SStatic Pricing in Dynamic Sales (cid:42)
Martino Banchio † Frank Yang ‡ February 16, 2021
Abstract
A monopolist sells items repeatedly over time to a consumer with persistent pri-vate information. The seller has limited commitment: she cannot commit to a long-term contract but always has the option to commit to posted prices for unsold items.We show that a static price path is the unique equilibrium outcome; that is, the sellercannot do better than simply posting the monopoly price for each item. The ratchete ff ect eliminates price discrimination gains for any degree of persistence of the pri-vate information. The paper also shows how dynamic mechanism design can helpderive new results in games with limited commitment. Keywords:
Dynamic Sales, Limited Commitment, Ratchet E ff ect, Mechanism Design JEL Classification Codes:
D82, D86, D42 (cid:42)
We thank Mohammad Akbarpour, Anirudha Balasubramanian, Gabriel Carroll, Daniel Chen, MatthewGentzkow, David Kreps, Paul Milgrom, Michael Ostrovsky, David Ritzwoller, Ilya Segal, Jesse Shapiro,Andy Skrzypacz, Takuo Sugaya, Bob Wilson, and Weijie Zhong for helpful comments. † Graduate School of Business, Stanford University. Email: [email protected] ‡ Graduate School of Business, Stanford University. Email: [email protected] a r X i v : . [ ec on . T H ] F e b Introduction
Consider a monopolist who sells nondurable goods over time to a forward-looking buyer.The monopolist would like to set prices dynamically to maximize profits. If the buyer’svalue stays constant over time, then a classic result on intertemporal price discrimina-tion shows a “false dynamic”: committing to a constant price path independent of thehistory is optimal. However, when the monopolist sells di ff erentiated products, or thebuyer’s tastes vary over time, then the optimal long-term contract is dramatically dif-ferent: it depends sensitively on the dynamics of the buyer’s value, is non-stationary,and has infinite memory, even if the type follows a Markov process and is highly persis-tent (Battaglini, 2005). Moreover, when the monopolist does not have any commitmentpower, there is often a plethora of equilibria, some of which involve excessive mixing andcomplex random price cycles (Kennan, 2001). In this paper, we consider a monopolist who has an intermediate form of commitmentpower. She cannot issue long term contracts but has the option to post irreversible pricesfor unsold products. We show that even with stochastic values, static monopoly pricesarise as the unique equilibrium outcome. The monopolist cannot secure better outcomesby negotiating than simply posting the monopoly price for each item.Such long-term relationships are natural in bilateral procurement negotiations. Amarket-facing retailer decides in each period whether to purchase some input from a mo-nopolist supplier. The retailer’s willingness to pay for the input varies over time depend-ing on the market conditions. It is often hard for the parties to sign a long-term contractin the face of these uncertainties. In many situations, instead of negotiating regularly, thesupplier o ff ers a price quote valid for an extended period of time or a catalog specifyingseasonal prices. These prices are not contingent on the retailer’s purchasing decisionsand are credible due to reputational e ff ects. Our result shows that it is optimal for thesupplier to commit to such static prices instead of engaging in periodic negotiations, eventhough she forgoes the possible gains from price discrimination.To fix ideas, suppose there are two periods and the buyer’s values for the good in eachperiod have the same marginal distribution. By posting the static monopoly price in eachperiod, the seller obtains some revenue 2 π ∗ independent of the joint distribution of thebuyer’s values for the two goods. If the seller engages in negotiation with the buyer, therevenue depends crucially on the joint distribution of the buyer’s values. This is also the See, for example, Stokey (1979), Baron and Besanko (1984), Hart and Tirole (1988) among others. Because the goods are nondurable, in repeated-sales models, the seller implicitly has commitmentpower within each period. To avoid confusion, we refer to these models as no-commitment since the sellercannot commit to future prices. In the procurement example, this arises naturally when there are idiosyncratic market demand shocksfor an individual retailer but no aggregate demand shock. Thus,revenue from long-term contracting is always higher, often by a large margin, than therevenue from posting static monopoly prices. Figure 1 illustrates how the revenue fromcontracting and negotiation may change as the correlation of willingness to pay varies. Much less is known when the seller does not have commitment power and engages innegotiation with the buyer. At one extreme, when the buyer’s values are independent,the seller gets exactly 2 π ∗ by backward induction. At the other extreme, when the buyer’svalues are constant, the seller gets at most 2 π ∗ since static monopoly pricing is the optimalcontract. However, it is far from clear what happens in between. One might expectthat the seller could do better for an intermediate range of correlations by engaging innegotiation and price discrimination; this possibility is illustrated in Figure 1a. Long-term contractNegotiation π ∗ ρ Rev (a) Long-term contractNegotiation
Rev2 π ∗ ρ (b) Figure 1: Expected revenue as a function of correlationOur main contribution is to show that this is not the case. The revenue from staticpricing dominates the revenue from negotiation across the correlation spectrum. In other See e.g. Baron and Besanko (1984), Battaglini (2005) and Pavan et al. (2014). Papadimitriou et al. (2016) show that the multiplicative gap between these two can be unbounded. For simplicity, consider a family of copulas parameterized by the correlation of willingnesses to pay. Note that the option of committing to posted prices for unsold items is vacuous in the last period. Ifthe game lasts more than two periods, then the distinction of whether the option of committing to staticprices is provided in every period or only in the beginning matters. Besides being a reasonable assumption,the former also provides tractability for a multi-period model. ff ective when the seller has long-term commitmentpower, become completely futile when the seller cannot commit? With evolving privateinformation, an important lesson from dynamic mechanism design is that early contract-ing at the initial stage prevents the buyer from capitalizing on his future informationrents. However, such early contracting requires substantial commitment power: it is ef-fectively ruled out by the seller’s sequential rationality in our setting. Because the typesare persistent over time, it is sequentially rational for the seller to post a high price fol-lowing an acceptance of the first-period o ff er and a low price following a rejection. As inthe constant-type case, the buyer anticipates the e ff ect of his present choice on future ne-gotiations: he could give up the purchase today in exchange for a better price tomorrow.Moreover, such a delay in the purchase would give further information rents to the buyeras the new private information arrives. If the seller could commit to history-contingentprices, the price schedule would never be time-consistent.However, the intuition is incomplete. In an environment where more informationarrives in the future, the buyer also worries less about his decision to reveal his currentprivate information. In other words, the ratchet e ff ect becomes weaker as the persistenceof private information decreases. After all, when types are independent across time,there is no ratchet e ff ect (nor gain from price discrimination). This key tradeo ff requiresa precise understanding of how equilibrium payo ff s depend on the joint distribution ofbuyer’s values, but characterizing all equilibria in games involving the ratchet e ff ect isoften intractable. Our methodological contribution is developing a mechanism design approach to study-ing the ratchet e ff ect in bargaining games with persistent private information. Ratherthan constructing equilibria and directly computing seller’s revenue, we pose a con-strained dynamic mechanism design problem as a relaxation. The seller’s revenue fromthe optimal constrained mechanism gives an upper bound for her revenue from any equi-librium. The delicate part of this construction is identifying the appropriate set of con-straints. The optimal revenue from the relaxed problem is bounded between the revenuesfrom seller-optimal equilibria and static pricing.We assume that the seller has distributional knowledge about how the buyer’s privateinformation evolves. In many applications, the seller might need to spend costly e ff ortto estimate the statistical dependency. Because static monopoly pricing only requires in-formation about the marginal distribution of the buyer’s values for each item, our resultsuggests that the seller may want to invest resources to better estimate demand in each See e.g. La ff ont and Tirole (1988). This work contributes to several branches of literature. There is an extensive literatureon the ratchet e ff ect and the limits of price discrimination based on purchase history, assurveyed by Fudenberg and Villas-Boas (2006, 2012) and Acquisti et al. (2016). Theclassic work by Hart and Tirole (1988) highlights the role of commitment and formalizesthe ratchet e ff ect’s intuition. In contrast to these works, we allow for stochastic typesand show that the ratchet e ff ect always eliminates the gain from price discrimination forany degree of persistence.Kennan (2001) studies a bargaining model with persistent private information. Theseller in his model engages in repeated negotiations, and the buyer has two types thatevolve according to a two-state Markov chain. He characterizes a particular class of equi-libria and shows that they always involve random price cycles. We study a similar modelbut assume the seller always has the option to commit to posted prices. With limitedcommitment, our game has a unique equilibrium outcome: the seller commits to postedprices in the beginning and never engages in bargaining.Liu et al. (2019) analyze a durable-good problem in an auction context. They provea generalization of the Coase conjecture by showing that an auctioneer without commit-ment power for future reserve prices would run an e ffi cient auction as the period lengthvanishes. To establish their result, they also consider an auxiliary mechanism designproblem and impose a constraint on the seller’s continuation payo ff to capture that theseller can always run an e ffi cient auction to end the game. The repeated-sales environ-ment we consider features a distinct commitment issue for the seller. There is no clearlower bound for her continuation payo ff , and we identify a set of constraints acting di-rectly on the allocation rules.Because the buyer has evolving private information, the solution to our auxiliarymechanism design problem builds on the recent advances in dynamic mechanism de-sign. We leverage a version of payo ff equivalence theorem in dynamic environmentsestablished in the seminal work by Pavan et al. (2014). We contribute to this literatureby providing a natural setting in which long-term commitment power is indispensable Recent work by Bonatti and Cisternas (2020) for example considers strategic consumers interactingwith a sequence of firms reduce their demand to influence their scores that aggregate purchase histories. The monopolist’s commitment issue is first recognized by Coase (1972) and central to the literature onCoase conjecture. See e.g. Stokey (1981), Bulow (1982), Fudenberg et al. (1985), Gul et al. (1986). Special cases for the dynamic payo ff equivalence theorem have been developed by Baron and Besanko(1984), Courty and Li (2000) and Eso and Szentes (2007). ffi cul-ties. In recent work, Doval and Skreta (2020) establish a version of the revelation prin-ciple in this setting. They show that any mechanism-selection game can be reduced toa constrained optimization problem in which the designer specifies not only the alloca-tion rule but also the information that the designer herself would receive subsequently.Even under this reduction, the designer needs to solve a joint mechanism and informa-tion design problem subject to sequential rationality constraints that are usually very in-volved. As a result, most models with limited commitment consider only binary types. In contrast, we restrict attention to price mechanisms but allow for stochastic types incontinuum type spaces. We illustrate one setting where such constraints can be replacedwith a set of more tractable constraints while maintaining the key forces imposed by se-quential rationality. Moreover, the literature has not been concerned with the designer’so ff -equilibrium-path beliefs. In our setting, we show that unreasonable o ff -path beliefscould grant the seller more and sometimes full commitment power. We suspect that sim-ilar issues could arise in designing general mechanisms as well. We consider a repeated-sales model with persistent but not necessarily permanent privateinformation. A seller wants to sell a nondurable good in each period t ∈ { , . . . , T } to abuyer, with cost normalized to 0. The buyer privately observes his willingness to pay θ t ∈ [ θ t , θ t ] ⊂ R + at the beginning of each period. Value θ is drawn from F ( θ ), andconditional on θ t , θ t +1 is drawn from F t ( θ t +1 | θ t ). Both the initial distribution and thetransition kernels are common knowledge. We assume F t ( θ t ) is an absolutely continuousdistribution with density f t ( θ t ) and F t +1 ( θ t +1 | θ t ) is continuously di ff erentiable in ( θ t , θ t +1 )with density f t +1 ( θ t +1 | θ t ). At the beginning of each period t the seller, if she has not yet committed, has theoption to commit to a static price path ( p t , · · · , p T ) for all unsold items or make a take-it-or-leave-it o ff er to the buyer for the product in period t . We say that a price path is staticwhen it is history-independent. The buyer decides whether to accept or not. After thebuyer makes his decision, the game moves to the next period. Let x t ∈ { , } denote the The notion of limited commitment here refers to the designer having commitment power within butnot across periods. This is always the case in a repeated-sales setting as the good is nondurable. For related work, see also Bester and Strausz (2001), Skreta (2006), Skreta (2015). See e.g. Doval and Skreta (2019), Strulovici (2017), Gerardi and Maestri (2020). We assume that the marginal and conditional distributions are supported on closed intervals. Weallow for shifting supports for the transition kernels, which can naturally happen for processes like AR(1). t , and p t ∈ R denote the price o ff ered by the seller. Utilitiesare additive across periods and quasilinear in each period, with a common discount factor δ ∈ (0 , t payo ff is u St ( x t , p t ) = δ t − x t p t and the buyer’s payo ff is u Bt ( θ t , x t , p t ) = δ t − x t ( θ t − p t ). Strategies and Beliefs
Let h t and ˆ h t denote the public history observed by the seller and the history observedby the buyer up to period t , respectively. A seller has no action at any history after shehas already committed. Otherwise, the seller’s strategy σ maps a public history h t to adistribution on ( { , } , R T − t +1 , R ) representing whether she commits or not, a static pricepath if she commits, and a price to post today if she does not commit. A buyer’s strategy τ maps a history ˆ h t to a distribution on { , } representing his purchase decision in thegiven period. The seller forms belief µ ( h t ) about θ T := ( θ , . . . , θ T ) given a public history. Equilibrium
The equilibrium concept we consider is perfect Bayesian equilibrium with a further re-striction on the seller’s o ff -path beliefs. A perfect Bayesian equilibrium satisfying the D1criterion (PBE-D) of this game is a tuple ( σ , τ, µ ) such that:• Given µ ( h t ) and τ , σ is sequentially rational at every public history h t .• Given σ , τ is sequentially rational at every history ˆ h t . • µ ( h t ) is derived via Bayes’ rule whenever possible.• Seller’s o ff -path beliefs satisfy a D1 criterion, formally defined in Appendix B.The only di ff erence from a standard PBE is in the last bullet point. We require thatan equilibrium satisfy a D1 criterion in the spirit of Banks and Sobel (1987), at any o ff -path history. The seller cannot update her belief via Bayes’ rule at some history h t whenall types of the buyer accept (or reject) in equilibrium, but she observes a rejection (oracceptance). Intuitively, the refinement requires that when observing an unexpected de-viation, the seller believes that the buyer is of the type that could benefit the most fromthe deviation. Even though our model is not a signaling game, Banks and Sobel (1987)’sidea can be extended to our setting, as formally shown in Appendix B. For the purposeof our results, we say that two equilibria are outcome-equivalent if for every realization ofthe type process they induce the same allocation and the same total discounted payment. Formally the buyer also forms beliefs about his future types, but those are simply the conditionaldistributions given by the environment. istributional Assumptions Our main result and one of the extensions focus on the model presented above with T =2. Unless stated otherwise, the remainder of the paper will consider only a two-periodmodel. Section 4.2 provides an extension for arbitrary finite time horizons under morerestrictive distributional assumptions.We impose the following assumptions for our main result. Assumption A1. (MLRP) For any θ (cid:48) > θ , The conditional likelihood ratio f ( θ | θ (cid:48) ) f ( θ | θ )is strictly increasing in θ .Assumption A1 is equivalent to strict a ffi liation of θ , θ as in Milgrom and Weber(1982). As an example, consider a AR(1) process of the form θ = αθ + (cid:15) , where (cid:15) isdrawn from a Gaussian distribution. Then Assumption A1 is equivalent to α > Assumption A2. (1 /δ -Lipschitz) For any θ (cid:48) > θ , E [ θ | θ (cid:48) ] − E [ θ | θ ] < δ ( θ (cid:48) − θ )Assumption A2 guarantees that a change in the first-period type leads to a comparablechange in the second-period type in expectation. For any AR(1) process, θ = αθ + (cid:15) , As-sumption A2 is equivalent to α < δ . Note that any AR(1) process satisfies this assumptionfor some δ >
0. To state our last assumption, recall the definition of an impulse responsefunction from Pavan et al. (2014), adapted to our setting.
Definition 1.
For a Markov type process, the impulse response function is I ( θ s , . . . , θ t ) := t (cid:89) τ = s +1 − ∂F τ ( θ τ | θ τ − ) /∂θ τ − f τ ( θ τ | θ τ − ) (1)Intuitively, the impulse response function measures the e ff ect of a small change in θ s on θ t . For any AR(1) process, θ = αθ + (cid:15) , I ( θ , θ ) = α . Assumption A3. (Regularity) The second-period virtual value function defined as ψ ( θ , θ ) := θ − − F ( θ ) f ( θ ) I ( θ , θ )is non-decreasing in θ and strictly increasing in θ .7ssumption A3 is the dynamic version of Myerson’s regularity condition and com-monly imposed in dynamic mechanism design for tractability (see e.g. Bergemann andVälimäki (2019) and references therein). For any AR(1) process, ψ ( θ , θ ) = θ − − F ( θ ) f ( θ ) α and hence the assumption holds whenever F has a monotone hazard rate.For marginals F , F , let Γ ( F , F ) denote the set of feasible joint distributions thatsatisfy Assumptions A1 to A3. Existence of Equilibrium
Our first result shows the existence of a PBE that satisfies the D1 criterion.
Proposition 1.
A PBE-D always exists for this game.Proof.
See Appendix A.6.Since this is a multi-stage game with continuous action spaces, incomplete informa-tion, and further restrictions on o ff -path beliefs, there are no standard existence resultsthat apply. We consider an auxiliary one-shot continuous game, for which Glicksberg(1952) applies, and construct a PBE using the continuity properties of the equilibria ofthe auxiliary game. We then show that the constructed PBE also satisfies the o ff -pathbelief restrictions. Remark . Under the distributional assumptions, D1 is satisfied if the seller believes thebuyer is of type θ when observing an o ff -path rejection, and of type θ when observingan o ff -path acceptance. Moreover, for any PBE that satisfies the D1 criterion, there existsan outcome-equivalent PBE with o ff -path beliefs of this form. See Appendix B for theformal statements and proofs. Therefore, it is without loss to focus on PBE with such o ff -path beliefs for all of our results. We discuss why o ff -path beliefs matter in Section 3.1. Remark . All of our results are proved using PBE-D as the equilibrium concept. Forsimplicity, we will refer to PBE-D as PBE, keeping in mind that the equilibria referencedneed to satisfy the restrictions on the o ff -path beliefs. Our main result states that static pricing is the unique equilibrium outcome of this game. Our assumption is slightly stronger than the standard one as we require ψ ( θ , θ ) to be strictly (insteadof weakly) increasing in θ which is needed for the uniqueness of equilibrium outcome. Even for games with continuous action spaces and almost perfect information, Harris et al. (1995)shows a subgame perfect equilibrium may not exist. heorem 1. Every PBE is outcome-equivalent to one in which the seller posts a monopoly pricewith respect to marginal F t in each period t . It is clear that if the seller chooses to commit to a static path of prices at the beginningof the game, she will commit to charging a monopoly price with respect to marginal F t in every period t . Let π ∗ t denote the revenue in period t under static monopoly pricing.Suppose the seller decides not to commit: let E ( γ ) be the set of PBE of the subgamefollowing that history, for a given joint distribution γ . Let Π ( E, γ ) denote the seller’sexpected revenue with joint distribution γ in equilibrium E . Theorem 1 follows if for any γ ∈ Γ ( F , F ) and any E ∈ E ( γ ):(1) The seller cannot do better than static pricing: Π ( E, γ ) ≤ (cid:88) t δ t − π ∗ t (2) She does strictly worse if E induces a di ff erent equilibrium outcome.Therefore, we focus our attention on the subgame after the history in which the sellerdecides not to commit. Recall that our solution concept allows mixing for both the buyerand the seller. Hence there could be many equilibria in E ( γ ). Nevertheless, the sellercannot do better than static pricing in any of them. The mechanics of our argument can be understood as follows. Let us suppose the sellercan commit to any dynamic price path in the first period, specifying the first-period price p , the price in the second period after buyer’s acceptance p A , and the price after buyer’srejection p R . What would the seller do then? Perhaps surprisingly, the key claim we willestablish is that it is in the seller’s interest to commit to some p A < p R . Before explainingthe intuition of this claim, let us illustrate why this implies that static pricing would bethe unique equilibrium outcome.When the seller observes an acceptance, she knows that the buyer is of a higher type.Because types are persistent, it is not sequentially rational for her to charge p A < p R in thesecond period. Lack of commitment power constrains the seller to set p A ≥ p R . But by ourearlier claim, a full-commitment seller prefers to set p A < p R . Thus the best that a sellerwith limited commitment can do is to decrease p A or increase p R until they are equal,but that is exactly what the seller can achieve by posting static prices. This argument isdepicted in Figure 2a. When types are constant, the optimal commitment prices p A , p R both equal to the static monopoly price p ∗ . When types become stochastic, as we will9xplain, the seller benefits from committing to a lower p A and a higher p R , but sequentialrationality constrains the seller in exactly the opposite direction. Hence the seller cannever do better than static pricing when she chooses to engage in negotiations.10 p R p A p ∗ ρ (a) Seller’s optimal commitment prices pθ θ θ − pp (b) Buyer and seller’s ex-post utilities Figure 2: Intuition of Theorem 1We now explain why the seller would prefer to set p A < p R if allowed to commit todynamic prices. It is helpful to consider the following simple example. Example 1.
The buyer has independent values across the two periods, and in each period, θ t is drawn from the uniform distribution on the interval [1 , ,
2] is 1, if the sellercommits to a static price path, she will post price 1 in both periods and collect a totalrevenue of 2. Now suppose the seller commits to charge p = 1 . p A = 1 , p R = 2 in the second period. Consider how the buyer of type θ decides in the firstperiod. If he rejects the first period o ff er, then he gets 0 payo ff because the second periodo ff er will be too high. If he accepts, then his expected payo ff is given by θ − p + E [( θ − p A ) + | θ ] = θ − p + E [ θ − p A | θ ] = θ − . E [ θ ] − θ − ≥ θ is always no less than p A = 1, and θ is independent of θ . Therefore, all typesof buyer will accept the first-period o ff er, and then accept the second-period o ff er. Theseller gets a total revenue of 1 . . > p = 1 . . p A and a high p R ,the seller adds a high surcharge on top of the monopoly price to sell the second good “inexpectation”. This allows her to extract the entire surplus of the second good from the10uyer: the buyer pays 1 + 1 . θ , as the second-period surplus E [ θ ] = 1 . p A < p R and adding the option price of having alow second-period price on top of the static monopoly price in the first period. Intuitively,the second-period ex-post utility ( θ − p ) + of the buyer is more “risk-seeking” than thesecond-period ex-post utility of the seller p θ ≥ p , as illustrated in Figure 2b. This allowsthe seller to benefit from selling the “option contract” to the buyer in the first period sothat the buyer bears the risk of having a low second-period value.This e ff ect echoes the power of “early contracting” in the literature of dynamic mecha-nism design: contracting before the agent’s future private information realizes allows thedesigner to extract more surplus. Naturally, techniques from dynamic mechanism designhelp us make a precise statement about when the seller prefers to commit to p A < p R .The sequential rationality constraint only comes into play if the seller is at an on-path history. Without any equilibrium refinement, the seller can have implausible beliefsat some o ff -path history in equilibrium. Such belief removes the sequential rationalityconstraint and e ff ectively grants the seller more and sometimes full commitment power.To see this concretely, consider the following example. Example 2.
Type θ is drawn from a uniform distribution over the interval [1 , θ = 2, then type θ = 2; otherwise, θ is drawn uniformly over [1 ,
2] independent of θ .Example 2 di ff ers from Example 1 only when θ = 2, but the seller can now obtainthe full-commitment outcome that we just described ( p = 1 . , p A = 1 , p R = 2) in a perfectBayesian equilibrium if there is no restriction imposed on the o ff -path belief. To seethis, note that all types solve the same problem in the first period as before, except thetype θ = 2, but clearly, this type also prefers accepting the first-period o ff er. So all typesaccept in both periods. Now suppose the seller holds a belief µ concentrated on θ = 2after a first-period rejection (which is an o ff -path history). This makes the price p R = 2sequentially rational. D1 eliminates such implausible beliefs by essentially requiring theseller to believe that the buyer who deviates must be of the type most likely to profit fromthe deviation. As noted in Remark 1, the seller would believe that the buyer is of type θ = 1 after observing rejection, forcing p A ≥ p R even at an o ff -path history. This outcome is actually optimal even if the seller can commit to any mechanism. To see this, notethat the seller can never do better than observing the realization of θ and then committing to a directmechanism mapping type reports ˆ θ to the allocation of the two goods and transfers. In that case, theoptimal mechanism is to allocate both goods and charge a price of 1 + θ , which generates a revenue of 2 . .2 Proof Outline We now sketch an outline of the proof for all pure-strategy PBE, assuming δ = 1 and γ having full support. All omitted details are in Appendix A.The proof proceeds in the following steps.1. Characterize necessary conditions satisfied in any PBE.2. Pose a full-commitment relaxation using necessary conditions as constraints.3. Characterize the solution to the relaxed problem.
Step 1:
Let p A , p R be the seller’s equilibrium choice of price posted after observing ac-ceptance and rejection in the first period. Let x ( θ ) be the equilibrium first-period allo-cation. We show that the key restriction imposed by sequential rationality is the followingset of monotonicity constraints. Proposition 2.
In any PBE of the game, x is a threshold rule and p A ≥ p R . Before proving this claim, we introduce two elementary lemmas.
Lemma 1.
For any k ∈ ( θ , θ ) , f ( θ | θ ≥ k ) f ( θ | θ < k ) is strictly increasing in θ , i.e. F ( · | θ ≥ k ) (cid:31) L F ( · | θ < k ) . This lemma is a direct consequence of Assumption A1. It sorts the seller’s posteriorbelief in the second period assuming a threshold rule in the first period. Since likeli-hood ratio ordering implies inverse hazard rate ordering, the next observation followsimmediately from monotone selection theorem in Topkis (1998).
Lemma 2. If F (cid:31) L F , and p i is an optimal monopoly price under F i , then p ≥ p . Now we provide a proof of Proposition 2. Allowing for δ < ffi culty, and we omit the proof. Allowing for mixed-strategy PBE andjoint distribution γ that may not have positive density everywhere on [ θ , θ ] × [ θ , θ ] requires attentionand is handled in the Appendix. For simplicity, we continue to refer pure-strategy PBE as PBE, and we willbe clear whenever refer to mixed-strategy PBE. We use the notation (cid:31) L to represent likelihood-ratio dominance: the likelihood ratio of two distribu-tions is a strictly increasing function. roof of Proposition 2. In the second period, the buyer accepts when the price is below hissecond-period type θ . Thus, in the first period, type θ accepts a price p only if( θ − p ) + E [( θ − p A ) + | θ ] ≥ E [( θ − p R ) + | θ ] (2)Let m ( θ ) := ( θ − p R ) + − ( θ − p A ) + , h ( θ ) := E [ m ( θ ) | θ ] and g ( θ ) := θ − p . So type θ accepts only if g ( θ ) ≥ h ( θ ) and always accepts if the inequality is strict. Note that m ( θ )is 1-Lipschitz and has a.e. derivative m (cid:48) ≤
1. For any θ (cid:48) > θ , using integration by parts, h ( θ (cid:48) ) − h ( θ ) = E [ m ( θ ) | θ (cid:48) ] − E [ m ( θ ) | θ ] = (cid:90) ∞ [ P ( θ > s | θ (cid:48) ) − P ( θ > s | θ )] m (cid:48) ( s ) ds (3) ≤ E [ θ | θ (cid:48) ] − E [ θ | θ ] < θ (cid:48) − θ = g ( θ (cid:48) ) − g ( θ )where we also used that θ | θ (cid:48) (cid:23) FOSD θ | θ (implied by Assumption A1) and the Lipschitzcondition on E [ θ | · ] (Assumption A2). So g crosses h at most once from below, and thusthe first-period allocation is characterized by a cuto ff type k , proving the first part.Suppose the cuto ff type k ∈ ( θ , θ ). By Lemma 1, F ( · | θ ≥ k ) (cid:31) L F ( · | θ < k ). Thenby Lemma 2, p A ≥ p R , since p A must be optimal under belief F ( · | θ ≥ k ) and p R must beoptimal under belief F ( · | θ < k ). Now note that if k = θ , then a rejection is o ff -pathand it is without loss to let the seller form a belief F ( · | θ = θ ) as noted in Remark 1. So p A ≥ p R as F ( · | θ ≥ θ ) (cid:31) L F ( · | θ = θ ). The same argument holds for k = θ . Step 2:
Since this set of monotonicity constraints must hold in any equilibrium, if theseller could commit to a price mechanism subject to these constraints, she could alwaysreplicate any PBE. This implies that providing her the commitment power subject to theseconstraints forms a relaxed problem. The seller’s revenue from the relaxed problem givesan upper bound for her revenue in any PBE. Specifically, the relaxation is done as follows.
Original :Select a seller-optimal PBE s.t.1. Buyer’s IC constraints2. Seller’s sequential rationality3. Belief consistency
Relaxed :Design a revenue-maximizing mechanism s.t.1. Buyer’s IC constraints2. x ( θ ) = θ ≥ k x ( θ , θ ) = x ( θ ) θ ≥ p A + (1 − x ( θ )) θ ≥ p R , p A ≥ p R Since we assume atomless distributions, it is without loss to let the threshold type accept for sure.
13n this relaxation, we replace the sequential rationality and belief consistency con-straints with much simpler constraints imposed on the allocation rules. The space ofprice mechanisms constrains the set of feasible allocation and payment rules jointly. Let X denote the set of all incentive-compatible dynamic direct mechanisms satisfying thefeasibility and monotonicity constraints. The seller then solves the following problem.max ( x,t ) ∈X E (cid:34) θ x ( θ ) + θ x ( θ , θ ) − V x,t ( θ ) (cid:35) (4)where V x,t ( θ ) is the expected ex-ante utility of a type θ buyer, under the mechanism( x, t ). As in static mechanism design, local IC constraints imply a form of payo ff equiv-alence in dynamic environments. However, the derivation of such payo ff equivalenceresult is much more delicate. We leverage directly from Pavan et al. (2014): Proposition 3 (Pavan, Segal, Toikka (2014)) . Under any incentive-compatible direct mecha-nism, V ( θ ) is Lipschitz continuous and has almost everywhere the derivative: V (cid:48) ( θ ) = E (cid:20) T (cid:88) t =1 I t ( θ t ) x t ( θ t ) (cid:12)(cid:12)(cid:12)(cid:12) θ (cid:21) (ICFOC) Proof.
Our assumptions guarantee that the environment satisfies the regularity condi-tions in Pavan et al. (2014). The claim follows from Theorem 1 in Pavan et al. (2014)adapted to our setting.With ICFOC and the usual integration by parts, we can rewrite objective (4) asmax ( x,t ) ∈X E [ θ x ( θ ) + θ x ( θ , θ )] − E − F ( θ ) f ( θ ) (cid:16) x ( θ ) + I ( θ , θ ) x ( θ , θ ) (cid:17) − V x,t ( θ )Importantly, with price mechanisms, one cannot extract all the ex-ante surplus from thelowest type. The lowest type can always reject the o ff er in the first period and proceedsto the second period facing price p R without any punishment. This implies V x,t ( θ ) ≥ E [( θ − p R ) + | θ = θ ]With these observations, we reach the final form of our relaxationmax ( x,t ) ∈X E [ ψ ( θ ) x ( θ )] + E [ ψ ( θ , θ ) x ( θ , θ )] − E [( θ − p R ) + | θ = θ ] (5) Unfortunately t denotes both the transfer and time. We hope its meaning will be clear from the context. ψ ( θ ) := θ − − F ( θ ) f ( θ ) , ψ ( θ , θ ) := θ − − F ( θ ) f ( θ ) I ( θ , θ )are the virtual value functions for period 1 and 2, respectively. Note that if a mechanism( x ∗ , t ∗ ) solving (5) satisfies V x ∗ ,t ∗ ( θ ) = E [( θ − p ∗ R ) + | θ = θ ], then the mechanism also solves(4) and the value of (5) is exactly its expected revenue. Step 3:
To solve (5), note that under the constraints, the relevant parameters for theallocation rule are just ( k, p A , p R ) with p A ≥ p R . We optimize directly over these:max k,p A ,p R : p A ≥ p R E [ ψ ( θ ) θ ≥ k ] + E [ ψ ( θ , θ )( θ ≥ k θ ≥ p A + θ Claim 1. For any k , the constraint p A ≥ p R must bind in any optimal solution to ( (cid:63) ).Proof. The mechanics of the argument are illustrated in Figure 3 (Assumption A3 impliesthat the boundary is downward sloping). Let z = D ( k ) be the point D crosses vertical lineat k . Fix any feasible solution ( p A , p R ) to program ( (cid:63) ). Suppose p R < z < p A . As Figure 3shows, by lowering p A the seller captures only types with positive ψ ( θ , θ ). Similarly, byraising p R , the seller foregoes types with negative ψ ( θ , θ ). Crucially, the second termin the objective, − E [( θ − p R ) + | θ = θ ], increases in p R as well. Since ψ ( θ , θ ) is strictlyincreasing in θ and γ has full support, ( z, z ) does strictly better than ( p A , p R ). Suppose z ≤ p R < p A . By the same reasoning, lowering p A to p R strictly increases the objective.Suppose p R < p A ≤ z . Raising p R to p A strictly increases the objective. Thus, p A = p R inany optimal solution. This is the standard pointwise maximization solution under the regularity condition. If for some θ , ψ ( θ , θ ) < θ , let D ( θ ) := θ . − z k θ θ D p A p R Figure 3: Allocation rule in the type spaceSince this claim holds for any k , we can further reduce (5) tomax k,p E [ ψ ( θ ) θ ≥ k ] + E [ ψ ( θ , θ ) θ ≥ p ] − E [( θ − p ) + | θ = θ ] (6)Let ( k ∗ , p ∗ ) be an optimal solution. Consider the indirect mechanism that posts a price k ∗ in period 1 followed by a price p ∗ in period 2. It implements the allocation rule x ( θ ) = θ ≥ k ∗ , x ( θ , θ ) = θ ≥ p ∗ that solves (6). Thus it solves (5). Moreover, the lowest type getsan expected payo ff V ( θ ) = E [( θ − p ∗ ) + | θ = θ ]. Therefore, it solves (4) and its expectedrevenue is an upper bound for what the seller can achieve in any PBE. But this is a staticprice mechanism, and hence the seller cannot do better than committing to the staticmonopoly prices. Suppose, for contradiction, the equilibrium induces an outcome thatcannot be replicated by any static price path. Then we must have p A > p R . But by Claim 1,the seller does strictly better by committing to the static price path ( k ∗ , p ∗ ). Contradiction.Q.E.D. Remark . The argument demonstrates that if γ has full support, then the on-path priceswould be static monopoly prices in any PBE of the original game. Without the full-support assumption, this does not always have to be the case, but any PBE is still outcome-equivalent to one with static monopoly prices (see Appendix A.3). In other words, theprices could be history-dependent but only in an immaterial way that maintains the sameex-post allocation and total payment. An optimal solution exists since we are maximizing a continuous function on a compact domain. Extensions The baseline model assumes that the buyer’s utility in the second period does not dependon the consumption of the first-period product, which precludes the possibility that theproducts may be complementary. This is especially relevant in the setting of price dis-crimination, as one might expect that purchase histories become even more important.Surprisingly we show that the same result also extends to this setting. This is becausethe ratchet e ff ect grows stronger precisely when the past information becomes more rele-vant. The static prices, however, may not be the monopoly prices anymore. When the twoitems are complements, the seller benefits from lowering the first-period price to increasethe second-period demand. Nevertheless, the seller does not benefit from exploiting theinformation in the buyer’s purchase history.To allow for such complementarity, we extend our model by letting the distribution of θ depend on the allocation of the first-period product. Specifically, the game proceedsexactly as in Section 3, but θ ∼ F ( · | θ , x ) given the realization of θ and the first-periodallocation x ∈ { , } . A natural assumption for complementarity is to assume that the distribution shiftsupward in the sense of likelihood-ratio dominance after consuming the first product. Assumption A4 (Complement) . For any θ , the conditional likelihood ratio f ( θ | θ , x = 1) f ( θ | θ , x = 0)is non-decreasing in θ .We adapt Assumptions A1, A2 and A3 in Section 3 to this setting. Assumption A1’ (MLRP’) . For any θ (cid:48) > θ and x ∈ { , } , the conditional likelihood ratio f ( θ | θ (cid:48) , x ) f ( θ | θ , x )is strictly increasing in θ .Instead of requiring Assumption A2 for each value of x , it is enough to assume itholds when the buyer does not consume the first good. An equivalent formulation is to assume a non-additively-separable utility: u ( θ, x ) = θ x + θ x + κ ( θ , θ ) x x , where κ governs the extent of complementarity. To see the equivalence, one can define˜ θ = θ if x = 0, and ˜ θ = θ + κ ( θ , θ ) if x = 1. Then the buyer’s utility would be additively separable in( θ , ˜ θ ), and the distribution of ˜ θ depends on θ and x . ssumption A2’ (1 /δ -Lipschitz’) . For any θ (cid:48) > θ , E [ θ | θ (cid:48) , x = 0] − E [ θ | θ , x = 0] < δ ( θ (cid:48) − θ )The impulse response function now takes the form I ( θ , θ , x ) = − ∂F ( θ | θ ,x ) ∂θ (cid:46) f ( θ | θ , x ). Assumption A3’ (Regularity’) . The second period virtual value function ψ ( θ , θ , x ) := θ − − F ( θ ) f ( θ ) I ( θ , θ , x )is non-decreasing in θ , strictly increasing in θ , and non-decreasing in x .We make a few remarks. If the distribution of θ conditional on θ does not depend on x as in Section 3, then Assumption A4 trivially holds, and Assumptions A1’, A2’ and A3’reduce to Assumptions A1, A2 and A3 respectively. In Assumption A3’, the virtual valuefunction being non-decreasing in x is equivalent to I ( θ , θ , ≤ I ( θ , θ , θ would lead to a smaller change in θ if the firstproduct is consumed.To get a concrete sense of these assumptions, suppose the type process follows anAR(1) process given first period allocation x ∈ { , } : θ = αθ + (cid:15) x , where (cid:15) x is drawnfrom a Gaussian distribution N ( µ x , σ ) truncated to some interval. The complementarityassumption A4 is satisfied here whenever µ ≥ µ . Assumptions A1’ and A2’ amountto 0 < α < /δ . Assumption A3’ is equivalent to that F ( θ ) has monotone hazard rateproperty. Thus if µ ≤ µ , < α < /δ and F ( θ ) has monotone hazard rate property, thenall the assumptions are satisfied. Theorem 2. Under Assumptions A1’, A2’, A3’ and A4,(i) there exists a PBE of the game, and(ii) every PBE is outcome-equivalent to one in which the seller posts static prices. Part (i) is proved along with Proposition 1 in Appendix A.6. The proof of part (ii) issimilar to the main proof in Section 3. We focus on the subgame after the seller choosesnot to commit to static prices. We first show that in any equilibrium, the buyer follows athreshold rule, which relies on Assumptions A1’ and A2’. It su ffi ces to show that h ( θ ) := E [( θ − p R ) + | θ , x = 0] − E [( θ − p A ) + | θ , x = 1]18atisfies h ( θ (cid:48) ) − h ( θ ) < θ (cid:48) − θ for any θ (cid:48) > θ (Lemma 3). Coupled with Assumption A4,the seller thinks the buyer is a higher type after observing acceptance and therefore al-ways posts a higher price (Lemma 4). Let us provide an upper bound for the seller’srevenue using the same relaxed problem with the price monotonicity constraint:max k,p A ,p R : p A ≥ p R E [ ψ ( θ ) θ ≥ k ] + E [ ψ ( θ , θ , θ 1) cross zero.Under the regularity condition in Assumption A3’, both D and D are well-defined anddownward sloping. Moreover, D always stays below D . For any cuto ff k , the e ff ectiveboundary D ( θ , k ) is of the form D ( θ , k ) = θ Under Assumption A5, for any finite T ,(i) there exists a PBE of the game, and(ii) every PBE is outcome-equivalent to one in which the seller posts static monopoly prices. We sketch the proof for part (ii) here. Omitted details and the proof of part (i) can befound in Appendix A.We first extend Proposition 2 to this setting by showing that at any history ˆ h t , thebuyer always adopts a threshold rule: x t ( ˆ h t ) = θ t ≥ k t where the threshold k t depends onlyon the public history. This statement is proved inductively by bounding the derivative ofthe buyer’s continuation payo ff s following an acceptance or a rejection, and then showingan appropriate single-crossing condition holds as in the proof of Proposition 2. It relieson the condition α t ∈ (0 , δ ) for all t in Assumption A5.The rest of the proof proceeds by induction on T . The base case T = 1 is trivial.For the inductive step, fix any equilibrium. Consider what action the seller decides totake in period 2. Because the buyer adopts a threshold rule in period 1 (say the cuto ff type is k ), the seller’s posterior belief about θ follows a truncated distribution of F afterobserving an acceptance or a rejection. Since log-concavity is preserved under truncation,convolution, and linear operations, her posterior belief about ( θ , θ , · · · , θ T ) in period 2also satisfies Assumption A5. Moreover, because the buyer uses a threshold strategy,information about θ does not matter to both players. Invoking the inductive hypothesis20hen implies that any equilibrium of the subgame must be outcome-equivalent to the onewhere the seller posts static monopoly prices from period 2 onward.Let { p At } Tt =2 denote the sequence of static monopoly prices with respect to the marginals { F t ( · | θ ≥ k ) } Tt =2 , and similarly define { p Rt } Tt =2 . We then show that under Assumption A5( θ , θ t ) satisfies a MLRP condition that implies p At ≥ p Rt for all t in this setting. This en-ables us to adopt the same approach as in the proof of Theorem 1. Using Proposition 3,the following relaxed problem gives an upper bound for the seller’s revenue:max ( x,t ) ∈X E (cid:20) T (cid:88) t =1 ψ t ( θ , · · · , θ t ) x t ( θ , · · · , θ t ) (cid:21) − V x,t ( θ ) (8)where X encodes the constraints that the seller selects a price mechanism subject to firstperiod allocation monotonicity, and price monotonicity for period t ≥ 2. Following simi-lar steps as before, the problem reduces tomax k, { p At } Tt =2 , { p Rt } Tt =2 p At ≥ p Rt ∀ t ≥ E [ ψ ( θ ) θ ≥ k ] + T (cid:88) t =2 E [ ψ t ( θ , · · · , θ t )( θ ≥ k θ t ≥ p At + θ Suppose T = 2, Θ = { , } and P ( θ t = 1) = P ( θ t = 2) = 0 . P ( θ = 2 | θ = 1) = P ( θ = 1 | θ = 2) = 1. An optimal monopoly price is 2 and posting this price for bothperiods generates revenue 2. Consider instead the following strategy profile. The sellerposts p = 2, and then posts p A = 1 following acceptance and p R = 2 following rejection.Type (2 , 1) accepts in both periods. Type (1 , 2) rejects in period 1 and accepts in period2. There is no profitable deviation for the buyer. This is also sequentially rational for theseller, since she knows θ = 2 after rejection and θ = 1 after acceptance. Therefore, aseller-optimal PBE generates revenue at least 2 . > ff ect completely disappears, and the seller capitalizes on theinformational gain. Intuitively, the low-type buyer in the first period can try to imitatethe high-type buyer to avoid future price discrimination, but doing so comes at a cost.22hen types are positively correlated, it is always the high type trying to imitate the lowtype to avoid discrimination, in which case the high type never su ff ers a negative payo ff in the first period.Our model assumes a particular form of limited commitment: the seller can commit tostatic prices but not history-contingent prices. There are multiple ways one can providemicrofoundations for this assumption. Static posted prices are easy to write down (e.g.,using a catalog), whereas history-contingent prices often lead to a complex contract asthe possible contingencies quickly explode. Deviations from static pricing are also easierto detect than deviations from dynamic pricing since one does not need to keep track ofthe history. This is especially relevant when the seller’s commitment power comes fromher long-term reputation. In this paper, we do not take a stand on why the seller canonly commit to static prices but ask what happens if we take this intuitive idea seriously.In standard models, the commitment power is often binary: the seller has either full orno commitment power. However, some classes of simple strategies may seem easier tocommit to than other strategies. We view it as a promising future direction to provide aformal model that links simplicity to “commitability”, and to study whether the gain inthe commitment power outweighs the loss in flexibility in other environments. 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(1998): Supermodularity and Complementarity , Princeton University Press.25 ppendix A Omitted Proofs A.1 Proof of Lemma 1 Fix k ∈ ( θ , θ ), note that f ( θ | θ ≥ k ) f ( θ | θ < k ) ∝ (cid:82) θ k f ( θ | θ = s (cid:48) ) f ( θ = s (cid:48) ) ds (cid:48) (cid:82) kθ f ( θ | θ = s ) f ( θ = s ) ds It then su ffi ces to show that for any s (cid:48) > k , one has f ( θ | θ = s (cid:48) ) (cid:82) kθ f ( θ | θ = s ) f ( θ = s ) ds is strictly increasing in θ . Equivalently, (cid:82) kθ f ( θ | θ = s ) f ( θ = s ) dsf ( θ | θ = s (cid:48) )is strictly decreasing in θ . It su ffi ces to show f ( θ | θ = s ) f ( θ | θ = s (cid:48) )is strictly decreasing in θ for every s (cid:48) > s , but that is implied by the MLRP property. A.2 Proof of Lemma 2 It is clear that p i ∈ argmax p [log( p ) + log(1 − F i ( p ))]The claim follows from Topkis’s monotone selection theorem, if we show that v ( p, i ) :=log(1 − F i ( p )) has strictly increasing di ff erences. Let ∆ ( p ) := log(1 − F ( p )) − log(1 − F ( p ))Note that ∆ (cid:48) ( p ) = f ( p )1 − F ( p ) − f ( p )1 − F ( p ) > .3 Proof of Theorem 1 We complete the proof for Theorem 1 by extending it to cover mixed-strategy equilibriaand allowing for γ that may not have positive density everywhere on the domain. Support of γ . As noted before, γ may not have full support on the domain [ θ , θ ] × [ θ , θ ], because theconditional distributions F ( · | θ ) could have supports varying with θ . This happens inour example of AR(1) process. Note that the assumption of γ having full support is onlyused in the proof of Claim 1. Without this assumption, the proof of Claim 1 still showsthat there is an optimal solution to ( (cid:63) ) such that p A = p R . We say that two allocation rulesare equivalent if they di ff er only on γ -null set of types. Then we have Claim 2. For any optimal solution to ( (cid:63) ), there is an equivalent allocation rule with p A = p R .Proof. Fix any optimal solution to ( (cid:63) ), with p A > p R . Suppose p R < z < p A , with z de-fined as in the main text. The set of types [ θ , k ] × [ p R , z ] must be a γ -null set becauseotherwise increasing p R up to z strictly increases the objective. Similarly, the set of types[ k, θ ] × [ z, p A ] must be a γ -null set as well. Thus the second period allocation rule θ ≥ z isequivalent to θ Instead of choosing two fixed prices p A , p R , the seller now chooses two random variables˜ p A , ˜ p R . We refer to their realizations as p A , p R . Step 1: We show allocation monotonicity continues to hold, and price monotonicityholds in the sense that p A ≥ p R for any realized p A , p R , i.e. inf { supp( ˜ p A ) } ≥ sup { supp( ˜ p R ) } .Note that (3) is preserved under averaging. So if we define h ( θ ) := E ˜ p A , ˜ p R [ E [( θ − p R ) + − ( θ − p A ) + | θ ]] Technically, Assumption A3 is only well-defined for types in the support of γ . To be more formal, let θ ( θ ) , θ ( θ ) be the left and right end of the interval support for F ( · | θ ). The maintained assumptions(along with Assumption A1) imply that θ ( θ ) , θ ( θ ) is continuous and non-decreasing in θ . For ourpurpose, it su ffi ces to let ψ ( θ , θ ) := ψ ( θ , θ ( θ )) if θ < θ ( θ ) and ψ ( θ , θ ) := ψ ( θ , θ ( θ )) if θ >θ ( θ ), and only require Assumption A3 for types in the support. It is without loss to just focus on mixing over p A , p R since mixing over p implies there always exists adeterministic p achieving the same revenue. h ( θ (cid:48) ) − h ( θ ) < θ (cid:48) − θ for any θ (cid:48) > θ . Type θ accepts if( θ − p ) + E ˜ p A [ E [( θ − p A ) + | θ ]] ≥ E ˜ p R [ E [( θ − p R ) + | θ ]]Then as before, g ( θ ) := θ − p crosses h at most once from below. Therefore, the firstperiod allocation is characterized by a cuto ff type k .Suppose k ∈ ( θ , θ ). By Lemma 1, F ( · | θ ≥ k ) (cid:31) L F ( · | θ < k ). Note that anyrealized p A must be an optimal price under belief F ( · | θ ≥ k ), as the seller only mixesamong optimal prices. Similarly, any realized p R must be an optimal price under belief F ( · | θ < k ). Hence p A ≥ p R for any realized p A , p R by Lemma 2. The case that k = θ or θ is handled similarly as before. Step 2: The same relaxation argument applies. The only di ff erence is that the sellernow commits to a randomized price mechanism in the second period withinf { supp( ˜ p A ) } ≥ sup { supp( ˜ p R ) } denoted by ˜ p A (cid:23) ˜ p R . The same steps go through, except the relaxation problem is nowmax k, ˜ p A , ˜ p R : ˜ p A (cid:23) ˜ p R E [ ψ ( θ ) θ ≥ k ] + E [ ψ ( θ , θ )( θ ≥ k θ ≥ ˜ p A + θ To solve this, we consider a further relaxation. We couple ˜ p A , ˜ p R in a commonprobability space ( Ω , F , P ). Consider the requirement that ˜ p A ( ω ) ≥ ˜ p R ( ω ) for all ω ∈ Ω ,denoted by ˜ p A ≥ ˜ p R . The disjoint support constraint clearly implies this, so this is arelaxation of the problem above. The seller’s problem is to maximize over k and two F -measurable functions:max k, ˜ p A , ˜ p R : ˜ p A ≥ ˜ p R E P (cid:20) E [ ψ ( θ ) θ ≥ k ] + E [ ψ ( θ , θ )( θ ≥ k θ ≥ ˜ p A ( ω ) + θ Lemma 3. h ( θ (cid:48) ) − h ( θ ) < θ (cid:48) − θ for any θ (cid:48) > θ .Proof. Fix θ (cid:48) > θ . Integration by parts implies E [( θ − p R ) + | θ (cid:48) , x = 0] − E [( θ − p R ) + | θ , x = 0]= (cid:90) ∞ [ P ( θ > s | θ (cid:48) , x = 0) − P ( θ > s | θ , x = 0)] s ≥ p R ds ≤ (cid:90) ∞ [ P ( θ > s | θ (cid:48) , x = 0) − P ( θ > s | θ , x = 0)] ds = E [ θ | θ (cid:48) , x = 0] − E [ θ | θ , x = 0] < θ (cid:48) − θ where we used FOSD (implied by Assumption A1’) and Assumption A2’. The claim fol-lows by noting that FOSD implies E [( θ − p A ) + | θ (cid:48) , x = 1] − E [( θ − p A ) + | θ , x = 1] ≥ Lemma 4. p A ≥ p R .Proof. Using the assumptions, we have F ( · | θ (cid:48) , x = 1) (cid:31) L F ( · | θ , x = 1) (cid:23) L F ( · | θ , x = 0)for any θ (cid:48) > θ . This then implies that for any k ∈ ( θ , θ ), F ( · | θ ≥ k, x = 1) (cid:31) L F ( · | θ To complete the proof for Theorem 2, we need to show that the samerevenue bound holds for any mixed-strategy PBE of the subgame after the seller choosesnot to commit. The proof is virtually the same as in Appendix A.3 and thus omitted. A.5 Proof of Part (ii) in Theorem 3 Lemma 5. For any seller’s strategy, the buyer’s optimal strategy is given by a threshold rule: atevery history ˆ h t , x t ( ˆ h t ) = θ t ≥ k t , where k t depends only on the public history.Proof. Note that at any history ˆ h t , the buyer solves the following problemmax x ∈{ , } [( θ t − p t ) + U At ( θ t , h † t )] x + U Rt ( θ t , h † t )(1 − x )where U At , U Rt are the continuation payo ff following acceptance and rejection, and wedecompose the history of the buyer ˆ h t into ( θ t , h † t ).We prove the following three claims together by backward induction:291) x t ( ˆ h t ) = θ t ≥ k t , where k t only depends on the public history.(2) U At ( · , h † t ) , U Rt ( · , h † t ) are 1-Lipschitz and non-decreasing function of θ t .(3) U At ( θ t , · ) , U Rt ( θ t , · ) depend on h † t only through public history h ot := ( h t , p t ). Base Case: At the last period, k T = p T and U AT = U RT = 0. Trivially, the claims hold. Inductive Step: Note that U At ( θ t , h † t ) = δ E (cid:20) max (cid:26) θ t +1 − p t +1 + U At +1 ( θ t +1 , h † t +1 ) , U Rt +1 ( θ t +1 , h † t +1 ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) θ t , h † t , x t = A (cid:21) p t +1 depends only on the public history by definition. Since the type process is Marko-vian, the distribution of θ t +1 only depends on θ t . By the inductive hypothesis, U At +1 ( θ t +1 , · ), U Rt +1 ( θ t +1 , · ) only depend on the public history. Therefore, fixing θ t , U At ( θ t , · ) depends on h † t only through the public history h ot . The same argument works for U Rt ( θ t , · ). Moreover,by the inductive hypothesis, for any fixed p ,max (cid:26) θ t +1 − p + U At +1 ( θ t +1 , h ot +1 ) , U Rt +1 ( θ t +1 , h ot +1 ) (cid:27) is 2-Lipschitz and non-decreasing in θ t +1 . Therefore, U t +1 ( θ t +1 , h ot ) := E p t +1 ∼ σ (cid:20) max (cid:26) θ t +1 − p t +1 + U At +1 ( θ t +1 , h ot +1 ) , U Rt +1 ( θ t +1 , h ot +1 ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) h ot , x t = A (cid:21) is also 2-Lipschitz and non-decreasing in θ t +1 , where σ is the seller’s strategy. Fix any θ (cid:48) t > θ t , integration by parts yields that U At ( θ (cid:48) t , h ot ) − U At ( θ t , h ot ) = δ (cid:90) ∞ (cid:104) P ( θ t +1 > s | θ (cid:48) t ) − P ( θ t +1 > s | θ t ) (cid:105) U (cid:48) t +1 ( s, h ot ) ds Under Assumption A5, P ( θ t +1 > s | θ (cid:48) t ) ≥ P ( θ t +1 > s | θ t ). Thus U At ( θ (cid:48) t , h ot ) − U At ( θ t , h ot ) ≤ δ (cid:90) ∞ (cid:104) P ( θ t +1 > s | θ (cid:48) t ) − P ( θ t +1 > s | θ t ) (cid:105) ds = 2 δ ( E [ θ t +1 | θ (cid:48) t ] − E [ θ t +1 | θ t ]) < θ (cid:48) t − θ t where the last inequality is due to the assumption α t ∈ (0 , δ ) for all t . This shows that U At ( · , h ot ) is 1-Lipschitz. It is clear from above that U At ( · , h ot ) is non-decreasing. The samearguments work for U Rt ( · , h ot ). Together these also imply that for any h † t , H ( θ t , h † t ) := ( θ t − p t ) + U At ( θ t , h † t ) − U Rt ( θ t , h † t ) = ( θ t − p t ) + U At ( θ t , h ot ) − U Rt ( θ t , h ot )is a strictly increasing function and thus crosses 0 at most once from below. Therefore,the buyer’s strategy is given by a threshold rule and the threshold k t only depends on thepublic history h ot . This proves the inductive step.30 emma 6. For any ≤ t ≤ T and any θ (cid:48) > θ , the conditional likelihood ratio f ( θ t | θ (cid:48) ) f ( θ t | θ ) isnon-decreasing in θ t .Proof. Fix any t . We can write θ t = α t θ t − + (cid:15) t = α t ( α t − θ t − + (cid:15) t − ) + (cid:15) t = · · · = ( t (cid:89) s =2 α s ) · θ + t (cid:88) s =2 ( t (cid:89) n = s +1 α n ) · (cid:15) s where we interpret (cid:81) tn = s +1 α n = 1 when s = t . Let α := (cid:81) ts =2 α s > 0. Since (cid:15) s are mutuallyindependent and each follows a log-concave distribution, (cid:15) := (cid:80) ts =2 ( (cid:81) tn = s +1 α n ) (cid:15) s followsa log-concave distribution. Therefore θ t = αθ + (cid:15) where α > (cid:15) follows some log-concave distribution G . Since G is a log-concave distribution, the conditional distributionof θ t has MLRP in αθ by standard result, i.e. f ( θ t | θ (cid:48) ) f ( θ t | θ ) = g ( θ t − αθ (cid:48) ) g ( θ t − αθ )is non-decreasing in θ t , for any αθ (cid:48) > αθ . Since α > 0, the latter is simply θ (cid:48) > θ ,proving the claim. Lemma 7. p At ≥ p Rt for all ≤ t ≤ T .Proof. Because F t ( · | θ < k ) and F t ( · | θ ≥ k ) are both log-concave, both admit a uniqueoptimal monopoly price. Given this, we no longer needs the monotone selection theo-rem used in Lemma 2. It su ffi ces to show that F t ( · | θ ≥ k ) weakly dominates F t ( · | θ < k )in hazard rate order, which is implied by weak likelihood-ratio dominance F t ( · | θ ≥ k ) (cid:23) L F t ( · | θ < k ). By the proof of Lemma 1, it su ffi ces to show ( θ , θ t ) satisfies the weak MLRPcondition, which is established in Lemma 6. The case where k = θ or θ follows by thesame argument as in the proof of Proposition 2. A.6 Proof of Proposition 1 and Part (i) in Theorem 2 As explained in Remark 1 and showed formally in Appendix B, it su ffi ces to show theexistence of a PBE with a specific restriction on the o ff -path beliefs. Definition 2. A PBE (cid:63) is a Perfect Bayesian Equilibrium with o ff -path beliefs:• After observing an o ff -path rejection at time t under belief µ t on θ t , the seller be-lieves that the period t type of the buyer was min { supp( µ t ) } . • After observing an o ff -path acceptance at time t under belief µ t on θ t , the sellerbelieves that the period t type of the buyer was max { supp( µ t ) } . To see this, note that if F is log-concave, then log( p ) + log(1 − F ( p )) is strictly concave and admits aunique maximizer. The minimum is well-defined here since the support is always bounded and closed. 31n what follows, we show the existence of a PBE (cid:63) in the subgame after the sellerchooses not to commit in the first period. It is then clear that there exists a PBE (cid:63) inthe whole game by backward induction: if the sum of monopoly revenues in the two pe-riods is higher than the revenue from the identified PBE (cid:63) of the subgame, let the sellercommit to the static monopoly prices, and otherwise let the seller choose not to commit.Moreover, we prove Part (i) in Theorem 2 which nests Proposition 1.Consider the following auxiliary complete information one-shot game parameterizedby p with 3 players: B , S A , S R . Player B chooses k ∈ [ θ , θ ] to maximize E [ { ( θ − p ) + ( θ − p A ) + } θ ≥ k | x = 1] + E [( θ − p R ) + θ ≤ k | x = 0]Player S A chooses p A ∈ [ θ , θ ] to maximize(1 − F ( p A | θ ≥ k, x = 1)) p A Player S R chooses p R ∈ [ θ , θ ] to maximize(1 − F ( p R | θ ≤ k, x = 0)) p R Note that this is a continuous game. By Glicksberg (1952), there exists a mixed-strategyNash equilibrium, denoted by measures ( η pB , η pA , η pR ).By Lemma 3 we know that Player B will not mix. Thus we can identify the equilib-rium strategy η pB by a cuto ff k p . For any fixed p , let E p be the set of equilibrium tuples( k p , π pA , π pR ), where π pA , π pR are equilibrium payo ff s of players S A and S R . We claim that E p is a closed and bounded set in R . Clearly, it is bounded. To show it is closed, takeany sequence ( k pn , π pA,n , π pR,n ) ∈ E p converging to ( k p , π pA , π pR ). Let ( η pB,n , η pA,n , η pR,n ) be the cor-responding equilibrium points. Note that this family is tight (each equilibrium point isviewed as a product measure on a compact subset of R ). Therefore, by Prokhorov’s The-orem, this family converges weakly along a subsequence to some ( η pB , η pA , η pR ). By Theorem2 in Milgrom and Weber (1985), ( η pB , η pA , η pR ) is an equilibrium point. Since the utilityfunctions are bounded and continuous, the corresponding tuple for this equilibrium is( k p , π pA , π pR ) and therefore ( k p , π pA , π pR ) ∈ E p . Now define the function G ( k, π A , π R ; p ) := ( p + π A )(1 − F ( k )) + π R F ( k )which is jointly continuous in ( k, π A , π R ) and hence attains maximum on the compact set E p . Let H ( p ) := max ( k,π A ,π R ) ∈E p G ( k, π A , π R ; p )We claim that H ( p ) is upper semi-continuous. To see that, fix any p . Let { p n } n convergingto p be a sequence along which we obtain lim sup p → p H ( p ). Let( k ∗ n , π ∗ A,n , π ∗ R,n ) ∈ argmax ( k,π A ,π R ) ∈E pn G ( k, π A , π R ; p n )Let ( η ∗ B,n , η ∗ A,n , η ∗ R,n ) be the corresponding equilibrium points. By the same argument as32efore, along a subsequence we know that ( η ∗ B,n , η ∗ A,n , η ∗ R,n ) converges weakly to some( η ∗ B, , η ∗ A , η ∗ R ), which by Milgrom and Weber (1985) is an equilibrium point of the gamewith parameter p . Then as before ( k ∗ n , π ∗ A,n , π ∗ R,n ) also converges (along the subsequence)to ( k ∗ , π ∗ A , π ∗ R ) given by the equilibrium ( η ∗ B, , η ∗ A , η ∗ R ). By continuity of G ,lim sup p → p H ( p ) = lim n →∞ G ( k ∗ n , π ∗ A,n , π ∗ R,n ; p n ) = G ( k ∗ , π ∗ A , π ∗ R ; p ) ≤ H ( p )where the last inequality is due to that ( k ∗ , π ∗ A , π ∗ R ) ∈ E p , proving the claim.Since H is upper semi-continuous, and the set [ − ( θ + θ ) , θ + θ ] is compact, thereexists a p ∗ that maximizes H ( p ). We now recover a PBE (cid:63) of the original game. Let theseller o ff er p ∗ in period 1. For any p o ff ered by the seller, let ( k p , π pA , π pR ) be a tuple in E p that maximizes G ( · ; p ). Let ( k p , η pA , η pR ) be the corresponding equilibrium point. Letthe buyer use strategy x ( θ , p ) = θ ≥ k p . Following history A, R , the seller o ff ers pricesaccording to the mixed strategy η pA , η pR , respectively. Her belief is simply µ A = F ( · | θ ≥ k p ) and µ R = F ( · | θ ≤ k p ). The buyer accepts in the second period if and only if his typeis above the price.By inspection, this is a PBE of the original game (use Lemma 3 for the buyer’s opti-mality). Note that the construction guarantees that whenever k = θ , η A maximizes theseller’s payo ff with respect to F ( · | θ = θ , x = 1) and whenever k = θ , η R maximizesthe seller’s payo ff with respect to F ( · | θ = θ , x = 0). Thus the PBE is in fact a PBE (cid:63) asin Definition 2. A.7 Proof of Part (i) in Theorem 3 As in Appendix A.6, we show that there exists a PBE (cid:63) of the subgame after the sellerchooses not to commit in the first period, which immediately implies that the whole gamealso has a PBE (cid:63) . Let us proceed in two steps. Step 1: We first show that the following game has a mixed-strategy PBE (cid:63) : for a fixed T ≥ 2, the seller posts a price p in period 1, and the buyer decides whether to accept.Following history A , R , the seller commits to a sequence of prices { p At } Tt =2 , { p Rt } Tt =2 respec-tively. The payo ff s and the evolution of buyer’s types are the same as in Section 4.2. Lemma 8. There exists a PBE (cid:63) of this game.Proof. Fix T ≥ 2, and consider the following auxiliary complete-information one-shotgame with 3 players: B , S A , S R . Player B chooses k ∈ [ θ , θ ] to maximize E (cid:20) { ( θ − p ) + T (cid:88) t =2 ( θ t − p At ) + } θ ≥ k (cid:21) + E (cid:20) T (cid:88) t =2 ( θ t − p Rt ) + θ ≤ k (cid:21) One can view ( η ∗ B,n , η ∗ A,n , η ∗ R,n , p n ) as a product measure on some compact subset of R , and the claimfollows by observing that the utility functions are bounded and jointly continuous in the actions and p . S A chooses { p At } Tt =2 ∈ T (cid:89) t =2 [ θ t , θ t ] to maximize T (cid:88) t =2 (1 − F t ( p At | θ ≥ k )) p At Player S R chooses { p Rt } Tt =2 ∈ T (cid:89) t =2 [ θ t , θ t ] to maximize T (cid:88) t =2 (1 − F t ( p Rt | θ ≤ k )) p Rt The rest of proof is virtually the same as in Appendix A.6 and thus omitted. Step 2: We now consider the original subgame and show the following by induction. Lemma 9. The subgame after the seller chooses not to commit has a PBE (cid:63) .Proof. Base case: When T = 1, the claim holds trivially. Inductive step: Using the objects in Lemma 8, we construct the strategies for thefirst two periods. Let the seller o ff er p ∗ in period 1. At any history, after the seller of-fers some price p in period 1, the buyer responds according to the threshold rule withthreshold k p . Then following history A, R , the seller o ff ers subsequent prices accordingto the distributional strategies η pA , η pR respectively, and her belief on ( θ , · · · , θ T ) will be F ( · | θ ≥ k p ) , F ( · | θ ≤ k p ) respectively. The buyer accepts the o ff er in period t wheneverthe price in period t is weakly below θ t for all 2 ≤ t ≤ T .To complete the construction, consider the o ff -path history where the seller choosesnot to commit in period 2. Note that the seller’s posterior belief about ( θ , · · · , θ T ) satisfiesAssumption A5 after any history ( p, x ) (since the buyer’s first-period strategy is a thresh-old rule). Thus, by the inductive hypothesis, there exists a PBE (cid:63) for the ( T − θ , · · · , ˜ θ T ) (conditionalon θ ≥ k p or θ ≤ k p ), after any history of ( p, x ) and the seller choosing not to commit.Use that PBE (cid:63) to complete the construction of the strategies for the seller and buyer.Given the seller’s strategy, all types of the buyer are playing optimally by construction.Does the seller have a profitable deviation? By part (ii) of Theorem 3 applying to the gamestarting from period 2, we know that after any history ( p, x ) the seller weakly preferscommitting to static pricing. Then by the construction of the objects in Lemma 8, theseller has no profitable deviation and has the desired o ff -path beliefs. This concludes theinductive step. 34 ppendix B D1 Criterion In this appendix, we provide a notion of D1 criterion in the spirit of Banks and Sobel(1987) for our game. Our goal is not to provide the right refinement for generic extensive-form games but to illustrate that a sensible refinement can preclude pathological caseslike Example 2. For clarity, in this appendix, we use PBE, PBE-D, PBE (cid:63) to refer tothe standard notion of PBE, PBE with D1 refinement (to be defined), and PBE with theparticular class of o ff -path beliefs as in Definition 2. B.1 Definition of D1 Criterion Suppose we are given a PBE. In any period t and at any buyer’s history ˆ h t , a price p t isposted by the seller. Type θ t = ( θ , · · · , θ t ) of buyer takes an action x t ∈ { , } . The sellersees this action, updates her belief and moves to the continuation game. Let U ∗ t ( θ t , ˆ h t )be type θ t ’s continuation utility in this PBE (including payo ff from this period). We willsuppress the dependency on ˆ h t whenever it is clear. For a given (potentially mixed) strat-egy α of the seller over the future plays in the continuation game, let U t ( θ t , α, ˆ h t ) denotethe continuation utility of type θ t best responding to α . As explained in Lemma 5, thedependency on θ t all reduces to dependency on θ t . The seller also only makes inferenceabout θ t from x t .Now fix any ˆ h t at which the PBE prescribes all types of the buyer to take the same ac-tion. Let µ t be the prior belief on θ t prescribed by the PBE for the seller before observingthe buyer’s action. Let x t be the o ff -path action that the buyer could take. Let BR ( Θ (cid:48) t , ˆ h t , x t )denote the set of possible future plays α that are best responses to some buyer’s strategy,under some belief ν t on θ t with the restriction that ν t is only supported on Θ (cid:48) t ⊆ supp( µ t ).Now denote D ( θ t , Θ (cid:48) t , x t ) := (cid:26) α ∈ BR ( Θ (cid:48) t , x t ) s.t. U ∗ t ( θ t ) < U t ( θ t , α, x t ) (cid:27) as the set of seller’s possible strategies for the future play under some belief concentratedon Θ (cid:48) t that makes type θ t strictly prefer x t to the equilibrium play. Similarly, define D ( θ t , Θ (cid:48) t , x t ) := (cid:26) α ∈ BR ( Θ (cid:48) t , x t ) s.t. U ∗ t ( θ t ) = U t ( θ t , α, x t ) (cid:27) Let ˆ Θ t = supp( µ t ). A type θ t is deleted if there is another type θ (cid:48) t ∈ ˆ Θ t such that (cid:20) D ( θ t , ˆ Θ t , x t ) ∪ D ( θ t , ˆ Θ t , x t ) (cid:21) ⊂ D ( θ (cid:48) t , ˆ Θ t , x t )Let Θ ∗ t ( x t ) be the set of surviving types until this process stops. We say that the PBE doesnot survive the D1 criterion if there exists some θ t ∈ Θ ∗ t ( x t ) such that U ∗ t ( θ t ) < inf α ∈A ( Θ ∗ t ( x t ) ,x t ) U t ( θ t , α, x t ) Cho (1987) extends the notion of the Intuitive Criterion to generic extensive-form games, but theIntuitive Criterion does not help much in our setting since we have a continuum of types. In the main text of the paper, we refer to PBE-D as PBE for the ease of notation. A ( Θ ∗ t ( x t ) , x t ) is the set of seller’s strategies for the future play that belong to a PBE-D of the continuation game with some prior belief about θ t supported only on Θ ∗ t ( x t ).Note that this definition is recursive but not circular since the continuation game is oneperiod shorter. In the last period, after the buyer’s choice, the seller no longer makes anydecision, this set is ∅ . Therefore, in the second to last period, this set is the set of seller’sstrategies that belongs to a standard PBE of the one period continuation game.We remark that in a signaling game, the above definition reduces to the usual defini-tion. This is because there is no continuation game. The set of mixed best responses BR ( · )no longer depends on what the sender does subsequently, and that the set of receiver’scontinuation equilibrium responses A ( · ) reduces to the set of mixed best responses. B.2 PBE-D and PBE (cid:63) In this section, we show that PBE-D are essentially PBE (cid:63) . This provides a foundationfor the particular class of o ff -path beliefs we focus on and confirms the intuition that theD1 criterion e ff ectively requires the seller to believe the buyer is of the type that wouldbenefit the most upon an unexpected deviation. Lemma 10. At any history ˆ h t , the set of D1 surviving types after an o ff -path rejection and ac-ceptance are given by Θ ∗ t (0) = { θ t ( µ t ) } , Θ ∗ t (1) = { θ t ( µ t ) } , where θ t ( µ t ) = min { supp ( µ t ) } , θ t ( µ t ) =max { supp ( µ t ) } and µ t is the seller’s belief on θ t at history h t .Proof. We prove the case for an unexpected o ff -path rejection. The other case follows bya symmetric argument. Note that the mixed best response set BR ( · ) for the seller containsall feasible future strategies since in the definition we allow for the buyer to use arbitrarystrategy for the continuation play. For any seller’s strategy α fixed, by Lemma 5, wehave that H ( θ t , α ) := U ∗ t ( θ t ) − U t ( θ t , α, θ t . This implies that for any ˆ Θ t andany θ (cid:48) t < θ t ∈ ˆ Θ t , (cid:20) D ( θ t , ˆ Θ t , ∪ D ( θ t , ˆ Θ t , (cid:21) ⊆ D ( θ (cid:48) t , ˆ Θ t , p . Note that H ( θ t , · ) is a continuous functionof ˆ p and for high enough H ( θ t , ˆ p ) = U ∗ t ( θ t ) ≥ 0. Moreover, for any ˆ p low enough (poten-tially negative), H ( θ t , ˆ p ) < 0. Therefore, there exists ˆ p ∗ ( θ t ) such that H ( θ t , ˆ p ∗ ( θ t )) = 0. Thisstrategy is in the set D ( θ t , ˆ Θ t , 0) for any θ t and ˆ Θ t . Fix any θ t and any θ (cid:48) t < θ t . Considerthe seller’s strategy of selling all future items at price ˆ p ∗ ( θ t ) + (cid:15) for (cid:15) > D ( θ t , ˆ Θ t , ∪ D ( θ t , ˆ Θ t , 0) by construction, but this strategy wouldbe in D ( θ (cid:48) t , ˆ Θ t , 0) for (cid:15) small enough. This is because H ( · , · ) is strictly increasing in the firstargument and continuous in the second. This argument then shows that for any ˆ Θ t and If the buyer rejects all future o ff ers, then any price path of the seller is in the BR ( · ) set. If the transition depends on x t as in Section 4.1, this follows from Lemma 3. θ (cid:48) t < θ t ∈ ˆ Θ t , (cid:20) D ( θ t , ˆ Θ t , ∪ D ( θ t , ˆ Θ t , (cid:21) ⊂ D ( θ (cid:48) t , ˆ Θ t , Θ ∗ t (0) = { θ t ( µ t ) } . Lemma 11. Every PBE (cid:63) is a PBE-D.Proof. We prove this by induction on T . Base case: When T = 1, these two concepts are both equivalent to standard PBE. Inductive step: Suppose for any ˜ T ≤ T − (cid:63) is a PBE-D. Consider a T -period game and fix a PBE (cid:63) . Fix any history ˆ h t where all types areprescribed to accept. Consider the o ff -path deviation x t = 0. Note that by Lemma 10, wehave Θ ∗ t (0) = θ t ( µ t ). By the definition of PBE, U ∗ t ( θ t ( µ t )) ≥ U t ( θ t ( µ t ) , ˜ σ ∗ , σ ∗ is the seller’s equilibrium continuation play. By the definition of PBE (cid:63) , ˜ σ ∗ ispart of a PBE (cid:63) for the continuation game under the belief concentrated on θ t ( µ t ). Sincethe continuation game has ≤ T − σ ∗ ∈ A ( θ t ( µ t ) , U ∗ t ( θ t ( µ t )) ≥ U t ( θ t ( µ t ) , ˜ σ ∗ , ≥ inf α ∈A ( θ t ( µ t ) , U t ( θ t ( µ t ) , α, (cid:63) passes the D1 test. The same argumentholds for an unexpected acceptance. This proves the inductive step. Lemma 12. For every PBE-D, there exists an outcome-equivalent PBE (cid:63) .Proof. We prove this by induction on T . Base case: When T = 1, these two concepts are both equivalent to standard PBE. Inductive step: Suppose for any ˜ T ≤ T − (cid:63) .Consider a T -period game and fix a PBE-D. Fix any history ˆ h t where all types areprescribed to accept. Consider the o ff -path deviation x t = 0. Note that by Lemma 10, wehave Θ ∗ t (0) = θ t ( µ t ). By the definition of PBE-D, U ∗ t ( θ t ( µ t )) ≥ inf α ∈A ( θ t ( µ t ) , U t ( θ t ( µ t ) , α, ≤ T − α ∈A ( θ t ( µ t ) , U t ( θ t ( µ t ) , α, 0) = inf β ∈B ( θ t ( µ t ) , U t ( θ t ( µ t ) , β, B ( θ t ( µ t ) , 0) is the collection of seller’s strategies that belong to a PBE (cid:63) of the con-tinuation game that is outcome-equivalent to a PBE-D in A ( θ t ( µ t ) , (cid:63) . It is also easyto check that the induction proof of part (ii) of Theorem 3 can also be done for PBE (cid:63) without referring to PBE-D. This implies that the PBE (cid:63) corresponding to B ( θ t ( µ t ) , 0) are37 ll outcome-equivalent. The existence of such a PBE (cid:63) follows from the proof in Ap-pendix A.6 and Appendix A.7 (and thus B ( θ t ( µ t ) , 0) and A ( θ t ( µ t ) , 0) are non-empty). Nowfix any such PBE (cid:63) of the continuation game. Replace the continuation strategy profilegiven in the original PBE-D by this PBE (cid:63) , with the seller’s o ff -path belief assigned to beconcentrated on θ t ( µ t ). We claim that the resulting strategy profile for the whole game isstill a PBE. To see this, note that by construction this replacement leaves θ t ( µ t ) a payo ff of inf α ∈A ( θ t ( µ t ) , U t ( θ t ( µ t ) , α, ≤ U ∗ t ( θ t ( µ t )). So it remains optimal for type θ t ( µ t ) to playthe prescribed strategy of acceptance. By the threshold-rule results, it is then optimalfor all types above θ t ( µ t ) to continue accepting the o ff er. Now consider one layer abovewhere the seller is choosing what price to o ff er in period t . Because this replacement onlyhappens at an o ff -path history and the buyer uses the same threshold rule as before forthis period, it is optimal for the seller to follow the originally prescribed strategy as well.It then follows that after this replacement the strategy profile is still a PBE of the wholegame, and has the same equilibrium outcome as the original PBE-D. The same procedurealso works for any history ˆ h t at which all types of the buyer are prescribed to reject.Now for each period t = 1 , · · · , T , for each ˆ h t , iteratively apply the above procedurewhenever (1) all types of the buyer are prescribed to accept or reject at ˆ h t and (2) theassigned o ff -path belief is not already concentrated on θ t ( µ t ) or θ t ( µ t ). It is evident thatwhen the process stops, we have found a PBE (cid:63)(cid:63)