following hold:1. Without loss of generality, if the buyer rejects the seller’s equilibrium choice of mech-anism at history h t , the seller assigns probability 1 to the buyer’s valuation beingv H .2. For every history on the path of play, h t , h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i| h t is incentive effi-cient.3. For all histories on the path of ( Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V ) , the buyer is indifferent betweenparticipating in the mechanism and not when her valuation is v L .4. For all histories on the path of ( Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V ) , the buyer is indifferent betweenreporting v H and v L when her valuation is v H . Part 1 implies that it is without loss of generality to assume that the buyer’svalue of rejecting the seller’s equilibrium choice of mechanism is 0: when theseller assigns probability 1 to v H , the seller asks for a payment of v H . Part 1 fol-lows from two observations: (i) By Lemma 1, the buyer’s rejection of the seller’sequilibrium offer is an off the path event, so that the seller’s beliefs are not pinneddown by Bayes’ rule, and (ii) by setting the beliefs of the seller to 1 on that event,we only relax the buyer’s participation constraint without affecting the seller’sincentives. Part 2 is a consequence of the result that it is without loss of generality to focuson Public PBE. While in general it is not true that h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i| h t is a PBEassessment of G ∞ ( µ ∗ ( h t )) , this is indeed the case when h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i is aPublic PBE assessment (see Proposition I.1 in Doval and Skreta (2019)). Part 2then says that along the equilibrium path the seller’s continuation payoff is the the seller, the definition in Bester and Strausz (2001) only considers Pareto improvements for theseller. Part 1 of Proposition 1 is not a consequence of using PBE as opposed to sequential equilibrium,since the buyer could always tremble and reject the mechanism with a higher probability whenher valuation is v H . consistent with the buyer’s equilibrium payoff. Thus,while the seller may not be earning u ∗ S ( µ ∗ ( h t )) for a history h t on the equilibriumpath, the payoffs starting from h t are the highest equilibrium payoffs consistentwith the buyer’s payoff at h t .Parts 3 and 4 imply that the standard observations from second-degree pricediscrimination hold along the path of play of h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i , namely, thatall surplus is extracted from the buyer when her valuation is v L and the buyer’spayoff coincides with her information rents when her valuation is v H . Clearly,they must hold at the initial history, because either of them failing implies theseller is “leaving money on the table.” That they hold after every history on thepath of play is a result of discounting and the linearity of payoffs in transfers,which allows us to distribute the players’ payoffs across time without affecting(and sometimes even increasing) payoffs at some history. It follows from parts 3 and 4 that the participation constraint for v L and theincentive compatibility constraint for v H bind along the equilibrium path. Recallthat when the seller has commitment to long-term mechanisms, these constraintsare the ones that we use to replace the transfers out of the seller’s payoffs, so thatthey are expressed solely in terms of the allocation. In what follows, we showthat the same can be done when analyzing the revenue-maximizing PBE whenthe seller can only commit to short-term mechanisms.Let u ∗ S ( µ ) denote the seller’s highest equilibrium payoff in G ∞ ( µ ) and let h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i denote the PBE assessment that achieves it. Lemma 1 impliesthat we can write u ∗ S ( µ ) as follows: u ∗ S ( µ ) = ∑ v ∈ V µ ( v ) ∑ µ ′ ∈ ∆ ( V ) β M ∗ ( µ ′ | v )[ x M ∗ ( µ ′ ) + δ ( − q M ∗ ( µ ′ )) U ∗ S ( M ∗ , 1, µ ′ , 0, x M ∗ )] , (2) Parts 2-4 may sound counterintuitive from a dynamic game perspective: there are gameswhere, to sustain high payoffs today, low continuation payoffs are needed, even on the equilib-rium path. Importantly, when we prove Proposition 1, we show that when we modify the strategyprofile from history h t onwards, we do not upset the equilibrium constraints at the histories thatprecede h t , thereby showing that the new assessment is also a PBE assessment. Although with binary types, the incentive constraint for v H binds as a result of revenue max-imization, with a continuum of types the adjacent downward-looking incentive constraints obtainbecause incentive compatibility implies the envelope representation of payoffs. In the online ap-pendix to Doval and Skreta (2018), we show how to obtain the envelope representation of payoffsin the abstract mechanism selection game we study there; it is immediate to show that it holdsin this game as well. Thus, with a continuum of types, we obtain a representation of the seller’spayoff similar to the one in this section. U ∗ S ( h ) denotes the seller’s payoffs under ( Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V ) from t = U ∗ S ( h ) is not necessarily u ∗ S ( µ ∗ ( h )) and hence the differencein notation.) By definition, u ∗ S ( µ ) is the highest payoff the seller can achieve,given the continuation strategy, from among all mechanisms M ∈ M C that sat-isfy the participation and incentive compatibility constraints at the initial history.Proposition 1 implies M ∗ satisfies the following three properties. First, the buyer’spayoff is 0 when her valuation is v L , so that the following holds: ∑ µ ′ ∈ ∆ ( V ) β M ∗ ( µ ′ | v L ) v L q M ∗ ( µ ′ ) = ∑ µ ′ ∈ ∆ ( V ) β M ∗ ( µ ′ | v L ) x M ∗ ( µ ′ ) , (3)where we have already replaced the buyer’s continuation value with 0. Second,when the buyer’s valuation is v H , she is indifferent between reporting v H and v L ,so that the following equality holds: ∑ µ ′ ∈ ∆ ( V ) ( β M ∗ ( µ ′ | v H ) − β M ∗ ( µ ′ | v L ))( v H q M ∗ ( µ ′ ) + ( − q M ∗ ( µ ′ )) U ∗ H | L ( h ))= ∑ µ ′ ∈ ∆ ( V ) ( β M ∗ ( µ ′ | v H ) − β M ∗ ( µ ′ | v L )) x M ∗ ( µ ′ ) , (4)where the notation U ∗ H | L ( h ) signifies that the continuation value of a buyer of val-uation v H is the utility that she would obtain from the allocation that correspondsto v L – these are her information rents. Finally, the induced distribution overposteriors averages out to µ . That is, letting τ M ∗ ( µ , µ ′ ) = ∑ v ∈ V µ ( v ) β M ∗ ( µ ′ | v ) denote the probability that posterior µ ′ is induced, we have ∑ µ ′ ∈ ∆ ( V ) τ M ∗ ( µ , µ ′ ) µ ′ = µ . (5) Replacing transfers out of seller’s payoff:
We can replace equations 3-5 in Equation 2to obtain u ∗ S ( µ ) = ∑ µ ′ ∈ ∆ ( V ) τ M ∗ ( µ , µ ′ ) (cid:2) q M ∗ ( µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ )) + ( − q M ∗ ( µ ′ )) × δ (cid:16) U ∗ S ( h ) + ( µ ′ − µ ′ − µ − µ )( − µ ′ ) U ∗ H | L ( h ) (cid:17)i , (6) where ˆ v L ( µ ) = v L − µ − µ ∆ v ,18s the buyer’s virtual value when her valuation is v L and the seller assigns proba-bility µ to the buyer’s valuation being v H .Equation 6 says that we can think of the seller’s optimal mechanism at t = τ M ∗ , and, for each posterior he in-duces, a probability of trade, q M ∗ . If at a given posterior, µ ′ , he sells the good(i.e., q M ∗ ( µ ′ ) = virtual surplus , where the expectation istaken with respect to µ ′ , but the low-valuation buyer’s virtual value is evaluatedat µ . That the low-valuation buyer’s virtual value is evaluated at µ instead of at µ ′ is intuititive. After all, the seller assigns probability µ to the buyer’s valuationbeing v H so that, whenever he sells to both buyer types (i.e., µ ′ > v H with probability µ . If at a given posterior, µ ′ , he delays trade (i.e., q M ∗ ( µ ′ ) = v H . Note thecontinuation rents enter with a term that depends on µ : the seller assigns proba-bility µ to v H , and hence that is the rate at which he pays rents (this time in termsof continuation values) to the buyer.It follows from Proposition 1 that an expression similar to that of Equation 6holds for any history h t on the path of ( Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V ) . That is, U ∗ S ( h t ) = ∑ µ ′ ∈ ∆ ( V ) τ M ∗ t ( µ ∗ ( h t ) , µ ′ ) (cid:2) q M ∗ t ( µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ ∗ ( h t ))) + ( − q M ∗ t ( µ ′ )) × δ (cid:16) U ∗ S ( h t + ) + ( µ ′ − µ ′ − µ ∗ ( h t ) − µ ∗ ( h t ) )( − µ ′ ) U ∗ H | L ( h t + ) (cid:17)i ,(7) where ∑ µ ′ ∈ ∆ ( V ) τ M ∗ t ( µ ∗ ( h t ) , µ ′ ) µ ′ = µ ∗ ( h t ) .Part 2 implies that this payoff is the best one the seller can achieve in E ∗ ( µ ∗ ( h t )) ,consistent with the buyer’s payoff from h t onwards. That is, from amongst all themechanisms that satisfy the participation and incentive compatibility constraintsgiven the buyer’s equilibrium payoffs, M ∗ t delivers the seller the highest payoff at h t .As in Equation 6, Equation 7 shows that the seller’s mechanism at h t can bethought of as a distribution over posteriors, τ M ∗ t , and a probability of trade, q M ∗ t . ( τ M ∗ t , q M ∗ t ) are chosen so as to maximize a version of the virtual surplus, evaluatedat the seller’s prior belief at h t , µ ∗ ( h t ) . As in the discussion of Equation 6, the Equation 7 is obtained by replacing the corresponding versions of equations 3-5 at history h t . v L is evaluated at µ ∗ ( h t ) because µ ∗ ( h t ) is the probability the sellerassigns to the buyer’s valuation being v H at h t .Moreover, by expanding the above expressions and using the Bayes’ consistencyconditions, we can show that if h t + = ( h t , M ∗ t , 1, µ ′ , ( x M ∗ t ( µ ′ ))) is a history onthe path of ( Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V ) , then: U ∗ S ( h t + ) + (cid:18) µ ′ − µ ′ − µ ∗ ( h t ) − µ ∗ ( h t ) (cid:19) ( − µ ′ ) U ∗ H | L ( h t + ) = ∑ ˜ µ ∈ ∆ ( V ) τ M ∗ t + ( µ ′ , ˜ µ ) h q M ∗ t + ( ˜ µ )( ˜ µ v H + ( − ˜ µ ) ˆ v L ( µ ∗ ( h t ))) + ( − q M ∗ t + ( ˜ µ )) × δ (cid:16) U ∗ S ( h t + ) + ( ˜ µ − ˜ µ − µ ∗ ( h t ) − µ ∗ ( h t ) )( − ˜ µ ) U ∗ H | L ( h t + ) (cid:17)i , (8) where we use that in history h t + the seller assigns probability µ ′ to the buyer’svaluation being v H .The above expression shows how the term (cid:16) µ ′ − µ ′ − µ ∗ ( h t ) − µ ∗ ( h t ) (cid:17) ( − µ ′ ) U ∗ H | L ( h t + ) adjusts the seller’s continuation values. If at history h t , the seller’s mechanismdoes not sell the good to the buyer with some probability, he understands thatthe buyer receives continuation rents U ∗ H | L ( h t + ) , which have to be reflected inthe payments of M ∗ t . Because the seller assigns probability µ ∗ ( h t ) to the buyer’svaluation being v H , he pays those rents with probability µ ∗ ( h t ) . Hence, he adjuststhe continuation values to reflect his perceived probability of paying these futurerents, so that from his perspective, the virtual value ˆ v L ( · ) is computed at µ ∗ ( h t ) atall periods after t .By contrast, Equation 7 evaluated at period t + µ ′ , he chooses his mechanism taking into account that he leaves rents withprobability µ ′ to the buyer. Whenever µ ′ = µ ∗ ( h t ) , the seller in period t + t does.One last implication of incentive efficiency is that, along the equilibrium path,the seller’s beliefs together with the buyer’s rents pin down the strategy profile: Corollary 1.
Let h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i be the PBE assessment that delivers payoff u ∗ S ( µ ) . If there are two histories, h τ , h t , on the path of play such that µ ∗ ( h t ) = µ ∗ ( h τ ) and U ∗ H | L ( h t ) = U ∗ H | L ( h τ ) , then Γ ∗ ( h τ ) and Γ ∗ ( h t ) are payoff equivalent forthe seller. 20orollary 1 implies that along the equilibrium path, we can think of U ∗ S ( h t ) and U ∗ H | L ( h t ) as functions of the seller’s belief µ ∗ ( h t ) . In a slight abuse of notation, wedenote them by U ∗ S ( µ ∗ ( h t )) and U ∗ H | L ( µ ∗ ( h t )) .Corollary 1, Equation 7, and Equation 8 are the basis for the recursive formula-tion we use to characterize the seller’s highest equilibrium payoff, u ∗ S ( µ ) . Indeed,let h t be a history such that the seller assigns probability µ ∗ to v H . Define R ∗ ( µ ′ , µ ∗ ) = U ∗ S ( µ ′ ) + ( µ ′ − µ ′ − µ ∗ − µ ∗ )( − µ ′ ) U ∗ H | L ( µ ′ ) ,and let R ∗ ( µ ∗ , µ ∗ ) = (9) ∑ µ ′ ∈ ∆ ( V ) τ M ∗ t ( µ ∗ , µ ′ )( q M ∗ t ( µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ ∗ )) + ( − q M ∗ t ( µ ′ )) δ R ∗ ( µ ′ , µ ∗ )) .Consider now the following problem: Problem 1.
Find a policy ( τ , q ) : ∆ ( V ) ∆ ( ∆ ( V )) × [
0, 1 ] ∆ ( V ) and a value func-tion R ( τ , q ) such that
1. For all µ ∈ ∆ ( V ) , R µ ′ τ ( µ , d µ ′ ) = µ ,2. For all µ , µ ′ ∈ ∆ ( V ) such that µ < R ( τ , q ) ( µ ′ , µ ) = Z h q ( µ ′ , ˜ µ )( ˜ µ v H + ( − ˜ µ ) ˆ v L ( µ )) + δ ( − q ( µ ′ , ˜ µ )) R ( τ , q ) ( ˜ µ , µ ) i τ ( µ ′ , d ˜ µ ) ,
3. For all 0 ≤ µ < R ( τ , q ) ( µ , µ ) ≥ Z h q ′ ( µ , µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ )) + δ ( − q ′ ( µ , µ ′ )) R ( τ , q ) ( µ ′ , µ ) i τ ′ ( µ , d µ ′ ) for all τ ′ ( µ , · ) ∈ ∆∆ ( V ) such that R µ ′ τ ′ ( µ , d µ ′ ) = µ and all q ′ ( µ , · ) : ∆ ( V ) [
0, 1 ] . For µ = q (
1, 1 ) = µ , a Bayes’ plausible distri-bution over posteriors (part 1) and a probability of trade for each induced poste-rior, and (ii) continuation values that are consistent with the policy (part 2). More-over, h ( τ , q ) , R ( τ , q ) i is a solution if for each prior belief the seller may hold, he In what follows, we abuse notation and denote q ( µ )( µ ′ ) ∈ [
0, 1 ] by q ( µ , µ ′ ) . ( τ , q ) , given the continuationvalues (part 3).Standard dynamic programming arguments imply that if we find a solution, h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i , to Problem 1, we will have found a solution to the system de-fined by Equation 7 for every h t on the path of play. Because this system of equa-tions satisfies the necessary conditions in Proposition 1, R ( τ ∗ , q ∗ ) ( µ , µ ) is an upperbound on u ∗ S ( µ ) . If we then show that there is a PBE that attains R ( τ ∗ , q ∗ ) ( µ , µ ) ,we have characterized the seller’s best PBE payoff using the solution to the recur-sive program. Problem 1 is not a decision problem, but a game . The players in this game repre-sent the seller holding different beliefs about the buyer’s valuation being v H . Eachplayer chooses a distribution over posteriors that averages out to his prior and, foreach posterior that is induced, a probability of trade. In this way, Problem 1 is aformal representation of an idea already present in the literature on the sale of adurable good: when the seller cannot commit not to lower the prices in the future,the seller today competes against his future selves. While at first glance Problem 1 looks like a dynamic programming formulation of the seller’sproblem in Equation 6, it is not. After all we cannot apply Bellman’s principle of optimalitysince the strategy that is optimal for the seller at t = McAfee and Wiseman (2008) put forward this intuition in the introduction to their paper. In asense, it already appears in Coase (1972) where Coase points out[. . . ] In these circumstances, why should the landowner continue to hold MQ offthe market? The original landowner could obviously improve his position by sellingmore land since he could by these means acquire more money. It is true that thiswould reduce the value of the land OM owned by those who had previously boughtland from him loss would fall on them, not on him. decision problem : in the event that he does not sell the good, theseller does not need to consider that his mechanism may not be optimal given theinformation he has now learned.Notwithstanding the aforementioned difference, Problem 1 shares a useful sim-ilarity with the usual approach when the designer has commitment: in both cases,the buyer’s behavior has been reduced to a system of equations and is no longer aplayer. Moreover, under the assumption of transferable utility, these equationsallow us to express the seller’s problem only in terms of choosing the allocationand the communication device. This result is a consequence of the applicationof the revelation principle in Doval and Skreta (2018), and thus, we expect that asimilar simplification is feasible in settings other than the one we study here.Problems like Problem 1 have a long tradition in economics. Indeed, it is an ex-ample of an intrapersonal equilibrium (see Pollak (1968); Peleg and Yaari (1973);Harris and Laibson (2001); Bernheim et al. (2015)). For future reference, we recordthis observation in Definition 3 below.
Definition 3.
A policy ( τ , q ) : ∆ ( V ) ∆ ( ∆ ( V )) × [
0, 1 ] ∆ ( V ) and a value function R ( τ , q ) constitute an intrapersonal equilibrium if they solve Problem 1.The main result of this section shows that an intrapersonal equilibrium existsin the game that we study. This is stated formally in Theorem 2 below (note thesequence in the statement is the one described after the statement of Theorem 1). Theorem 2.
There exists a unique intrapersonal equilibrium, h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i . Indeed,there exists a sequence = µ < µ = v L / v H < · · · < µ n < . . . such that if µ ∈ [ µ i , µ i + ) , Contrast this to the approach in Freixas et al. (1985); Laffont and Tirole (1987, 1988);Bester and Strausz (2001); Gerardi and Maestri (2018) among others, where the designer can offermenus of contracts and observes the agent’s choice out of the menu. In these papers, the offeredmenu together with the agent’s choice out of the menu determine the designer’s beliefs about theagent’s type. This makes it difficult to separate the design of the allocation from the “design” ofthe information that is revealed by the choice out of the menu. Indeed, the system defined by Equation 7 and Equation 8 has analogues in the work ofHarris and Laibson (2001) and Bernheim et al. (2015) on hyperbolic discounting. . and i = , then τ ∗ ( µ , µ ) = q ∗ ( µ , µ ) = , while2. if i ≥ , then τ ∗ ( µ , 1 ) = − τ ∗ ( µ , µ i − ) = ( µ − µ i − ) / ( − µ i − ) andq ∗ ( µ , 1 ) = = − q ∗ ( µ , µ i − ) . Theorem 2 says that when the seller’s prior is low (i.e., µ < µ ), he sells withprobability 1 and transmits no information, whereas when the seller’s prior is high(i.e., µ ≤ µ ), he induces two posteriors: one in which trade happens ( µ ′ =
1) andone in which trade is delayed ( µ ′ = µ i − ).Theorem 2 is the basis for the equilibrium constructed in Theorem 1. Indeed, thecommunication device and the probability of trade used by the seller to achievehis maximum PBE payoff are those from the intrapersonal equilibrium. Theorem 2,however, says nothing about the transfers or the buyer’s behavior in the PBE as-sessment. After all, in the intrapersonal game, the seller is the only player.The proof of Theorem 2 is constructive. The rest of this section sketches out themain steps and provides intuition for Theorem 2. The reader interested in under-standing how we move from the intrapersonal equilibrium to a PBE assessmentthat delivers payoff u ∗ S ( µ ) may skip straight to Section 3.3, where we tackle thelast step of our construction. Necessary conditions:
To show that an intrapersonal equilibrium exists andis unique, we proceed as follows. Proposition 2 and Proposition 3 below de-scribe necessary conditions that an intrapersonal equilibrium must satisfy. We usethese properties to build a candidate policy, ( τ ∗ , q ∗ ) , and its continuation values, R ( τ ∗ , q ∗ ) , and show they constitute an equilibrium. Proposition 2.
Let h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i denote an intrapersonal equilibrium. Then, thefollowing hold:1. If µ < µ , then q ∗ ( µ , µ ′ ) = for all µ ′ ∈ ∆ ( V ) ,2. If µ > µ , then the seller places positive probability on two beliefs, { µ D ( µ ) , 1 } ,and sets q ∗ ( µ , µ D ( µ )) = and q ∗ ( µ , 1 ) = . An incomplete intuition for Proposition 2 is as follows. To understand part 1,recall that the solution when the seller has commitment and his prior is below µ is to sell to both types of the buyer with probability 1. This is still a solution under More precisely, the communication device used by the seller is derived from the distributionover posteriors in the intrapersonal equilibrium. R ( τ ∗ , q ∗ ) ( µ , µ ) = max τ , q Z (cid:16) q ( µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ )) + ( − q ( µ ′ )) δ R ( τ ∗ , q ∗ ) ( µ ′ , µ ) (cid:17) τ ( d µ ′ )= max τ Z max { µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) , δ R ( τ ∗ , q ∗ ) ( µ ′ , µ ) } τ ( d µ ′ ) . (10) Indeed, given the continuation values, the seller can obtain whatever is best be-tween selling today (with value µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) ) and delaying trade (withvalue δ R ( τ ∗ , q ∗ ) ( µ ′ , µ ) ), by choosing q ( · ) appropriately.Equation 10 shows the seller’s problem is like an information design problem. Starting from an equilibrium h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i , it follows that the seller alwayshas a best response in which he uses at most two posteriors. Conditional on in-ducing two posteriors, it follows that the seller sets q = q = q = v H .By setting q = The reason this intuition is incomplete is that we have only argued that thereis a best response in which the seller uses at most two posteriors, not that every best response involves two posteriors. While the seller with belief µ is indiffer-ent between all his best responses, this does not mean that a seller with belief µ ′ is indifferent between all the solutions to Equation 10. Thus, starting froman intrapersonal equilibrium h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i , we cannot just replace the policy ( τ ∗ ( µ , · ) , q ∗ ( µ , · )) for the one that solves the maximization in Equation 10 andthat induces at most two posteriors. This may affect the best response conditionof a seller with belief µ ′ = µ that either (i) assigns positive probability to µ inthe candidate equilibrium and sees his payoff changed as a result of the changein the policy for µ , or (ii) does not assign positive probability to µ , but may nowprefer to do so.Key to our result is then to show that the seller uses at most two posteriorsfor any prior µ in any solution to Problem 1 (Proposition III.2 in Appendix III However, we cannot just invoke concavification-style arguments because the continuation val-ues, R ( τ ∗ , q ∗ ) , may fail to be upper semi-continuous (see the discussion at the end of this section). The linearity of the seller’s payoff in the allocation and in the posterior beliefs in the eventof trade matters for this argument. The same linearity obtains when the buyer’s valuations aredrawn from a continuum.
25n Doval and Skreta (2019)). Proposition C.1 implies that whenever the seller in-duces a posterior at which the allocation is q =
0, his beliefs drop, i.e., the inducedposterior is below the prior. That is, whenever the seller does not trade, he be-comes more pessimistic about the buyer’s valuation being high, which increaseshis temptation to lower the price to v L .Policies ( τ , q ) where the seller uses more than two posteriors could help theseller slow down the rate at which his beliefs drop. Consider, for instance, aseller with belief µ who induces three posteriors µ ′ < µ ′ < µ <
1. Any sellerwith prior µ ′ > µ who induces a posterior equal to µ prefers the ”three-way”split over the policy in which the seller with belief µ just induces posteriors µ ′ and 1, since the three-way split implies a slower decay in beliefs. This kind ofslow and probabilistic rationing exploits analogous forces as those in the construc-tion in Ausubel and Deneckere (1989) and is behind the optimality of rationing inDenicolo and Garella (1999). Ruling out that these more complex policies can bepart of an intrapersonal equilibrium is probably where the assumption of binaryvaluations matters the most. The proof of Proposition III.2 exploits the propertiesof intrapersonal equilibria where for each prior, µ , the seller induces at most twoposteriors listed in Proposition 3 below, to which we turn next.The result in Proposition 2 implies we can reduce the construction of the intrap-ersonal equilibrium to the construction of the belief µ D ( µ ) at which the sellersets q = µ > µ . Suppose h µ D ∗ , R µ D ∗ i is an intrapersonal equilib-rium, where R µ D ∗ are the continuation values implied by the policy that sets q ∗ ( µ , µ D ∗ ( µ )) =
0. The martingale property of beliefs implies τ ∗ ( µ , µ D ∗ ( µ )) = − µ − µ D ∗ ( µ ) ,so that part 2 of Problem 1 implies R µ D ∗ ( µ , µ ) = µ − µ D ∗ ( µ ) − µ D ∗ ( µ ) v H + − µ − µ D ∗ ( µ ) R µ D ∗ ( µ D ∗ ( µ ) , µ ) . (11)Proposition 3 characterizes the properties of µ D ∗ , under the assumption that suchan equilibrium exists: Proposition 3.
Suppose h µ D ∗ , R µ D ∗ i is an intrapersonal equilibrium. Then, the follow-ing hold:1. If µ < µ is such that µ D ∗ ( µ ) < µ , then µ D ∗ ( µ ) = . . If µ < µ , then there exists N µ < ∞ such that the seller’s belief drops below µ after N µ periods, i.e., µ ( N µ ) D ∗ ( µ ) ≤ µ .3. If µ < µ ′ , then µ D ∗ ( µ ) ≤ µ D ∗ ( µ ′ ) .4. If N µ = N µ ′ , then µ D ∗ ( µ ) = µ D ∗ ( µ ′ ) . The proof is in Appendix C. In what follows, we provide intuition for the re-sult, which in turn also provides intuition for the structure of the intrapersonalequilibrium in Theorem 2.Part 1 is immediate. Recall from Proposition 2 that a seller with prior µ ′ < µ trades immediately. Thus, if a seller with prior µ < µ sets µ D ∗ ( µ ) = ˜ µ ∈ ( µ ) ,he knows that he trades with v L in the next period. That is, R µ D ∗ ( ˜ µ , µ ) = ˜ µ v H +( − ˜ µ ) ˆ v L ( µ ) . By setting µ D ∗ ( µ ) =
0, the seller with prior µ maximizes theprobability of trading with the high-valuation buyer today.Part 2 says that trade with the buyer when her valuation is v H happens in finitelymany periods. It does not say, however, that the same is true when the buyer’svaluation is v L , unless 0 < µ . We now explain why the seller taking infinitelymany periods for his belief to update below µ , conditional on the event of notallocating the good, cannot be part of an intrapersonal equilibrium. Starting fromany prior, µ < µ , the repeated application of the function µ D ∗ induces a decreas-ing sequence of beliefs, µ m = µ ( m ) D ∗ ( µ ) , all of them above µ . But if µ m remainsabove µ for every finite m , the probability τ ∗ ( µ m , 1 ) becomes small. Furthermore,a seller with a prior µ m always has the possibility of placing weight µ m on 1 andthe remaining weight on 0. We show that for m large enough, this deviation isprofitable. Thus, whereas the seller with prior belief, µ , would benefit from hissuccessors never splitting their beliefs below µ , taking infinitely many periods toupdate below µ cannot be part of an intrapersonal equilibrium.Part 2 echoes the results in Fudenberg et al. (1985), Gul et al. (1986), and Ausubel and Deneckere(1989) that in the “gap” case, trade happens with both types of the buyer in finitelymany periods. We do not phrase it in these terms because when v L =
0, whichcorresponds to the no-gap case in our model, part 2 holds; however, in the uniqueintrapersonal equilibrium, the seller with prior µ = v L . Because once we reach a posterior of 0 beliefsdo not change, trade never happens with the low-valuation buyer. The formal arguments are reminiscent to those in De Fraja and Muthoo (2000) andCondorelli et al. (2016). µ < µ ′ , the seller with belief µ ′ chooses a higher posterior at which to delay trade than the seller with belief µ . This is intuitive. When 0 < µ , the rents for the high-valuation buyer aredetermined by how long it takes to trade with v L : the longer it takes, the lowerthe rents. The seller’s prior is also the probability with which he pays rents to thebuyer. Thus, the higher the prior, the higher the incentive for the seller to delaytrade with v L so as to make the rents for v H smaller. Part 4 is important in the construction of the intrapersonal equilibrium when0 < µ . We can classify the priors, µ , according to how many periods it takes for µ to drop below µ . That is, define D n = { µ ∈ ∆ ( V ) : µ ( n ) D ∗ ( µ ) = } , (12)where we appeal to Proposition 3 to say that the “final” belief at which the sellerupdates is 0 and we define D = { } . Part 4 says that if µ , µ ′ ∈ D n , they delaytrade at the same posterior. Otherwise, by setting µ D ∗ ( · ) to be min { µ D ∗ ( µ ) , µ D ∗ ( µ ′ ) } ,the seller (weakly) increases the probability of immediately trading with v H with-out affecting how long it takes to trade with v L (both µ D ∗ ( µ ) , µ D ∗ ( µ ′ ) correspondto beliefs for which the seller trades in n − v L .)Part 4 suggests the “smallest” element of D n is of relevance: this prior, denotedby µ n in what follows, is the “largest” prior in D n − and the “smallest” in D n .The equilibrium policy, ( τ ∗ , q ∗ ) , in Theorem 2 selects the policy of µ n so that it isan element of D n . We break the indifference between n and n − n . The sequence { µ n } n ≥ : We now construct the cutoffs µ n . We set µ = µ denotes the ratio v L / v H . Note that it satisfies that µ v H + ( − µ ) ˆ v L ( µ ) = µ v H + ( − µ ) δ ˆ v L ( µ ) | {z } R µ D ∗ ( µ ) , (13)because ˆ v L ( µ ) =
0. Equation 13 shows that, when his prior is µ , the seller isindifferent between trading today with both types of the buyer, or trading today Note that when µ =
0, the seller sets µ D ( µ ) = < µ : in this case, the seller neverpays rents to the buyer, so that for each prior, the seller sells to the high-valuation buyer with themaximum possible probability, µ . Technically, at this point, an infimum rather than a minimum. v H and waiting until tomorrow to trade with thebuyer whose type is v L . Set µ D ∗ ( µ ) = = µ and τ ( µ , 0 ) = − µ . That is,we break the indifference of the seller with belief µ between trading immediatelywith v L and delaying trade with v L for one period in favor of delaying trade.For n ≥
1, define inductively µ n + to be the prior of the seller such that µ n + − µ n − µ n v H + − µ n + − µ n δ R µ D ∗ ( µ n , µ n + ) = µ n + − µ n − − µ n − v H + − µ n + − µ n − δ R µ D ∗ ( µ n − , µ n + ) .(14) That is, when the seller’s prior is µ n + , he is indifferent between taking n + v L or taking n periods to trade with v L . Note that Equation 14only depends on the policy for µ m , m ≤ n , which has already been defined.Lemma C.3 in the Appendix shows { µ n } n ≥ is an increasing sequence. Thesecutoffs are precisely those in the statement of Theorem 2. Indeed, we set D n =[ µ n , µ n + ) for n ≥ µ D ∗ ( µ ) = µ n − for µ ∈ D n . The policy ( τ ∗ , q ∗ ) satisfiesall the properties of Proposition 3. We verify in Section C.3 that it constitutes anintrapersonal equilibrium. Uniqueness follows from the results in Appendix Cand Appendix III in Doval and Skreta (2019), and in particular, Proposition III.1,where we show that although seller incarnations with beliefs { µ n } n ≥ have mul-tiple best responses, the one specified above is the only one that can be part of anintrapersonal equilibrium.Although the intrapersonal game has a unique equilibrium, the game betweenthe seller and the buyer does not (see Appendix V in Doval and Skreta (2019)).This finding is a reflection of a deeper observation: we derived the intrapersonalgame from necessary conditions for revenue maximization, whereas not all equi-libria in the game between the seller and the buyer give the seller his best equilib-rium payoff. Tie-breaking and upper semicontinuity:
Before proceding with the construc-tion of the PBE assessment, h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i , we make a final observation aboutthe properties of the intrapersonal equilibrium. Figure 1 below illustrates thatwhen µ ∈ [ µ n , µ n + ) , the continuation values R ( τ ∗ , q ∗ ) ( · , µ ) fail to be upper semi-continuous at beliefs { µ k : k ≥ n + } : Lemma IV.2 in Doval and Skreta (2019) shows formally that upper semicontinuity fails at thesebeliefs. ′ δ R ( τ ∗ , q ∗ ) ( · , µ ) µ µ µ µ Figure 1: Illustration of the failure of upper-semicontinuity for µ ∈ [ µ , µ ) : blueempty circles indicate the discontinuityConceptually, the failure of upper semicontinuity illustrates that sellers withdifferent prior beliefs disagree about how ties should be broken. If µ ∈ [ µ n , µ n + ) ,the seller with belief µ finds that sellers with beliefs µ k , k ≥ n + v L . However, inducing posteriorsgreater than µ n + when µ < µ n + is never optimal: any weight on such posteriorscan be split between µ n − and 1, which are the optimal choices for the seller whenhis prior µ ∈ [ µ n , µ n + ) . This observation, together with the property that sellerincarnations with priors lower than µ break ties in favor of the seller with prior µ , guarantees an intrapersonal equilibrium exists.Note that conflicts in tie-breaking are usually an issue for equilibrium existencein the literature on intrapersonal games. A similar issue arises in sequential votinggames (see Duggan (2006) and the references therein). Two properties of our gameseem important in managing to sidestep these issues: (i) the agreement betweena seller with belief µ with how sellers with lower priors break ties, and (ii) the30eller with belief µ ∈ [ µ n , µ n + ) can always choose not to put a seller with belief µ ′ ≥ µ n + on the move. Section 3.3 describes how we use the intrapersonal equilibrium policy, ( τ ∗ , q ∗ ) ,to construct an assessment, h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i , that gives the seller his highestequilibrium payoff in G ∞ ( µ ) .Theorem 2 delivers an allocation rule and a distribution over posteriors for eachprior belief the seller may hold. We now use the seller’s policy in the intrapersonalequilibrium to construct a mapping, γ ∗ : ∆ ( V )
7→ M C , that assigns a mechanismto each belief the seller may have about the buyer (omitted details of this construc-tion are in Section D.1). Given the seller’s prior, µ , we can use the distributionover posteriors, τ ∗ ( µ , · ) , to construct a communication device β ∗ µ , which underthe assumption of truthtelling, implements the same distribution over posteriorsas τ ∗ . Indeed, if µ ∈ D n , β ∗ µ ( µ n − | v L ) = β ∗ µ ( µ n − | v H ) = µ n − τ ∗ ( µ , µ n − ) µ .Moreover, we can use the allocation rule q ∗ ( µ , · ) to construct the probability oftrade: q ∗ µ ( µ ′ ) = q ∗ ( µ , µ ′ ) for µ ′ ∈ { µ n − , 1 } .Finally, using Equations 3 and 4, we can use the allocation rule, q ∗ ( µ , · ) , and thecommunication device, β ∗ µ , to construct the transfers. Indeed, if µ ∈ D n , x ∗ µ ( µ n − ) = x ∗ µ ( ) = v L + ( − δ n ) ∆ v . (15)The mapping γ ∗ then maps each prior, µ , to h ( V , β ∗ µ , ∆ ( V )) , ( q ∗ µ ( · ) , x ∗ µ ( · )) i .Moreover, we construct the buyer’s rents under this mechanism as a function ofthe seller’s prior. Indeed, if µ ∈ D n , the buyer’s continuation payoff when hervaluation is v H is given by: u ∗ H ( µ ) = δ n ∆ v . (16)To construct a PBE assessment, we need to specify the buyer’s and the seller’sstrategy after every history. We do so in Sections E.1 and E.2 in Appendix E. In31articular, Section E.2 shows how to jointly construct a strategy for the seller, Γ ∗ ,and a system of beliefs, µ ∗ , such that after every history, the seller’s choice ofmechanism is determined by Γ ∗ ( h t ) = γ ∗ ( µ ∗ ( h t )) . Moreover, µ ∗ ( h t ) is consistentwith the buyer’s strategy constructed in Section E.1.The PBE assessment we construct has the following property at all histories h t . Ifat history h t , the seller has beliefs µ ∗ ( h t ) , the seller’s payoff is R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) and the buyer’s payoff is 0 if her valuation is v L and is u ∗ H ( µ ∗ ( h t )) if her valuationis v H (recall equation (16)). Section E.3 verifies that given these continuation pay-offs, at history h t − , neither the buyer nor the seller have a one-shot deviation fromthe equilibrium strategy profile. In the language of Abreu et al. (1990), the strat-egy at h t − , together with the continuation payoffs at h t , decompose the payoffs at h t − . However, without knowing whether ( R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) , 0, u ∗ H ( µ ∗ ( h t ))) is itself an equilibrium payoff in G ∞ ( µ ∗ ( h t )) , this is not enough to conclude h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i is a PBE assessment.In Doval and Skreta (2019), we lay out the definitions and statements needed toshow that self-generation techniques apply to our setting, so that the above stepsverify h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i is a PBE assessment. Ours is a dynamic incomplete in-formation game with persistent types, like the one studied in Athey and Bagwell(2008). The authors show that under a restriction on the solution concept, whichthey term Public PBE, equilibrium payoffs can be characterized using techniqueslike in Abreu et al. (1990). More precisely, a Public PBE is a PBE assessmentwhere each player’s strategy depends on the public history of the game and theircurrent private information.Although in general the set of Public PBE payoffs is a strict subset of the PBEpayoffs, it is not in the game we study. As we discussed in Section 3.1, it fol-lows from Doval and Skreta (2018) that it is without loss of generality to focus onPBE assessments where the buyer’s strategy depends only on her valuation andthe public history. Thus, self-generation techniques can be applied to our gameto characterize not only the seller’s best equilibrium payoff, but all equilibriumpayoffs in our game. To be sure, in Athey and Bagwell (2008), types are persistent, but not fully persistent as in ourgame. Cole and Kocherlakota (2001) introduce similar techniques, but in dynamic games with fullsupport . That is, the information structure in the game is such that no player can infer from hissignal realizations that another player has deviated. Notwithstanding this difficulty, Athey and Bagwell (2008) show that, under some conditions,the most collusive outcome can be achieved by using Public PBE strategy profiles.
32e close Section 3 by illustrating some qualitative features of the model bymeans of pictures and discussing the case v L = Illustrations:
Although the recursive nature of the equations defining the cut-offs { µ n } n ≥ make performing comparative statics somewhat difficult, in whatfollows, we reproduce the price path and the seller’s payoff as a function of theseller’s prior and the discount factor for a specific parameterization of the model.In particular, the figures are plotted using v L = v H = µ = Figure 2: Price dynamics as a function of seller’s prior – different discountfactors.Figure 2 plots the price dynamics as a function of the discount factor. Note thatfor low values of the discount factor ( δ = v H as long as his prior is above µ . At the other extreme, when the seller’s discount factor is high ( δ = More precisely, Figure 2 plots the seller’s initial price offer as a function of his prior. Given thestructure of the PBE assessment that we characterize, the equilibrium price dynamics can be readfrom the figure.
Figure 3: Seller’s payoff as a function of seller’s prior – different discount factors.Figure 3 may give the impression that the seller’s revenue is monotonically de-creasing in the discount factor for a given prior. It is not, however, as Figure 4shows: 34
Figure 4: Seller’s payoff as a function of the discount factor – different priors.
The case v L = v L = no gap case inAusubel and Deneckere (1989). The ‘no gap’ case refers to the possibility that theseller’s value is above the lowest possible buyer’s value (in our case this means v L ≤ v L =
0, the seller can sustain the full commit-ment profit even with limited commitment.In particular, at the seller optimal equilibrium revenue is µ v H , regardless ofthe discount factor (and not just in the limit as in Ausubel and Deneckere (1989)).From the perspective of the intrapersonal equilibrium, regardless of his belief, µ ,as long as µ >
0, the seller prefers that the seller with belief 0 breaks ties in favorof no trade. In that case, each seller with belief µ > v H because the “threat” of no further trade is credible.The reader may then be tempted to draw these conclusions more broadly. How-ever, that the seller can achieve the commitment profits when v L = Conclusions and further directions
This paper makes two contributions to the literature on mechanism design withlimited commitment. First, we characterize the optimal mechanism for the saleof a durable good when the seller has limited commitment and interacts with thebuyer for infinitely many periods and show that posted prices are optimal.Second, we provide a recipe for solving problems of mechanism design with lim-ited commitment with transferable utility, which applies to settings other than theones we study here. Indeed, the steps followed in the proof of our main resultclosely follow the standard steps in classical mechanism design. To wit, the pro-totypical procedure to analyze problems of mechanism design with commitmentand transferable utility follows these steps: (i) Apply the revelation principle tosimplify the space of mechanisms and the features of the agent’s strategy profile(participation and truthtelling), (ii) appeal to quasilinearity and single-crossing toderive an envelope representation of payoffs that replaces the transfers out of thedesigner’s payoff, (iii) solve the designer’s decision problem , and (iv) verify that thesolution satisfies all constraints that (may) have been ignored.Note that these steps are mainly the ones we followed in Section 3. Indeed,thanks to the result in our previous work, Doval and Skreta (2018), we are ableto reduce the buyer’s behavior to a series of participation and truthtelling con-straints, just as we do in the case of full commitment. The main difference is step(iii): with limited commitment, the designer’s problem is an intrapersonal game .This is a reflection of a deeper observation: optimal mechanisms under full com-mitment often fail to be sequentially rational, whereas the best response conditionof the intrapersonal equilibrium captures the restrictions imposed by sequentialrationality on the designer’s behavior.The analysis so far has focused on the case of binary valuations, which is con-sistent with recent papers in the literature that studies infinite-horizon principal-agent problems with limited commitment, like Strulovici (2017) and Gerardi and Maestri(2018). The case of binary valuations allows us to bring to the forefront the con-ceptual innovations that arise in an infinite horizon game between two long-runplayers, where one of them, the seller, has a rich action space. Nevertheless, theextension to the case in which buyer valuations are drawn from a continuum is ofinterest, and we plan to address it in future work.While the optimality of posted prices maybe specific to our setting, some of theforces we illustrate in the paper are more broadly applicable. Indeed, the richer set36f mechanisms we endow the seller with allow him to design how much he learnsabout the buyer’s type and also how to use this information (subject, of course, tothe buyer’s incentives). The seller’s ability to design his beliefs, and therefore, thedemand he faces is relevant in other settings, as is the case, for instance, when theseller interacts repeatedly with the consumer across different transactions. Whileour setting is not rich enough to capture some of the nuances introduced by re-peated purchases, we believe that future work can leverage the framework wehave developed to study these issues.
References A BREU , D., D. P
EARCE , AND
E. S
TACCHETTI (1990): “Toward a theory of dis-counted repeated games with imperfect monitoring,”
Econometrica , 58, 1041–1063.A
CHARYA , A.
AND
J. O
RTNER (2017): “Progressive learning,”
Econometrica , 85,1965–1990.A
THEY , S.
AND
K. B
AGWELL (2008): “Collusion with persistent cost shocks,”
Econometrica , 76, 493–540.A
USUBEL , L. M.
AND
R. J. D
ENECKERE (1989): “Reputation in bargaining anddurable goods monopoly,”
Econometrica , 511–531.B
ECCUTI , J.
AND
M. M ¨
OLLER (2018): “Dynamic adverse selection with a patientseller,”
Journal of Economic Theory , 173, 95–117.B
ERNHEIM , B. D., D. R AY , AND
S¸ . Y
ELTEKIN (2015): “Poverty and self-control,”
Econometrica , 83, 1877–1911.B
ESTER , H.
AND
R. S
TRAUSZ (2001): “Contracting with imperfect commitmentand the revelation principle: the single agent case,”
Econometrica , 69, 1077–1098.——— (2007): “Contracting with imperfect commitment and noisy communica-tion,”
Journal of Economic Theory , 136, 236–259.B J ¨ ORK , T.
AND
A. M
URGOCI (2014): “A theory of Markovian time-inconsistentstochastic control in discrete time,”
Finance and Stochastics , 18, 545–592.37
OARD , S.
AND
M. P
YCIA (2014): “Outside options and the failure of the Coaseconjecture,”
American Economic Review , 104, 656–71.B
ULOW , J. I. (1982): “Durable-goods monopolists,”
Journal of Political Economy , 90,314–332.B
URGUET , R.
AND
J. S
AKOVICS (1996): “Reserve prices without commitment,”
Games and Economic Behavior , 15, 149–164.C
AILLAUD , B.
AND
C. M
EZZETTI (2004): “Equilibrium reserve prices in sequen-tial ascending auctions,”
Journal of Economic Theory , 117, 78–95.C
OASE , R. H. (1972): “Durability and monopoly,”
The Journal of Law and Eco-nomics , 15, 143–149.C
OLE , H. L.
AND
N. K
OCHERLAKOTA (2001): “Dynamic games with hidden ac-tions and hidden states,”
Journal of Economic Theory , 98, 114–126.C
ONDORELLI , D., A. G
ALEOTTI , AND
L. R
ENOU (2016): “Bilateral trading in net-works,”
The Review of Economic Studies , 84, 82–105.C
ONLISK , J., E. G
ERSTNER , AND
J. S
OBEL (1984): “Cyclic pricing by a durablegoods monopolist,”
The Quarterly Journal of Economics , 99, 489–505.D E F RAJA , G.
AND
A. M
UTHOO (2000): “Equilibrium partner switching in a bar-gaining model with asymmetric information,”
International Economic Review , 41,849–869.D EB , R. AND
M. S
AID (2015): “Dynamic screening with limited commitment,”
Journal of Economic Theory , 159, 891–928.D
ENICOLO , V.
AND
P. G. G
ARELLA (1999): “Rationing in a durable goodsmonopoly,”
The RAND Journal of Economics , 44–55.D
ILM ´ E , F. AND
F. L I (Forthcoming): “Revenue Management without Commit-ment: Dynamic Pricing and Periodic Flash Sales,” The Review of Economic Stud-ies .D OVAL , L.
AND
V. S
KRETA (2018): “Mechanism Design with Limited Commit-ment,” arXiv preprint arXiv:1811.03579 .——— (2019): “Supplement to “Optimal mechanism for the sale of a durablegood”,” Click here. 38
UGGAN , J. (2006): “Endogenous voting agendas,”
Social Choice and Welfare , 27,495–530.F
IOCCO , R.
AND
R. S
TRAUSZ (2015): “Consumer standards as a strategic deviceto mitigate ratchet effects in dynamic regulation,”
Journal of Economics & Man-agement Strategy , 24, 550–569.F
REIXAS , X., R. G
UESNERIE , AND
J. T
IROLE (1985): “Planning under incompleteinformation and the ratchet effect,”
The Review of Economic Studies , 52, 173–191.F
UDENBERG , D., D. L
EVINE , AND
J. T
IROLE (1985): “Infinite horizon models ofbargaining with one-sided uncertainty,” in
Game Theoretic Models of Bargaining ,Cambridge University Press, vol. 73, 79.F
UDENBERG , D.
AND
E. M
ASKIN (1986): “The Folk Theorem in Repeated Gameswith Discounting or with Incomplete Information,”
Econometrica , 54, 533–554.F
UDENBERG , D.
AND
J. T
IROLE (1991):
Game theory , MIT Press.G
ARRETT , D. F. (2016): “Intertemporal price discrimination: Dynamic arrivalsand changing values,”
American Economic Review , 106, 3275–99.G
ERARDI , D.
AND
L. M
AESTRI (2018): “Dynamic Contracting with Limited Com-mitment and the Ratchet Effect,” Tech. rep.G UL , F., H. S ONNENSCHEIN , AND
R. W
ILSON (1986): “Foundations of dynamicmonopoly and the coase conjecture,”
Journal of Economic Theory , 39, 155 – 190.H
ARRIS , C.
AND
D. L
AIBSON (2001): “Dynamic choices of hyperbolic con-sumers,”
Econometrica , 69, 935–957.H
ART , O. D.
AND
J. T
IROLE (1988): “Contract renegotiation and Coasian dynam-ics,”
The Review of Economic Studies , 55, 509–540.H
OLMSTR ¨ OM , B. AND
R. B. M
YERSON (1983): “Efficient and durable decisionrules with incomplete information,”
Econometrica , 1799–1819.L
AFFONT , J.-J.
AND
J. T
IROLE (1987): “Comparative statics of the optimal dy-namic incentive contract,”
European Economic Review , 31, 901–926.——— (1988): “The dynamics of incentive contracts,”
Econometrica , 1153–1175.39—— (1990): “Adverse selection and renegotiation in procurement,”
The Reviewof Economic Studies , 57, 597–625.L IU , Q., K. M IERENDORFF , X. S HI , AND
W. Z
HONG (2019): “Auctions with lim-ited commitment,”
American Economic Review , 109, 876–910.M C A FEE , R. P.
AND
D. V
INCENT (1997): “Sequentially optimal auctions,”
Gamesand Economic Behavior , 18, 246–276.M C A FEE , R. P.
AND
T. W
ISEMAN (2008): “Capacity choice counters the Coaseconjecture,”
The Review of Economic Studies , 75, 317–331.M
YERSON , R. B. (1982): “Optimal coordination mechanisms in generalizedprincipal–agent problems,”
Journal of Mathematical Economics , 10, 67–81.O
RTNER , J. (2017): “Durable goods monopoly with stochastic costs,”
TheoreticalEconomics , 12, 817–861.P
ELEG , B.
AND
M. E. Y
AARI (1973): “On the existence of a consistent course ofaction when tastes are changing,”
The Review of Economic Studies , 40, 391–401.P
ESKI , M. (2019): “Alternating-offer bargaining with incomplete information andmechanisms,” .P
OLLAK , R. A. (1968): “Consistent planning,”
The Review of Economic Studies , 35,201–208.S
IMON , L. K.
AND
W. R. Z
AME (1990): “Discontinuous games and endogenoussharing rules,”
Econometrica , 861–872.S
KRETA , V. (2006): “Sequentially optimal mechanisms,”
The Review of EconomicStudies , 73, 1085–1111.——— (2015): “Optimal auction design under non-commitment,”
Journal of Eco-nomic Theory , 159, 854–890.S
OBEL , J. (1991): “Durable goods monopoly with entry of new consumers,”
Econo-metrica , 59, 1455–1485.S
OBEL , J.
AND
I. T
AKAHASHI (1983): “A multistage model of bargaining,”
TheReview of Economic Studies , 50, 411–426.40
TOKEY , N. L. (1979): “Intertemporal price discrimination,”
The Quarterly Journalof Economics , 355–371.——— (1981): “Rational expectations and durable goods pricing,”
The Bell Journalof Economics , 112–128.S
TROTZ , R. H. (1955): “Myopia and inconsistency in dynamic utility maximiza-tion,”
The Review of Economic Studies , 23, 165–180.S
TRULOVICI , B. (2017): “Contract negotiation and the Coase conjecture: A strate-gic foundation for renegotiation-proof contracts,”
Econometrica , 85, 585–616.V
IEILLE , N.
AND
J. W. W
EIBULL (2009): “Multiple solutions under quasi-exponential discounting,”
Economic Theory , 39, 513.
A Omitted formal statements and equations
A.1 Bayes’ rule where possible
In this section, we define formally what we mean by Bayes’ rule where possiblein Section 2. We do so using G ∞ ( µ ) , but it should be clear that the same appliesto G ∞ M ( µ ) . Note that in the game under consideration, the public history h t rep-resents an information set for the seller with nodes h tB ∈ H tB ( h t ) . In what follows,we use the notation y , y ′ to denote nodes in the game.Fix a strategy profile ( Γ , ( π v , r v ) v ∈ V ) , and two nodes y and y ′ such that y pre-cedes y ′ . We can use the strategy profile to define a probability, P ( Γ , ( π v , r v ) v ∈ V ) ( y ′ | y ) ,of reaching node y ′ conditional on being at node y . Extend this probability to allnodes by making it 0 for nodes y ′ that do not succeed y .Say that information set h t precedes information set h t + , or that h t , h t + areconsecutive information sets if there exists a mechanism, M , such that either ofthe following hold:(a) there is a posterior, µ ′ , such that ∑ v ∈ V β M ( µ ′ | v ) > and h t + = ( h t , M , 1, µ ′ , ( x M ( µ ′ )) , ω t + ) , or In Doval and Skreta (2018), we argue that it is without loss of generality to prune from the treeall histories that correspond to posteriors that cannot be generated with positive probability by themechanism. h t + = ( h t , M , 0, ∅ , (
0, 0 ) , ω t + ) .Fix an assessment, h Γ , ( π v , r v ) v ∈ V , µ i , and two consecutive information sets h t , h t + .Say that h t + is reached with positive probability from h t under h Γ , ( π v , r v ) v ∈ V , µ i ,if P h Γ , ( π v , r v ) v ∈ V , µ i ( h t + | h t ) ≡ ∑ y ∈ h t , y ′ ∈ h t + µ ∗ ( y | h t ) P ( Γ , ( π v , r v ) v ∈ V ) ( y ′ | y ) > h t + can be reached from h t through a deviation by the seller if there exists Γ ′ such that P h Γ ′ , ( π v , r v ) v ∈ V , µ i ( h t + | h t ) > Definition 4.
An assessment h Γ , ( π v , r v ) v ∈ V , µ i satisfies Bayes’ rule where possibleif for all t ≥ h t , h t + , µ ( y ′ | h t + ) is obtained via Bayes’ rulefrom µ ( ·| h t ) and ( Γ , ( π v , r v ) v ∈ V ) if either1. P h Γ , ( π v , r v ) v ∈ V , µ i ( h t + | h t ) >
0, or2. h t + can be reached from h t through a deviation by the seller. A.2 Participation, truthtelling, and Bayes’ plausibility constraints
Lemma 1 implies that for any payoff ( u S , u H , u L ) ∈ E ∗ ( µ ) , we can find a PBEassessment, h Γ , ( π v , r v ) v ∈ V , µ i , such that the following constraints are satisfied.For each period t and each public history h t , the buyer must find it optimal toparticipate in the mechanism chosen by the seller: ∑ µ ′ ∈ ∆ ( V ) β M ∗ t ( µ ′ | v )( vq M ∗ t ( µ ′ ) − x M ∗ t ( µ ′ ) + ( − q M ∗ t ( µ ′ )) δ U v ( h t , M ∗ t , 1, µ ′ , ( x M ∗ t ))) ≥ δ U v (( h t , M ∗ t , ∅ , (
0, 0 ))) , (PC v , h t )where we use U v ( h t + ) to denote the buyer’s continuation payoffs at history h t + when her valuation is v under strategy profile ( Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V ) . Second, the buyermust find it optimal to report her type truthfully. That is, for all v ∈ V and v ′ = v ,we have that ∑ µ ′ ∈ ∆ ( V ) β M ∗ t ( µ ′ | v )( vq M ∗ t ( µ ′ ) − x M ∗ t ( µ ′ ) + ( − q M ∗ t ( µ ′ )) δ U v ( h t , M ∗ t , 1, µ ′ , ( x M ∗ t ))) ≥ ∑ µ ′ ∈ ∆ ( V ) β M ∗ t ( µ ′ | v ′ )( vq M ∗ t ( µ ′ ) − x M ∗ t ( µ ′ ) + ( − q M ∗ t ( µ ′ )) δ U v ( h t , M ∗ t , 1, µ ′ , ( x M ∗ t ))) .(IC v , h t )42inally, if the mechanism outputs posterior µ ′ , it must be that µ ′ [ ∑ v ∈ V β M ∗ t ( µ ′ | v ) µ ( h t )( v )] = µ ( h t ) β M ∗ t ( µ ′ | v H ) . (BC µ ( h t ) )That is, from the communication device, we can infer a distribution over posteri-ors τ M ∗ t ( µ ( h t ) , · ) ∈ ∆ ( ∆ ( V )) such that τ M ∗ t ( µ ( h t ) , µ ′ ) = ∑ v ∈ V µ ( h t )( v ) β M ∗ t ( µ ′ | v ) . B Proof of Proposition 1
Part 1:
The proof of this part is immediate, and hence we omit it.
Part 2:
For this proof, we rely on several facts and results in Doval and Skreta(2019). First, by Proposition I.1, we know that if h Γ , ( π v , r v ) v ∈ V , µ i is a PBEassessment of G ∞ ( µ ) , then the continuation payoffs at history h t are equilib-rium payoffs of G ∞ ( µ ( h t )) . Second, because of the public randomization de-vice, it is without loss of generality to assume that the seller does not randomizehis choice of mechanism. Let h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i denote the PBE assessmentthat gives the seller payoff u ∗ S ( µ ) . Toward a contradiction, suppose that h t =( h t − , ω , M t , 1, µ ′ , ( x M t )) is a history on the path of play such that h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i| h t is not incentive-efficient. Thus, U ∗ S ( h t ) < max { u S : ( u S , U ∗ H ( h t ) , U ∗ L ( h t )) ∈ E ∗ ( µ ∗ ( h t )) } .Then, there is a strategy profile in G ∞ ( µ ∗ ( h t )) that gives the seller a higher payoffand the buyer the same payoff at h t . Consider then modifying h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i at h t so that the latter strategy profile is used. Clearly, this increases the seller’spayoff without changing the buyer’s payoff. Moreover, at history ( h t − , ω , M t ) ,the buyer still has an incentive to participate and truthfully report her type, sincewe have not modified her continuation payoffs in the mechanism. Finally, notethat at ( h t − , ω ) , the seller does not have an incentive to deviate: we have notchanged the buyer’s strategy at ( h t − , ω , M ′ t ) , for M ′ t = M t . Therefore, since theseller had no incentive to deviate in the original assessment, he also has no in-centive to deviate at the new one. The new assessment is thus a PBE assessmentin G ∞ ( µ ) and the seller obtains a higher payoff than u ∗ S ( µ ) , a contradiction. Part 3:
We first show that in the first period, v L has to be indifferent between par-ticipating or not in the mechanism. Let U ∗ L ( h , M ∗ ) denote the buyer’s equilib-rium payoff at the initial history in the PBE when the seller offers mechanism Technically, we just showed that there cannot be a positive measure of realizations of the cor-relating device for which h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i is not incentive-efficient. ∗ according to Γ ∗ ( h ) . If U ∗ L ( h , M ∗ ) >
0, consider the following modificationto the seller’s strategy profile. At the initial history, instead of offer mechanism M ∗ , the seller offers mechanism M ′ that coincides with mechanism M ∗ ≡ Γ ∗ ( ∅ ) ,except that all transfers x M ∗ ( µ ′ ) such that ∑ v ∈ V β M ∗ ( µ ′ | v ) >
0, they are raised by U ∗ L ( h , M ∗ ) . Modify the buyer’s strategy at h so that π ∗ v ( h , M ′ ) = π ∗ v ( h , M ∗ ) and r ∗ v ( h , M ′ ) = r ∗ v ( h , M ∗ ) ; leave the buyer’s strategy everywhere else un-changed. Finally, modify the seller’s beliefs so that when the buyer rejects M ′ ,he assigns probability 1 to the buyer’s valuation being v H . It is standard tocheck that the seller’s and buyer’s sequential rationality constraints hold andthe seller’s payoff increases by U ∗ L ( h ) > h t denote theshortest-length public history on the path of play such that a mechanism M ∗ t isoffered by the seller at h t such that U ∗ L ( h t , M ∗ t ) >
0. Following similar steps asabove, we can modify the mechanism at h t , so that the buyer’s payoff startingfrom h t is 0 when her valuation is v L and the seller’s payoff at h t goes up by U ∗ L ( h t , M ∗ t ) ; adjusting the strategies of the buyer so that it is a best response toparticipate and truthfully report her type in the modified mechanism. Letting h t = ( h t − , M ∗ t − , µ ′ , 0, x M ∗ t − ( µ ′ )) , modify M ∗ t − so that transfers are lowered by δ ( − q M ∗ t − ( µ ′ )) U ∗ L ( h t , · ) . Clearly, the change is revenue neutral for the seller,it satisfies that U ∗ L ( h t , · ) = U ∗ L ( h t − , · ) =
0, and it is standard to check that thebuyer’s incentives and payoffs remain unchanged.
Part 4:
Let h Γ ∗ , ( π ∗ v , r ∗ v ) v ∈ V , µ ∗ i denote the PBE assessment that delivers the high-est payoff for the seller. Clearly, if the mechanism used in the initial historysatisfies that | supp µ ( v ) β M ∗ ( ·| v ) | = v L and v H at h .Consider then the case in which | supp µ ( v ) β M ∗ ( ·| v ) | >
1. Then, Bayes’ consis-tency implies that two beliefs, µ ′ H , µ ′ L , exist such that β M ∗ ( µ ′ H | v H ) > β M ∗ ( µ ′ H | v L ) Because the original assessment is a PBE, sequential rationality of the seller’s strategy profileimplies that he does not have an incentive to deviate to mechanisms different from M ∗ and M ′ .Clearly, given the buyer’s strategy profile, offering M ′ dominates offering M ∗ . As for the buyer,note that conditional on participating in M ′ , reporting her valuation truthfully is a best responsebecause it was a best response in the original PBE. Because in the new assessment, participat-ing in M ′ yields a non-negative payoff for the buyer, whereas rejection guarantees a payoff of 0,participating is a best response for the buyer. β M ∗ ( µ ′ L | v H ) < β M ∗ ( µ ′ L | v L ) . Let ǫ H = U ∗ H ( h , M ∗ ) − U ∗ H | L ( h , M ∗ ) , where U ∗ H ( h , M ∗ ) denotes the equilibrium payoff of the high type at history h and U ∗ H | L denotes the buyer’s payoff when her type is v H , she reports v L into themechanism M ∗ , and then plays according to her equilibrium strategy. By as-sumption, ǫ H > ∆ µ ′ H , ∆ µ ′ L to be such that β M ∗ ( µ ′ H | v L ) ∆ µ ′ H = β M ∗ ( µ ′ L | v L ) ∆ µ ′ L , ǫ H = β M ∗ ( µ ′ H | v H ) ∆ µ ′ H − β M ∗ ( µ ′ L | v H ) ∆ µ ′ L .Modify the transfers of mechanism M ∗ so that x M ′ ( µ ′ L ) = x M ∗ ( µ ′ L ) − ∆ µ ′ L , x M ′ ( µ ′ H ) = x M ∗ ( µ ′ H ) + ∆ µ ′ H . Modify the buyer’s strategy at h so that π ∗ v ( h , M ′ ) = π ∗ v ( h , M ∗ ) and r ∗ v ( h , M ′ , 1 ) = r ∗ v ( h , M ∗ , 1 ) ; leave the buyer’s strategy everywhere else un-changed. Finally, modify the seller’s beliefs so that when the buyer rejects M ′ , heassigns probability 1 to the buyer’s valuation being v H . The seller’s revenue goesup by µ ǫ H and the buyer’s best response is still to participate in the mechanismand truthfully report her type. To see that along the path of play, v H must be indifferent between truthfully re-porting and reporting v L , let h t denote the shortest-length history on the pathof play such that the seller offers a mechanism M ∗ t such that the buyer strictlyprefers to report v H than to report v L when her type is v H . As before, at leasttwo posteriors must be generated in the mechanism. Let ǫ H ( h t ) > v H and reporting v L . Following the con-struction in the previous paragraph, modify the strategy profile at h t so that theseller, instead of offering M ∗ t , offers the modified version M ′ t . Mechanism M ′ t isas M ∗ t , except that the transfers have been modified as we did for the case t = v H and v L , when her val-uation is v H . Modify the buyer’s strategy at h t so that π ∗ v ( h t , M ′ t ) = π ∗ v ( h t , M ∗ t ) and r ∗ v ( h t , M ′ t , 1 ) = r ∗ v ( h t , M ∗ t , 1 ) ; for now, leave the buyer’s strategy everywhere To see that participating with probability one and truthtelling are best responses for the buyer,note that when the buyer announces v L , her expected payment (averaging out over posteriors) isunchanged, whereas when she announces v H her expected payment goes up by ǫ H . This meansthat when her valuation is v H the buyer is now indifferent between reporting v L and v H , whereasthe buyer now has a strict incentive to report v L when her valuation is v L . Because without loss of generality we can focus on assessments where the buyer is truthful, ifshe is not indifferent, she must strictly prefer to tell the truth. M ′ t , he assigns probability 1 to the buyer’s valuation being v H . To keepthe incentives at the histories preceding h t the same, we proceed as follows. Let h t = ( h t − , M ∗ t − , 1, µ ′ , ( x M ∗ t − ( µ ′ ))) . Modify the strategy at h t − so that insteadof offering M ∗ t − , the seller offers M ′ t − . This mechanism coincides with M ∗ t − except that x M ∗ t − ( µ ′ ) is lowered by ∆ µ ′ = δ ( − q M ∗ t − ( µ ′ )) ǫ H ( h t ) . Note that thisis payoff neutral for the seller: his increase in payoff at ( h t , M ′ t ) is exactly offsetby his payoff decrease at h t − , ∆ µ ′ . However, by lowering x M ∗ t − ( µ ′ ) , we mayhave relaxed the participation constraint of v L if 1 − µ ∗ ( h t − ) >
0, in which case,we increase all transfers by ǫ L = β M ∗ t ( µ ′ | v L ) ∆ µ ′ >
0. Again, modify the buyer’sstrategy so that her best response at ( h t − , M ′ t − ) is the same as at ( h t − , M ∗ t − ) and modify the seller’s beliefs so that when the buyer rejects M t − , he assignsprobability 1 to the buyer’s valuation being v H . The best response conditionsfrom the original assessment imply the new assessment is also a PBE assessment.In this new assessment, the seller’s payoff (weakly) increases, which contradictsincentive efficiency if 1 − µ ∗ ( h t − ) > C Proofs of Section 3.2
Instead of proving Proposition 2, Proposition 3, and Theorem 2 chronologically , weproceed as follows:1. Proposition C.1 in Section C.1 shows that in any intrapersonal equilibrium(a) If µ < µ , then q ( µ , · ) = µ > µ , theni. If the seller induces a posterior at which he sets q ( µ , µ ′ ) =
1, then µ ′ = ( µ , 1 ) .2. Proposition C.2 shows that if there is an intrapersonal equilibrium such that(a) If µ < µ , then q ( µ , · ) = µ ≥ µ , there exists µ D ∗ ( µ ) < µ such that τ ∗ ( µ , µ D ∗ ( µ )) = − τ ∗ ( µ , 1 ) and q ∗ ( µ , 1 ) = = − q ∗ ( µ , µ D ∗ ( µ )) .46hen it satisfies the properties stated in Proposition 3.3. Theorem C.1 shows that the policy described in Theorem 2 is an intrapersonalequilibrium.To complete the proofs of the results in Section 3.2 it only remains to show that theintrapersonal equilibrium is unique. This is done in Appendix III in Doval and Skreta(2019). Section III.1 shows that there is a unique equilibrium in which the selleruses at most two posteriors. Section III.2 shows that there is no equilibrium inwhich the seller uses more than two posteriors. C.1 Proof of Proposition C.1
Proposition C.1.
Let h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i be an intrapersonal equilibrium. Then,1. For all µ < µ , q ∗ ( µ , · ) = ,2. For all µ > µ , if q ∗ ( µ , µ ′ ) = and the seller induces µ ′ with positive probability,then µ ′ = .3. For all µ > µ , R µ τ ∗ ( µ , d µ ′ ) = τ ∗ ( µ , 1 ) . The proof of Proposition C.1 uses the following lemma:
Lemma C.1 (Properties of the continuation values, R ( τ , q ) ) . Let R ( τ , q ) denote thecontinuation values induced by a policy ( τ , q ) . Then,1. If µ < µ , then R ( τ , q ) ( µ ′ , µ ) ≤ µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) ,2. If µ > µ , then R ( τ , q ) ( µ ′ , µ ) ≤ µ ′ v H .Proof. Note the contraction mapping theorem implies R ( τ , q ) is well defined forany policy, ( τ , q ) . Fix a prior µ , and consider the operator T µ that takes boundedfunction w : ∆ ( V ) R into T µ ( w )( µ ′ ) = Z (cid:0) q ( µ ′ , ˜ µ )( ˜ µ v H + ( − ˜ µ ) ˆ v L ( µ )) + ( − q ( µ ′ , ˜ µ )) δ w ( ˜ µ ) (cid:1) τ ( µ ′ , d ˜ µ ) To show part 1 holds, let w be a bounded function such that w ( ˜ µ ) ≤ ˜ µ v H + ( − µ ) ˆ v L ( µ ) . Then, T µ ( w ( µ ′ )) ≤ Z (cid:0) q ( µ ′ , ˜ µ )( ˜ µ v H + ( − ˜ µ ) ˆ v L ( µ )) + ( − q ( µ ′ , ˜ µ )) δ ( ˜ µ v H + ( − ˜ µ ) ˆ v L ( µ )) (cid:1) τ ( µ ′ , d ˜ µ ) ≤ µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) , where the inequalities follow from (i) the bound on w , (ii) ˆ v L ( µ ) >
0, and (iii)Bayes’ plausibility. This implies that for all µ ′ ∈ ∆ ( V ) , R ( τ , q ) ( µ ′ , µ ) ≤ µ ′ v H +( − µ ′ ) ˆ v L ( µ ) .Similarly, to show part 2 holds, let w be a bounded function such that w ( ˜ µ ) ≤ ˜ µ v H . Then, T µ ( w ( µ ′ )) ≤ Z (cid:0) q ( µ ′ , ˜ µ )( ˜ µ v H + ( − ˜ µ ) ˆ v L ( µ )) + ( − q ( µ ′ , ˜ µ )) δ ˜ µ v H (cid:1) τ ( µ ′ , d ˜ µ ) ≤ µ ′ v H ,where the inequalities follow from (i) the bound on w , (ii) ˆ v L ( µ ) <
0, and (iii)Bayes’ plausibility. This implies that for all µ ′ ∈ ∆ ( V ) , R ( τ , q ) ( µ ′ , µ ) ≤ µ ′ v H . Proof of Proposition C.1.
To prove part 1, note Lemma C.1 implies that R ( τ ∗ , q ∗ ) ( µ , µ ) ≤ µ v H + ( − µ ) ˆ v L ( µ ) ,and this bound is achieved by setting q ∗ ( µ , · ) = µ > µ , for any posterior µ ′ it follows that µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) < µ ′ v H + ( − µ ′ ) δ ˆ v L ( µ ) , (C.1)because ˆ v L ( µ ) < δ <
1. Thus, instead of setting q ∗ ( µ , µ ′ ) =
1, which yieldsthe payoff on the left-hand side of Equation C.1, the seller can split µ ′ between 0and 1, setting q ∗ ( µ , 1 ) = = − q ∗ ( µ , 0 ) , which yields the payoff on the right-hand side of Equation C.1.To prove part 3, let A = ( µ , 1 ) and suppose that R A τ ∗ ( µ , d µ ′ ) >
0. Then, Z A δ R ( τ ∗ , q ∗ ) ( µ ′ , µ ) τ ∗ ( µ , d µ ′ ) ≥ Z A R ( τ ∗ , q ∗ ) ( µ ′ , µ ) τ ∗ ( µ , d µ ′ ) ,otherwise, the seller with prior µ would be better off by generating µ ′ ∈ A andimitating at that point the policy that the seller with belief µ ′ uses (rather thangenerating µ ′ and q ∗ ( µ , µ ′ ) =
0, which has payoff R ( τ ∗ , q ∗ ) ( µ ′ , µ ) .48ince A has positive measure under τ ∗ ( µ , · ) , it follows that the set of posteriors µ ′ > µ such that R ( τ ∗ , q ∗ ) ( µ ′ , µ ) ≤ µ ′ , we have: µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) ≤ δ R ( τ ∗ , q ∗ ) ( µ ′ , µ ) ≤ µ ′ and set q ∗ ( µ , µ ′ ) = q ∗ ( µ , µ ′ ) =
1. Since µ < µ ′ , we then have v L ≡ µ v H + ( − µ ) ˆ v L ( µ ) < µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) ≤ τ ∗ ( µ , A ) >
0, which completes the proof.
C.2 Proof of Proposition C.2
Proposition C.2.
Suppose h µ D ∗ , R µ D ∗ i is an intrapersonal equilibrium with the proper-ties that:1. For all µ < µ , q ∗ ( µ , µ ′ ) = for all µ ′ ∈ ∆ ( V ) .2. For all µ ≥ µ , there exists µ D ∗ ( µ ) < µ such that τ ∗ ( µ , µ D ∗ ( µ )) = − τ ∗ ( µ , 1 ) and q ∗ ( µ , 1 ) = = − q ∗ ( µ , µ D ∗ ( µ )) .Then, it satisfies parts 1-4 of Proposition 3. Part 1:
Note that if µ D ∗ ≡ µ D ∗ ( µ ) < µ , then R µ D ∗ ( µ D ∗ , µ ) = ( − µ D ∗ ) ˆ v L ( µ ) + µ D ∗ v H . Consider the following alternative policy for µ , where µ D ′ ( µ ) =
0. Thisis achieved by τ ′ ( µ , 1 ) = τ ∗ ( µ , 1 ) + µ D ∗ and τ ′ ( µ , 0 ) = − τ ∗ ( µ , 1 ) − µ D ∗ ,where τ ∗ ( µ , · ) are the weights on µ D ∗ and 1 in the original best response. Thedifference in payoffs is given by: µ D ∗ v H + ( − τ ∗ ( µ , 1 )) δ [ ˆ v L ( µ ) − µ D ∗ v H − ( − µ D ∗ ) ˆ v L ( µ )] − µ D ∗ δ ˆ v L ( µ ) == µ D ∗ v H + ( − τ ∗ ( µ , 1 ) δµ D ∗ [ ˆ v L ( µ ) − v H ] − µ D ∗ δ ˆ v L ( µ )= µ D ∗ ( v H − δ ˆ v L ( µ )) − ( − τ ∗ ( µ , 1 )) µ D ∗ δ ( v H − ˆ v L ( µ ))= µ D ∗ v H ( − τ ∗ ( µ , 1 ) δ ) − µ D ∗ δ ˆ v L ( µ ) τ ∗ ( µ , 1 ) > v L ( µ ) <
0. This contradicts that µ D ∗ ( µ ) is a best response. Hence, part 1 holds.49 art 2: Let µ ≤ µ denote the seller’s prior. Toward a contradiction, assume thatfor all finite n , µ n = µ ( n ) D ∗ ( µ ) > µ . Fix ǫ >
0. Note that there exists M ǫ such thatfor all M ǫ ≤ m , we have µ m − µ m + < ǫ ( − µ m + ) , (C.2)so that as m goes to infinity, the seller puts smaller weight on posterior 1. To seethat Equation C.2 holds, assume to the contrary that for all m , we have ǫ ( − µ m + ) ≤ µ m − µ m + .Adding up from m = m = N −
1, we obtain ǫ N ∑ i = ( − µ i ) ≤ µ − µ N . (C.3)Now, the right-hand side of Equation C.3 is bounded above by µ − µ because µ < µ N . The left-hand side of Equation C.3 is bounded below by N ǫ ( − µ ) because µ ≥ µ i for all i ≥
0. We then obtain the following chain of inequalities: N ǫ ( − µ ) < ǫ N ∑ i = ( − µ i ) ≤ µ − µ N < µ − µ ,and this cannot possibly hold for all N . Thus, Equation C.2 holds.Now fix ǫ > ǫ − δ ( − ǫ ) < µ ( − δ ) . (C.4)Note the right-hand side of Equation C.4 is bounded above by µ m ( − δ ) for all m ≥
0. Take M ǫ ≤ m and note R µ D ∗ ( µ m , µ m ) < ǫ v H + ( − ǫ ) δ R µ D ∗ ( µ m + , µ m ) . (C.5)To see that Equation C.5 holds, note R µ D ∗ ( µ m , µ m ) = µ m − µ m + − µ m + v H + − µ m − µ m + δ R µ D ∗ ( µ m + , µ m )= µ m − µ m + − µ m + ( v H − δ R µ D ∗ ( µ m + , µ m )) + δ R µ D ∗ ( µ m + , µ m ) < ǫ v H + ( − ǫ ) δ R µ D ∗ ( µ m + , µ m ) ,50here the inequality follows from v H > δ R µ D ∗ ( µ m + , µ m ) and M ǫ ≤ m . BecauseEquation C.5 holds for all M ǫ ≤ m , applying it recursively, we obtain R µ D ∗ ( µ m , µ m ) < v H ǫ − δ ( − ǫ ) + lim N → ∞ ( δ ( − ǫ )) N R µ D ∗ ( µ m + N , µ m ) = v H ǫ − δ ( − ǫ ) ,because R µ D ∗ ( · , µ m ) is bounded and δ ( − ǫ ) <
1. Now, the following policy isalways feasible for µ m : place probability µ m on 1 and 1 − µ m on 0. This yields µ m ( − δ ) v H + v L δ ,when µ > µ m v H ,when µ =
0. Suppose 0 < µ (the other case is similar). Because splittingbetween 0 and 1 is not optimal, we must have µ m ( − δ ) v H + v L δ ≤ R µ D ∗ ( µ m , µ m ) < ǫ − δ ( − ǫ ) v H < v H ( − δ ) µ ,where the last inequality follows from Equation C.4. This is a contradiction to h µ D ∗ , R µ D ∗ i being an intrapersonal equilibrium. Thus, starting from µ , a finite N exists such that µ ( N ) D ∗ ( µ ) ≤ µ .Lemma C.2 below is used to show that if an intrapersonal equilibrium like theone described in Proposition C.2 exists, then it satisfies part 3 of Proposition 3,and it is the first step in the induction in the proof that it also satisfies part 4: Lemma C.2.
Let µ < µ < µ ′′ be such that µ D ∗ ( µ ) = µ D ∗ ( µ ′′ ) = . Then, for all µ ′ ∈ ( µ , µ ′′ ) , µ D ∗ ( µ ′ ) = .Proof. Suppose µ < µ ′ < µ ′′ are such that µ D ∗ ( µ ) = µ D ∗ ( µ ′′ ) = µ D ≡ µ D ∗ ( µ ′ ) ≥ µ . Then, the following must be true: µ ′ − µ D − µ D v H + − µ ′ − µ D δ R µ D ∗ ( µ D , µ ′ ) ≥ µ v H + ( − µ ′ ) δ ˆ v L ( µ ′ ) .Rewriting the above expression, we obtain: g ( µ ′ ) ≡ (cid:20) µ ′ − µ D − µ D − µ ′ (cid:21) v H + − µ ′ − µ D δ R µ D ∗ ( µ D , µ ′ ) + ( − µ ′ ) δ ( − ˆ v L ( µ ′ )) ≥ By Proposition C.2, part 1, if µ D ( µ ′ ) =
0, then µ D ( µ ′ ) ≥ µ .
51n Section IV.1 in Doval and Skreta (2019), we show the continuation values R µ D ∗ can be written as: R µ D ∗ ( µ D , µ ′ ) = v H ( µ D − µ − µ + ( − µ D ) N D − ∑ i = δ i − µ i µ i − µ i + − µ i + ) | {z } α ( µ D ) + ( − µ D ) δ N D | {z } γ ( µ D ) ˆ v L ( µ ′ ) ,where µ > µ > · · · > µ N D − ≥ µ > = µ N D are the posteriors generatedby the policy at which trade is delayed. Note R µ D ∗ ( µ D , µ ′ ) is differentiable in µ ′ and ∂∂µ R µ D ∗ ( µ D , µ ) | µ = µ ′ = − γ ( µ D ) ∆ v ( − µ ′ ) .Therefore, g is differentiable with respect to µ ′ , and we obtain the following ex-pression for g ′ ( µ ′ ) : g ′ ( µ ′ ) = µ D − µ D v H − δ R µ D ∗ ( µ D , µ ′ ) − µ D + δ ˆ v L ( µ ′ ) + δ ( − µ ′ ) (cid:20) − γ ( µ D ) − µ D (cid:21) ∆ v ( − µ ′ ) = v H µ D − µ D − δ − µ D ( α ( µ D ) v H + γ ( µ D ) ˆ v L ( µ ′ )) + δ ˆ v L ( µ ′ ) + δ ∆ v − µ ′ (cid:20) − γ ( µ D ) − µ D (cid:21) = v H (cid:20) µ D − δα ( µ D ) − µ D (cid:21) + δ (cid:18) ˆ v L ( µ ′ ) + ∆ v − µ ′ (cid:19) (cid:20) − γ ( µ D ) − µ D (cid:21) = v H (cid:20) µ D − δα ( µ D ) − µ D (cid:21) + δ v H (cid:20) − γ ( µ D ) − µ D (cid:21) .Now, note α ( µ D ) ≤ µ D . Intuitively, the largest probability that µ D can trade with v H is µ D . To see this formally, we use that α ( µ D ) = µ D − µ − µ + ( − µ D ) N D − ∑ i = δ i − µ i µ i − µ i + − µ i + ≤ µ D − µ − µ + ( − µ D ) µ − µ N D − ∑ i = δ i ,where the inequality follows from noting that the expression inside the sum isincreasing in µ i and decreasing in µ i + , and that µ D − µ + ( − µ D ) µ ∑ N D − i = δ i ≤ µ D ( − µ ) . See Section IV.1 in Doval and Skreta (2019).
52t then follows that g ′ ( µ ′ ) ≥
0, which contradicts that µ ′′ finds it optimal to set µ D ∗ ( µ ′′ ) = Part 3:
We use the following property of a policy where µ D ∗ ( · ) is weakly increas-ing. If we let N µ denote the smallest value of n such that µ ( n ) D ∗ ( µ ) ≤ µ , then N µ is weakly increasing in µ . To see this, let µ < µ ′ , and inductivelydefine µ i + = µ D ∗ ( µ i ) and similarly, µ ′ i + = µ D ∗ ( µ ′ i ) . Clearly, we have that µ > µ > · · · > µ N > . . . and similarly, µ ′ > µ ′ > · · · > µ ′ N > . . . . Moreover,monotonicity implies µ = µ D ∗ ( µ ) ≤ µ ′ = µ D ∗ ( µ ′ ) and inductively µ N < µ ′ N .Lemma C.2 implies that if µ D ∗ ( µ ′ N ) =
0, then letting n be the smallest m suchthat µ m ≤ µ ′ N , we have µ D ∗ ( µ m ) =
0. Hence, N µ ≤ N µ ′ .Let µ denote the smallest prior above µ such that there exists µ ′ > µ with µ D ∗ ( µ ) > µ D ∗ ( µ ′ ) . Now consider a policy for µ that splits the weight τ ( µ , µ D ∗ ( µ )) between µ D ∗ ( µ ′ ) and 1. The payoff from this policy is: v H (cid:20) µ − µ D ∗ ( µ ) − µ D ∗ ( µ ) + − µ − µ D ∗ ( µ ) µ D ∗ ( µ ) − µ D ∗ ( µ ′ ) − µ D ∗ ( µ ′ ) (cid:21) + δ − µ − µ D ∗ ( µ ′ ) R µ D ∗ ( µ D ∗ ( µ ′ ) , µ ) .Consider the difference between the payoff of the above policy and that of thepolicy that the seller employs when his prior is µ : − µ − µ D ∗ ( µ ) µ D ∗ ( µ ) − µ D ∗ ( µ ′ ) − µ D ∗ ( µ ′ ) v H + δ − µ − µ D ∗ ( µ ′ ) R µ D ∗ ( µ D ∗ ( µ ′ ) , µ ) − δ − µ − µ D ∗ ( µ ) R µ D ∗ ( µ D ∗ ( µ ) , µ ) . (C.7) Optimality of µ ′ ’s policy implies µ ′ − µ D ∗ ( µ ′ ) − µ D ∗ ( µ ′ ) v H + δ − µ ′ − µ D ∗ ( µ ′ ) R µ D ∗ ( µ D ∗ ( µ ′ ) , µ ′ ) ≥ µ ′ − µ D ∗ ( µ ) − µ D ∗ ( µ ) v H + δ − µ ′ − µ D ∗ ( µ ) R µ D ∗ ( µ D ∗ ( µ ) , µ ′ ) ,or, equivalently, δ [ R µ D ∗ ( µ D ∗ ( µ ′ ) , µ ′ ) − µ D ∗ ( µ ′ ) − R µ D ∗ ( µ D ∗ ( µ ) , µ ) − µ D ∗ ( µ ) ] ≥ − ( µ D ∗ ( µ ) − µ D ∗ ( µ ′ ))( − µ D ∗ ( µ ))( − µ D ∗ ( µ ′ )) v H . Proposition C.2 implies µ D ∗ ( µ ( N µ − ) D ∗ ) = Note µ > µ because we know µ D ∗ ( µ ) =
53o show µ can improve by using the new policy, we only need to show R µ D ∗ ( µ D ∗ ( µ ′ ) , µ ) − µ D ∗ ( µ ′ ) − R µ D ∗ ( µ D ∗ ( µ ) , µ ) − µ D ∗ ( µ ) ≥ R µ D ∗ ( µ D ∗ ( µ ′ ) , µ ′ ) − µ D ∗ ( µ ′ ) − R µ D ∗ ( µ D ∗ ( µ ) , µ ′ ) − µ D ∗ ( µ ) ⇔ ( δ N µ D ∗ ( µ ′ ) − δ N µ D ∗ ( µ ) )( − ˆ v L ( µ )) ≤ ( δ N µ D ∗ ( µ ′ )) − δ N µ D ∗ ( µ ) )( − ˆ v L ( µ ′ )) ,where recall that N · denotes the first time at which updating leads to a poste-rior below µ for a given prior. Note that a sufficient condition for the aboveinequality to hold is that N µ D ∗ ( µ ′ ) < N µ D ∗ ( µ ) . That this is indeed the case followsfrom the observation before the proof because the policy below µ (and hence for µ D ∗ ( µ ) , µ D ∗ ( µ ′ ) ) satisfies monotonicity.Because N µ D ∗ ( µ ′ ) < N µ D ∗ ( µ ) implies the expression in Equation C.7 is non-negative,we see that without loss of generality, we can have µ D ∗ ( µ ) ≤ µ D ∗ ( µ ′ ) andstrictly so when N µ D ∗ ( µ ) = N µ D ∗ ( µ ′ ) . Part 4:
The proof is by induction on n ≥
1. Define D = { } , and for n ≥ D n = { µ : µ ( n ) D ∗ = } ,to be the set of priors, µ , such that the seller updates his prior to 0 in n periodswhen his prior is µ . Let P ( n ) denote the following inductive statement: P ( n ) : If µ , µ ′ ∈ D n , then µ D ∗ ( µ ) = µ D ∗ ( µ ′ ) ∈ D n − .Lemma C.2 implies the inductive statement is true for n =
1, i.e., P ( ) = P ( n ) =
1, and we show that P ( n + ) =
1. Let µ , µ ′ ∈ D n + andassume without loss of generality that µ < µ ′ . Toward a contradiction, suppose µ D ∗ ( µ ) = µ D ∗ ( µ ′ ) ; by part 3, it follows that µ D ∗ ( µ ) < µ D ∗ ( µ ′ ) . The payoff forthe seller when his prior is µ ′ from following his policy is µ ′ − µ D ∗ ( µ ′ ) − µ D ∗ ( µ ′ ) v H + δ − µ ′ − µ D ∗ ( µ ′ ) (cid:20) µ D ∗ ( µ ′ ) − µ n − − µ n − v H + δ − µ D ∗ ( µ ′ ) − µ n − R µ D ∗ ( µ n − , µ ′ ) (cid:21) ,where µ n − = µ D ∗ ( µ D ∗ ( µ ′ )) . By the inductive hypothesis, µ D ∗ ( µ D ∗ ( µ ′ )) = µ D ∗ ( µ D ∗ ( µ )) = µ n − .Instead, if the seller follows µ ’s policy when his posterior is µ ′ , his payoff wouldbe µ ′ − µ D ∗ ( µ ) − µ D ∗ ( µ ) v H + δ − µ ′ − µ D ∗ ( µ ) (cid:20) µ D ∗ ( µ ) − µ n − − µ n − v H + δ − µ D ∗ ( µ ) − µ n − R µ D ∗ ( µ n − , µ ′ ) (cid:21) ,54here we use the inductive statement that µ D ∗ ( µ ) , µ D ∗ ( µ ′ ) ∈ D n − and hence µ D ∗ ( µ D ∗ ( µ )) = µ D ∗ ( µ D ∗ ( µ ′ )) = µ n − .Taking differences, we have: v H (cid:20) µ ′ − µ D ∗ ( µ ′ ) − µ D ∗ ( µ ′ ) + δ − µ ′ − µ D ∗ ( µ ′ ) µ D ∗ ( µ ′ ) − µ n − − µ n − − µ ′ − µ D ∗ ( µ ) − µ D ∗ ( µ ) − δ − µ ′ − µ D ∗ ( µ ) µ D ∗ ( µ ) − µ n − − µ n − (cid:21) = v H ( − µ ′ )( − µ D ∗ ( µ ))( − µ D ∗ ( µ ′ )) (cid:0) ( δ − )( µ D ∗ ( µ ′ ) − µ D ∗ ( µ )) (cid:1) < which contradicts the optimality of µ ′ ’s policy. Thus, µ D ∗ ( µ ) = µ D ∗ ( µ ′ ) (bypart 3 we cannot have µ D ∗ ( µ ′ ) < µ D ∗ ( µ ) ). Hence, P ( n + ) = C.3 Proof of Theorem C.1
Theorem C.1.
The policy described in Theorem 2 defines an intrapersonal equilibrium.Proof of Theorem C.1.
As in the proof of Proposition C.2, define D n = { µ : µ ( n ) D ∗ ( µ ) = } ,to be the set of seller priors such that trade with v L happens in n periods.The proof of Theorem C.1 is structured as follows. First, Lemma C.3 shows thecutoffs { µ n } n ≥ form a strictly increasing sequence. Second, we construct the pay-offs from ( τ ∗ , q ∗ ) . Finally, we show h ( τ ∗ , q ∗ ) , R ( τ ∗ , q ∗ ) i is an intrapersonal equilib-rium. Lemma C.3.
The sequence { µ n } n ≥ is strictly increasing in n for n ≥ .Proof. Recall that we are defining µ = µ = v L / v H . Now let n ≥ µ n ∈ D n and µ n − ∈ D n − . Fix a prior, µ ≥ µ n . We claim that the difference ∆ n ( µ ; µ n , µ n − ) = (cid:20) µ − µ n − µ n v H + − µ − µ n δ R µ D ∗ ( µ n , µ ) (cid:21) − (cid:20) µ − µ n − − µ n − v H + − µ − µ n − δ R µ D ∗ ( µ n − , µ ) (cid:21) , is increasing in µ for any monotone simple policy µ D ∗ .55he proof follows similar steps to those in the proof of Lemma C.2. The argu-ments in that proof imply ∆ n is differentiable in µ . Then, ∂∂µ ∆ n ( µ ; µ n , µ n − ) == v H ( µ n − µ n − )( − µ n )( − µ n − ) − δ [ R µ D ∗ ( µ n , µ ) − µ n − R µ D ∗ ( µ n − , µ ) − µ n − ] + δ ( − µ )[ δ n − − δ n ] ∆ v ( − µ ) = v H ( µ n − µ n − )( − µ n )( − µ n − ) − δ [ v H α ( µ n ) − µ n − v H α ( µ n − ) − µ n − ] + δ ˆ v L ( µ )( δ n − − δ n ) + δ [ δ n − − δ n ] ∆ v ( − µ )= v H ( µ n − µ n − )( − µ n )( − µ n − ) − δ v H [ µ n − µ n − ( − µ n )( − µ n − ) ] + δ ( δ n − − δ n ) v H > where the last equality follows from using the definition of α ( · ) .Now recall Equation 14, which we reproduce below for easy reference: µ n + − µ n − µ n v H + − µ n + − µ n δ R ( τ ∗ , q ∗ ) ( µ n , µ n + ) = µ n + − µ n − − µ n − v H + − µ n + − µ n − δ R ( τ ∗ , q ∗ ) ( µ n − , µ n + ) .Note that the cutoff µ n + is defined by ∆ n ( µ n + ; µ n , µ n − ) = µ n + = µ n if n ≥
1. If µ n = µ n + , then0 = ∆ n ( µ n + ; µ n , µ n − ) = δ R ( τ ∗ , q ∗ ) ( µ n , µ n ) − R ( τ ∗ , q ∗ ) ( µ n , µ n ) ≤ δ =
1. Because ∆ n ( · ; µ n , µ n − ) is increasing, µ n + > µ n .This completes the proof.An implication of Lemma C.3 is that µ n + = inf { µ : µ ∈ D n + } = sup { µ : µ ∈ D n } . The equilibrium we construct makes it so that µ n + = min { µ : µ ∈ D n + } .Given ( τ ∗ , q ∗ ) as in the statement of Theorem 2, we construct the continuationpayoffs, R ( τ ∗ , q ∗ ) , for the seller for each of his priors, µ , and posteriors he mayinduce, µ ′ . If µ ′ < µ , R ( τ ∗ , q ∗ ) ( µ ′ , µ ) = µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) , (C.8)whereas if µ ′ ∈ [ µ n , µ n + ) for n ≥
1, we have R ( τ ∗ , q ∗ ) ( µ ′ , µ ) = µ ′ − µ n − − µ n − v H + δ − µ ′ − µ n − R ( τ ∗ , q ∗ ) ( µ n − , µ ) (C.9) R ( τ ∗ , q ∗ ) ( µ n , µ ) = v H " µ n − µ n − − µ n − + ( − µ n ) n − ∑ i = δ n − i − µ i µ i − µ i − − µ i − + δ n ( − µ n ) ˆ v L ( µ )
56e now verify that for each µ ≥ µ , h τ ∗ ( µ , · ) q ∗ ( µ , · ) i , is optimal given R ( τ ∗ , q ∗ ) ( · , µ ) .Recall that given R ( τ ∗ , q ∗ ) ( · , µ ) , the seller solves:max τ , q Z [ q ( µ , µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ )) + δ ( − q ( µ , µ ′ )) R ( τ ∗ , q ∗ ) ( µ ′ , µ )] d τ ( µ , µ ′ ) (C.10)subject to the constraints that q ( µ , µ ′ ) ∈ [
0, 1 ] and R µ ′ d τ ( µ , µ ′ ) = µ .Consider first the case in which µ = µ . Note that µ ′ v H + ( − µ ′ ) ˆ v L ( µ ) = µ ′ v H ≥ δ R ( τ ∗ , q ∗ ) ( µ ′ , µ ) = δα ( µ ′ ) v H strictly so when µ ′ >
0. Thus, q ∗ ( µ , µ ′ ) =
1, except possibly at µ =
0. Moreover, the payoff of setting q ∗ ( µ , 0 ) = q ∗ ( µ , 1 ) = µ v H and is exactly thesame as the payoff of setting q ∗ ( µ , · ) = τ ∗ ( µ , 0 ) = − µ = − τ ∗ ( µ , 1 ) is optimal.Consider next µ ∈ D n for n ≥
1, if n =
1, then µ > µ . Part 3 of Proposition C.1implies that the seller never places positive probability on ( µ , 1 ) . Moreover,Proposition C.2 implies that τ ( µ , · ) has finite support since it cannot assign posi-tive probability to two posteriors µ ′ , µ ′ ∈ D m for some m ≤ n . It also implies thatif µ ′ ∈ D m and τ ( µ , µ ′ ) >
0, then µ ′ = µ m . Finally, it must be that τ ( µ , µ n ) = µ ∈ D n , then ∆ n ( µ ; µ n , µ n − ) < µ − µ n − − µ n − v H + − µ − µ n − δ R ( τ ∗ , q ∗ ) ( µ n − , µ ) ≥ τ ( µ , 1 ) v H + n − ∑ m = τ ( µ , µ m ) δ R ( τ ∗ , q ∗ ) ( µ m , µ ) (C.11)for any τ ( µ , · ) ∈ ∆ ( V ) such thatsupp τ ( µ , · ) ⊂ { µ , . . . , µ n − , 1 } τ ( µ , 1 ) + n − ∑ m = τ ( µ , µ m ) µ m = µ .Using the properties of τ ( µ , · ) , one can write the RHS of Equation C.11 as follows: v H " µ − µ n − + ∑ n − m = τ ( µ , µ m )( µ n − − µ m ) − µ n − + δ − µ − µ n − R ( τ ∗ , q ∗ ) ( µ n − , µ )+ n − ∑ m = τ ( µ , µ m )( − µ m ) " δ R ( τ ∗ , q ∗ ) ( µ m , µ ) − µ m − δ R ( τ ∗ , q ∗ ) ( µ n − , µ ) − µ n − n − ∑ m = τ ( µ , µ m )( − µ m ) " δ R ( τ ∗ , q ∗ ) ( µ n − , µ ) − µ n − − δ R ( τ ∗ , q ∗ ) ( µ m , µ ) − µ m − v H ( µ n − − µ m )( − µ n − )( − µ m ) ≥ m ≤ n − µ m − µ m − − µ m v H + − µ m − µ m − δ R ( τ ∗ , q ∗ ) ( µ m − , µ ) == µ m − µ m − − µ m − v H + − µ m − µ m − δ R ( τ ∗ , q ∗ ) ( µ m − , µ m ) + δ m ( ˆ v L ( µ ) − ˆ v L ( m ))= µ m − µ m − − µ m − v H + − µ m − µ m − δ R ( τ ∗ , q ∗ ) ( µ m − , µ m ) + δ m ( ˆ v L ( µ ) − ˆ v L ( m ))= µ m − µ m − − µ m − v H + − µ m − µ m − δ R ( τ ∗ , q ∗ ) ( µ m − , µ ) + δ m − ( ˆ v L ( m ) − ˆ v L ( µ )) + δ m ( ˆ v L ( µ ) − ˆ v L ( m ))= µ m − µ m − − µ m − v H + − µ m − µ m − δ R ( τ ∗ , q ∗ ) ( µ m − , µ ) + δ m − ( ˆ v L ( m ) − ˆ v L ( µ ))( − δ ) ,where the third line uses the indifference condition that defines µ m . This is if, andonly if, for all m ≤ n − δ R ( τ ∗ , q ∗ ) ( µ m − , µ ) − µ m − − δ R ( τ ∗ , q ∗ ) ( µ m − , µ ) − µ m − ≥ v H ( µ m − − µ m − )( − µ m − )( − µ m − ) . (C.13)Note that the inequality is strict whenever m < n −
1. Successive application ofEquation C.13 implies that Equation C.12 holds. Indeed, this can be verified bywriting the term inside the summation in Equation C.12 as: δ R ( τ ∗ , q ∗ ) ( µ n − , µ ) − µ n − − δ R ( τ ∗ , q ∗ ) ( µ m , µ ) − µ m = n − ∑ l = m δ R ( τ ∗ , q ∗ ) ( µ l + , µ ) − µ l + − δ R ( τ ∗ , q ∗ ) ( µ l , µ ) − µ l ! ,noting that Equation C.13 implies that the RHS of the above expression is boundedbelow by n − ∑ l = m v H ( µ l + − µ l )( − µ l + )( − µ l ) = v H µ n − − µ m ( − µ m )( − µ n − ) .This completes the proof. 58 Collected objects from intrapersonal equilibrium
In this section, we construct the following objects using the ingredients from theintrapersonal equilibrium:1. The mapping γ ∗ : ∆ ( V )
7→ M C that associates to each belief the seller mayhold, µ , a mechanism γ ∗ ( µ ) ∈ M C ,2. The mapping u ∗ H : ∆ ( V ) R that associates to each belief the seller mayhold, µ , the buyer’s payoff when her valuation is v H and the seller employsthe mechanism derived from the intrapersonal equilibrium,3. The correspondence U ∗ H : ∆ ( V ) ⇒ R that associates to each belief the sellermay hold, µ , the set of feasible buyer’s payoffs when her valuation is v H , theseller employs the mechanism derived from the intrapersonal equilibrium,but does not necessarily break ties as in the intrapersonal equilibrium. Thatis, U ∗ H ( µ ) = u ∗ H ( µ ) whenever µ = µ i , i =
0, 1, . . . .
D.1 Mechanism and buyer’s payoffs in the intrapersonal equi-librium
In this section, we construct the transfers and the continuation rents for the buyer,when her type is v H , implied by the intrapersonal equilibrium. (The communica-tion device and the probability of trade are the ones specified in Section 3.3.) Weproceed “backwards,” starting from µ < µ and then µ ∈ D n for n ≥ µ < µ . Using the binding participation constraint for the buyer when hertype is v L , we have x ∗ µ ( µ ) = v L .Note that the utility of the buyer of type v H when the seller has belief µ < µ is u ∗ H ( µ ) = ∆ v .Fix n ≥
1. Consider now µ ∈ [ µ n , µ n + ) . Using the binding participationconstraint for the buyer when her valuation is v L , we have x ∗ µ ( µ n − ) = x ∗ µ ( ) , we use the incentive compatibility constraint for the hightype: β ∗ µ ( | v H )( v H − x ∗ µ ( )) + β ∗ µ ( µ n − | v H ) δ u ∗ H ( µ n − ) = δ u ∗ H ( µ n − ) x ∗ µ ( ) = v H − δ u ∗ H ( µ n − ) .We then construct x ∗ µ ( ) recursively. Set n =
1. Then, for µ ∈ [ µ , µ ) , x ∗ µ ( ) = v H − δ ∆ v = v L + ( − δ ) ∆ vu ∗ H ( µ ) = δ ∆ v ,and for n ≥
2, we obtain that for µ ∈ [ µ n , µ n + ) , x ∗ µ ( ) = v H − δ n ∆ v = v L + ( − δ n ) ∆ vu ∗ H ( µ ) = δ n ∆ v .Note the construction of the transfers highlights that if the seller posts a price of x ∗ µ ( ) when his belief is µ , the buyer, when her type is v H , is indifferent betweenaccepting this price and rejecting it; whereas the low type is always better offrejecting this price because it lies above v L .Summing up, the intrapersonal equilibrium determines a map γ ∗ : ∆ ( V ) C such that γ ∗ ( µ ) = h ( V , β ∗ µ , ∆ ( V )) , ( q ∗ µ , x ∗ µ ) i . (D.1)The buyer’s payoffs when her valuation is v H are given by: u ∗ H ( µ ) = δ i ∆ v , µ ∈ D i , i =
0, . . . .Let U ∗ H denote the following correspondence: U ∗ H ( µ ) = (cid:26) u ∗ H ( µ ) if µ = µ i , i ≥ (cid:2) δ i ∆ v , δ i − ∆ v (cid:3) if µ = µ i . (D.2)For future use, note U ∗ H is upper-hemicontinuous, convex-valued, and compact-valued. 60 Proofs of Section 3.3
Appendix E is organized in three parts. Section E.1 constructs the buyer’s strat-egy profile so as to specify her best responses after every history in the game.Section E.2 performs the same exercise for the seller, using the strategy from theintrapersonal equilibrium (recall Equation D.1). Finally, in Section E.3, we showthat if continuation values are specified as in the intrapersonal equilibrium, andthe buyer’s strategy is completed as in the first step, the seller has no one-shot de-viations from the equilibrium strategy. We also argue the buyer has no one-shotdeviations. Appendix I in Doval and Skreta (2019) lays out the formalisms thatjustify the use of self-generation arguments in our setting; this, in turn, impliesthat guaranteeing the absence of one-shot deviations given the continuation val-ues is enough to conclude we have indeed constructed a PBE of G ∞ ( µ ) , whichgives the seller the same payoff as in the intrapersonal equilibrium. E.1 Completing the buyer’s strategy
Fix a public history h t and let µ denote the seller’s beliefs at that public history. To complete the buyer’s strategy, we classify mechanisms, M , in four categories:
0. Mechanisms that given the intrapersonal equilibrium continuation valuessatisfy participation and truthtelling, that is, M such that the following hold: ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v H )[ v H q M ( µ ′ ) − x M ( µ ′ ) + δ ( − q M ( µ ′ ) u ∗ H ( µ ′ )] ≥ max { ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v L )[ v H q M ( µ ′ ) − x M ( µ ′ ) + δ ( − q M ( µ ′ ) u ∗ H ( µ ′ )] } ,and ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v L )[ v L q M ( µ ′ ) − x M ( µ ′ )] ≥ max { ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v H )[ v L q M ( µ ′ ) − x M ( µ ′ )] } .Denote the set of these mechanisms M C . This prior will, of course, be an equilibrium object, but we suppress this from the notation tokeep things simple. Gerardi and Maestri (2018) use a similar trick to complete the worker’s strategy in their paper.
61. Mechanisms not in M C such that the buyer, when her type is v H , has no re-porting strategy, ρ ∈ ∆ ( V ) , such that ∑ v ′ β M ( µ ′ | v ′ ) ρ ( v ′ )[ v H q M ( µ ′ ) − x M ( µ ′ )] ≥
0; let M C denote the set of these mechanisms.2. Mechanisms not in M C such that the buyer, when her type is v H , has a re-porting strategy, ρ ∈ ∆ ( V ) , such that ∑ v ′ β M ( µ ′ | v ′ ) ρ ( v ′ )[ v H q M ( µ ′ ) − x M ( µ ′ )] ≥
0, but not when her type is v L ; let M C denote the set of these mechanisms.3. Mechanisms not in M C such that both types have such a reporting strategy;let M C denote the set of these mechanisms.If M ∈ M C , specify that the buyer rejects the mechanism for both types. Hence,under this strategy, the seller does not update his beliefs after observing a rejec-tion. If, however, the buyer accepts, the seller believes v = v H . Note that, in thiscase, continuation payoffs for the buyer are 0 from then on, regardless of her type.For each type v , let r ∗ v ( M , 1 ) denote an element of arg max ρ ∈ ∆ ( V ) ∑ v ′ ∈ V ρ ( v ′ ) ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ′ )( vq M ( µ ′ ) − x M ( µ ′ )) .If M ∈ M C , specify that the buyer rejects when her type is v L . Hence, withoutloss of generality, we can specify that if the seller observes that the buyer acceptsthe mechanism, the buyer’s type is v H . For type v H , let r ∗ v H ( M , 1 ) satisfy that if r ∗ v H ( M , 1 )( v ) >
0, then ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v )[ v H q M ( µ ′ ) − x M ( µ ′ )] ≥ ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ′ )[ v H q M ( µ ′ ) − x M ( µ ′ )] for all v ′ ∈ V . Note we are using that the buyer’s continuation payoffs are 0conditional on her accepting the mechanism.Then, v H ’s payoff from participating in the mechanism M is given by U v H ( M ) = ∑ v ′ ∈ V r ∗ v H ( M , 1 )( v ′ ) ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ′ )[ v H q M ( µ ′ ) − x M ( µ ′ ) + δ ( − q M ( µ ′ )) × ] ,whereas the payoff from rejecting is U v H ( π v H , f ) = δ f ( ν ( µ , π v H )) . Even if the buyer does not participate on the equilibrium path, we still need to guarantee thereporting strategy is sequentially rational. ν ( µ , π v H ) = µ ( − π v H ) µ ( − π v H ) + − µ .is the seller’s belief that the buyer is of type v H when observing a rejection, accord-ing to Bayes’ rule, and f ( ν ( µ , π v H )) is a measurable selection from U ∗ H ( ν ( µ , π v H )) .Note the payoff from rejecting is specified under the assumption that in the contin-uation, the equilibrium path coincides with that of the intrapersonal equilibriumwhen beliefs are ν ( µ , π v H ) . We use, however, a selection from U ∗ H to ensure thatif needed, the seller randomizes between the posted prices when indifferent tohelp make the buyer’s continuation problem well-behaved.Now, ( π ∗ v H ( M ) , f ∗ ( M )) are chosen so that π ∗ v H ∈ arg max p ∈ [ ] ( − p ) U v H ( π ∗ v H , f ∗ ( M )) + pU v H ( M ) The main result in Simon and Zame (1990) implies a solution to the above prob-lem exists, given the properties of U ∗ H and the linearity in p of the objective. Let π ∗ v H ( M ) denote this fixed point. Now, if π ∗ v H ( M ) < ν ( µ , π ∗ v H ( M )) = µ i forsome i ≥
1, then there exists a weight φ ( µ i , M ) ∈ [
0, 1 ] that solves the following: f ∗ ( M , µ i ) = ( − φ ) δ i − ∆ v + φδ i ∆ v . (E.1)This weight captures the probability with which the seller, when his prior is µ i ,mixes between γ ∗ ( µ i ) and γ ∗ ( µ i − ) .For a mechanism in M C , let r ∗ v L ( M , 1 ) denote an element inarg max ρ ∈ ∆ ( V ) ∑ v ′ ∈ V ρ ( v ′ ) ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ′ )( v L q M ( µ ′ ) − x M ( µ ′ )) .Finally, if M ∈ M C , specify that the buyer accepts for both types. If the sellerobserves that the buyer rejects the mechanism, he assigns probability 1 to thebuyer’s valuation being v H . Thus, upon rejection, continuation payoffs are 0 forboth types. Note that mechanisms in M C satisfy the participation constraint giventhe continuation values of the intrapersonal equilibrium. Now, let m ∗ L satisfy ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | m ∗ L )( v L q M ( µ ′ ) − x M ( µ ′ )) ≥ ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v )( v L q M ( µ ′ ) − x M ( µ ′ )) , We cannot ensure that truthtelling will hold for mechanisms in M C , which is why we needthe extra piece of notation. v ∈ V . Set r ∗ v L ( M , 1 )( m ∗ L ) =
1. Let { m ∗ H } = V \ { m ∗ L } and define ∆ ( V ) H = { µ ′ : β M ( µ ′ | m ∗ H ) > } ∆ ( V ) L = { µ ′ : β M ( µ ′ | m ∗ L ) > } .Let µ denote the seller’s belief about the buyer’s valuation being v H . Let r ∈ [
0, 1 ] denote the weight the buyer assigns to m ∗ L when her valuation is v H : ν ( µ , µ ′ , r , δ m ∗ L ) = µ ′ ∈ ∆ ( V ) H \ ∆ ( V ) L µ ( r β M ( µ ′ | m ∗ L )+( − r ) β M ( µ ′ | m ∗ H )) µ ( r β M ( µ ′ | m ∗ L )+( − r ) β M ( µ ′ | m ∗ H ))+( − µ ) β M ( µ ′ | m ∗ L ) if µ ′ ∈ ∆ ( V ) H ∩ ∆ ( V ) L µ r µ r +( − µ ) if µ ′ ∈ ∆ ( V ) L \ ∆ ( V ) H ,where we assume that if the seller observes an output message that is not con-sistent with m ∗ L , he believes it was generated by the high-valuation buyer. Thisspecification of beliefs does not conflict with Bayes’ rule where possible: either m ∗ H has positive probability in the optimal reporting strategy of the buyer whenher valuation is v H , in which case, ν would be consistent with Bayes’ rule, orit does not, in which case, Bayes’ rule where possible places no restrictions on ν ( µ , µ ′ , · ) for µ ′ ∈ ∆ ( V ) H \ ∆ ( V ) L .Given ν , the buyer when her valuation is v H obtains a payoff of U v H ( r , m , f ) = ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | m )( v H q M ( µ ′ ) − x M ( µ ′ ) + δ ( − q M ( µ ′ )) f ( ν ( µ , µ ′ , r , δ m ∗ L ))) ,when she reports m ∈ V , where f is a selection from U ∗ H . We want to find ( r ∗ v H , f ∗ ( M )) so that r ∗ v H ∈ arg max r ∈ [ ] rU v H ( r ∗ v H , m ∗ L , f ∗ ( M )) + ( − r ) U v H ( r ∗ v H , m ∗ H , f ∗ ( M )) . (E.2)Again, we appeal to the result in Simon and Zame (1990) to argue that the proper-ties of U ∗ H and the objective function above imply the existence of such an ( r ∗ , f ∗ ( M )) .Set r ∗ v H ( M , 1 )( m ∗ L ) = r ∗ v H .As we did before, whenever ν ( µ , µ ′ , r ∗ v H , δ m ∗ L ) = µ i for i ≥
1, we can define φ ( M , µ i ) as the weight on γ ∗ ( µ i ) implied by f ∗ ( M , µ i ) .Summing up, the buyer’s strategy at any history h tB is given by π ∗ v ( h tB , M ) = M ∈ M C ∪ M C π ∗ v H ( M ) if M ∈ M C and v = v H r ∗ v ( h tB , M , 1 ) = (cid:26) δ v if M ∈ M C r ∗ v ( M , 1 ) otherwise . (E.4) E.2 Full specification of the PBE assessment
To complete the PBE assessment for G ∞ ( µ ) , we now specify the seller’s strategyand the belief system. We introduce two pieces of notation: the first one allows usto keep track of the last payoff-relevant event; the second one allows us to keeptrack of how the seller’s beliefs evolve given the buyer’s strategy.From the beginning of period t of the game until the end of that period, thefollowing things are determined: (i) the seller’s choice of mechanism, M ; (ii) thebuyer’s participation decision, p ∈ (
0, 1 ) ; and (iii) if the buyer participates of themechanism, the allocation and the output message. We let z ∅ ( M ) = ( M , 0, ∅ , 0, 0 ) to denote the outcome when mechanism M is chosen, the buyer does not partici-pate, no output message is produced, and no trade and no transfers occur. We let z q ( M , µ ′ ) = ( M , 1, µ ′ , q , x ) denote the outcome at the end of period t when mech-anism M is chosen, the buyer participates (1), the output message is µ ′ , and theallocation is ( q , x ) . Of course, if q =
1, the game ends. Note that any public his-tory h t can be written as ( h t − , z ) for some z as defined above. Given h t , let z ( h t ) denote the corresponding outcome z . Given any prior µ , define T ( µ , z ) = µ ′ if z = z q ( M , µ ′ ) and M ∈ M C z = z ∅ ( M ) and M ∈ M C ∪ M C or z = z q ( M , µ ′ ) and M ∈ M C ∪ M C µ if z = z ∅ ( M ) and M ∈ M C ν ( µ , π ∗ v H ( M )) if z = z q ( M , µ ′ ) and M ∈ M C ν ( µ , µ ′ , r ∗ v H ( M , 1 ) , r ∗ v L ( M , 1 )) if z = z q ( M , µ ′ ) and M ∈ M C (E.5)Let µ denote the seller’s prior in G ∞ ( µ ) . Define µ ∗ ( ∅ ) = µ and Γ ∗ ( ∅ ) = γ ∗ ( µ ) , where γ ∗ is the strategy in the intrapersonal equilibrium as defined inEquation D.1. For any public history h t , define µ ∗ ( h t ) = T ( µ ∗ ( h t − ) , z ( h t ))) . (E.6)65f either (i) z = z q ( M , · ) and M ∈ M C ∪ M C ∪ M C , (ii) z = z ∅ ( M ) , µ ∗ ( h t ) / ∈{ µ i } i ≥ and M ∈ M C , (iii) z = z q ( M , · ) , µ ∗ ( h t ) / ∈ { µ i } i ≥ and M ∈ M C , or (iv) z = z ∅ ( M ) and M ∈ M C ∪ M C ∪ M C , define Γ ∗ ( h t )( M ) = [ M = γ ∗ ( T ( µ ∗ ( h t − ) , z ( h t ))))] . (E.7)If z ( h t ) = z ∅ ( M , · ) for M ∈ M C and µ ∗ ( h t ) = µ i , i ≥
1, define Γ ∗ ( h t )( M ) = φ ( M , µ i ) M = γ ∗ ( µ i ) − φ ( M , µ i ) M = γ ∗ ( µ i − ) z ( h t ) = z q ( M , · ) for M ∈ M C and µ ∗ ( h t ) = µ i , i ≥
1, define Γ ∗ ( h t )( M ) = φ ( M , µ i ) M = γ ∗ ( µ i ) − φ ( M , µ i ) M = γ ∗ ( µ i − ) E.3 Seller’s and buyer’s sequential rationality
We now check that at all histories, neither the seller nor the buyer have a one-shot deviation from the prescribed strategy profile, given the continuation valuesfor the seller and the buyer constructed using the intrapersonal equilibrium. Thisimplies that the payoffs in the intrapersonal equilibrium together with the beliefoperator defined in Equation E.5 belong in the self-generating set. Appendix I inDoval and Skreta (2019) lays out the formalisms that justify the use of techniques´a la Abreu et al. (1990) for the game under consideration.Let µ ∗ ( h t ) denote the seller’s belief at history h t that the buyer’s type is v H . Wenow show the seller cannot achieve a payoff higher than R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) .Note that among mechanisms of type 0, the one that corresponds to the intrap-ersonal equilibrium is, by definition, the best that the seller can do. Hence, toshow that there are no deviations, we need to show the seller cannot benefit fromoffering mechanisms of types 1-3. 66iven the buyer’s strategy, the payoff from offering a mechanism in M C is δ R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) ≤ R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) , and hence this deviation is notprofitable. Let M denote a mechanism in M C and let π ∗ v H ( h tB , M ) , r ∗ v H ( h tB , M , 1 ) denote the buyer’s best responses as constructed in Section E.1. Denote by ν ≡ ν ( µ ∗ ( h t ) , π ∗ v H ( h tB , M )) the seller’s belief that he is facing a buyer with valuation v H when he observes non-participation. The seller’s payoff is then µ ∗ ( h t ) π ∗ v H ( h tB , M ) ∑ µ ′ ∈ ∆ ( V ) ∑ v ∈ V β M ( µ ′ | v ) r ∗ v H ( h tB , M , 1 )( v ) ! [ x M ( µ ′ ) + δ ( − q M ( µ ′ )) v H ]+ ( − µ ∗ ( h t ) π ∗ v H ( h tB , M )) R ( τ ∗ , q ∗ ) ( ν , ν )) , (E.10)where the continuation values after rejection are constructed using the policy fromthe intrapersonal equilibrium once the seller has posterior ν ( µ ∗ ( h t ) , π ∗ v H ( h tB , M )) . Now, consider the following alternative mechanism, M ′ : β M ′ ( | v H ) = π ∗ v H ( M ) ∑ v ∈ V , µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ) r ∗ v H ( M )( v ) β M ′ ( ν | v H ) = ( − π ∗ v H ( M )) β M ′ ( ν | v L ) = q M ′ ( ν ) = x M ′ ( ν ) = x M ′ ( ) = ∑ v ∈ V ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ) r ∗ v H ( M )( v ) x M ( µ ′ ) q M ′ ( ) = ∑ v ∈ V ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ) r ∗ v H ( M )( v ) q M ( µ ′ ) Note that if the buyer participates and truthfully reports her type, M ′ gives theseller a payoff equal to the expression in Equation E.10. Moreover, participatingand truthfully reporting her type is a best response for the buyer. Then, we canwrite the payoff to mechanism, M ′ , as ∑ µ ′ ∈{ ν ,1 } τ M ′ ( µ ∗ ( h t ) , µ ′ )[ x M ′ ( µ ′ ) + δ ( − q M ′ ( µ ′ )) R ( τ ∗ , q ∗ ) ( µ ′ , µ ′ )] ,where, as in the main text, τ M ′ ( µ ∗ ( h t ) , µ ′ ) = ∑ v ∈ V µ ∗ ( h t )( v ) β M ′ ( µ ′ | v ) . To keep notation simple, we do not use that the seller in the continuation may randomize inhis choice. x M ′ ( ) ≤ v H q M ′ ( ) − δ u ∗ H ( µ ′ ( π ∗ v H ( M ))) , so that ∑ µ ′ ∈{ ν ,1 } τ M ′ ( µ ∗ ( h t ) , µ ′ )[ x M ′ ( µ ′ ) + δ ( − q M ′ ( µ ′ )) R ( τ ∗ , q ∗ ) ( µ ′ , µ ′ )] ≤ ∑ µ ′ ∈{ ν ,1 } τ M ′ ( µ ∗ ( h t ) , µ ′ ) n q M ′ ( µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ ∗ ( h t )))+ δ ( − q M ′ ( µ ′ )) (cid:20) R ( τ ∗ , q ∗ ) ( µ ′ , µ ′ ) + ( − µ ′ )( µ ′ − µ ′ − µ ∗ ( h t ) − µ ∗ ( h t ) ) u ∗ H ( µ ′ ) (cid:21)(cid:27) , which, by definition, is weakly lower than R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) .Similar steps imply the seller does not have a one-shot deviation to a mechanismof type 3.Finally, note the construction of π ∗ vv
We now check that at all histories, neither the seller nor the buyer have a one-shot deviation from the prescribed strategy profile, given the continuation valuesfor the seller and the buyer constructed using the intrapersonal equilibrium. Thisimplies that the payoffs in the intrapersonal equilibrium together with the beliefoperator defined in Equation E.5 belong in the self-generating set. Appendix I inDoval and Skreta (2019) lays out the formalisms that justify the use of techniques´a la Abreu et al. (1990) for the game under consideration.Let µ ∗ ( h t ) denote the seller’s belief at history h t that the buyer’s type is v H . Wenow show the seller cannot achieve a payoff higher than R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) .Note that among mechanisms of type 0, the one that corresponds to the intrap-ersonal equilibrium is, by definition, the best that the seller can do. Hence, toshow that there are no deviations, we need to show the seller cannot benefit fromoffering mechanisms of types 1-3. 66iven the buyer’s strategy, the payoff from offering a mechanism in M C is δ R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) ≤ R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) , and hence this deviation is notprofitable. Let M denote a mechanism in M C and let π ∗ v H ( h tB , M ) , r ∗ v H ( h tB , M , 1 ) denote the buyer’s best responses as constructed in Section E.1. Denote by ν ≡ ν ( µ ∗ ( h t ) , π ∗ v H ( h tB , M )) the seller’s belief that he is facing a buyer with valuation v H when he observes non-participation. The seller’s payoff is then µ ∗ ( h t ) π ∗ v H ( h tB , M ) ∑ µ ′ ∈ ∆ ( V ) ∑ v ∈ V β M ( µ ′ | v ) r ∗ v H ( h tB , M , 1 )( v ) ! [ x M ( µ ′ ) + δ ( − q M ( µ ′ )) v H ]+ ( − µ ∗ ( h t ) π ∗ v H ( h tB , M )) R ( τ ∗ , q ∗ ) ( ν , ν )) , (E.10)where the continuation values after rejection are constructed using the policy fromthe intrapersonal equilibrium once the seller has posterior ν ( µ ∗ ( h t ) , π ∗ v H ( h tB , M )) . Now, consider the following alternative mechanism, M ′ : β M ′ ( | v H ) = π ∗ v H ( M ) ∑ v ∈ V , µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ) r ∗ v H ( M )( v ) β M ′ ( ν | v H ) = ( − π ∗ v H ( M )) β M ′ ( ν | v L ) = q M ′ ( ν ) = x M ′ ( ν ) = x M ′ ( ) = ∑ v ∈ V ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ) r ∗ v H ( M )( v ) x M ( µ ′ ) q M ′ ( ) = ∑ v ∈ V ∑ µ ′ ∈ ∆ ( V ) β M ( µ ′ | v ) r ∗ v H ( M )( v ) q M ( µ ′ ) Note that if the buyer participates and truthfully reports her type, M ′ gives theseller a payoff equal to the expression in Equation E.10. Moreover, participatingand truthfully reporting her type is a best response for the buyer. Then, we canwrite the payoff to mechanism, M ′ , as ∑ µ ′ ∈{ ν ,1 } τ M ′ ( µ ∗ ( h t ) , µ ′ )[ x M ′ ( µ ′ ) + δ ( − q M ′ ( µ ′ )) R ( τ ∗ , q ∗ ) ( µ ′ , µ ′ )] ,where, as in the main text, τ M ′ ( µ ∗ ( h t ) , µ ′ ) = ∑ v ∈ V µ ∗ ( h t )( v ) β M ′ ( µ ′ | v ) . To keep notation simple, we do not use that the seller in the continuation may randomize inhis choice. x M ′ ( ) ≤ v H q M ′ ( ) − δ u ∗ H ( µ ′ ( π ∗ v H ( M ))) , so that ∑ µ ′ ∈{ ν ,1 } τ M ′ ( µ ∗ ( h t ) , µ ′ )[ x M ′ ( µ ′ ) + δ ( − q M ′ ( µ ′ )) R ( τ ∗ , q ∗ ) ( µ ′ , µ ′ )] ≤ ∑ µ ′ ∈{ ν ,1 } τ M ′ ( µ ∗ ( h t ) , µ ′ ) n q M ′ ( µ ′ )( µ ′ v H + ( − µ ′ ) ˆ v L ( µ ∗ ( h t )))+ δ ( − q M ′ ( µ ′ )) (cid:20) R ( τ ∗ , q ∗ ) ( µ ′ , µ ′ ) + ( − µ ′ )( µ ′ − µ ′ − µ ∗ ( h t ) − µ ∗ ( h t ) ) u ∗ H ( µ ′ ) (cid:21)(cid:27) , which, by definition, is weakly lower than R ( τ ∗ , q ∗ ) ( µ ∗ ( h t ) , µ ∗ ( h t )) .Similar steps imply the seller does not have a one-shot deviation to a mechanismof type 3.Finally, note the construction of π ∗ vv , r ∗ vv