Optimal mechanism for the sale of a durable good

Doval, Laura; Skreta, Vasiliki Article Optimal mechanism for the sale of a durable good Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Doval, Laura; Skreta, Vasiliki (2024) : Optimal mechanism for the sale of a durable good, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 2, pp. 865-915, https://doi.org/10.3982/TE4485 This Version is available at: https://hdl.handle.net/10419/320255 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 865–915 1555-7561/20240865 Optimal mechanism for the sale of a durable good Laura Doval Economics Division, Columbia Business School, Columbia University and CEPR Vasiliki Skreta Department of Economics, University of Texas at Austin, Department of Economics, University College London, and CEPR A buyer wishes to purchase a durable good from a seller who in each period chooses a mechanism under limited commitment. The buyer’s value is binary and fully persistent. We show that posted prices implement all equilibrium outcomes of an infinite-horizon, mechanism-selection game. Despite being able to choose mechanisms, the seller can do no better and no worse than if he chose prices in each period, so that he is subject to Coase’s conjecture. Our analysis marries insights from information and mechanism design with those from the literature on durable goods. We do so by relying on the revelation principle in Doval and Skreta (2022). Keywords. Mechanism design, limited commitment, information design, public PBE, posted prices, Coase conjecture. JEL classification. D84, D86. 1. Introduction We characterize the equilibrium outcomes of an infinite-horizon, mechanism-selection game between a durable-good seller and a privately informed buyer under limited commitment, so that the seller can commit to today’s mechanism, but not to the mechanism he will offer if no sale occurs. Theorem 1shows that all equilibrium outcomes can be implemented via posted prices. We construct a perfect Bayesian equilibrium of the mechanism-selection game, which achieves the seller’s unique equilibrium payoff, and we show that it implements the essentially unique equilibrium outcome.1In this equilibrium, as long as a sale has not occurred, the seller will choose a mechanism that can be implemented as a posted price. Despite being able to choose from a rich set of mechLaura Doval: [email protected] Vasiliki Skreta: [email protected] We thank three anonymous referees for excellent comments, which substantially improved the paper. We would like to thank Rahul Deb, Frederic Koessler, Dan Quigley, Pablo Schenone, and especially Max Stinchcombe, as well as audiences at Cowles, SITE, and Stony Brook, for thought-provoking questions and illuminating discussions. Vasiliki Skreta is grateful for generous financial support through the ERC consolidator grant 682417 “Frontiers in design.” This research was supported by grants from the National Science Foundation (Doval: SES-2131706; Skreta: SES-1851729). 1Whenever the equilibrium outcome is not unique, all equilibrium outcomes are achieved also via a sequence of posted prices. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4485 866 Doval and Skreta Theoretical Economics 19 (2024) anisms, the seller can do no better and no worse than if he could only choose prices in each period. In our game, an uninformed seller faces a privately informed buyer, whose valuation is binary, fully persistent, and strictly above the seller’s marginal cost. In each 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. A mechanism consists of (i) a set of input messages for the buyer, and (ii) for each input message, a distribution over output messages and allocations. Whereas the seller observes the output message and the allocation, he does not observe the input message the buyer submits to the mechanism. Thus, when designing the mechanism, the seller gets to design how much he observes about the buyer’s choices, and hence design his beliefs about the buyer’s value. The combination of mechanism design and information design elements is key to our characterization. Our analysis bridges the literatures on mechanism design and on the durable-good monopolist, especially the work of Gul, Sonnenschein, and Wilson (1986). To see this, it is useful to review the main steps involved in the proof of Theorem 1. First, we construct an assessment that is identical along the path to that in Hart and Tirole (1988), which we dub the posted-prices assessment. In this assessment, along the path of play, the seller sells the good using a decreasing sequence of prices, which reflect that conditional on the good not being sold the seller assigns less probability to the buyer’s value being high. Second, we argue that the seller’s payoff under the posted-prices assessment is an upper bound on the seller’s equilibrium payoff in the mechanism-selection game. To do so, we rely on an auxiliary program, that only involves the seller (see (OPT)inSection4). In this program, the seller maximizes the dynamic analogue of the virtual surplus,by choosing a Bayes’ plausible distribution over posteriors and for each posterior (i) a probability of trade and (ii) a vector of equilibrium continuation payoffs. We arrive at the program defined in (OPT) by relying on the tools in our previous work, Doval and Skreta (2022). The main theorem in Doval and Skreta (2022) allows us to simplify the class of mechanisms the seller offers in any equilibrium of the game and the buyer’s equilibrium behavior. This step reduces the search for the optimal sequence of mechanisms to those that satisfy, loosely speaking, a sequence of participation and truth-telling constraints, allowing us for the most part to ignore the buyer as a player. Like in standard mechanism design, the low-valuation buyer’s utility and the high-valuation buyer’s truth-telling constraint determine an upper bound on the revenue the seller can extract within a period (Lemma 2). Replacing this upper bound in the seller’s payoff provides us with a dynamic analogue of the virtual surplus (Equation (4)), where the seller’s payoff is written as a function of the allocation, but also the continuation payoffs. We show that the value of (OPT) coincides with the seller’s payoff in the postedprices assessment, and hence that the seller cannot do better than in the posted-prices equilibrium. Because the auxiliary program (OPT) ignores the truth-telling constraint of the low-valuation buyer (i.e., it corresponds to the relaxed program in mechanism design), our result implies that the solution to (OPT) satisfies the remaining constraints and can thus be implemented as an equilibrium outcome. As we discuss in the conclusions, we expect that in settings with transferable utility, the study of the analogous 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 867 problem to (OPT) provides a natural benchmark to understand the properties of the principal’s optimal mechanism, even if in some settings the solution to the analogue of (OPT) may not deliver an implementable outcome. Finally, following the logic in Gul, Sonnenschein, and Wilson (1986), we show that the seller’s payoff in the posted-prices assessment is a lower bound on the seller’s equilibrium payoff. Underlying the argument in Gul, Sonnenschein, and Wilson (1986)that a unique equilibrium payoff exists in the gap case is the property that the minimum price the seller chooses in equilibrium imposes an upper bound on the maximum payoff the buyer can obtain. Relying once again on (OPT), we establish that guarantees on the seller’s equilibrium payoff translate into upper bounds on the high-valuation buyer’s payoff. Armed with this result, we show that the seller can always undercut the price in the posted-prices assessment and earn close to his payoff in that assessment. The significance of our results is two-fold. First, to the best of our knowledge, this is the first paper to characterize optimal mechanisms under limited commitment and persistent private information in an infinite-horizon setting. Because the set of tools available to tackle the difficulties with the revelation principle under limited commitment do not readily apply to infinite-horizon settings (see, e.g., the seminal work of Bester and Strausz (2001,2007), and the discussion in the related literature), such characterization has proved elusive. In Doval and Skreta (2022), we provide a revelation principle for mechanism-selection games under limited commitment that applies to a broad class of games, including infinite-horizon ones. It is the application of this tool that allows us to argue that the mechanism we characterize is the optimal one among all mechanisms the seller could have offered the buyer under limited commitment. Second, the optimality of posted prices should not be taken for granted, even if it is evocative of Skreta (2006). First, our model is not an infinite-horizon version of that in Skreta (2006), since we consider a larger class of mechanisms than Skreta (2006). Indeed, the mechanisms in Skreta (2006) presume that the seller must observe the buyer’s input message (cf. Laffont and Tirole (1988), Bester and Strausz (2001)), whereas we consider mechanisms in which the seller gets to design how much he observes about the buyer’s input message. In Doval and Skreta (2022), we study a two-period version of the model in Skreta (2006), but we allow the seller to offer mechanisms like those in this paper. We show that when the seller is sufficiently patient it is not an equilibrium for the seller to post a price in each period (Remark 3explains why posted prices may fail to be optimal in Doval and Skreta (2022)). It follows that we cannot take limits using the equilibrium outcome in Skreta (2006) to analyze the equilibrium outcomes of the game we study, even after showing that the game ends in finite time. Second, Breig (2022)showsin a binary-value model with a perishable good that posted prices may not be optimal: Indeed, the seller may benefit from using random delivery contracts. Related literature: The paper contributes mainly to three strands of literature. The first strand, similar to this paper, derives optimal mechanisms when the designer has limited commitment. Most papers in that literature examine either finite-horizon settings (see Laffont and Tirole (1988), Skreta (2006,2015), Deb and Said (2015), Fiocco and Strausz (2015), Beccuti and Möller (2018)), or infinite-horizon settings, imposing restrictions on the class of contracts that can be offered (e.g., Gerardi and Maestri (2020)), or on the 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 868 Doval and Skreta Theoretical Economics 19 (2024) solution concept (e.g., Acharya and Ortner (2017)).2Underlying these restrictions are that the results in both Bester and Strausz (2001)andSkreta (2006) do not readily extend to infinite-horizon settings. For instance, the result in Bester and Strausz (2001) applies only if the principal is earning his highest payoff consistent with the agent’s payoff (see Lemma 1 in Bester and Strausz (2001)). Thus, implicit in their multistage extension is a restriction to equilibria of the mechanism-selection game that possess a Markov structure, which as shown by Ausubel and Deneckere (1989), may not be enough to characterize the principal’s best equilibrium payoff. The approach in Skreta (2006)hastheadvantage that the set of incentive feasible outcomes has a well-understood structure. It is not clear, however, how to incorporate the principal’s sequential rationality constraints in infinite-horizon settings in a tractable way. The second strand is the literature that follows the observation in Coase (1972)that the durable-good monopolist faces a time-inconsistency problem, which in turn limits his monopoly power. The papers in the durable-good monopolist literature (Stokey (1981), Bulow (1982), Gul, Sonnenschein, and Wilson (1986), Sobel (1991), Ortner (2017)) study price dynamics and establish (under some conditions) Coase’s conjecture.3Related to this literature is the problem of dynamic bargaining with one-sided incomplete information4(e.g., Sobel and Takahashi (1983), Fudenberg, Levine, and Tirole (1985), Ausubel and Deneckere (1989)). In all these papers, the uninformed party’s inability to commit limits his bargaining power. Finally, as will become clear from the analysis, the paper contributes to the literature on information design (Aumann, Maschler, and Stearns (1995)andKamenica and Gentzkow (2011)), highlighting its potential to provide tractable characterizations of equilibrium outcomes in games. Contrary to most of these papers, however, the seller aims to persuade his future self, as opposed to another player, highlighting the role of information as a commitment device (see, e.g., Carrillo and Mariotti (2000), and recently, Habibi (2020)). Organization The rest of the paper is organized as follows. Section 2describes the model; Section 2.1 summarizes the results in Doval and Skreta (2022) used to simplify the analysis that follows. Section 3presents the main result of the paper, Theorem 1. Section 4introduces the auxiliary program (OPT) and studies its properties. Section 5 reviews the main steps of the proof of Theorem 1.Section6concludes. All proofs not in the main text are in the Appendix. 2. Model Primitives: Two players, a seller and a buyer, interact over infinitely many periods. The seller owns one unit of a durable good to which he attaches value 0. The buyer has 2Beccuti and Möller (2018) lies somewhat in between these two strands because they take limits of their finite-horizon results to draw conclusions about the infinite-horizon game. They do not show that this limit corresponds to the seller’s revenue-maximizing equilibrium in the infinite-horizon game. 3Other papers, like Wolinsky (1991), McAfee and Wiseman (2008), and Board and Pycia (2014), study variations on Coase’s original problem and their implications for Coase’s conjecture. Relatedly, Brzustowski, Georgiadis-Harris, and Szentes (2023) show that smart contracts help the seller avoid the Coase conjecture. 4Recently, Peski (2022) studies alternating bargaining games where players can offer menus. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 869 private information: before her interaction with the seller starts, she observes her value v∈{vL,vH}≡V,with0<v L<v H.Letv ≡vH−vLdenote the difference in values. Let μ0denote the probability that the buyer’s value is vHat the beginning of the game. In what follows, we denote by (V)the set of distributions on V. An allocation in period tis a pair (q,x)∈{0, 1}×R,whereqindicates whether the good is traded (q=1) or not (q=0), and xis a payment from the buyer to the seller. Let Adenote the set of allocations.5The game ends the first time the good is sold. Payoffs are as follows. If in period t, the allocation is (q,x), the flow payoffs are uB(q,x,v)=vq −xand uS(q,x)=xfor the buyer and the seller, respectively. The seller and the buyer maximize the expected discounted sum of flow payoffs. They share a common discount factor δ∈(0, 1). Mechanisms: To introduce the timing of the game, we first define the seller’s action space. In each period, the seller offers the buyer a mechanism. Following Doval and Skreta (2022), we define a mechanism as follows. A mechanism, M=(MM,SM,ϕM), consists of a set of input messages MM,asetofoutput messages SM, and a transition probability ϕMfrom MMto SM×A.6For instance, MMcan be the set of buyer values, V, and SMbe the set of seller beliefs about the buyer’s value, (V). In this case, the mechanism associates to each report a distribution over beliefs and allocations. We endow the seller with a collection (Mi,Si)i∈Iof input and output messages in which each Mi contains at least two elements, and each Sicontains (V).7Denote by MIthe set of all mechanisms with message sets (Mi,Si)i∈I.8 Mechanism-selection game: The seller’s prior μ0and the collection (Mi,Si)i∈Idefine a mechanism-selection game, denoted GI(μ0), as follows. In each period t,aslong as the good has not been sold, the game proceeds as follows. First, the seller and the buyer observe the realization of a public randomization device, ω∼U[0, 1].Second,the seller offers the buyer a mechanism, M. Observing the mechanism, the buyer decides whether to participate or not. If she does not participate in the mechanism, the good is not sold and no payments are made. If she instead chooses to participate, she sends a message m∈MM, which is unobserved by the seller. An output message and an allocation, (s,q,x),aredrawnfromϕM(·|m)and are observed by both the seller and the buyer. If the good is not sold, the game proceeds to period t+1. Histories: The game GI(μ0)has two types of histories: public and private. Public histories capture what the seller knows through period t: the past realizations of the 5Even if the set of allocations is {0, 1}×R, we allow the seller to offer randomizations on A, and hence induce fractional assignments of the good. 6Throughout, we assume that MMand SMare Polish spaces, that is, they are separable, completely metrizable topological spaces. Note that the set of allocations Ais also a Polish space and, therefore, SM×A is a Polish space. For a Polish space X,let(X)denote the set of Borel measures on X.Weendow(X) with the weak∗topology. Thus, (X)is also a Polish space (Aliprantis and Border (2013)). For any two measurable spaces Xand Y, a mapping ζ:X→ (Y)is a transition probability from Xto Yif for any measurable C⊆Y,ζ(C|x)is a measurable real valued function of x∈X. 7Because Vis finite, taking MMto be finite is without loss of generality (Doval and Skreta,2022). 8We restrict the seller to choose mechanisms with input and output messages in (Mi,Si)i∈Ito have a well-defined action space for the seller. This allows us to have a well-defined set of deviations, avoiding set-theoretic issues related to self-referential sets. The analysis in Doval and Skreta (2022) shows that the choice of the collection plays no further role in the analysis. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 870 Doval and Skreta Theoretical Economics 19 (2024) public randomization device, his past choices of mechanisms, the buyer’s participation decisions, and the realized output messages and allocations. We let htdenoteapublic history through period tand let Htdenote the set of all such histories. Instead, private histories capture what the buyer knows through period t. First, the buyer knows the public history of the game and her input messages into the mechanism (henceforth, a buyer history). Second, the buyer also knows her private information. We let ht Bdenote a buyer history through period tand let Ht B(ht)denote the set of buyer histories consistent with public history ht.Thus,V×Ht B(ht)denotes the set of private histories consistent with public history ht. Strategies and beliefs: A behavioral strategy for the seller is a collection of measurable mappings ≡(t)∞ t=0, where for each period tand each public history ht,t(ht)describes the seller’s (possibly random) choice of mechanism at ht.9Similarly, a behavioral strategy for the buyer is a collection of measurable mappings (πt(v,·),rt(v,·))∞ t=0,where for each period t, each private history (v,ht B), and each mechanism, Mt,πt(v,ht B,Mt) describes the buyer’s participation decision, whereas rt(v,ht B,Mt)describes the buyer’s choice of input messages in the mechanism, conditional on participation. We denote the tuple (πt(v,·),rt(v,·)) by (πtv,rtv), and the collection (πtv,rtv)∞ t=0by (πv,rv). A belief for the seller at the beginning of time t,historyht, is a distribution μt(ht)∈ (V×Ht B(ht)). The belief system, (μt)∞ t=0, is denoted by μ. Solution concept: We are interested in studying the perfect Bayesian equilibrium (henceforth, PBE) payoffs of this game, where PBE is defined informally as follows. An assessment, ,(πv,rv)v∈V,μ, is a PBE if the following hold: 1. ,(πv,rv)v∈V,μsatisfies sequential rationality, and 2. μsatisfies Bayes’ rule where possible. Appendix Econtains the formal statement. For now, we note that if the seller’s strategy space was finite and the mechanisms used by the seller had finite support, then this coincides with the definition in Fudenberg and Tirole (1991b).10 The prior μ0together with the strategy profile (,(πv,rv)v∈V)induce a distribution over the terminal nodes V×H∞ B. We are interested instead on the distribution it induces over the payoff-relevant outcomes, V×A∞. We say that a distribution η∈(V×A∞)is a PBE outcome if a PBE assessment ,(πv,rv)v∈V,μexists that induces η.Wedenote by O∗ I(μ0)the set of PBE outcomes and by E∗ I(μ0)⊆R3the set of PBE payoffs of GI(μ0). We denote a generic element of E∗ I(μ0)by u≡(uL,uH,uS),whereuSis the seller’s payoff and uL,uHdenote the buyer’s payoff when her value is vL,vH, respectively. Theorem 1characterizes O∗ I(μ0)and E∗ I(μ0). In particular, we show that the essentially unique equilibrium outcome can be achieved via a sequence of posted prices so that the seller of a durable good can do no better and no worse than by using posted prices. 9As we explain in Appendix E,MIis a Polish space—which we endow with its Borel σ-algebra—and we can follow Aumann (1964) when defining the seller’s strategy. 10The only difference between Bayes’ rule where possible and consistency in sequential equilibrium is the following. Under PBE, the seller can assign zero probability to one of the buyer’s values and then, after the buyer deviates, can assign positive probability to that same value. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 871 2.1 Revelation principle The game GI(μ0)is not simple to analyze for at least two reasons. First, the seller’s action space is large, and a priori, it is not clear which mechanisms could be ruled out from consideration. Second, fixing a seller’s strategy, and hence a sequence of mechanisms faced by the buyer, we still need to understand the buyer’s best response in the game induced by the sequence of mechanisms. Let G(μ0)denote the same game in the previous section, except that in each period the seller’s action space is the set of canonical mechanisms, denoted by MC, and defined as follows. MCis the set of all mechanisms where the set of input and output messages are the set of buyer’s values and of seller’s beliefs about the buyer’s value, respectively. That is, (M,S)=(V,(V)). Let O∗(μ0)denote the set of PBE outcomes and E∗(μ0)denote the set of PBE payoffs of G(μ0). In what follows, a subset of the set of canonical mechanisms has special significance: the set of direct Blackwell mechanisms. A direct Blackwell mechanism is a canonical mechanism ϕ:V→ ((V)×A)that can be decomposed into a Blackwell experiment, β:V→ ((V)),andanallocation rule,α:(V)→ (A).11 Lemma 1summarizes the key implications of Doval and Skreta (2022) for our analysis, which we explain below. Lemma 1(Doval and Skreta (2022)). For any PBE outcome of any mechanism-selection game GI(μ0), an outcome-equivalent PBE of game G(μ0)exists. That is, IO∗ I(μ0)= O∗(μ0). Moreover, let η∈O∗(μ0). Then a PBE assessment ,(πv,rv)v∈V,μof G(μ0)exists that induces ηand satisfies the following properties: (a) For all histories ht, the buyer participates in the mechanism offered by the seller at that history and truthfully reports her type, with probability 1; (b) For all histories ht, if the mechanism offered by the seller at htoutputs posterior μ, the seller’s updated equilibrium beliefs about the buyer’s value coincide with μ; (c) For all histories ht, the mechanism offered by the seller at htis a direct Blackwell mechanism; (d) The buyer’s strategy depends only on her private value and the public history. Lemma 1has several implications. Part (a) of Lemma 1implies the mechanisms chosen by the seller in equilibrium must satisfy a participation constraint and an incentive compatibility constraint for each buyer value and each public history. As in the case of commitment to long-term mechanisms, part (a) simplifies the analysis of the buyer’s 11That is, for all measurable subsets U⊂(V)and A⊂A, we have that for all v∈V, ϕU×A|v=U αA|μβ(dμ|v). 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 872 Doval and Skreta Theoretical Economics 19 (2024) behavior, by reducing it to a series of constraints the seller’s equilibrium offer of a mechanism must satisfy (see Equations (PCht,v)and(ICht,v,v)inSection4.1). Part (b) implies that the mechanism’s output message encodes all of the information that the seller has in equilibrium about the buyer’s value. In particular, conditional on observing the output message, the allocation carries no more information about the buyer’s value. As a consequence, conditional on the output message, the allocation can be drawn independently of the buyer’s report. This, in turn, delivers the decomposition of ϕMtas a direct Blackwell mechanism described in part (c). Part (c) implies that the choice of mechanism at history htcan be equivalently thought of as the choice of a Blackwell experiment, βMt, and an allocation rule, αMt. Direct Blackwell mechanisms allow us to separately optimize on the allocation given a particular experiment, and then optimize on the experiment. As in the literature on information design, it is convenient to work with the distribution over posteriors induced by the experiment βMt,whichwedenotebyτMtand is defined as follows. For all Borel subsets U⊆(V), U τMt(dμt+1)= v∈V μtht(v)βMtU|v,(BC μt(ht)) where μt(ht)∈(V)istheseller’sbeliefaboutthebuyer’svalueatht. Furthermore, as in the literature on mechanism design with quasilinear utilities, we can write αMt(·|μ)as an expected payment, xMt(μ), and a probability of trade, qMt(μ). Part (d) implies the set of PBE payoffs of G(μ0)coincides with the set of Public PBE payoffs of G(μ0)(Athey and Bagwell (2008)). Relying on Abreu, Pearce, and Stacchetti (1990), Athey and Bagwell (2008) show that Public PBE have a recursive structure and we use this property to argue the assessment we define in Section 3is indeed a PBE assessment. The rest of the paper studies the equilibrium outcomes and payoffs of G(μ0)and when we refer to a PBE assessment, we mean one that satisfies the conditions of Lemma 1. Remark 1. Below, we abuse notation in the following two ways. First, because values are binary, we can think of an element in (V)(a distribution over vLand vH)asanelement of the interval [0, 1](the probability assigned to vH). We use the latter formulation in what follows. That is, whereas the mechanism outputs a distribution over vLand vH,we index this distribution by the probability of vH. Second, even though β(·|v)is a measure over (V)(in this case a c.d.f.), we sometimes write β(μ|v)when βhas an atom at μ. 3. Main result Section 3contains the main result of the paper: Theorem 1characterizes the equilibrium outcomes and payoffs of G(μ0). To state Theorem 1, we proceed as follows. First, we informally describe the posted-prices assessment, ∗,(π∗ v,r∗ v)v∈V,μ∗, which is singled out by the proof of Theorem 1. Second, we explain why the outcome induced by ∗,(π∗ v,r∗ v)v∈V,μ∗can be implemented via a sequence of posted prices. Finally, we state Theorem 1. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 879 Program (OPT) allows us to circumvent this difficulty: Because in (OPT) the seller can choose both the mechanism and the continuation payoffs, the seller does not need to consider how his choice of mechanism may adversely affect his continuation payoffs. Section 4.1 contains the details of the construction of the virtual surplus defined in Equation (4) and verifies that it is an upper bound on the seller’s equilibrium payoff (Lemma 2). The reader interested in the properties of the solution to (OPT)andhowitis used in the proof of Theorem 1can proceed to Section 4.2 with little loss of continuity. 4.1 Derivation of the virtual surplus We now derive the virtual surplus defined in Equation (4) by relying on Lemma 1and show that it is an upper bound on the seller’s equilibrium payoff. To do so, consider a PBE assessment, ,(πv,rv)v∈V,μ. Fix a history htand let Mtdenote the mechanism offered by the seller at htunder the assessment. Let (τMt,qMt)denote the distribution over posteriors and the probability of trade associated with Mt. Furthermore, the PBE assessment specifies continuation payoffs uMtwhen the seller offers mechanism Mtat history ht. In what follows, we show the following. Lemma 2. The virtual surplus of mechanism Mtis an upper bound on the sum of the seller and the low-valuation buyer’s payoffs. That is, USht+ULht≤VS τMt,qMt,uMt,μtht. Lemma 2is the analogue of the result in mechanism design that the mechanism’s allocation together with the lowest type’s utility in the mechanism pin down the seller’s maximum revenue. Because the low-valuation buyer’s payoff is nonnegative, Lemma 2 also implies that the virtual surplus of Mtis an upper bound on the seller’s payoff. The inequality in Lemma 2showsthatthemechanism’svirtualsurplusisthemaximumpayoff the seller and the low-valuation buyer can share. As we explain in Section 5,once we show the seller captures the entirety of the virtual surplus, the inequality in Lemma 2 implies low-valuation buyer’s payoff is 0 in any equilibrium. To see why Lemma 2holds, note that Lemma 1implies that the seller’s equilibrium payoff at ht,US(ht), can be written as USht=(V)xMt(μt+1)+1−qMt(μt+1)δuMt S(μt+1)τMt(dμt+1),(7) where uMt S(μt+1)is shorthand notation for the seller’s continuation payoff when at history ht, he offers Mtand the output message is μt+1.15 Equation (7) uses Lemma 1as follows. First, the seller’s payoff from offering Mtis written under the assumption that the buyer participates in the mechanism and truthfully reports her value. Second, it uses Lemma 1to write the mechanism in terms of the distribution over posteriors τMt and the allocation (qMt,xMt). 15This continuation payoff can also depend on ht, but we omit this dependence to simplify notation. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 880 Doval and Skreta Theoretical Economics 19 (2024) In particular, the mechanism Mttogether with the continuation payoffs (uMt L,uMt H) satisfy the following constraints. First, the buyer prefers to participate in the mechanism for both her values, that is, for v∈{vL,vH}the following holds: (V)vqMt(μt+1)−xMt(μt+1)+1−qMt(μt+1)δuMt v(μt+1)βMt(dμt+1|v) ≥uMt v(∅),(PC ht,v) where the left-hand side of Equation (PCht,v)isthebuyer’spayoffatht,Uv(ht),and uMt v(∅)is shorthand notation for the buyer’s continuation payoff when at history ht,the seller offers Mtand the buyer rejects. Also, the buyer prefers to truthfully report her value to the mechanism, that is, for v∈{vL,vH}and v= v, the following holds: (V)vqMt(μt+1)−xMt(μt+1) +1−qMt(μt+1)δuMt v(μt+1)βMt(dμt+1|v)−βMtdμt+1|v ≥0. (ICht,v,v) The above expressions implicitly use Lemma 1in one more way. By Lemma 1,the assessment ,(πv,rv)v∈V,μis a public PBE, so that the continuation payoff vector uMt(μt+1)≡(uMt L(μt+1),uMt H(μt+1),uMt S(μt+1)) is an equilibrium payoff vector of G(μt+1). Formally, uMt(μt+1)∈E∗(μt+1). Equations (PCht,v)and(ICht,v,v) are analogous to the participation and incentive compatibility constraints one would obtain in mechanism design except for the following: The participation constraint is potentially type-dependent (the right-hand side is uMt v(∅)). To sidestep this challenge, we ignore the right-hand side of the low-valuation buyer’s participation constraint. Instead, we use the following identity to solve for the transfers given the low-valuation buyer’s utility in the mechanism, UL(ht): (V)vLqMt(μt+1)−xMt(μt+1) +1−qMt(μt+1)δuMt L(μt+1)βMt(dμt+1|vL)=ULht.(8) Equation (8) can be used to rewrite Equation (ICht,v,v)forv=vHas follows: (V)vHqMt(μt+1)−xMt(μt+1)+1−qMt(μt+1)δuMt H(μt+1))βMt(dμt+1|vH) ≥(V)vqMt(μt+1)+1−qMt(μt+1)δuMt H(μt+1)−uMt L(μt+1)βMt(dμt+1|vL) +ULht.(9) Equations (8)and(9) already impose constraints on the maximum revenue the seller can make in period twhen the low-valuation buyer’s payoff is UL(ht). Indeed, like in standard mechanism design, the utility of vLand the truth-telling constraint for vHdetermine the maximum expected transfer the seller can extract from the buyer in period 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 881 t. Replacing the upper bound on the expected transfers obtained from these equations in the seller’s payoff (Equation (7)), we obtain an upper bound on the seller’s revenue at htwhen he offers mechanism Mt. It is immediate to check that this upper bound corresponds to VS ((τMt,qMt),uMt,μt(ht))−UL(ht). Lemma 2then follows. Having established that (OPT) is an upper bound on the seller’s equilibrium payoff, Section 4.2 studies properties of its solution, which are important for the proof of Theorem 1. 4.2 Properties of the solution to (OPT) The main result in this section, Proposition 1, shows two properties of the solution to (OPT) that are important to prove that the seller’s unique equilibrium payoff is u∗ S(μ0). As we explain next, a key difficulty that (OPT) allows us to circumvent is the analysis of the belief dynamics in the game. An important result in the bargaining literature is the skimming lemma,which states that incentive compatibility of the buyer’s behavior implies that the expected discounted probability of trade of vHis higher than that of vL(see, e.g., Fudenberg, Levine, and Tirole (1985)). This property immediately implies that along the path of play the seller’s beliefs fall conditional on no trade. As a consequence, prices must fall along the path of play. Contrast this to the game we analyze in which the seller offers mechanisms that enable him to design how much he observes about the buyer’s choices. In particular, the seller can choose how fast he learns about the buyer’s value conditional on no trade; he could even choose to become more optimistic about the buyer’s value conditional on no trade.16 On the one hand, this would allow the seller to avoid the belief dynamics associated with posted prices, and hence avoid the temptation to trade more often with vLin future rounds. On the other hand, this comes at a cost. Lemma 1implies that the mechanism’s allocation is measurable with respect to the information generated by the mechanism and this information is, in turn, subject to the Bayes’ plausibility constraint. This implies that for the seller’s beliefs conditional on no trade to fall slowly (or not fall at all), it must be that the seller is selling the good to vHwith small probability. It turns out that (OPT) is useful to discipline belief dynamics. Whereas it may not be obvious how to rule out that an equilibrium in which the seller’s beliefs may sometimes go up conditional on no trade exists, it turns out that this is never the case in a solution to (OPT). Indeed, as we establish in Proposition 1below, it is never optimal to not sell the good and induce a belief above the prior. Furthermore, whenever μ0> μ1, conditional on selling the good with positive probability, the seller sells the good only to the highvaluation buyer. 16Whereas the truth-telling equations (ICht,v,v) can be used to derive a “monotonicity” condition analogous to that in the skimming lemma, this condition only implies that on average the expected probability of trade of vHmust be higher than that of vL: (V)vqMt(μt+1)+1−qMt(μt+1)δuMt H(μt+1)−uMt L(μt+1)βMt(dμt+1|vH)−βMt(dμt+1|vL)≥0. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 882 Doval and Skreta Theoretical Economics 19 (2024) Proposition 1. Suppose that μ0≥μ1and let (τ0,q0),udenote a solution to (OPT). Then the following hold: (a) It is never optimal to induce a belief μ1≥μ0and not sell the good. That is, [μ0,1]1−q0(μ1)τ0(dμ1)=0. (b) Furthermore, if μ0> μ1and the seller induces μ1and sells the good (i.e., q0(μ1)> 0), then μ1=1. The proof is in Appendix B. In what follows, we provide intuition for Proposition 1, starting from part (a). To see why not selling the good and at the same time induce a belief μ1≥μ0is not optimal, note the following. First, associated to any continuation payoff, u(μ1), there is a mechanism chosen by the seller when his belief is μ1, and continuation payoffs for the seller and the buyer in the event that the good is not sold. This implies that, conditional on inducing a belief μ1, the seller with belief μ0could always choose today the mechanism and the continuation payoffs associated with u(μ1)in a solution to (OPT). Second, the seller with belief μ0below μ1pays rents to vHwith lower probability than the seller with belief μ1(after grouping terms, the term pre-multiplying uH(μ1)−uL(μ1)in Equation (6) is positive). It follows that the seller with belief μ0 prefers to accrue today the payoff from the mechanism (and continuation payoffs) that induce u(μ1), contradicting that it is optimal to induce μ1and not sell the good. Part (b) follows from the observation that a seller with prior above μ1prefers to trade with vLin at least two periods (recall that ˆ vL(μ0)<0whenμ0> μ1). Thus, it is never optimal to sell the good to vLwith positive probability today. It follows that if q0(μ1)>0, then the seller must assign the good to vH, and hence μ1=1. Proposition 1implies that a solution to (OPT) never induces posteriors in [μ0,1 ). This, in turn, delivers the following expression for the value of (OPT), which in a slight abuse of notation, we denote by VS (μ0): VS (μ0)≡max τ0,uτ0{1}vH+[0,μ0) δuS(μ1)+uL(μ1) +μ1−μ0 1−μ0uH(μ1)−uL(μ1)τ0(dμ1). (10) Equation (10) simply states that the solution to (OPT) can be described by the probability of selling to vHtoday (the probability of inducing a belief μ1=1) and the probability with which the good is not sold and a belief below the prior is induced. One distribution over posteriors is of particular interest in what follows: the one that splits μ0between 1(withq0(1)=1) and μ1<μ 0(with q0(μ1)=0). Bayes’ plausibility implies that the weights on 1 and μ1are μ0−μ1 1−μ1 and 1−μ0 1−μ1 , 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 883 respectively. Note that this is precisely the kind of mechanism that the seller uses in the posted-prices assessment. Corollary 2shows that these are essentially the distributions over posteriors that solve the problem in Equation (10). Corollary 2. The value of (OPT), VS (μ0), equals the value of max G∈([0,μ0))max u[0,μ0)μ0−μ1 1−μ1 vH+1−μ0 1−μ1 δuS(μ1)+uL(μ1) +μ1−μ0 1−μ0uH(μ1)−uL(μ1)G(dμ1). (11) The proof is in Appendix Band is a consequence of the constraint that τ0is Bayes’ plausible for μ0. Given the preceding discussion, the term in the square brackets inside the integral in Equation (11) is the payoff from splitting μ0between 1 and μ1<μ 0. Corollary 2implies that, for a fixed choice of continuation payoffs, the solution to the problem in Equation (10) is as if the seller were randomizing over posterior distributions that split the prior between 1 (and selling the good) and μ1<μ 0(and not selling the good). In other words, if posteriors μ1and μ 1are on the support of τ0, then the seller is indifferent between splitting μ0between μ1and 1 and splitting μ0between μ 1and 1. As a consequence, to determine the optimal τ0it is enough to compare the payoffs of the splittings of μ0between μ1and 1 for different μ1. Because conditional on not selling the good the seller induces beliefs below the prior, Equation (11) shows that for a fixed distribution G, the seller trades off the sum uS+uL against the high-valuation buyer’s rents, uH−uL, when choosing continuation payoffs. Indeed, if continuation payoffs u,u∈E∗(μ1)exist such that (uH−uL,uS+uL)(u H− u L,u S+u L), the solution to the program in Equation (11) would choose uover uif the value of the integrand is higher for u. That is, the seller may forgo maximizing uS+uL— and hence, by Lemma 2potentially forgo the maximum continuation virtual surplus— if this could lead to lower rents for the high-valuation buyer. As we explain in Section 5 below, we circumvent this trade-off in the proof of Theorem 1by showing that for each μ1<μ 0the sum uS+uLis unique, which in turn relies on excluding the existence of payoffs like uand u. 5. Proof of Theorem 1:Key steps Section 5overviews the main steps of the proof of Theorem 1. Taking as given that ∗,(π∗ v,r∗ v)v∈V,μ∗is a PBE assessment, Section 5.1 reviews the main steps to show that it achieves the unique equilibrium payoff for the seller and the low-valuation buyer, and that except for the threshold beliefs, {μn}n≥1, a unique equilibrium outcome exists, and hence a unique equilibrium payoff for the high-valuation buyer as well. Section 5.2 reviews the main steps to show that ∗,(π∗ v,r∗ v)v∈V,μ∗is a PBE assessment. 5.1 Characterization of the equilibrium payoffs of G(μ0) We show u∗ S(μ0) is both a lower bound and an upper bound on the seller’s equilibrium payoff. That is, we show the seller can never do better nor worse than if he were limited 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 884 Doval and Skreta Theoretical Economics 19 (2024) to choose prices in each period so that having access to a richer action space does not increase nor decrease the seller’s payoff. Program (OPT)iskeytoshowu∗ S(μ0) is the seller’s unique equilibrium payoff. The proof that u∗ S(μ0) is an upper bound on the seller’s equilibrium payoff follows from showing that u∗ S(μ0) is the value of (OPT). Instead, the proof that u∗ S(μ0) is a lower bound on the seller’s payoff uses a constrained version of (OPT) to then apply the logic in Gul, Sonnenschein, and Wilson (1986): If an equilibrium in which the seller earns less than u∗ S(μ0) exists, the seller can always undercut the price in the posted-prices assessment and earn close to u∗ S(μ0). Using the property in the posted-prices assessment that along the path of play beliefs fall, the proof of Theorem 1proceeds by induction on the interval the seller’s prior belongs to. For each n≥0, we establish two results. First, for all μ0∈[μn,μn+1),theset of equilibrium payoffs corresponds to that in Theorem 1(cf. Equation (3)). Second, for all μ0≥μn, the seller can guarantee a certain payoff, denoted u∗ S(μ0,n), which coincides with u∗ S(μ0)forμ0∈[μn,μn+1). This payoff is obtained by the seller emulating the strategy that sells the good to vLin nperiods in the posted-prices assessment (Figure 1(a)): The seller uses a mechanism that splits his beliefs between 1 and μn−1; when the belief is 1, the good is sold at a price of vL+(1−δn)v, and when the belief is μn−1,the good is not sold and play proceeds according to the posted-prices assessment. As we explain below, this second result is key to establish in the (n+1)th step of the induction that the seller can guarantee the payoff from the posted-prices assessment when μ0∈[μn+1,μn+2). To illustrate the main steps of the proof that u∗ S(μ0) is the seller’s unique equilibrium payoff, fix n≥1.17 Suppose that for all m<nwe have already shown that (i) Theorem 1 holds for all μ0∈[μm,μm+1)and (ii) the seller can guarantee u∗ S(μ0,m)for all μ0≥μm. In what follows, we argue that (i) Theorem 1holds for μ0∈[μn,μn+1)and (ii) the seller’s payoff is at least u∗ S(μ0,n)for μ0≥μn. Posted prices maximize the virtual surplus: To show that the value of (OPT)corresponds to the seller’s payoff in the posted-prices assessment, we argue that splitting the prior between 1 and μn−1dominates all other splittings. Corollary 2implies that this is enough to show that u∗ S(μ0)=VS (μ0). In what follows, we first argue that, conditional on inducing a belief μ1< μn, the seller places weight on at most μn−1(and on μn−2only if μ0=μn). We then argue it is never optimal to induce a belief μ1∈[μn,μ0). The inductive hypothesis—which identifies the continuation payoffs below μn—and the properties of the posted-prices assessment imply that conditional on inducing a belief in [0, μn), the solution to the problem in Equation (11)placesweightonatmost {μn−2,μn−1}. First,exceptatthethresholdbeliefs{μm}m≤n−1, the inductive hypothesis pins down the value of the integrand in Equation (11)forμ1< μn. Second, because at the threshold beliefs the seller’s and the low-valuation buyer’s payoff are unique, the seller then chooses the continuation payoff that minimizes the high-valuation buyer’s rents, u∗ H(·), as is the case in the posted-prices assessment. Third, relying on the properties of the posted-prices assessment, Lemma C.2 shows that inducing beliefs in [0, μn) other than {μn−2,μn−1}is not optimal. Furthermore, μn−2can be in the support of τ0 17The proof also verifies that Theorem 1holds for μ0∈[0, μ1)(see Appendix C.2.1). 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 885 only if μ0=μn. This is intuitive. On the one hand, it can only be optimal to induce the threshold beliefs {μm}m≤n−1:Form≤n−1, both μ1∈(μm,μm+1)and μmimply that trade happens with vLin mperiods; however, inducing μmallows the seller to trade with vHwith a higher probability. On the other hand, the indifference condition that defines the threshold beliefs implies that because μ0≥μn, inducing beliefs below μn−2cannot be optimal (see Lemma C.1). To conclude the proof that u∗ S(μ0)is an upper bound on VS(μ0), we show in Lemma C.3 that inducing posteriors in [μn,μ0)is not optimal. Whereas Proposition 1 implies that at the solution to (OPT) the seller’s beliefs go down conditional on no trade, it does not say how fast they go down. Indeed, it could be optimal to induce beliefs in [μn,μ0)if at the induced beliefs continuation equilibria exist where the seller somehow manages to slowly trade with vHso as to maximally delay trade with vL.Asweshowin Appendix C.2, this cannot be optimal for μ0close to μn:Theclosertoμn, the smaller the probability that the seller with belief μ0can trade with vHif conditional on no trade, his beliefs must remain above μn. It follows that μ∗ 0small enough exists such that if the seller’s prior μ0is in [μn,μ∗ 0], it is better to trade with vLin nperiods in exchange of increasing the probability of trading with vHtoday. More formally, note that Lemma C.2 implies that for μ0∈[μn,μn+1), the value of (OPT), VS(μ0), is bounded above by VS (μ0)≤maxu∗ S(μ0),μ0−μn 1−μn vH+1−μ0 1−μn δVS [μn,μ0), (12) where VS [μn,μ0)is the supremum of the value function VS (μ)on [μn,μ0].Toseewhy Equation (12) holds, note the following. First, if the solution to (OPT) places positive mass below μn, Lemma C.2 implies that VS (μ0)=u∗ S(μ0), since the seller places weight on μn−1(or μn−2if μ0=μn). The equality follows from Corollary 2and that u∗ S(μ0)corresponds to splitting μ0between 1 and μn−1. Second, if the solution to (OPT)places weight on [μn,μ0), the second term on the right-hand side of Equation (12) is an upper bound to VS(μ0). After all, (i) (μ0−μn)/(1−μn)is the largest weight that can be assigned to vHwhile still remaining on [μn,μ0)and (ii) the remaining weight corresponds to some μ1∈[μn,μ0)with payoff uS(μ1)+uL(μ1)+(1−μ1)μ1 1−μ1 −μ0 1−μ0uH(μ1)−uL(μ1) ≤uS(μ1)+uL(μ1)≤VS [μn,μ0), where the first inequality follows from μ1<μ 0and the second from Lemma 2and the definition of VS [μn,μ0)together with μ1∈[μn,μ0). For μ∗ 0close to μn, the probability of trading with vHtoday and at the same time remaining above μnis small and u∗ S(μ0)attains the maximum on the right-hand side of Equation (12)forμ0∈[μn,μ∗ 0](see Lemma C.3 for details). In other words, μ∗ 0small enough exists such that for all μ0∈[μn,μ∗ 0], the value of (OPT), VS (μ0), coincides with the seller’s payoff in the posted-prices assessment, u∗ S(μ0). Replacing μnwith μ∗ 0in Equation (12), one can argue that for beliefs μ0close enough to μ∗ 0,u∗ S(μ0)is also an 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 886 Doval and Skreta Theoretical Economics 19 (2024) upper bound on the seller’s payoff. Proceeding this way, one establishes that u∗ S(μ0)is an upper bound on the value of (OPT) for all beliefs μ0∈[μn,μn+1). Seller can guarantee the payoff from selling to vLin nperiods: We show in Proposition C.2 that the seller can guarantee the payoff u∗ S(μ0,n)for μ0≥μn, and hence the payoff of the posted-prices assessment for μ0∈[μn,μn+1). The logic is similar to that in Gul, Sonnenschein, and Wilson (1986): We show that the seller can always undercut the price in the posted-prices assessment—say, by offering a mechanism that sells the good for vL+(1−δn)v −δF for some small F>0—and earn close to u∗ S(μ0,n). Provided the high-valuation buyer participates in the mechanism with positive probability (note that the low-valuation buyer always rejects), we show that Fsmall enough can be chosen so that the seller’s beliefs conditional on rejection are μn−1.That beliefs conditional on rejection are μn−1, in turn, implies the seller earns a payoff close to u∗ S(μ0,n): First, it implies that the high-valuation buyer’s acceptance probability coincides with the probability of selling the good in the posted-prices assessment. Second, because μn−1< μn, the inductive hypothesis implies the seller’s continuation payoffs coincide with those in the posted-prices assessment. Thus, it suffices to show that the seller can guarantee that the high-valuation buyer participates in the mechanism. Proposition 2is key to showing that the high-valuation buyer does not reject the mechanism with probability 1. Proposition 2 (Buyer’s maximal rents for μ0≥μn). For all μ0≥μn,uH≤δn−1v. The proof is in Appendix C.3. Proposition 2implies the high-valuation buyer cannot reject the price of vL+(1−δnv)−δF with probability 1, as doing so can yield a payoff of at most δδn−1v. Together with the argument in the preceding paragraph, we conclude that as Fbecomes small the seller can secure u∗ S(μ0,n)for μ0≥μn. To prove Proposition 2, we show that if the high-valuation buyer makes at least δn−1v, the seller makes at most u∗ S(μ0,n−1)(Lemma C.5). Because by the inductive hypothesis, the seller can guarantee u∗ S(μ0,n−1), and we conclude the high-valuation buyer can make at most δn−1v. To show Lemma C.5, we first argue that whenever the high-valuation buyer makes at least δn−1v, the seller’s payoff is bounded above by the value of a constrained version of (OPT) stated in Lemma C.4. In this program, the seller maximizes the virtual surplus subject to the constraint that the high-valuation buyer obtains at least δn−1v. Relying on Proposition 1and that the value of (OPT)atμnis the seller’s payoff in the posted-prices assessment, Lemma C.5 shows that the value of this constrained program is exactly u∗ S(μ0,n−1). The upper and lower bound results for μ0∈[μn,μn+1)imply that u∗ S(μ0) is the seller’s unique equilibrium payoff. This is key to show that the buyer’s payoff is as in Theorem 1. Buyer’s payoff for μ0∈[μn,μn+1):We first argue that the low-valuation buyer’s payoff is 0 in any equilibrium. Proposition 3(uL=0forμ0∈[μn,μn+1)). Let μ0∈[μn,μn+1)and suppose the following hold. First, uS≥u∗ S(μ0)for all u∈E∗(μ0).Second,u∗ S(μ0)=VS (μ0). Then, for all u∈E∗(μ0),uL=0. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 887 In other words, when, as we have argued above, the seller can capture the entirety of the maximum virtual surplus, there is nothing left for the low-valuation buyer. Proof. Wehavethatforallu∈E∗(μ0), the following holds: u∗ S(μ0)+uL≤uS+uL≤u∗ S(μ0), where the first inequality follows from uS≥u∗ S(μ0), and the second inequality follows from Lemma 2and that u∗ S(μ0)is the value of (OPT). It follows that uL=0forallu∈ E∗(μ0). As we show in Appendix C.2, that the seller can capture the maximum virtual surplus also implies the uniqueness of the high-valuation buyer’s payoff except at μn,asthe solution to (OPT) is unique except at μn. 5.2 ∗,(π∗ v,r∗ v)v∈V,μ∗is an equilibrium assessment The analysis so far has relied on the observation that (0, u∗ H(μ0),u∗ S(μ0)) is an equilibrium payoff. The rest of the proof of Theorem 1shows that ∗,(π∗ v,r∗ v)v∈V,μ∗is an equilibrium assessment (see Appendix D).Todothis,wefirstcompletetheequilibrium assessment by specifying the seller’s and the buyer’s strategy after every history (see Appendices D.1–D.2). We then show that given beliefs and continuation payoffs, neither the buyer nor the seller have a one shot deviation (Appendix D.3). The results in Athey and Bagwell (2008) imply that this is enough to conclude that ∗,(π∗ v,r∗ v)v∈V,μ∗is an equilibrium assessment. Seller’s strategy: Except for the cutoff beliefs {μn}n≥1, we specify that the seller plays the mechanism described in the posted-prices assessment on and off the path of play. Instead, the seller’s strategy off the path of play when his beliefs are in {μn}n≥1needs to be determined jointly with the buyer’s strategy, to which we turn next. Buyer’s strategy (Appendix D.1)To complete the buyer’s strategy, we first classify mechanisms according to whether they satisfy the participation and truth-telling constraints for the buyer given the continuation payoffs under ∗,(π∗ v,r∗ v)v∈V,μ∗.Formechanisms that satisfy these constraints, we specify that the buyer indeed participates and truthfully reports her value to the mechanism. To specify the buyer’s strategy for mechanisms that fail to satisfy either constraint, one needs to determine simultaneously the buyer’s best response and the seller’s beliefs conditional on observing either the buyer reject the mechanism, or the buyer accept the mechanism and the output message that results from the buyer’s report. On the one hand, the buyer’s continuation payoff depend on the seller’s beliefs, which are determined by the buyer’s strategy. On the other hand, whether the buyer’s strategy is a best response depends on her continuation payoff. We use the results in Simon and Zame (1990) to solve for this fixed point. It is at this point where the possibility that the seller randomizes when indifferent between trading with vLin nor n−1 periods arises to ensure that the buyer’s best response is well-defined. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 888 Doval and Skreta Theoretical Economics 19 (2024) By specifying the buyer’s strategy in the way described above, we ensure that the buyer is best responding to the seller’s strategy given her continuation payoff. It remains to show that the seller has no one-shot deviations. The seller has no one-shot deviations (Appendix D.3): To show that the seller does not have one-shot deviations, we rely once again on (OPT). For concreteness, suppose we are at a history htand let μtdenote the seller’s belief at ht. We show in Appendix D.3 that the payoff from any deviation at history htis bounded above by the value of (OPT) evaluated at μ0=μtsubject to the constraint that for posterior beliefs μt+1below μt, the continuation payoffs for the buyer and the seller are given by (0, u∗ H(μt+1),u∗ S(μt+1)). The results in Section 5.1 imply that the value of this program coincides with u∗ S(μt). Since this is the seller’s payoff under the equilibrium strategy at ht, it follows that the seller has no one-shot deviations at ht. Two observations are key to show that this upper bound holds. The first is one we have already used in the solution to (OPT): For threshold beliefs μnbelow μt, the seller with prior belief μtprefers the continuation payoff vector in which the high-valuation buyer receives her lowest equilibrium payoff, u∗ H(μn). Thus, by selecting continuation payoffs in this way, we exaggerate the payoff that the seller can guarantee from a deviation. The second is that any mechanism Mttogether with the buyer’s best response to Mtdefine a new mechanism, M t, that satisfies the buyer’s participation and truth-telling constraint given the continuation payoffs associated to Mt. Thus, we can use mechanism M ttogether with the continuation payoffs specified by the assessment when Mtis offered to bound the revenue from Mtby its virtual surplus. This completes the description of the main steps in the proof of Theorem 1. 6. Conclusions This is the first paper to characterize all equilibrium outcomes in an infinite-horizon, mechanism-selection game between an uninformed designer and a privately informed agent with persistent private information under limited commitment. We do so by marrying insights from the literatures on bargaining and mechanism design. Following the results in our previous work, Doval and Skreta (2022), we endow the seller with a class of mechanisms that enables the seller to design his posterior beliefs about the buyer’s value. The combination of mechanism design and information design elements was key in obtaining a tractable characterization. The revelation principle in Doval and Skreta (2022) also allows us to simplify the buyer’s equilibrium behavior, so that for the most part we were able to focus on the strategic considerations that pertain to the seller. Properties of the solution to (OPT) can provide a useful benchmark for the analysis of the designer’s best equilibrium payoff in other settings with quasilinear utility. In such settings, the virtual surplus is an upper bound on the designer’s equilibrium payoff. Furthermore, when there is a continuum of types, the application of the envelope theorem implies that the designer’s payoff can be represented as the virtual surplus. Program (OPT) is exactly like the relaxed program in standard mechanism design: If a solution to (OPT) satisfies the ignored constraints, then a PBE of the mechanism-selection game 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 895 Proof of Lemma C.1. Recall that μ0=0 and as argued below Equation (C.2), μ1= vL/vH. Suppose we have shown that μn−1> μn−2. We show that μn> μn−1. To do so, fix μ≥μn−1. We claim that the difference n(μ;μn−1,μn−2) =μ−μn−1 1−μn−1 vH +1−μ 1−μn−1 δu∗ S(μn−1)+(1−μn−1)μn−1 1−μn−1 −μ 1−μu∗ H(μn−1) −μ−μn−2 1−μn−2 vH+1−μ 1−μn−2 δu∗ S(μn−2) +(1−μn−2)μn−2 1−μn−2 −μ 1−μu∗ H(μn−2),(C.4) is increasing in μ.Notethatnis differentiable in μ,and ∂ ∂μn(μ;μn−1,μn−2)=vH(μn−1−μn−2) (1−μn−1)(1−μn−2) −δvH μn−1−μn−2 (1−μn−1)(1−μn−2)+δδn−2−δn−1vH>0. The cutoff μnis defined by n(μn;μn−1,μn−2)=0. Note that μn= μn−1if n≥1. If μn= μn−1,then 0=n(μn;μn−1,μn−2)=δu∗ S(μn−1)−u∗ S(μn−1)<0, since δ<1. Because n(·;μn−1,μn−2)is increasing, we conclude that μn> μn−1. Having established this, we can now define the seller’s payoff at any history htunder ∗,(π∗ v,r∗ v)v∈V,μ∗.Ifμ∗ t(ht)∈[μn,μn+1),then U∗ Sht =μ∗ tht−μn−1 1−μn−1 vH+1−μ∗ tht 1−μn−1 δu∗ S(μn−1) +(1−μn−1)μn−1 1−μn−1 −μ∗ tht 1−μ∗ thtu∗ H(μn−1) ≡u∗ Sμ∗ tht.(C.5) In what follows, we simplify notation by denoting for any μ0,μ1∈(V): R(τ∗,q∗)(μ1,μ0)=u∗ S(μ1)+(1−μ1)μ1 1−μ1 −μ0 1−μ0u∗ H(μ1), and note that R(τ∗,q∗)(μ0,μ0)=u∗ S(μ0). 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 896 Doval and Skreta Theoretical Economics 19 (2024) Equation (C.3) implies that under the specification of equilibrium play, when the seller’s belief is μn, he is indifferent between a posted price of vL+(1−δn)v and a price of vL+(1−δn−1)v. This, in turn, implies that the high-valuation buyer may obtain any payoff in [δnv,δn−1v], if the seller were to randomize between these two posted prices. This randomization is important for the specification of the buyer’s and the seller’s strategies off the path of play. For future reference, let U∗ Hdenote the following correspondence: U∗ H(μ0)=u∗ H(μ0)if μ0= μn,n≥1, δnv,δn−1vif μ0=μnfor some n≥1. Note U∗ His upper hemicontinuous, convex-valued, and compact-valued. C.2 Omitted proofs from Section 5.1 Appendix C.2 completes the steps to show that the set of equilibrium payoffs of G(μ0) is as described in Equation (3). In what follows, we first state the inductive hypothesis. We then prove the base case, which establishes Equation (3)forμ0∈[0, μ1)(Appendix C.2.1). We then prove the inductive step in Appendix C.2.2.Alongtheway,we provide the omitted proofs of the statements in Section 5.1 regarding the properties of the solution to (OPT) (Lemmas C.2 and C.3). In what follows, uS(μ0),uS(μ0)denote the seller’s maximum and minimum equilibrium payoffs when his belief is μ0.Moreover, for n≥0andμ0≥μn,weletu∗ S(μ0,n)denote the seller’s payoff from “emulating” the strategy in the posted-prices assessment that sells the good to the low-valuation buyer in nperiods from now. Formally, u∗ S(μ0,n)=μ0−μn−1 1−μn−1 vH+1−μ0 1−μn−1 δR(τ∗,q∗)(μn−1,μ0).(C.6) Inductive hypothesis: Fix n∈N0. The inductive hypothesis P(n)is given by P(n).The following hold: (n.1) For all μ0≥μn,uS(μ0)≥u∗ S(μ0,n). (n.2) For all μ0∈[μn,μn+1),Equation(3)inTheorem1holds. C.2.1 Base case We first show that P(0)=1. Proposition C.1 (Seller payoffguarantee for μ0≥μ0). For all μ0≥μ0,uS(μ0)≥ u∗ S(μ0,0 ). Proof of Proposition C.1. Define MS={μ0∈(V):uS(μ0)<v L}.Towardacontradiction, suppose that MSis nonempty and let μS=infMS. We consider two cases. Case 1: μS∈MS.Let uL(μS)=sup{uL:u∈E∗(μS)}. Suppose the seller with prior μSoffers mechanism MFthat sells the good at price vL−δF for F>0. We argue that if MFis rejected with positive probability in a PBE assessment, then the seller’s beliefs 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 897 conditional on rejection, μR, coincide with μS. Suppose that MFis rejected with positive probability. Then Lemma A.4 implies that μR≤μS. Furthermore, it cannot be that μR=0 (Lemma A.2, (a) implies that μS>0) because δuL(μR)=0<δF.Thus,the high-valuation buyer must reject the mechanism with positive probability. We finally rule out that μR∈(0, μS). In that case, we would need that δF +v ≤δuH(μR)and δF ≤δuL(μR), contradicting that uS(μR)≥vLfor μR<μ S(Lemma A.3). Thus, μR=μS,sothatforF=uL(μS)+/δ, the low-valuation buyer accepts MF with probability 1 and so does the high-valuation buyer (cf. Lemma A.2, (b)). Then uS(μS)≥vL−δuL(μS).(C.7) If uL(μS)=0, we arrive at a contradiction. Suppose then that uL(μS)>0. We now argue that if the seller with prior μSoffers MFfor F=δuL(μS)+/δ, then the buyer accepts this mechanism with probability 1. By the same logic as above, if MFis rejected with positive probability, then μR=μSand δF +v ≤δuH(μR),δF ≤δuL(μR).(C.8) Lemma A.3 implies that uS(μS)<v L−δuL(μS), a contradiction to Equation (C.7). Proceeding iteratively, we conclude that for all n,uS(μS)≥vL−δnuL(μS),sothatasn→∞ we have that uS(μS)≥vL, contradicting the definition of μS. Case 2: μS/∈MS.Fix η>0andletuL(η)=sup{uL:u∈E∗(μ0),μ0∈(μS,μS+η)}.Fix μ0∈(μS,μS+η). Suppose the seller with prior μ0offers MFthat sells the good at price vL−δF,F>0. Similar logic to Case 1 implies that if MFis rejected with positive probability in a PBE assessment, then μR∈(μS,μ0]. In particular, Lemma A.2, (a) implies that [0, μS]is nonempty, and hence we can rule out that only the low-valuation buyer rejects MF. We conclude that if F=uL(η)+/δ, the low-valuation buyer accepts with probability 1 and so does the high-valuation buyer. We conclude that for all μ0∈(μS,μS+η), uS(μ0)≥vL−δuL(η).(C.9) If uL(η)=0, we arrive at a contradiction since Equation (C.9)holdsforallμ0∈(μS,μS+ η)contradicting the definition of μS. Suppose then that uL(η)>0. We now argue that for μ0∈(μS,μS+η)the mechanism MFmust be accepted with probability 1 for F=δuL(η)+/δ.Toseethis,note that if MFis rejected with positive probability, by the same logic as above μR∈(μS,μ0], so that Equation (C.8) holds. Lemma A.3 implies that μR∈(μS,μ0]exists such that uS<v L−δuL(η), a contradiction to Equation (C.9). Proceeding iteratively, we conclude that for all nand all μ0∈(μS,μS+η), uS(μ0)≥vL−δnuL(η), (C.10) so that as n→∞we have that uS(μ0)≥vLfor all μ0∈(μS,μS+η), contradicting the definition of μS. It follows that for all μ0∈(V),uS(μ0)≥vL=u∗ S(μ0,0 ). We conclude that part (n.1) holds. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 898 Doval and Skreta Theoretical Economics 19 (2024) We now show that part (n.2) holds for μ0∈[μ0,μ1), starting from the seller’s payoff: uS≤u∗ S(μ0)for μ0∈[μ0,μ1):NotethatuS(μ0)≤vLas vLis the seller’s payoff in the commitment solution for μ0∈[0, μ1). Since vLis the seller’s payoff in the posted-prices assessment, then uS(μ0)=u∗ S(μ0)=vL. Together with Proposition C.1, this implies that the seller’s equilibrium payoff is unique and coincides with vL. uL=0in [μ0,μ1): Proposition C.1 and the above argument imply that the assumptions in Proposition 3hold for μ0∈[μ0,μ1).Hence,uL=0forallu∈E∗(μ0) uH=v in [μ0,μ1): It remains to show that the high-valuation buyer’s payoff is unique in [μ0,μ1). That the seller’s payoff is at least vLfor μ0∈[μ0,μ1)and Lemma A.3 imply that uH≤v. Moreover, that the seller’s payoff is at most vLimplies that uH=v. C.2.2 Inductive step Fix n≥1 and suppose that P(k)=1forallk∈{0, ,n−1}. Part (n.1): WenowshowthatP (n)=1, starting from the lower bound on the seller’s payoff. Proposition C.2 (Seller payoffguarantee for μ0≥μn). For all μ0≥μn,uS(μ0)≥ u∗ S(μ0,n). The proof of Proposition C.2 relies on Proposition 2, the proof of which is in Appendix C.3. Proof of Proposition C.2.Fixμ0≥μnand assume u∈E∗(μ0)exists such that uS< u∗ S(μ0,n).Let,(πv,rv)v∈V,μdenote a PBE assessment with payoff uand consider the following deviation for the seller. The seller offers a mechanism that sells the good at price vL+(1−δn)v −δF,thatis,forv∈{vL,vH},β(μ0|v)=1, (q(μ0),x(μ0)) =(1, vL+ (1−δn)v −δF ),whereF>0 satisfies that for n≥2, δn−1v < δn−1v +F<δ n−2v. Because −v(1−δn)+δF < −v(1−δn−1)<0, the low-valuation buyer rejects the mechanism with probability 1, so that the seller’s beliefs upon rejection μRare determined by Bayes’ rule and satisfy that μR∈[0, μ0]. For n=1, the high-valuation buyer must accept the mechanism with probability 1: Proposition 2implies δv is the largest payoff from rejection and the high-valuation buyer gets strictly more by accepting the mechanism. The seller’s payoff is then μ0(vH− δv−δF )+(1−μ0)δvL=u∗ S(μ0,1 )−δF, where we use Lemma A.2, (a) to determine the seller’s payoff conditional on rejection. Letting F→0, proves Proposition C.2 for n=1. For n≥2, we argue the high-valuation buyer must randomize between accepting and rejecting the mechanism. It cannot be the case that the high-valuation buyer rejects the mechanism with probability 1: Proposition 2implies that δuH(μR)≤δδn−1v < δ(δn−1v +F),whereμR=μ0. Similarly, it cannot be the case that the high-valuation buyer accepts the mechanism with probability 1: In that case, rejection reveals the lowvaluation buyer (i.e., μR=0) and the high-valuation buyer’s continuation payoff is δv, which is strictly larger than δnv +δF (cf. Lemma A.2, (a)). It follows that the high-valuation buyer must be indifferent between accepting and rejecting so that the continuation payoffs after rejection satisfy that δn−1v +F= uH(μR), which can only be the case if μR=μn−1. Indeed, μR∈(μn−1,μ0)would yield 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 899 apayoffofatmostδn−1v (cf. Proposition 2)andμR< μn−1wouldyieldapayoffofat least δn−2v. This pins down the high-valuation buyer’s acceptance probability to be (μn−1/μ0)(μ0−μn−1/(1−μn−1)). Simple algebra shows that the seller’s payoff from offering the above mechanism is μ0−μn−1 1−μn−1 vH+δ1−μ0 1−μn−1u∗ S(μn−1)+μn−1−μ0 1−μ0u∗ H(μn−1)+F =u∗ S(μ0,n)−δμ0−μn−1 1−μn−1 F. Letting F→0 completes the proof of Proposition C.2 for n≥2. Part (n.2): We now prove that part (n.2) holds for μ0∈[μn,μn+1). Because we have already established that the seller can guarantee the payoff from the posted-prices assessment for μ0∈[μn,μn+1), it remains to show that u∗ S(μ0)is an upper bound on the seller’s payoff. Define Mn={μ0∈[μn,μn+1):Equation(3)doesnothold }. Toward a contradiction, suppose Mn= ∅ and let μ=inf Mn. Then, for all >0, μ 0∈[μ,μ+)exists such that either uS(μ 0)>u ∗ S(μ 0)or the buyer’s payoff is not as in Equation (3). Lemmas C.2 and C.3 below deliver that the seller’s payoff in the posted-prices assessment is the seller’s maximal equilibrium payoff in [μn,μn+1).Fixμ0∈[μ,μn+1).By definition, for μ1∈[0, μ), the seller’s and the low-valuation buyer’s payoffs are given by u∗ S(μ1)and 0, respectively. Furthermore, since μ0≥μn, the seller prefers to minimize the high-valuation buyer’s continuation payoff at {μm}m≤n−1.Hence,forμ0∈[μ,μn+1), the objective function in Equation (11)equals [0,μ)μ0−μ1 1−μ1 vH +1−μ0 1−μ1 δu∗ S(μ1)+(1−μ1)μ1 1−μ1 −μ0 1−μ0u∗ H(μ1)  R(τ∗,q∗)(μ1,μ0) G(dμ1) +[μ,μ0)μ0−μ1 1−μ1 vH+1−μ0 1−μ1 δuS(μ1)+uL(μ1) +(1−μ1)μ1 1−μ1 −μ0 1−μ0uH(μ1)−uL(μ1)G(dμ1). (C.11) Lemma C.2. Fix μ0∈[μ,μn+1)and suppose Gmaximizes the expression in Equation (C.11) and is such that G([0, μ)) >0.ThenGplaces positive probability on at most {μn−2,μn−1}. Proof. It is immediate to see that no μ1∈n−1 m=0(μm,μm+1)can be on the support of G:ifμ1∈(μm,μm+1)for m≤n−1, this is dominated by choosing μm: μ0−μm 1−μm vH+1−μ0 1−μm δR(τ∗,q∗)(μm,μ0)−μ0−μ1 1−μ1 vH+1−μ0 1−μ1 δR(τ∗,q∗)(μ1,μ0) 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 900 Doval and Skreta Theoretical Economics 19 (2024) =vHμ0−μm 1−μm +δ1−μ0 1−μm μm−μm−1 1−μm−1 −μ0−μ1 1−μ1 −δ1−μ0 1−μ1 μ1−μm−1 1−μm−1>0. A similar argument implies μ1∈[μn,μ)is dominated by choosing μn−1. Thus, if it is optimal to set G([0, μ)) >0, we can reduce the problem of finding the optimal such G to max G∈({μ0,,μn−1}) n−1  m=0μ0−μm 1−μm vH+1−μ0 1−μm δR(τ∗,q∗)(μm,μ0)G{μm}. (C.12) However, for m≤n−2, Lemma C.1 implies that m+1(μ0;μm,μm−1) =μ0−μm 1−μm vH+1−μ0 1−μm δR(τ∗,q∗)(μm,μ0) −μm−1−μ0 1−μm−1 vH+1−μ0 1−μm−1 δR(τ∗,q∗)(μm−1,μ0)>0, since μ0> μn−1. Thus, any solution to the problem in Equation (C.12) satisfies that α∈ [0, 1]exists such that the value of this problem is given by αμ0−μn−1 1−μn−1 vH+1−μ0 1−μn−1 δR(τ∗,q∗)(μn−1,μ0) +(1−α)μ0−μn−2 1−μn−2 vH+1−μ0 1−μn−2 δR(τ∗,q∗)(μn−2,μ0) =αn(μ0;μn−1,μn−2)+μ0−μn−2 1−μn−2 vH+1−μ0 1−μn−2 δR(τ∗,q∗)(μn−2,μ0), so that unless μ0=μnit is not optimal to set α<1. Instead, for μ0=μnany α∈[0, 1]is a maximizer. Lemma C.3. A real number >0exists such that for all μ0∈[μ,μ+),VS (μ0)= u∗ S(μ0). Proof.For∈(0, μn+1−μ),letVS denote the supremum of VS (·)over [μ,μ+). The same arguments as the ones after Equation (12) imply that for all μ0∈[μ,μ+), the following holds: VS (μ0)≤maxu∗ S(μ0),μ0−μ 1−μ vH+1−μ0 1−μ δVS . (C.13) Taking the supremum over μ0∈[μ,μ+)on both sides of Equation (C.13), we obtain VS ≤maxuS, 1−μ (vH−δVS )+δVS , (C.14) 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 901 where uS is the supremum of u∗ S(μ0)overμ0∈[μ,μ+), and the expression in the second term follows from noting that μ0−μ<and vH>δVS . We claim >0existssuchthatforall∈(0, ), the right-hand side of Equation (C.14) equals uS. Toward a contradiction, suppose not. Then, for all ∈(0, μn+1−μ),f()∈ (0, )exists such that uSf ()<f() 1−μ (vH−δVS f())+δVS f(). (C.15) Since f()<,thenf()→0as→0. Equation (C.14) together with Equation (C.15) imply VS f()≤f() 1−μ (vH−δVS f())+δVS f()⇒lim →0VS f()≤δlim →0VS f(), a contradiction, since for all ,VS ≥vL>0.22 It follows that >0existssuchthat ∀∈(0, ),VS =uS. We now claim that VS (μ0)=u∗ S(μ0)for all μ0∈[μ,μ+). Toward a contradiction, suppose μ0∈[μ,μ+)exists such that VS (μ0)>u ∗ S(μ0). By continuity of u∗ Son [μn,μn+1)(see Equation (C.5)), η>0 exists such that letting =μ0+η−μ,wehave u∗ S(μ0+η)=uS <VS (μ0)≤VS , where the first equality follows from u∗ Sbeing increasing on [μn,μn+1)(cf. Equation (C.5)). This is a contradiction. Thus, for all μ0∈[μ,μ+)we have that u∗ S(μ0)isan upper bound on the seller’s payoff.23 Lemmas C.2 and C.3 imply that uS(μ0)=u∗ S(μ0)for all μ0∈[μ,μ+). By definition of μ,μ0∈[μ,μ+)exists such that E∗(μ0)fails to satisfy Equation (3) because of the buyer’s payoff. In what follows, we rule this out by showing that for all μ0∈[μ,μ+), the buyer’s payoff is as in Theorem 1for μ0∈[μ,μ+). Low-valuation buyer’s payoff is 0in [μ,μ+): Proposition C.2 and VS (μ0)= u∗ S(μ0)for μ0∈[μ,μ+)imply that the assumptions in Proposition 3hold for μ0∈ [μ,μ+).Hence,uL=0forallu∈E∗(μ0)and all μ0∈[μ,μ+). High-valuation buyer’s payoff in [μ,μ+):Foranysuchμ0, suppose u∈E∗(μ0) exists such that u S=u∗ S(μ0),butu H= u∗ H(μ0). This equilibrium payoff, u, is associated to a mechanism in period 0, (τ 0,q 0), and continuation payoffs, u. We argue that u∗ S(μ0)=VS τ 0,q 0,u,μ0. (C.16) To see this, note that we must have VS ((τ 0,q 0),u,μ0)≥u∗ S(μ0);otherwiseu S≤ VS ((τ 0,q 0),u,μ0)<u ∗ S(μ0), where the first inequality follows from Lemma 2and the 22Note that the limit lim→0VS f()exists up to a convergent subnet because VS f()is bounded. 23TheargumentaboveisreminiscenttothatinFudenberg and Tirole (1991a)’s treatment of the equilibrium in the posted-prices game with binary values. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 902 Doval and Skreta Theoretical Economics 19 (2024) second inequality is by assumption. Since u S=u∗ S(μ0), we have a contradiction. Furthermore, it cannot be the case that VS ((τ 0,q 0),u,μ0)>u ∗ S(μ0), because (τ 0,q 0)together with the continuation payoffs uare feasible choices in (OPT). Equation (C.16) then follows. Thus, ((τ 0,q 0),u)is also a solution to (OPT). However, the proof that u∗ S(μ0) is the value of (OPT) implies that the solution to (OPT) is unique, unless μ0=μn. It then follows that μ0=μn, so that there is a continuum of solutions to (OPT), with vH’s payoff ranging from u∗ H(μ0)tou∗ H(μ0)/δ.This completes the proof that uHis as in Equation (3)forallμ0∈[μ,μ+). We conclude that Equation (3)holdsforallμ0∈[μ,μ+), contradicting the definition of μ. It follows that Mnis empty, and hence part (n.2) of the inductive statement holds. C.3 Proof of Proposition 2 We now prove Proposition 2, which provides the bound on the high-valuation buyer’s payoff. For n=1, the result follows from Proposition C.1 and Lemma A.3,asuS(μ0)≥vL implies that uH≤v. It remains to establish Proposition 2for n≥2. Thus, fix n≥2and assume that for k≤n−1, P(k)=1. The proof of Proposition 2relies on Lemmas C.4 and C.5: Lemma C.4 (Maximal seller’s payoffgiven high-valuation buyer’s rents). Fix n≥2and μ0≥μn.Let>0be such that an equilibrium payoff u∈E∗(μ0)exists such that uH≥ δn−1v +δ. Then the seller’s payoff is bounded above by VS (μ0,n−1, ),where VS (μ0,n−1, )≡max (q,τ,u)VS (τ,q),u,μ0(OPT(μ0,n−1, )) such that (V)vq(μ)+1−q(μ)δuH(μ)−uL(μ)β(dμ|vL) ≥δn−1v +δ,(R (n−1)) Eτ[μ]=μ0,(BP) ∀μ∈(V)u(μ)∈E∗(μ0).(Eqbm) Note that (OPT(μ0,n−1, )) does not reduce to (OPT)when=0andμ0∈ (μn,μn+1). Indeed, for μ0∈(μn,μn+1), the solution to (OPT) delivers rents δnv to the high-valuation buyer, which violates (R(n−1)). Instead, when μ0=μn, a solution to (OPT) exists that delivers rents δn−1v to the high-valuation buyer. Proof of Lemma C.4.Letμ0≥μn.Letu∈E∗(μ0)be such that uH≥δn−1v +δ.Letting Mdenote the mechanism offered by the seller in the first period, we have that M satisfies at least the following constraints: (V)vHqM(μ)+1−qM(μ)δuM H(μ)βM(dμ|vH)−xH 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 903 ≥δn−1v +δ,(PCH) (V)vHqM(μ)+1−qM(μ)δuM H(μ)βM(dμ|vH)−βM(dμ|vL) ≥xH−xL,(ICH) (V)vLqM(μ)+1−qM(μ)δuM L(μ)βM(dμ|vL) ≥xL,(PCL) where xv=(V)xM(μ)βM(dμ|v). Equations (PCH), (ICH), and (PCL)areonlyasubset of the constraints the mechanism Mmust satisfy. Indeed, we are ignoring the lowvaluation buyer’s truth-telling constraint and (PCL) is only a necessary condition for the low-valuation buyer to participate in the mechanism. Program (). The seller’s payoff, uS, is bounded above by the solution to the following program: Maximize the seller’s payoff by choosing (i) transfers xH,xL, (ii) trade probabilities q:(V)→ [0, 1], (iii) a Bayes’ plausible posterior distribution τ, and (iv) continuation payoffs u(·)∈E∗(·), subject to the constraints (PCH), (ICH), and (PCL). Because we allow the seller to choose xLand xHinstead of x(μ), we give the seller more degrees of freedom than in the game. In what follows, we argue that the value of ()equals VS (μ0,n−1, ). It is immediate to see that xLis chosen so that (PCL) binds. We can then write (ICH) as follows: (V)vHq(μ)1−q(μ)δuH(μ)β(dμ|vH)−xH ≥(V)vq(μ)+1−q(μ)δuH(μ)−uL(μ)β(dμ|vL).(ICH ) Wenowshowthat(ICH) must bind at the solution to (),whichinturnimpliesthat (R(n−1)) holds. Toward a contradiction, suppose that (ICH) does not bind. Then xH must be chosen so that (PCH) binds. The binding constraints (PCH)and(PCL)imply that the value of () obtains from maximizing (V) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ q(μ)μvH+(1−μ)vL−μ0 1−μ0δn−1v +δ +1−q(μ)δuS(μ)+μuH(μ) +(1−μ)uL(μ)−μ0 1−μ0δn−2v + ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ τ(dμ), (C.17) subjectto(BP)and(Eqbm). We argue that the solution to the above problem is not feasible for (). Indeed, the optimal value of the objective in Equation (C.17)is μ0vH+(1−μ0)vL−μ0 1−μ0vδn−1+δ. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 904 Doval and Skreta Theoretical Economics 19 (2024) To see this, note that Lemma A.1 and δ<1 imply that, for all μ, the following holds: δuS(μ)+μuH(μ)+(1−μ)uL(μ)−μ0 1−μ0vδn−2+ ≤δμvH+(1−μ)vL−μ0 1−μ0vδn−2+ <μv H+(1−μ)vL−μ0 1−μ0δn−1v +δ. Thus, for all μ, setting q(μ)=1 is preferred to setting q(μ)=0. The linearity in μof the term associated to qimplies the result. However, this is not feasible since q(μ)=1forall μand the binding (PCH)violates(ICH). It follows that (ICH) must hold with equality at a solution to (). Replacing the binding (PCL)and(ICH) in the seller’s payoff yields that the objective in ()isVS ((τ,q),u,μ0). Moreover, replacing the binding (ICH)in(PCH)yields Equation (R(n−1)). We conclude that () coincides with (OPT(μ0,n−1, )). Lemma C.5 (Value of (OPT(μ0,n−1, ))at=0). Fix n≥2and μ0≥μn.Then VS (μ0,n−1, 0)=u∗ S(μ0,n−1). Proof of Lemma C.5.Letλdenote the Lagrange multiplier on the constraint (R(n−1)) in the program (OPT(μ0,n−1, ))for=0. In a slight abuse of notation, define μ0(λ)= μ0−λ 1−λ. The Lagrangian is given by L(τ,q),u;λ=VS (τ,q),u,μ0(λ)−λδn−1v.(L) That is, up to a constant, the Lagrangian corresponds to the virtual surplus of a seller with belief μ0(λ).Letλ∗be such that μ0(λ∗)=μn. By the upper bound proof, we know that one of the solutions to maximizing the virtual surplus for a seller with belief μndelivers rents δn−1v. However, this solution satisfies the Bayes’ plausibility constraint of a seller with belief μn, whereas (OPT(μ0,n−1, )) requires that the distribution over posteriorsaveragestoμ0. Note that the policy that delivers a payoff of u∗ S(μ0,n−1)to the seller gives rents δn−1v to the high-valuation buyer and satisfies the Bayes’ plausibility constraint at μ0. In what follows, we describe the policy and argue that it is a solution to (OPT(μ0,n−1, )). Important for this argument is that the seller with belief μnfinds it optimal to give the high-valuation buyer rents equal to δn−1v. Let (ˆτ,ˆ q)denote the following mechanism: ˆτ(1)=μ0−μn−2 1−μn−2 =1−ˆτ(μn−2),ˆ q(1)=1=1−ˆ q(μn−2), and continuation payoffs ˆ u(μn−2)=(0, δn−2v,u∗ S(μn−2)). The mechanism (ˆτ,ˆ q)together with the equilibrium continuation payoffs ˆ u(μn−2)satisfies the constraints. Furthermore, we now argue that that for all ((τ,q),u)that satisfy the constraints in (OPT(μ0,n−1, )), L(τ,q),u;λ∗≤L(ˆτ,ˆ q),ˆ u;λ∗. (C.18) 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 911 is, u∗ H(μt+1). The last inequality follows by definition. Proposition 1and the results in Appendix C.2 imply the last equality. We conclude that the seller has no one-shot deviations. Appendix E: Perfect Bayesian equilibrium:Formal statement We introduce in this section the necessary formalisms to define PBE. To simplify notation, we assume in what follows that MIis such that Miis finite for all i∈I.Thisis without loss of generality when Vis finite (cf. Doval and Skreta (2022)). It affords two important simplifications. First, MIis itself a Polish space, which means that we can condition the buyer’s strategy directly on the mechanism, M, chosen by the seller (cf. Aumann (1964)). Second, given a public history ht, the set of buyer histories consistent with ht,Ht B(ht), is finite and, therefore, the support of μt(ht)∈(V×Ht B(ht)) is finite. Given the buyer’s participation and reporting strategy and a mechanism Mtdefine a distribution over (SMt×{0, 1}×R)such that, for all measurable subsets S×A⊆ SMt×{0, 1}×R, ρ(π,r)S×A|v,ht B,Mt=πtvht B,Mt m∈MMt ϕMtS×A|mrtvht B,Mt(m). Fix an assessment ,(πv,rv)v∈V,μ, a public history ht, and a mechanism, Mt.The seller’s payoff is given by US,(πv,rv)v∈V,μ|ht,Mt = (v,ht B) μthtv,ht B1−πtvht B,MtδEUS,(πv,rv)v∈V,μ|ht,z∅(Mt),· + (v,ht B) μthtv,ht BSMt×{0,1}×Rx+(1−q)δEUS,(πv,rv)v∈V,μ|ht, z(st,(0,x))(Mt),·ρ(π,r)d(st,q,x)|v,ht B,Mt. Similarly, the buyer’s payoff when her value is v, the history is ht B, and the seller offers mechanism Mtis given by Uv,(πv,rv),μ|ht B,Mt =1−πtvht B,MtδEUv,(πv,rv),μ|ht B,z∅(Mt),·+πtvht B,Mt × m∈MMt rtvht B,Mt(m)SMt×Avq −x+(1−q)δEUv,(πv,rv)v∈V,μ|ht B,m, z(st,(q,x))(Mt),·ϕMtd(st,q,x)|m. Definition 1. The assessment ,(πv,rv)v∈V,μsatisfies sequential rationality if for all periods t, and all public histories ht, we have: 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 912 Doval and Skreta Theoretical Economics 19 (2024) 1. For all mechanisms Mtin the support of t(ht),US(,(πv,rv)v∈V,μ|ht,Mt)≥ US(,(πv,rv)v∈V,μ|ht,M t)for all M t= Mt, 2. For all v∈V,allbuyerhistoriesht B∈Ht B(ht), and all mechanisms Mt,Uv(,(πv,rv), μ|ht B,Mt)≥Uv(,(π v,r v),μ|ht B,Mt)for all alternative strategies (π v,r v). Definition 2. The belief system μsatisfies Bayes’ rule where possible if for all public histories ht, and mechanisms Mt, the following hold: μt+1ht,z∅(Mt)vH,ht B,z∅(Mt) v∈V,ht B∈Ht B(ht) μthtv,ht B1−πtvht B,Mt =μthtvH,ht B1−πtvHht B,Mt, and, for all measurable subsets S×Aof SMt×A,  (v,ht B) μthtv,ht BS×A μt+1ht,·v,ht B,z(st,(qt,xt))(Mt),mρ(π,r)d(st,q,x)|v,ht B =μthtv,ht Bπtvht B,Mtrtvht B,Mt(m)ϕMtS×A|m. Definition 3. An assessment ,(πv,rv)v∈V,μis a perfect Bayesian equilibrium if it is sequentially rational and satisfies Bayes’ rule where possible. References Abreu, Dilip, David Pearce, and Ennio Stacchetti (1990), “Toward a theory of discounted repeated games with imperfect monitoring.” Econometrica, 58, 1041–1063. [872] Acharya, Avidit and Juan Ortner (2017), “Progressive learning.” Econometrica, 85, 1965– 1990. [868] Aliprantis, Charalambos and Kim Border (2013), Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer-Verlag, Berlin. [869] Athey, Susan and Kyle Bagwell (2008), “Collusion with persistent cost shocks.” Econometrica, 76, 493–540. [872,887,905] Aumann, Robert J. (1964), “Mixed and behavior strategies in infinite extensive games.” Tech. rep., Princeton University. [870,911] Aumann, Robert J., Michael Maschler, and Richard E. Stearns (1995), “Repeated games with incomplete information.” MIT press. [868] Ausubel, Lawrence M. and Raymond J. Deneckere (1989), “Reputation in bargaining and durable goods monopoly.” Econometrica, 511–531. [868] Beccuti, Juan and Marc Möller (2018), “Dynamic adverse selection with a patient seller.” Journal of Economic Theory, 173, 95–117. [867,868] 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 913 Bester, Helmut and Roland Strausz (2001), “Contracting with imperfect commitment and the revelation principle: The single agent case.” Econometrica, 69, 1077–1098. [867, 868] Bester, Helmut and Roland Strausz (2007), “Contracting with imperfect commitment and noisy communication.” Journal of Economic Theory, 136, 236–259. [867] Board, Simon and Marek Pycia (2014), “Outside options and the failure of the Coase conjecture.” American Economic Review, 104, 656–671. [868] Breig, Zachary (2022), “Repeated contracting without commitment.” Journal of Economic Theory, 204, 105514. [867] Brzustowski, Thomas, Alkis Georgiadis-Harris, and Balázs Szentes (2023), “Smart contracts and the Coase conjecture.” American Economic Review, 113, 1334–1359. [868] Bulow, Jeremy I. (1982), “Durable-goods monopolists.” Journal of Political Economy, 90, 314–332. [868] Carrillo, Juan D. and Thomas Mariotti (2000), “Strategic ignorance as a self-disciplining device.” The Review of Economic Studies, 67, 529–544. [868] Coase, Ronald H. (1972), “Durability and monopoly.” The Journal of Law and Economics, 15, 143–149. [868] Deb, Rahul and Maher Said (2015), “Dynamic screening with limited commitment.” Journal of Economic Theory, 159, 891–928. [867] Denicolo, Vincenzo and Paolo G. Garella (1999), “Rationing in a durable goods monopoly.” The RAND Journal of Economics, 44–55. [875,876] Doval, Laura and Vasiliki Skreta (2022), “Mechanism design with limited commitment.” Econometrica, 90, 1463–1500. [865,866,867,868,869,871,876,888,891,911] Dworczak, Piotr and Giorgio Martini (2019), “The simple economics of optimal persuasion.” Journal of Political Economy, 127, 1993–2048. [889] Fiocco, Raffaele and Roland Strausz (2015), “Consumer standards as a strategic device to mitigate ratchet effects in dynamic regulation.” Journal of Economics & Management Strategy, 24, 550–569. [867] Fudenberg, Drew, David Levine, and Jean Tirole (1985), “Infinite horizon models of bargaining with one-sided uncertainty.” In Game Theoretic Models of Bargaining,volume 73, 79. Cambridge University Press. [868,873,881] Fudenberg, Drew and Jean Tirole (1991a), Game Theory. MIT Press. [901] Fudenberg, Drew and Jean Tirole (1991b), “Perfect Bayesian equilibrium and sequential equilibrium.” Journal of Economic Theory, 53, 236–260. [870] Gentzkow, Matthew and Emir Kamenica (2016), “A Rothschild–Stiglitz approach to Bayesian persuasion.” American Economic Review, 106, 597–601. [889] 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 914 Doval and Skreta Theoretical Economics 19 (2024) Gerardi, Dino and Lucas Maestri (2020), “Dynamic contracting with limited commitment and the ratchet effect.” Theoretical Economics, 15, 583–623. [867,906] Gul, Faruk, Hugo Sonnenschein, and Robert Wilson (1986), “Foundations of dynamic monopoly and the Coase conjecture.” Journal of Economic Theory, 39, 155–190. [866, 867,868,884,886] Habibi, Amir (2020), “Motivation and information design.” Journal of Economic Behavior & Organization, 169, 1–18. [868] Hart, Oliver D. and Jean Tirole (1988), “Contract renegotiation and Coasian dynamics.” The Review of Economic Studies, 55, 509–540. [866,873] Kamenica, Emir and Matthew Gentzkow (2011), “Bayesian persuasion.” American Economic Review, 101, 2590–2615. [868] Kolotilin, Anton (2018), “Optimal information disclosure: A linear programming approach.” Theoretical Economics, 13, 607–635. [889] Laffont, Jean-Jacques and Jean Tirole (1988), “The dynamics of incentive contracts.” Econometrica, 1153–1175. [867] Laffont, Jean-Jacques and Jean Tirole (1993), A Theory of Incentives in Procurement and Regulation. MIT press. [889] McAfee, R. Preston and Thomas Wiseman (2008), “Capacity choice counters the Coase conjecture.” The Review of Economic Studies, 75, 317–331. [868] Ortner, Juan (2017), “Durable goods monopoly with stochastic costs.” Theoretical Economics, 12, 817–861. [868] Peski, Marcin (2022), “Bargaining with mechanisms.” American Economic Review, 112, 2044–2082. [868] Rockafellar, Ralph Tyrell (1970), Convex Analysis. Princeton University Press. [889] Simon, Leo K. and William R. Zame (1990), “Discontinuous games and endogenous sharing rules.” Econometrica, 861–872. [875,887,907] Skreta, Vasiliki (2006), “Sequentially optimal mechanisms.” The Review of Economic Studies, 73, 1085–1111. [867,868] Skreta, Vasiliki (2015), “Optimal auction design under non-commitment.” Journal of Economic Theory, 159, 854–890. [867] Sobel, Joel (1991), “Durable goods monopoly with entry of new consumers.” Econometrica, 59, 1455–1485. [868] Sobel, Joel and Ichiro Takahashi (1983), “A multistage model of bargaining.” The Review of Economic Studies, 50, 411–426. [868] Stokey, Nancy L. (1981), “Rational expectations and durable goods pricing.” The Bell Journal of Economics, 112–128. [868] 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 19 (2024) Optimal mechanism for the sale of a durable good 915 Wolinsky, Asher (1991), “Durable-good monopoly with inventories.” Economics Letters, 37, 339–343. [868] Co-editor Simon Board handled this manuscript. Manuscript received 25 August, 2020; final version accepted 26 July, 2023; available online 1 August, 2023. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE4485 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License