Information Design in the Hold-up Problem

Information Design in the Hold-up Problem Daniele Condorelli and Balázs Szentes May 4, 217 Abstract We analyze a bilateral trade model where the buyer can choose a cumulative distribution function (CDF) from a set of distributions supported on [, 1]. This CDF then determines her valuation. The seller, after observing the buyer s choice of the CDF but not its realization, makes a take-it-or-leave-it offer. If the buyer can choose any CDF, the price and the payoffs of both the buyer and the seller are shown to be 1/e in the unique equilibrium outcome. The equilibrium CDF of the buyer generates a unit-elastic demand and trade takes place with probability one. We consider various extensions by introducing additional constraints on the buyer s choice set and examine the robustness of our basic results. For example, we analyse models where the buyer s CDF must satisfy finitely many general moment restrictions. We also study scenarios where the buyer s value distribution is given but she can reshape it by either adding risk or by destroying values. In all these cases, the buyer s equilibrium demand is shown to be unit-elastic and trading is efficient. We have benefited from discussions with Dirk Bergemann, Ben Brooks, Jeremy Bulow, Sylvain Chassang, Eddie Dekel, Andrew Ellis, Jeff Ely, Faruk Gul, Yingni Guo, Sergiu Hart, Johannes Horner, Francesco Nava, Santiago Oliveros, Ludovic Renou and Phil Reny. The comments of several seminar audiences are also gratefully aknowledged. Department of Economics, University of Essex, Colchester, UK. E-mail: d.condorelli@gmail.com. Department of Economics, London School of Economics, London, UK. E-mail: b.szentes@lse.ac.uk. 1

1 Introduction A central result in Information Economics is that the distribution of economic agents private information is a key determinant of their welfare. In auctions, for example, asymmetric information between the seller and the bidders regarding bidders valuations affects all agents payoffs. In the context of labour markets, asymmetric information between the employers and workers regarding workers productivities affects workers wages. When analysing environments with asymmetric information, most microeconomic models assume that the distribution of types is exogenous. However, if an agent s payoff depends on her information rent then she is likely to take actions in order to generate this rent optimally. This paper contributes towards this problem by reconsidering the hold-up problem from the perspective of optimal information design. The literature on hold-up typically focuses on environments where the buyer s investment decision has a deterministic impact on her valuation for the seller s good. If the seller has full bargaining power, he appropriates all the surplus generated by the investment. Hence, if investment is costly, the buyer does not invest at all and ends up with a payoff of zero. In contrast, if investments have privately observed stochastic returns, the buyer s investment decision induces asymmetric information at the contracting stage. Therefore, the buyer may capture some of the surplus through receiving information rent which, in turn, could partially restore her incentives to invest on the first place. Our objective is to analyze the buyer s problem of optimally designing the stochastic returns with various constraints on the available distributions. For an example, consider a worker s decision to invest in human capital. Prior to going to the job market, an aspiring freelancer chooses the type and length of her education. This decision is observed by future employers but it is conceivable that the freelancer has private information regarding her productivity generated by her education. When the freelancer is about to be hired for a project, the offered contract can depend on her education but not on her actual productivity. Of course, if the choice of education perfectly reveals the worker s productivity, the employer s optimal contract might hold down the worker to her reservation payoff. Therefore, this freelancer may chose an education which is only a noisy signal of her productivity. In our baseline model, we do not impose any restriction on the set of distributions, except that the maximum valuation is bounded. Specifically, the buyer can choose any CDF supported on [, 1] which then determines her valuation for the object. After observing the buyer s CDF, the seller sets a price at which either the buyer trades or the game ends. The central trade-off arising in this model can be explained as follows. If the price of the good was given, the buyer would be better off the higher is her valuation. However, the seller who knows that the buyer s valuation is 2

likely to be high will charge a higher price. As a result, when choosing her value-distribution, the buyer faces a trade-off between the higher payoff she would receive from the good and the higher price she would have to pay. We show that this game has a unique equilibrium outcome. In each equilibrium, the price is 1/e and the buyer s distribution is supported on [1/e, 1], so that trade always occurs and the seller s payoff is 1/e. The buyer s value-distribution is a combination of a continuous distribution on [1/e, 1) defined by the CDF F (v) = 1 1/ (ev) and an atom of size 1/e at v = 1. The total equilibrium surplus is found to be shared equally between the seller and the buyer, so that the buyer also receives an equilibrium payoff of 1/e. The efficiency loss that results from the buyer s desire to generate information rent is found to be around one quarter of the first-best social surplus. The demand function generated by F maximizes consumer surplus in a monopoly market where the maximum willingness-to-pay cannot exceed one and production is costless. This is because each subgame induced by the buyer s CDF is equivalent to the problem of a monopolist with zero marginal cost who faces a unit mass of consumers whose aggregate demand is generated by the CDF. As a result, the buyer s problem of choosing the payoff-maximizing CDF is identical to the problem of identifying the demand function which maximizes consumer surplus in this market. In Section 7, we also characterize the demand curve which maximizes consumer surplus in a market where many firms are competing a la Cournot. Our baseline model provides a rationale for the endogenous emergence of unit-elastic demands. Indeed, the equilibrium CDF of the buyer, F, generates a unit-elastic demand function on (1/e, 1), such that the probability that the buyer is willing to buy the good at price p is 1 F (p) = 1/ (ep). When faced with this demand function, the seller is indifferent between charging any price in [1/e, 1], since any price in this range will result in an expected profit of p [1/ (ep)] = 1/e. We demonstrate that this result arises because the buyer can always increase her payoff by choosing a different CDF unless the seller is indifferent between any price on the support of the buyer s CDF. If the latter does not hold, the buyer can move probability mass from suboptimal prices to higher valuations without changing the seller s incentive to set a certain price. Our second main insight is that the equilibrium outcome is ex-post efficient, i.e., trade occurs with probability one. As we explained above, given the buyer s equilibrium CDF F, the seller finds it optimal to set the price at the lowest possible valuation of the buyer. In contrast, if the buyer s valuation was given exogenously, the seller would often find it optimal to set a price which excludes low-value buyers from trade. However, if a distribution induces a price above the smallest valuation on the support, the buyer can increase her payoff by choosing the same distribution conditional on the value being higher than the price. The reason is that, conditional on trade, the buyer s payoff 3

is the same but she trades more often if she chooses the conditional distribution. We consider various extensions of our benchmark model and examine the robustness of our results. Perhaps most importantly, we analyse scenarios where the set of distributions available for the buyer is restricted. First, we assume that these distributions must satisfy general moment restrictions, e.g. the buyer s expected valuation is bounded. Second, we characterize equilibria in models where the buyer starts with a prior value distribution but she can take certain actions in order to reshape this prior. For example, the buyer can add arbitrary risk so her final distribution will be a mean preserving spread of the prior CDF. We also consider cases where the buyer can stochastically destruct value, that is, her chosen distribution must be first-order stochastically dominated by the original distribution. We then introduce a cost associated to the buyer s choice of a CDF. We show that, in most of these cases, the buyer s demand is unit elastic and the trading decisions are ex post efficient. Finally, we discuss how the equilibria of our baseline model are modified if there is a positive production cost, the seller does not have full bargaining power and the buyer s choice is unobservable. There are papers on the hold-up problem which also consider the buyer s ability to generate asymmetric information. In the model by Gul (21) and Lau (28), the buyer can generate asymmetric information by randomizing over imperfectly observable investment decisions. Gul (21) shows that if investment is unobservable, the hold-up problem is as severe as in the fully observable case. In the unique mixed-strategy equilibrium, the efficiency gain from the buyer s randomization over positive investments is fully offset by the ex-post efficiency loss due to bargaining disagreement. Lau (28) assumes that the seller observes the buyer s investment perfectly with a given probability, and receives an uninformative signal otherwise. Her main result is that efficiency is maximized when this probability is strictly between zero and one. Nevertheless, the buyer does not earn information rent and ends up with a payoff of zero. Hermalin and Katz (29) consider the hold-up problem incorporating asymmetric information after the buyer s investment decision. In particular, a larger investment results in a first-order stochastic shift in the buyer s value distribution. The authors consider several scenarios in which the degree of observability of the investment is varied and find that the buyer is made worse off if her investment is unobservable. 1 Similarly, we demonstrate that in our model, the buyer s equilibrium payoff is zero if the seller is unable to observe the buyer s CDF. Finally, Skrzypacz (24) also emphasizes, as we do, that the desire to generate information rent can shape the buyer s incentives to invest. However, the author s analysis focuses on dynamic bargaining, where Coasian dynamics imply that the buyer 1 The consequences of the observability assumption in hold-up problems were also explored in Tirole (1986) and Gul (21) under various bargaining protocols. 4

obtains full bargaining power as the discount factor vanishes. Unit-elastic demands also appears in a literature on non-bayesian monopoly pricing. Bergemann and Schlag (28) consider a monopolist with the min-max regret criterion. The seller s ex-post regret is defined as the buyer s payoff if a trade occurs and the buyer s valuation otherwise. The authors argue that the optimal pricing policy coincides with the seller s equilibrium strategy in a zero-sum game played against nature in which nature chooses the buyer s valuation to maximize the seller s regret. The authors show that nature s equilibrium strategy in this case is the same as our equilibrium CDF and the seller fully randomizes on its support. 2 Neeman (23) considers second-price auctions with private values and addresses the problem of finding the value-distribution which minimizes the ratio between the seller s profit and the expected value. The author shows that the solution generates a unit-elastic demand function. Unit-elastic demands also play a role in models where randomization by the seller is required for the existence of an equilibrium, since this form of demand function is required to ensure that the seller is indifferent on the support of valuations. Renou and Schlag (21) consider models with min-max regret and imperfect competition and Hart and Nisan (212) approximate the seller s maximum revenue in a multiple-item auction. 3 (215). Related results also appear in Brooks (213) and in Kremer and Snyder Finally, our paper also relates to a recent literature on information design, see for example Kamenica and Gentzkow (211). Bergemann and Pesendorfer (27) consider the seller s problem of designing the information structure to determine how buyers learn about their valuations prior to participating in an auction. Bergemann, Brooks and Morris (215) analyze a model where the buyer s value distribution is given and the seller receives a signal about this value. The authors characterize the entire set of payoff outcomes that can arise from some signal structure. Unit-elastic demand plays a crucial role in their construction. Roesler and Szentes (215) analyze a setup where the buyer has a given value distribution and designs a signal structure to learn about her valuation. After observing the signal structure, the seller sets a price and the buyer trades if the expected valuation conditional on her signal exceeds the price. The buyer in Roesler and Szentes (215) faces a trade-off between signal precision improving the efficiency of her purchase decision for a given price, but potentially increasing the price since the buyer s demand is determined by her signal. 2 Since the seller s regret is the buyer s payoff if there is trade, it is not a coincidence that nature chooses the CDF which maximizes the buyer s expected surplus. 3 In a different context, Ortner and Chassang (214) show that, in order to eliminate collusion between an agent and the monitor, the principal benefits from introducing asymmetric information between the agent and the monitor by making the monitor s wage random. The optimal wage scheme is determined by a distribution similar to our equilibrium CDF. 5

Roesler and Szentes (215) show that the latter determines the buyer s equilibrium signal structure so that the equilibrium price is the lowest across all signal structures and the buyer always trades. As in our equilibrium, the seller is indifferent between any prices in the signal s support. 2 Model There is a seller who has an object to sell to a single buyer. Prior to interacting with the seller, the buyer can choose the distribution of her valuation subject to the constraint that the valuation is below one. Formally, the buyer can choose any F F where F is the set of CDFs supported on a subset of [, 1]. The seller observes the choice of the buyer and gives a take-it-or-leave-it price offer to the buyer, p. Finally, the buyer s valuation, v, is realized and she trades with the seller if and only if v p. 4 If trade takes place, the payoff of the seller is p and that of the buyer is v p; both receive a payoff of zero otherwise. The seller and the buyer are expected payoff maximizers. We restrict attention to Subgame Perfect Nash Equilibria of this game. We introduce a few pieces of notation. For each F F and p R, let D (F, p) denote the demand at price p generated by F, that is, the probability of trade. Observe that D (F, p) = 1 F (p)+ (F, p), where (F, v) denotes the probability of v according to F. 5 If the buyer chooses a CDF F F, then the seller s profit is Π (F ) = max p pd (F, p). 6 The seller might be indifferent between charging different prices which result in different payoffs to the buyer. For each F F, let P (F ) denote the set of profit maximizing prices, that is, P (F ) = arg max p pd (F, p). Finally, let U (F, p) denote the buyer s payoff if the seller sets price p, that is, U (F, p) = p v pdf (v). 3 Main result We solve our problem in two steps. First, for a given profit of the seller π, we compute the CDF F π F which maximizes the buyer s payoff subject to the constraint that the seller s profit is π. This step essentially reduces our search for an equilibrium to a one-dimensional problem, since the buyer s equilibrium CDF must be in the set {F π } π [,1]. In the second step, we characterize the profit π and the corresponding CDF F π which maximizes the buyer s payoff. To this end, we next define the CDF F π and show that it maximizes the buyer s payoff subject to the constraint that 4 Assuming that the buyer is non-strategic in the final stage of the game and buys the object whenever her valuation weakly exceeds the price has no effect on our results but makes the analysis simpler. 5 That is, (F, v) = F (v) sup z<v F (z). If F does not specify an atom at v then (F, v) =. 6 This maximum is well-defined because the buyer trades when she is indifferent. 6

the seller s profit is π. Equal-revenue distributions. For each π [, 1], let F π F be defined as follows: if v < π, F π (v) = 1 π v if v [π, 1), 1 if v 1. (1) Since F π (π) =, the valuation of the buyer is never below π. The function F π is continuous and strictly increasing on [π, 1). On this interval, F π is defined by the decreasing density f π (v) = π/v 2. Finally, there is an atom of size π at v = 1, that is, (F π, 1) = π. Note that the seller is indifferent between setting any price on the support of F π, that is, P (F π ) = [π, 1]. To see this, suppose first that the seller sets price p [π, 1). Then the seller s payoff is pd (F π, p) = p [1 F π (p)] = p (π/p) = π. If the seller sets a price of one then his payoff is (F π, 1) = π. Therefore, the seller s payoff is π as long as he sets a price in [π, 1], that is, Π (F π ) = π. As a consequence these distributions exhibit unit-elastic demand, and are also known as equal revenue distributions. Lemma 1 Suppose that G F and p P (G) and let π denote Π (G). (i) Then F π first-order stochastically dominates G, that is, for all v [, 1] F π (v) G (v). (2) (ii) Furthermore, U (F π, π) U (G, p) and the inequality is strict if F π G. Part (i) of the lemma implies that the expected value induced by F π is larger than that induced by G. Recall that π P (F π ), that is, the seller finds it optimal to set price π in the subgame generated by F π. In addition, Π (G) = π = Π (F π ). Therefore, part (ii) implies that the maximum payoff the buyer can achieve subject to the constraint that the seller s profit is π is U (F π, π). Proof. To prove part (i), note that since p P (G), it must be that for all v [, 1], or equivalently, 1 vd (G, v) pd (G, p). pd (G, p) v + (G, v) G (v). (3) Since (G, p) [, 1] and π = pd (G, p), the previous inequality implies (2). 7

To see part (ii), note that U (F π, π) = π v πdf π (v) π v πdg (v) p v pdg (v) = U (G, p), where the first inequality follows from F π first-order stochastically dominating G and the second one from π = pd (G, p) p. In addition, the first inequality is strict unless F π = G (see Proposition 6.D.1 of Mas-Colell et al (1995)). Figure 1: Improving on the Uniform Distribution demand 1 D(Fπ, v) π p 1 D(G, v) price Figure 1 illustrates the statement and the proof of Lemma 1 for the case when G is the uniform distribution, that is, G (v) = v on [, 1]. In this case, D (G, v) = 1 v, the seller s optimal price, p, is 1/2 and the seller s profit, π, is 1/4. The consumer surplus, U (G, 1/2), is the integral of the demand curve on [p, 1], represented by the darker shaded triangle. If the buyer s valuation is determined by the CDF F π, the seller s profit is π irrespective of which price he charges on the interval [1/4, 1]. This implies that D (F π, p) = D (G, p) and D (F π, v) > D (G, v) at v p. In other words, the demand curve generated by F π is above that of G and the two curves are tangent to each other s at p. This explains part (i) of the lemma. To see part (ii), note that switching from G to F π increases the buyer s payoff for two reasons. First, since D (F π, v) > D (G, v), the integral of the new demand curve on [p, 1] goes up. Second, if the seller charges π instead of p, the consumer 8

surplus is the integral of the demand on a larger range than before. The increase in the buyer s payoff from moving (G, p) to (F π, π) is the lighter shaded area on the figure. Let F and p denote the equilibrium CDF of the buyer and the on-path equilibrium price of the seller, respectively. We are now ready to state our main result. Theorem 1 In the unique equilibrium outcome, F = F 1/e and p = U (F, p ) = Π (F ) = 1/e. The seller s equilibrium strategy is not determined uniquely off the equilibrium path. He might charge different prices if he is indifferent after observing out-of-equilibrium CDFs. Proof. First, we show that there is an equilibrium in which the buyer chooses F 1/e and the seller responds by setting price 1/e. Recall that P (F π ) = [π, 1] for all π, so the seller optimally charges the price 1/e in the subgame generated by F 1/e. Next, we argue that the buyer has no incentive to deviate either. If the buyer does not deviate, her payoff is U ( F 1/e, 1/e ). By part (ii) of Lemma 1, any deviation payoff of the buyer is weakly smaller than U (F π, π) for some π [, 1]. So, it is sufficient to show that U (F π, π) is maximized at π = 1/e. Note that U (F π, π) = π v πdf π (v) = π vf π (v) dv + (F π, 1) π = π π dv = π log π, (4) v where the second equality follows from f π (v) = π/v 2 and (F π, 1) = π. The function π log π is indeed maximized at 1/e. From the equality chain (4), the buyer s payoff is U ( F 1/e, 1/e ) = 1/e log 1/e = 1/e in this equilibrium. Finally, note that Π ( F 1/e ) = 1/e. It remains to prove the uniqueness of the equilibrium outcome. Part (ii) of Lemma 1 implies that the equilibrium payoff of the buyer, U (F, p ), is weakly smaller than U ( F Π(F ), Π (F ) ). We showed that U ( F Π(F ), Π (F ) ) U ( F 1/e, 1/e ) in the previous paragraph, so the buyer s payoff cannot exceed 1/e in any equilibrium. Since 1/e is the unique maximizer of U (F π, π), part (ii) of Lemma 1 also implies that in order for the buyer to achieve this payoff, she must choose F 1/e and the seller must charge 1/e. Therefore, to establish uniqueness, it is sufficient to show that the buyer s equilibrium payoff is at least 1/e. To this end, for each ε (, 1/e), let F ε 1/e be ε + F 1/e on [p, 1) and F 1/e otherwise. The CDF F ε 1/e is constructed from F 1/e by moving a mass of size ε from the atom at v = 1 to v = 1/e. If the buyer chooses F1/e ε then the seller strictly prefers to set price 1/e. 7 In addition, the buyer s payoff is ε-close to U ( F 1/e, 1/e ) = 1/e. Therefore, the buyer can achieve a payoff arbitrarily close to 1/e irrespective of the seller s strategy to resolve ties and hence the buyer s equilibrium payoff cannot be smaller than 1/e. 7 Note that, by setting price 1/e, the seller achieves the same payoff as if the buyer chose F 1/e. For any price in (1/e, 1], the seller s payoff is strictly smaller than after the choice of F 1/e because the probability of trade is smaller by ε. 9

From the proof of Theorem 1 it follows that the buyer s expected valuation generated by F 1/e is 2/e. Indeed, from (4), 1/e vdf 1/e (v) = 1/e vf 1/e (v) dv + ( F 1/e, 1 ) = 1/e 1 ev dv + 1 e = 2 e. Note that efficiency requires the buyer to choose a distribution which would specify v = 1 with probability one, and the seller to quote a price which is weakly less than one. The total surplus would be one. In contrast, the total surplus in equilibrium is only U ( F 1/e, 1/e ) + Π ( F 1/e ) = 2/e. Hence, the efficiency loss due to the buyer s desire to generate information rent is 1 2/e.26. As mentioned earlier, requiring the upper bound of the buyer s CDF s support to equal one is just a normalization. If this upper bound is k then the equilibrium price, the payoff of the buyer and the payoff of the seller are k/e. As k goes to infinity, the payoffs also converge to infinity. 4 Moment Restrictions and Costly Investment In our baseline model, the only constraint faced by the buyer is that the upper bound of the valuation-support must be smaller than one. This section considers various extensions of our game by imposing moment restrictions on the set of distributions from which the buyer can choose. First, we consider the case where the expectation of the buyer s CDF is bounded. Then we analyse environments where the choice set of the buyer is the set of those CDFs which satisfy finitely many arbitrary moment restrictions. We demonstrate that, in all these situations, trade is efficient and the seller is indifferent between setting any price on the support of the buyer s equilibrium CDF. 4.1 Bounded Expectation In this section, we assume that the buyer is perfectly flexible at choosing the riskiness of her value distribution but has limited ability to influence its expectation. To be more precise, suppose that the buyer faces an additional constraint when choosing the distribution F, namely that the expected value must lie in the interval I = [µ, µ] (, 1], that is, the buyer s choice set is F I = { F F : } vdf (v) I. Of course, if 2/e I then the new constraint does not bind and the unique equilibrium outcome is identical to the one identified by Theorem 1. Next, we show that, if 2/e / I, the equilibrium CDF of the buyer is going to be again an equal-revenue distribution which generates an expected value of either µ or µ, depending which of these values is closer to 2/e. To this end, 1

we first argue that the range of means induced by the class of equal-revenue distributions defined by (1) is the interval (, 1]. Recall from the proof of Theorem 1 that the expected value of the equal-revenue distribution F π is π log π + π. This function converges to zero as π goes to zero, its value is one at π = 1 and it is continuous and strictly increasing on (, 1]. Therefore, for each µ (, 1] there exists a unique π µ (, 1] such that the expectation of F πµ is µ, that is, π µ log π µ + π µ = µ. We now characterize the equilibrium CDF of the buyer, F I, and the equilibrium price, p I. Proposition 1 If the buyer s action space is F I then, in the unique equilibrium outcome, F I = F π µ and p I = π µ where µ if µ < 2/e, µ = 2/e if 2/e I, µ if 2/e < µ. Since the seller charges π µ, trade always occurs and U ( F πµ, π µ ) = µ πµ. Proof. We only show that there exists an equilibrium outcome with the properties described in the statement of the theorem. The uniqueness proof is essentially identical to that in the proof of Theorem 1. If 2e I, the statement of the proposition follows from Theorem 1. ) If µ 2/e then µ = µ, so we need to show that (F πµ, π µ constitutes an equilibrium. If the ) [ ] buyer chooses F πµ, then the seller optimally responds by setting price π µ since P (F πµ = π µ, 1. We have to show that the buyer has no incentive to deviate. Let G F I, p P (G) and let π ) denote Π (G). It is enough to show that U (G, p) U (F πµ, π πµ. First, observe that vdf πµ (v) = µ vdg (v) vdf π (v), where the equality is implied by the definition of π µ, the first inequality follows from G F I and the second one follows from part (i) of Lemma 1. Since the expectation of F π is increasing in π, the previous inequality chain implies that π µ π. Finally, note that U (G, p) U (F π, π) U (F πµ, π πµ ), where the first inequality follows from part (ii) of Lemma 1. The second inequality follows from the observations that 1/e π µ (since 2/e µ) and that U (F π, π) is strictly decreasing in π on [1/e, 1] (see (4)). 11

Suppose now that µ < 2/e. If the buyer chooses F πµ, then the seller optimally responds by setting price π µ since P ( F πµ ) = [πµ, 1]. We have to show that the buyer has no incentive to deviate. Let G F I, p P (G) and let π denote Π (G). We show that U (G, p) U ( F πµ, π µ ). Note that U (G, p) = p v pdg (v) = p vdg (v) pd (G, p) µ π, where the inequality follows from G F I and π = pd (G, p). If π π µ then this inequality chain implies U (G, p) U ( F πµ, π µ ) because U ( Fπµ, π µ ) = µ πµ. If π π µ then U (G, p) U (F π, π) U ( F πµ, π µ ), where the first inequality follows from part (ii) of Lemma 1. The second inequality follows from the observations that π µ < 1/e (since 2/e µ) and that U (F π, π) is strictly increasing in π on (, 1/e] (see (4)). In the special case where µ = µ = µ, the previous proposition implies that the CDF F πµ optimal for the buyer among the CDFs with mean µ. Next, we show that F πµ is is also the CDF which minimizes the seller s profit among those CDFs which generate an expected valuation of exactly µ. Remark 1 If vdg (v) = µ then Π (G) Π ( F πµ ) = πµ. This result is also reported in Neeman (23) and Kremer and Snyder (216). Proof. Let π denote Π (G). Note that vdf π (v) vdg (v) = µ = vdf πµ (v), where the inequality follows from part (i) of Lemma 1, the first equality is the hypothesis of this remark, and the second equality follows from the definition of π µ. Since vdf p (v) is strictly increasing in p, the previous inequality chain implies that π π µ. Again, if the upper bound of the CDF s support is required to be k instead of one, the equilibrium price and the profit would be kπµ/k and the equilibrium payoff of the buyer would be ( ) ku F π µ/k, πµ/k. What happens if k goes to infinity for a fixed µ? Remark 2 For all µ R ++, lim k ku ( ) F π µ/k, πµ/k = µ and lim k kπµ/k =. 12

This remark implies that if the upper bound k can be arbitrarily large, the buyer extracts the full surplus, µ, by choosing a CDF which induces a price arbitrarily close to zero. Proof. By Proposition 1, πµ/k log π µ/k + π µ/k = µ/k or equivalently, log π µ/k + 1 = µ π µ/k k. Since πµ/k goes to zero as k goes to infinity, the left-hand side converges to infinity. Therefore, the right-hand side must also go to infinity. Hence, the price kπµ/k must converge to zero, which ( ) implies that the seller s payoff goes to zero and the buyer s payoff, ku F π µ/k, πµ/k, goes to µ. 4.2 General Moment and Variance Restrictions In this section, we assume that the buyer faces constraints on the variance and general moments of her value distribution. Previously, we have shown that if the buyer faces only constraints regarding the expectation of her valuation, she still chooses an equal-revenue distribution, F π, in equilibrium. Recall that the upper bound of the support of this distribution is the largest possible valuation, which we normalized to be one. In addition, the CDF F π specifies an atom of size π at one. Therefore, F π generates relatively large higher moments. Furthermore, if π is small then F π also has a sizable variance. One way to tailor an equal-revenue distribution F π to reduce its variance and higher moments is to truncate it at a threshold value which is smaller than one, that is, to move all the probability mass from above the threshold to the threshold itself. Below, we show that the buyer s equilibrium distribution will be in this class of truncated equal-revenue distributions. In what follows, we introduce the additional constraints and formally define the action space of the buyer. With a slight abuse of notations, for each F F, let V ar (F ) denote the variance of the random variable which is distributed according to the CDF F, that is, V ar (F ) = (v µ) 2 df (v), where µ denotes the expectation of F, that is, vdf (v). Next, we introduce general moment restrictions. To this end, let α i : [, 1] [, 1] be a continuous, strictly increasing, and convex function for all i = 1,..., n. We normalize α i () to be zero and α i (1) to be one. We refer to the function α i as a moment function. In addition, let γ i (, 1) for i = 1,..., n. The CDF F F satisfies the ith moment restriction if α i (v) df (v) γ i. 13

For example, if α i (v) = v i for i N then the ith constraint is a restriction on the ith moment of the distribution. In this section, the buyer is required to choose a CDF which satisfies all the moment restriction and a constraint on the variance of the distribution. To be more precise, the choice set of the buyer is { F α = F F : V ar (F ) σ 2 and } α i (v) df (v) γ i for all i = 1,..., n. The special case where the variance is not restricted (σ 2 = ) and there is only a mean constraint, i.e., n = 1 and α 1 (v) = v, was discussed in the previous section. Next, we define a set of distributions and show that the buyer s equilibrium choice is in this class. For each π [, 1] and B [π, 1] let if v [, π), Fπ B (v) = 1 π v if v [π, B), 1 if v [π, 1]. Note that the support of F B π is [π, B] and it specifies an atom of size π/b at B. Again, the seller is indifferent between charging any price on its support. Notice that F π π which specifies an atom of size one at π and F 1 π = F π. is a degenerate distribution The next lemma states that for each CDF G there is an element in this class which secondorder stochastically dominates G and makes the buyer weakly better off while generating the same profit to the seller as G. Lemma 2 Suppose that G F, p P (G) and let π denote pd (G, p). Then there exists a unique B [π, 1] such that (i) G is a mean-preserving spread of Fπ B and (ii) U (G, p) U ( Fπ B, π ). Proof. See Roesler and Szentes (216). We now characterize the equilibrium CDF of the buyer, F α, and the equilibrium price, p α. Proposition 2 Suppose that the buyer s action space is F α. outcome, the buyer chooses F α = F B π and the seller sets price p α = π where Then, in the unique equilibrium (π, B { ( ) = arg max U F B π, π ) : F B } π F α. (5) π (,1) B [π,1] 14

Note that the buyer s equilibrium CDF, Fπ B, again generates a unit elastic demand. In addition, since the seller charges price π, trade occurs with probability one. Proof. We only show that there exists an equilibrium outcome with the properties described in the statement of the theorem. The uniqueness proof is essentially identical to that in the proof of Theorem 1. If the buyer chooses F B π then the seller optimally responds by setting price π since P (F π ) = [π, 1]. We have to show that the buyer has no incentive to deviate. Let G F α, p P (G) and let π denote Π (G). It is enough to show that U (G, p) U ( Fπ B, π ). By part (i) of Lemma 2, there exists a B such that G is a mean-preserving spread of Fπ B. Next, we show that Fπ B F α. First, note that V ar ( Fπ B ) V ar (G) σ 2, where the first inequality holds because G is a mean-preserving spread of F B π and the second one follows from G F α. In addition, since α i is convex and G is a mean-preserving spread of Fπ B, it follows that for each i = 1,..., n α i (v) df B π (v) α i (v) dg (v), see Proposition 6.D.2 of Mas-Colell et al (1995)). Furthermore, since G F α, α i (v) dg (v) γ i. The previous three displayed inequalities imply that F B π F α. Finally, note that U (G, p) U ( F B π, π ) U ( F B π, π ), where the first inequality follows from part (ii) of Lemma 2 and the second one from F B π F α and (5). This inequality means that choosing G instead of F B π is not a profitable deviation. 5 Costly Investment In this section, we characterize equilibria in models where the buyer s investment is costly. We show that the buyer s problem of choosing a costly CDF is the dual of her problem when choosing a costless CDF with a single moment restriction. Then we use the arguments developed in the previous section to show that the equilibrium CDF of the buyer again generates a unit elastic demand. 15

We now assume that the buyer has to incur an additive cost C (F ) if she chooses F. That is, if the buyer chooses F and her valuation is v then her payoff is v p C (F ) is she trades at price p and C (F ) otherwise. We assume that each particular valuation has a deterministic cost and the cost of a CDF F is the expected cost generated by F. 8 To be more precise, let c : R + R + be an increasing, convex function such that c () = and c (1) = 1. Then the cost of F is given by C (F ) = c (v) df (v). (6) Since c (1) = 1, the buyer only chooses CDFs supported on [, 1]. The problem of the buyer here is the dual of her problem considered in the previous section with a single moment restriction in the following sense. Suppose that the buyer s optimal choice is G. Then G must remain optimal even if investment is costless but the buyer is restricted to choose from the set of those distributions which are cheaper than G, that is, { } F C = F F : c (v) df (v) C (G). To see this, note that if the buyer s payoff induced by a CDF F F C was higher than that by G then F would generate a higher payoff to the buyer even if the cost is taken into account because F is cheaper than G. Observe that the buyer s problem of choosing from F C at no cost is analyzed in the previous section. Indeed, the constraint defining F C is a convex moment restriction with α 1 c and γ 1 = C (G). Proposition 2 then suggests that the equilibrium choice of the buyer is a truncated equal-revenue distribution. Proposition 3 Suppose that the cost of each CDF F F is given by (6). Then, in the unique equilibrium outcome, the buyer chooses Fπ B sets price p = π. for some π (, 1) and B (π, 1] and the seller (π, B { ( ) = arg max U F B π, π ) C ( F B ) } π : F B π F α. (7) π (,1) B [π,1] Proof. We only show that there exists an equilibrium outcome with the properties described in the statement of the theorem. The uniqueness proof is essentially identical to that in the proof of Theorem 1. If the buyer chooses F B π then the seller optimally responds by setting price π since P (F π ) = [π, 1]. We have to show that the buyer has no incentive to deviate. Let G F α, p P (G) and let π denote Π (G). It is enough to show that U (G, p) C (G) U ( F B π, π ) C ( F B π ). By part (i) 8 In Section 8, we consider other possible specifications of the buyer s cost function. 16

of Lemma 2, there exists a B such that G is a mean-preserving spread of Fπ B. Next, we show that Fπ B is cheaper than G, that is, C ( Fπ B ) C (G). Note that C ( Fπ B ) 1 = c (v) dfπ B (v) c (v) dg (v) = C (G), (8) where the inequality follows from c being convex and G being a mean-preserving spread of F B π Proposition 6.D.2 of Mas-Colell et al (1995)). Therefore, U (G, p) C (G) U ( F B π, π ) C ( F B π ) ( ) U (Fπ B, π ) C Fπ B, (see where the first inequality follows part (ii) of Lemma 2 and (8), and the second one from (7). 6 Reshaping a Given Prior In this section, we consider scenarios where the buyer s valuation is given by a CDF H and she can take certain actions to tailor this distribution. We maintain the assumptions that the buyer s decision is observable and that there is no asymmetric information at the moment when this action is taken. In other words, the prior distribution H is common knowledge and the buyer takes an action before observing its realization. We analyze various environments depending how the buyer can reshape the prior H. First, we consider a situation where the buyer can add arbitrary amount of risk to the distribution H. Second, we analyse environments where all the buyer can do is stochastically destroy (or create) value. In neither of these cases, we explicitly model the set of actions which the buyer can take in order to modify the CDF H. Instead, we identify the buyer s action space with set of distributions that can arise from tailoring H. For example, if the buyer can add risk then this space is going to be set of those CDFs which are mean-preserving spreads of H. If the buyer can destroy (or create) value, she is required to select a CDF that is first-order stochastically dominated by (or first-order stochastically dominates) H. 6.1 Designing Riskiness Here, we characterize the buyer s problem of optimally adding risk to her prior value distribution H. We do not impose any restriction on the type of risk the buyer can induce but we continue to assume that the maximum valuation of the buyer cannot exceed one. If the buyer adds risk to H then the resulting distribution will be a mean-preserving spread of H. Therefore, we assume that 17

the buyer can choose any CDF which is a mean-preserving spread of H. 9 Formally, we assume that the choice set of the buyer is H = { F F : where µ denotes vdh (v). vdf (v) = µ and x F (v)dv x } H(v)dv x [, 1], Next, we introduce a class of distributions and then show that there always exists an equilibrium where the buyer chooses a CDF from this class. For each π (, 1] and l [π, 1] define 1 π l if v < l, F π,l (v) = 1 π v if v [l, 1). 1 if v 1 The CDF F π,l is a convex combination of an atom at zero and the equal-revenue distribution F l. Indeed, if the buyer s valuation is zero with probability 1 (π/l) and it is determined by F l with probability π/l then the resulting CDF is exactly F π,l. Of course, when the seller decides what price to set, he conditions on the event that the buyer s value is determined by F l. Hence, P (F π,l ) = [l, 1] and Π (F π,l ) = (π/l) l = π. Observe that F π,π = F π and the CDF F π,1 specifies an atom of size π at one and places the rest of the probability mass at zero. The next lemma shows that for each CDF G there is an element in this class which is a mean-preserving spread of G, induces the same profit for the seller and a weakly higher payoff to the buyer. Lemma 3 Suppose that G F, p P (G) and let π denote pd (G, p). Then there exists a unique l [π, 1] such that (i) F π,l is a mean-preserving spread of G, (ii) U (F π,l, l) = µ π and (iii) U (G, p) U (F π,l, l) and the inequality is strict if D (G, p) < 1 G (). Part (i) states that the valuation distributed according to F π,l is more risky than the one distributed according to G. Recall that Π (F π,l ) = π, so part (iii) says that this more risky distribution generates the same payoff to the seller and a higher payoff to the buyer. 9 We refer to Johnson and Myatt (26) for an extensive discussion of environments, such as product design and advertisement, where the shifts in demand are modeled as mean-preserving spreads. 18

Proof. First, we show that there exists an l [π, 1] such that the expectation of F π,l is µ. Note that vdf π,1 (v) = π µ = vdg (v) vdf π (v) = vdf π,π (v), where the second equality follows from G H, the second inequality follows from part (i) of Lemma 1, and the last equality from F π,π = F π. Since vdf π,l (v) is continuous and strictly decreasing in l on [π, 1], the Intermediate Value Theorem implies that there exists an l such that vdf π,l (v) = µ. (9) Next, we prove that there exists a z (, l) such that F π,l (v) G (v) if v z and F π,l (v) G (v) if v z. To this end, let z = inf {y [, 1] : F π (v) G (v) for all v y}. We have to show that F π,l (v) G (v) if v z. Note that if v l then F π,l (v) = F π (v) G (v), where the inequality follows from part (i) of Lemma 1. Therefore, z l. Suppose that y < z and F π,l (y) < G (y). Since F π,l is constant on [, l] and G is increasing, it follows that G (v) > F π,l (v) for all v [y, z] [y, l] which contradicts to the definition of z. We conclude that F π,l (v) G (v) on [, z]. We are ready to prove that F π,l is a mean-preserving spread of G. If x z then F π,l (v) G (v) for all v [, x] and hence, If x z then x F π,l (v)dv = 1 µ x x F π,l (v)dv x G(v)dv. F π,l (v)dv 1 µ x G(v)dv = x G(v)dv, where the inequality follows from F π,l (v) G (v) on [z, 1]. The previous two displayed inequalities imply part (i). To see part (ii), note that U (F π,l, l) = l v ldf π,l (v) = l vdf π,l (v) ld (F π,l, l) = vdf π,l (v) ld (F π,l, l) = µ π, where the third equality follows the fact that F π,l (v) = F π,l () for all v [, l] and the last equality follows from (9) and π = Π (F π,l ). 19

To prove part (iii), note that µ π p v pdg (v) = U (G, p), where the inequality follows from the fact that the buyer s payoff cannot exceed the first-best total surplus,µ, minus the seller s profit, π. The previous two displayed equations imply that U (G, p) U (F π,l, l). The last inequality is strict whenever the total surplus in the outcome (G, p) is strictly less than µ, that is, trading is ex-post inefficient. Trading is ex-post inefficient if and only if the probability that v (, p) is positive, that is, D (G, p) < 1 G (). Suppose that there is an equilibrium where the buyer chooses G and the seller sets price p. Applying Lemma 3 to this profile (G, p) yields the result that there is also an equilibrium where the buyer chooses a CDF from the set {F π,l }. If the buyer chooses F π,l, it is optimal for the seller to set price l which, in turn, provides the buyer with a payoff of µ π (see part (ii) of Lemma 3). The distribution which maximizes the buyer s payoff is therefore defined by the lowest π for which F π,l H for some l [π, 1]. Let us define π as this lowest value, that is, π = min{π [, 1] : l [π, 1] such that F π,l H} (1) and let l denote the unique value for which F π,l H. Proposition 4 If the buyer s choice set is H then there is an equilibrium where the buyer chooses F π,l and the seller sets price l. If there is also an equilibrium where the buyer chooses G and the seller sets price p then (i) U (G, p) = U (F π,l, l ) = µ π, (ii) D (G, p) = 1 G (), (iii) Π (G) = π and (iv) F π,l is a mean-preserving spread of G. The distribution F π,l again generates unit elastic demand for prices in [l, 1]. However, trade only occurs with probability π/l. Nevertheless, since only those buyers don t trade whose valuation is zero, trading decisions are ex-post efficient. Parts (i) and (iii) of this proposition imply that the payoffs of the buyer and seller are the same across all equilibria. In addition, by part (ii), trading decisions are efficient. In general, the buyer s equilibrium CDF is not determined uniquely because there might be many CDFs which are mean-preserving spreads of H and generate payoffs µ π and π to the buyer and the seller, respectively. However, according to part (iv), the CDF F π,l 2 is a mean-preserving spread of any

other equilibrium CDF. This means that F π,l induces more risk in the buyer s valuation than any other equilibrium CDF. Proof. First we show that there is an equilibrium where the buyer chooses F π,l and the seller sets price l. We have already argued that P (F π,l ) = [l, 1], so the the seller optimally responds to F π,l by setting price π. It remains to argue that the buyer has no incentive to deviate. Let G H, let p P (G) and let π denote Π (G). Part (i) of Lemma 3 implies that there exists an l [π, 1] such that F π,l is a mean-preserving spread of G. Observe that U (G, p) U (F π,l, l) = µ π µ π = U (F π,l ), (11) where the first inequality follows from part (iii) of Lemma 3, the first and third equality from part (ii) of Lemma 3 and the second inequality from (1). Suppose now that there is an equilibrium where the buyer chooses G H and the seller sets price p. Again, let π denote Π (G). Note that equation (11) is still valid. Similarly to the uniqueness proof of Theorem 1, it is easy to show that for each ε >, there is a CDF, F ε H, close to F π,l and p ε [, 1] such that p ε = P (F ε ) and U (F π,l, l) < U (F ε, p ε ) + ε. (12) Then for each ε > U (F ε, p ε ) U (G, p) U (F π,l ) < U (F ε, p ε ) + ε, where the first inequality holds because otherwise F ε would be a profitable deviation, the second inequality follows from (11) and the third one from (12). This inequality chain implies that U (G, p) = U (F π,l ) which proves part (i) of the proposition. In addition, this implies that all the inequalities in (11) are equalities. Hence, by part (iii) of Lemma 3, D (G, p) = 1 G (), which is just part (ii) of the proposition. In addition, the second inequality in (11) is strict unless π π, which proves part (iii). To prove part (iv), first note that F π,l is a mean-preserving spread of F π,l because π π and the expectation of both F π,l and F π,l is µ. In addition, F π,l is a mean-preserving spread of G by part (i) of Lemma 3. Since second-order dominance is a transitive relation, it follows that F π,l is indeed a mean-preserving spread of G. We observe that the key result to the arguments of this section, Lemma 3, is the counterpart of Lemma 2. Recall that for each G F, Lemma 2 identified a CDF, F B π, such that G is a meanpreserving spread of F B π and the buyer s payoff generated by F B π exceeds that generated by G. In contrast, Lemma 3 identifies a CDF F π,l F for each G such that F π,l is a mean-preserving spread 21

of G and the buyer is better off by choosing F π,l than G. Recall that Lemma 2 was used to show that the equilibrium CDF of the buyer is in the set { Fπ B } if she faces convex moment restrictions (see Proposition 2). Similarly, Lemma 3 can be used to show that the buyer s equilibrium CDF is in the class {F π,l } if she faces concave moment restrictions. 6.2 Destruction and Creation In this section, we continue to assume that a prior CDF, H F, is given. First, suppose that the buyer can take any action which decreases her valuation in the first-order stochastic dominance sense. That is, the set of distributions available for the buyer is H = {F F : F (v) H(v) v [, 1]}. The next proposition states that the equilibrium choice of the buyer is the pointwise maximum of her prior H and some equal-revenue distribution. Proposition 5 If the buyer s choice set is H then there exist an equilibrium and a π (, 1], such that the buyer s equilibrium choice is max {F π, H}. In this case, the buyer s euilibrium CDF is not unique, in general. Nevertheless, it is easy to show that the equilibrium payoffs are unique. In addition, it is without loss of generality to assume that trading is ex-post efficient because the buyer can always choose a CDF which has no mass on (, p), that is, only those buyers don t trade whose valuation is zero. However, in general, trade does not occur with probability one. Proof. Let G H, let p P (G) and let π denote Π (G). It is enough to show that if the buyer chooses H π = max {F π, H}, the seller still finds it optimal to set p and U (H π, p) U (G, p). First, we show that H π first-order stochastically dominates G. Note that part (i) of Lemma 1 implies that G (v) F π (v). Since G H, G (v) H (v). The previous two inequalities imply that G (v) max {F π (v), H (v)} = H π (v). (13) Next, we argue that p P (H π ). Observe that pd (G, p) = π = pd (F π, p) pd (H π, p) pd (G, p), where the second equality follows from π = Π (F π ), the first inequality follows from the fact that F π first-order stochastically dominates H π, and the last inequality follows from (13). From this 22