Robust contracting under common value uncertainty

Auster, Sarah Article Robust contracting under common value uncertainty Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Auster, Sarah (2018) : Robust contracting under common value uncertainty, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 13, Iss. 1, pp. 175-204, https://doi.org/10.3982/TE2385 This Version is available at: https://hdl.handle.net/10419/197144 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 13 (2018), 175–204 1555-7561/20180175 Robust contracting under common value uncertainty Sarah Auster Department of Decision Sciences and IGIER, Bocconi University A buyer makes an offer to a privately informed seller for a good of uncertain quality. Quality determines both the seller’s valuation and the buyer’s valuation, and the buyer evaluates each contract according to its worst-case performance over a set of probability distributions. This paper demonstrates that the contract that maximizes the minimum payoff over all possible probability distributions of quality is a screening menu that separates all types, whereas the optimal contract for any given probability distribution is a posted price, which induces bunching. Using the ε-contamination model, according to which the buyer’s utility is a weighted average of his single prior expected utility and the worst-case scenario, the analysis further shows that for intermediate degrees of confidence, the optimal mechanism combines features of both of these contracts. Keywords. Ambiguity, optimal contracting, lemons problem. JEL classification. D81, D82, D86. 1. Introduction The optimal design of contracts in the presence of asymmetric information has been the subject of investigation for several decades and its theoretical analysis has generated an array of powerful results. Most of this literature adopts the subjective expected utility model according to which contracting parties have a single subjective prior belief about the fundamentals. In real-life contracting situations, e.g., when buying a house or when investing in a foreign country, the involved parties rarely have a precise idea about the underlying probability distribution, either because they do not have enough experience or because they do not have sufficient information. It is well established that a lack of knowledge about the probability distribution can have important behavioral implications that are incompatible with the subjective expected utility hypothesis. This gives rise to the question of how the presence of uncertainty over probabilistic scenarios affects the optimal design of contracts and the implemented allocation. This paper analyzes a bilateral trade model that allows for common values—an important Sarah Auster: [email protected] This work was undertaken as part of my PhD thesis at the European University Institute. It previously circulated under the title “Bilateral Trade Under Ambiguity.” I am very grateful to my supervisor Piero Gottardi. I would also like to thank Arpad Abraham, Subir Bose, Itzhak Gilboa, Emeric Henry, Peter Klibanoff, Frederic Koessler, Nenad Kos, David Levine, Massimo Marinacci, Matthias Messner, Sujoy Mukerji, Andrew Postlewaite, David Pothier, Tomazs Strzalecki, Balazs Szentes, and the participants of various seminars for very helpful comments. Copyright ©2018 The Author. Theoretical Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://econtheory.org. https://doi.org/10.3982/TE2385 176 Sarah Auster Theoretical Economics 13 (2018) feature in many real-life contracting situations—and an ambiguous trading environment. In the environment considered, there is a risk-neutral buyer (she) who makes an offer to a risk-neutral seller (he), who is privately informed about the quality of his good. The paper first presents an introductory example in which quality is either high or low and then studies the richer environment in which quality belongs to an interval of values. In contrast to the classic setting, it is assumed that the buyer has ambiguous beliefs about the distribution of quality, which determines the valuation of both trading parties. The buyer’s preferences are represented by the maxmin expected utility model (Gilboa and Schmeidler 1989), according to which the buyer evaluates her choices with the most pessimistic probability distribution in a set of distributions. For the case where this set is a singleton, Samuelson (1984) shows that the optimal mechanism is a take-it-or-leave-it price. The results of this paper demonstrate that if the extent of ambiguity the buyer faces is sufficiently large and if values are interdependent, the optimal mechanism is a screening menu rather than a posted price. To show this, the paper first studies the case of Knightian uncertainty, where the buyer considers the worst-case payoff over all possible probability distributions of quality. Here the buyer optimally proposes a contract that equalizes her payoff across all seller types and thereby hedges against the ambiguity she perceives. This contract can be interpreted as maximally robust, since it yields the same expected payoff across all possible probabilistic scenarios. The analysis shows that the nature of the robust mechanism crucially depends on the relation between the buyer’s and the seller’s valuations: if the buyer’s valuation strictly increases in the seller’s type, the optimal mechanism is a screening menu that perfectly separates all seller types, whereas if the buyer’s and seller’s valuations are independent, the maximally robust mechanism is the pooling price. The intuition is that a separating menu allows the price of the good to increase with the buyer’s valuation of the good, thereby balancing her payoff across the different realizations of her and the seller’s valuation. The paper also studies the case in which the buyer’s ignorance is less extreme. To parameterize the buyer’s demand for robustness, her preferences are represented by the ε-contamination model, a special case of the maxmin expected utility model, which nests the case of Knightian uncertainty on the one hand and the case of a single subjective prior belief on the other hand. According to this representation, the buyer has a reference distribution but entertains some doubt regarding that distribution, captured by the parameter ε. The buyer’s confidence in the model distribution determines the nature of the optimal mechanism. If εis sufficiently small, the optimal mechanism is a posted price as in Samuelson (1984), whereas if εis sufficiently large, the optimal mechanism is a screening menu that perfectly hedges against ambiguity. For intermediate values of ε, the optimal contract combines features of both of these mechanisms: low quality sellers are bunched at a base price, while high quality sellers are separated by the mechanism. This hybrid contract solves the trade-off between maximizing the buyer’s expected utility evaluated at the reference distribution and limiting her minimal payoff in the worst-case scenario. In particular, by screening sellers with a valuation above the Theoretical Economics 13 (2018) Robust contracting 177 base price, the buyer avoids the possibility of trading with probability 0 in some states of the world. Given the longstanding debate on the economic implications of ambiguity as opposed to risk, I discuss how the optimal contract in the proposed setting differs from the benchmark model in which the buyer is ambiguity neutral but risk averse. Although the optimal contract under risk aversion may also be a separating menu, the cases in which it coincides with the optimal mechanism under ambiguity aversion are nongeneric. Moreover, the discussion demonstrates that there are situations in which a higher degree of risk aversion makes the pooling price optimal, while a higher degree of ambiguity aversion generally favors separation. Finally, in some applications of the maxmin expected utility model, results crucially rely on the feature that preferences are kinked (e.g., Dow and Ribeiro da Costa Werlang 1992;Condie and Ganguli 2017). This is not the case in the model studied here because the optimality of separating menus in the considered environment is driven by the buyer’s desire to hedge against ambiguity, a feature of all decision models that capture ambiguity averse behavior. This is illustrated by extending the characterization of the optimal contract for the binary type case to the smooth ambiguity model, introduced by Klibanoff et al. (2005). The characterization shows that the solution of the buyer’s optimization problem under smooth ambiguity aversion is a convex combination of the solution under maxmin expected utility and ambiguity neutrality. Related literature This paper is part of a growing literature on robust contracting in an uncertain environment, which includes work on procurement contracts (Garrett 2014), optimal delegation mechanisms (Frankel 2014;Carrasco and Moreira 2013), and optimal incentive contracts in the presence of moral hazard (Carroll 2015;Carroll and Meng 2016;Anti´ c 2014). Crucially, and in contrast to most prior work in mechanism design, the principal in these models evaluates contracts according to their worst-case performance, e.g., over the agent’s preferences or over the set of available technologies. There is also a small number of papers that study the maxmin optimal contract in the canonical principal–agent problem with hidden information, as this one does. In contrast to this work, existing papers assume that the principal’s and agent’s valuations are independent. Bergemann and Schlag (2011) show that under the assumption of independent private values, the maxmin optimal mechanism is a posted price. Intuitively, if the principal only faces ambiguity over the agent’s acceptance decision, the worst-case probabilistic scenario is always the one that maximizes the probability that the agent rejects. This implies that the optimization problem under maxmin preferences is equivalent to the optimization problem under a single pessimistic prior and Samuelson’s (1984) result applies. Bergemann and Schlag (2011) also consider the mechanism that minimizes maximal regret. In contrast to maxmin expected utility, the minimax regret criterion generates a regret trade-off that makes randomization across prices optimal. In recent work, also Carrasco et al. (2017) consider a monopoly pricing 178 Sarah Auster Theoretical Economics 13 (2018) model with independent private values and a principal with maxmin preferences. In their environment, the principal has partial probabilistic information about the agent’s valuation, such as the mean or the variance, which makes a random pricing rule optimal. There exists a complementary line of literature that studies mechanism design problems in which the agent rather than the principal faces uncertainty about the underlying probabilistic environment. This literature includes the work of Bose and Mutuswami (2012)andWolitzky (2016), both of which investigate the implementability of efficient trade in the canonical Myerson and Satterthwaite (1983) environment with the assumption that agents perceive uncertainty about the probability distribution of the opponent’s type. Also Bose et al. (2006), Bose and Daripa (2009), and Bodoh-Creed (2012) introduce ambiguity on the agent’s side, but in contrast to the work mentioned before, the focus of these papers lies on revenue maximization. Finally, Bose and Renou (2014) and di Tillio et al. (2017) show that the designer may benefit from introducing ambiguity via the mechanism. The rest of the paper is organized as follows. Section 2 presents the introductory example, which demonstrates the main features of robust contracting in the considered environment. Section 3 then introduces the main model. I first characterize the optimal contract for the case in which the buyer considers the worst-case scenario over all possible probability distributions. Next, I study the case of moderate ambiguity and show how the features of the optimal contract depend on the buyer’s attitude toward the ambiguity she faces. Section 4 discusses the differences to risk aversion and explains how the findings of the model extend when preferences are smooth rather than kinked. Section 5 concludes. 2. Introductory example A risk-neutral buyer makes an offer to a risk-neutral seller who possesses one unit of an indivisible good. The seller is privately informed about the quality of the object, which can be either high or low. Quality determines both the seller’s and the buyer’s valuations, implying that the seller knows both his own valuation c∈{clch}as well as the buyer’s valuation v∈{vlvh}, whereas the buyer knows neither of these values. I assume that both the seller and the buyer value high quality more than low quality, i.e., ch>c l,vh> vl, and that the buyer’s value always exceeds the seller’s value: vi>c i,i=lh. The buyer proposes a menu of contracts, {(x(c)t(c))}c∈{clch}, consisting of a trading probability x(c) and a transfer t(c) for each type of good. If the seller reveals truthfully his type, the buyer’s and the seller’s payoffs as a function of the seller’s type are given by πb(c) = x(c)v(c) −t(c) and πs(c) =t(c)−x(c)c, respectively. The buyer faces ambiguity over the quality distribution and has maxmin expected utility preferences (Gilboa and Schmeidler 1989). Under this representation, a decision maker evaluates her choices with the worst probability distribution in a convex set of distributions. Letting σdenote the probability that the quality of the object is high, this amounts to the buyer minimizing over an interval of values of σ.LetE σ Theoretical Economics 13 (2018) Robust contracting 179 denote the expectation operator with respect σ. The buyer’s payoff function is then given by inf σ∈[σσ]Eσπb(c) The buyer maximizes this payoff function subject to the incentive and individual rationality constraints of each type of seller: t(cl)−x(cl)cl≥t(ch)−x(ch)cl t(ch)−x(ch)ch≥t(cl)−x(cl)ch t(ci)−x(ci)ci≥0i=lh Since the seller knows his type, the constraints of the buyer’s optimization problem are not affected by the presence of ambiguity in this environment. This, and the fact that the buyer’s objective is weakly decreasing in t(cl)and t(ch), implies that the solution to the buyer’s optimization problem satisfies some well established properties (see, for example, Salanié 2005, Chapter 2): the incentive compatibility constraint of the low type seller and the individual rationality constraint of the high type seller are binding, while the remaining constraints are slack. Furthermore, the low type seller trades with probability 1. With these properties, the menu of contracts is completely characterized by the trading probability of the high type seller x(ch). For notational convenience, let this probability be denoted by α: x(cl)t(cl)x(ch) t(ch)=1αch+(1−α)cl(ααch) Remark. Note that there are two alternative interpretations of the model. In the interpretation followed throughout the paper, the good is indivisible and αis the probability of trade. In an alternative interpretation, the seller possesses one unit of a perfectly divisible good, utility functions are multiplicatively linear, and αis a quantity. Under this interpretation, the menu is a nonlinear pricing schedule with a quantity discount. Low quality is traded in large quantity at a low price, while high quality is traded in small quantity at a high price. Given the properties stated above, the buyer’s payoff can be stated as a function of α and her optimization problem becomes max α∈[01]inf σ∈[σσ]σα(vh−ch)+(1−σ)vl−αch+(1−α)cl To derive the solution to this problem, consider first the case in which the set of probability distributions is a singleton, so that the buyer is a subjective expected utility maximizer with a single prior σ.Samuelson (1984) shows that the buyer’s optimal contract is a posted price, which the seller either accepts or rejects. In the two-type setting, this result can easily be seen by considering the buyer’s objective function. Since Eσ[πb]is linear in α, the optimization problem has a corner solution. The mechanism characterized by α=0is a separating price equal to cl, while that characterized by α=1is the 180 Sarah Auster Theoretical Economics 13 (2018) pooling price equal to ch. Pooling is optimal if the probability that the seller’s type is high is large enough, which is the case if σvh+(1−σ)vl−ch≥(1−σ)(vl−cl)or, equivalently, σ≥ch−cl vh−cl  Proposition 2.1 shows that if the set of probability distributions is not a singleton, the buyer’s optimal contract is a posted price if and only if this price is optimal under all probability distributions in the interval [σσ]. Otherwise, the buyer optimally offers a separating menu. Proposition 2.1. Let ˜σ:= ch−cl vh−cl. The buyer optimally proposes a menu characterized by α∗=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0if σ≤˜σ 1if σ≥˜σ vl−cl vh−cl otherwise. See Appendix A for the proofs of most of the propositions. The parameter ˜σis the subjective prior under which all values of αyield the same payoff. If σ≥˜σ, the optimal value of αis equal to 1, since the contract that maximizes the buyer’s expected payoff is the pooling price chfor all σ∈[σσ]. Similarly, if σ≤˜σ, the optimal value of αis equal to 0, because the separating price clis optimal for all σ∈[σσ]. Ambiguity aversion thus affects the buyer’s utility but not her choice of contract. If σ<˜σ<σ, the buyer optimally proposes a separating menu, characterized by α∗=vl−cl vh−cl.Thisvalueofαis the contracting parameter under which the buyer’s payoff conditional on the seller’s type being high equals her payoff conditional on the seller’s type being low. Intuitively, offering a separating menu rather than a posted price allows the buyer to limit the extent of ambiguity over the probability with which the seller accepts and the net gain when trade occurs, thereby balancing her payoff across the different states of the world. The menu characterized by α∗=vl−cl vh−clcan be viewed as a robust contract because it makes the buyer’s payoff independent of the underlying type distribution and thereby yields a “safe” payoff equal to (vh−ch)(vl−cl) vh−cl. Offering a separating menu thus hedges against ambiguity in this environment. Hedging is optimal if and only if the environment is sufficiently ambiguous and the buyer is sufficiently ambiguity averse, i.e., if [σ σ]is large enough. This is illustrated in Figure 1. Under the assumption σ<˜σ<σ, the expected payoff of a buyer with subjective prior σis downward sloping in α(solid line), while the expected payoff of a buyer with subjective prior σis upward sloping in α(dashed line). All expected payoff functions Eσ[πb],σ∈(σσ) lie in between these two benchmarks and intersect at α∗=vl−cl vh−cl.Forα< vl−cl vh−cl, the worst case for the buyer is that the probability with which the seller has a high quality good is large, whereas for α> vl−cl vh−cl,theworst case is that the probability of this event is small. Since the buyer’s expected payoff evaluated at σis upward sloping in α, while her expected payoff evaluated at σis downward sloping in α, her maxmin expected payoff (thick curve) is maximized at α∗=vl−cl vh−cl. Theoretical Economics 13 (2018) Robust contracting 181 Figure 1. Expected payoffs Eσ[πb]for σ∈[σσ]. 3. The main model Consider now the case where the seller’s type takes a value in the interval [01].Asbefore, the seller knows his own valuation for the object cand also the buyer’s valuation v, whereas the buyer is uncertain about these values. Let the differentiable function v(c) define the relation between the two values, and assume v(c) ≥0and v(c) > c for all c∈[01]. That is, both the seller’s and the buyer’s value for the object are increasing in its quality, and the buyer’s value strictly exceeds the seller’s value. In this environment a menu of contracts is defined as {(x(c)t(c))}c∈[01],andthe incentive and individual rationality constraints of the seller are given by t(c)−x(c)c ≥t(˜ c) −x(˜ c)c ∀c ˜ c t(c)−x(c)c ≥0∀c Samuelson (1984, p. 997) shows that this set of constraints implies that x(c) weakly decreases in cand that π s(c) =−x(c) almost everywhere. With these properties, the buyer’s and seller’s payoffs can be derived as a function of x(c) only. Condition π s(c) = −x(c) together with πs(1)=0yields πs(c) =1 c x(u)du The term 1 cx(u)duis the information rent paid to type c. The buyer’s payoff as a function of the seller’s type, πb(c), is then given by the difference between the expected value of the realized gains from trade and the information rent paid to the seller. That is, πb(c) =x(c)(c) −1 c x(u)du where (c) ≡v(c) −cdenotes the gains from trade when the seller’s type is c. 182 Sarah Auster Theoretical Economics 13 (2018) 3.1 Unique subjective prior Consider first the benchmark case, where the buyer has a single prior belief F.This specification corresponds to the standard Bayesian setting, for which Samuelson (1984) shows that the optimal mechanism for the buyer is a posted price. Assume that Fis twice differentiable and let f(c)=F(c) denote the density function. Defining as the set of nonincreasing functions x:[01]→[01], the buyer solves the problem max x∈EFπb(c)=1 0x(c)(c) −1 c x(u)duf(c)dc (I) After an integration by parts, the buyer’s expected payoff can be written as EF[πb(c)]= 1 0((c)f(c) −F(c))x(c) dc,wherethefunctionH(c) ≡(c)f(c) −F(c) captures the marginal benefit of increasing x(c).IfH(c) is strictly decreasing in c, the optimal mechanism is characterized by the step function x∗(c) =1if c≤c∗ 0if c>c ∗ where c∗is such that H(c∗)=0if H(1)<0and c∗=1otherwise. This mechanism corresponds to a posted price equal to c∗. The buyer’s payoff associated to the posted price varies with the seller’s type c. In particular, the minimum payoff is given by ¯πmin ≡{(0)−c∗0}, which is obtained either when the seller is of the lowest type so that the buyer’s valuation is (0)or when the seller rejects the buyer’s offer. 3.2 Knightian uncertainty Consider now the polar case, where the buyer evaluates possible contracts by their worst-case performance over all probability distributions on [01]. This case of Knightian uncertainty has received a lot of attention in the literature on robust contracting, e.g., Frankel (2014), Garrett (2014), and Carroll (2015). Under this specification, the buyer solves the problem max x∈inf c∈[01]x(c)(c) −1 c x(u)du(II) The following proposition characterizes the solution to this problem. Proposition 3.1. The unique solution of problem (II)isgivenby x∗(c) =exp−c 0 v(t) (t) dtfor all c∈[01] The optimal mechanism for a buyer who minimizes over all types c∈[01]is a mechanism that equalizes her payoff across all types. Noting that π b(c) =x(c)(c) + x(c)v(c), the mechanism that yields a constant payoff across csolves the differential equation x(c)(c) +x(c)v(c) =0 Theoretical Economics 13 (2018) Robust contracting 189 •Posted Price. If ε≤1−1 δe−1−δ δ, x∗(c) =1if c≤δ 0if c>δ •Partial Separation. If ε∈(1−1 δe−1−δ δ1−1 δe−1 δ), x∗(c) =⎧ ⎨ ⎩ 1if c≤ˆ c δ−ˆ c δe−c−ˆ c δif c>ˆ c where ˆ c=1+δln(δ(1−ε)). •Perfect Separation. If ε≥1−1 δe−1 δ, x∗(c) =e−c δ∀c Under the assumption (c) =δand F(c)=c, the optimal mechanism in the benchmark case ε=0is a posted price equal to δ. The specification thus satisfies ¯πmin ≥0and there are three regions of εto be distinguished. If ε≤ε=1−1 δe−1−δ δ, the optimal mechanism is a posted price equal to δ,whereasifε≥ε=1−1 δe−1 δ, the optimal mechanism is a separating menu that perfectly hedges against ambiguity and yields a safe payoff equal to ¯πmax =δe−1 . Alternatively, if εlies between these two thresholds, the optimal mechanism bunches types below the threshold ˆ c=1+δln(δ(1−ε)) and separates the remaining seller types. The optimal threshold ˆ cis strictly decreasing in the buyer’s confidence parameter ε, reflecting the hedging function of separating menus in this environment. Figure 4 illustrated the optimal trading probability x∗(c) and the associated payoff πb(c) for different values of ε. The example helps one to understand why, for intermediate values of ε, the buyer optimally bunches low type sellers and separates high type sellers. Since (c) =δfor all c∈[01], the marginal gain of increasing the trading probability with a particular seller, given by the respective gains from trade, is the same across all types. The marginal cost of increasing the trading probability, alternatively, is strictly increasing in the seller’s type because the buyer has to pay the additional information rent to all lower types. As a result, the buyer’s expected payoff evaluated at Fis maximized when trading with low type sellers but not with high type sellers. Also the optimal mechanism for a buyer who demands robustness maximizes the trading probability with low type sellers—by letting them trade with probability 1at a base price—but minimizes the trading probability of high type sellers only up to the point where the minimum payoff is reached. 4. Discussion 4.1 Ambiguity aversion versus risk aversion Given the longstanding debate on the implications of ambiguity, an important question in this contracting problem is how the effects of ambiguity aversion differ from those of risk aversion. It is well known that if utility functions are not quasilinear, separating 190 Sarah Auster Theoretical Economics 13 (2018) Figure 4. Trading probability x∗(c) and payoff πb(c) for different values of ε∈[01]. menus can be optimal, even when there is no ambiguity. The following discussion shows that the optimality conditions that determine the mechanism under risk aversion are different from those under ambiguity aversion, and provides some intuition for why this is the case. As an example, suppose the buyer’s utility function uis concave in the difference between her valuation v(c) and the price she pays in exchange for the good. Given the arguments put forward in Section 3, the seller’s payoff as a function of his type and the trading probability x(c) can be written as πs(c) =1 cx(t) dtso that the price conditional on trading, denoted by p(c),isgivenbyp(c) =c+1 cx(t)dt x(c) if x(c) > 0and p(c) =0oth- erwise. Normalizing the buyer’s outside option to zero, the risk-averse buyer with single prior belief Fthus maximizes 1 0 x(c)u(c) −1 c x(t) x(c) dtf(c)dc Theoretical Economics 13 (2018) Robust contracting 191 subject to the monotonicity constraint on x(c). There exist parameter constellations under which the function x(c) that solves this problem corresponds to a separating menu rather than a posted price, as can be verified. However, when the buyer’s valuation depends on the seller’s type, the cases in which the solution x(c) corresponds exactly to the optimal mechanism under ambiguity aversion, whether in the form of Knightian uncertainty or the ε-contamination model, are nongeneric. More generally, when the buyer’s type increases in the seller’s type, the way in which the desire to hedge against ambiguity and to hedge against risk affect the optimal mechanism is typically different. The previous section showed that ambiguity aversion favors the separation of sellers through the mechanism. In particular, the maximally robust mechanism, which is optimal when the buyer is sufficiently ambiguity averse, is such that the trading probability strictly decreases in the seller’s type. Such a mechanism still exposes the buyer to risk since the buyer faces a strictly positive probability of not trading. In fact, there does not exist a mechanism that yields a riskless payoff. However, in situations where the buyer’s type does not depend too much on the seller’s type so that the pooling price yields a positive payoff with all types of seller, the buyer can limit her downside risk by offering the pooling price. If the buyer is sufficiently risk averse, the pooling price is then indeed optimal, implying that risk aversion can favor pooling rather than separation. As seen in the discussion after Proposition 3.1, the assumption that types are interdependent is crucial for the robust mechanism to be a separating menu. If, alternatively, the buyer’s valuation does not depend on the seller’s private information, the maximally robust mechanism characterized in Proposition 3.1 is the pooling price. Interestingly, when vis constant, offering the pooling price not only yields an unambiguous but also a riskless payoff for the buyer. Thus, in the independent private value case, hedging against ambiguity can take the same form as hedging against risk. 4.2 Smooth ambiguity aversion For a given set of measures, the maxmin expected utility model may be seen as a special case of the smooth ambiguity model, developed by Klibanoff et al. (2005), with infinite ambiguity aversion. This section illustrates that the main characteristics of the optimal contract extend to the case of smooth ambiguity aversion and thus do not hinge on the kink property of maxmin expected utility. To keep the analysis tractable, I return to the case of binary types, where the probability distribution of seller types is captured by a single parameter σ∈[01]. In the smooth ambiguity model, the buyer’s utility function is given by EμEσ[πb] where μ:[01]→[01]is a subjective prior on a set of probability measures, here captured by σ∈[01],and:R→Ris a function that weighs realizations of the decision maker’s expected utility Eσ[πb]. Ambiguity is captured by the second-order belief μ, which measures the buyer’s belief about a particular σbeing the “correct” probability distribution, while ambiguity attitude is captured by the function .Ifis linear, the 192 Sarah Auster Theoretical Economics 13 (2018) buyer is ambiguity neutral and her preferences are observationally equivalent to those of a subjective expected utility maximizer. If, alternatively, is concave, the buyer is ambiguity averse and prefers known risks over unknown risks. The degree of ambiguity aversion is measured by the coefficient of absolute ambiguity aversion −(x) (x) . As in Section 2, the buyer proposes a menu of the form {(1αch+(1−α)cl) (α αch)}. The optimization problem of the buyer thus amounts to max α∈[01]Eμσα(vh−ch)+(1−σ)vl−αch−(1−α)cl To make ambiguity matter, assume that μhas positive mass on both [0˜σ) and (˜σ1], and assume that (·)>0and (·)<0. The first-order condition of the buyer’s optimization problem is given by Eμσα(vh−ch)+(1−σ)vl−αch−(1−α)cl(˜σ−σ)|σ<˜σ   ≡MC(α) =Eμσα(vh−ch)+(1−σ)vl−αch−(1−α)cl(σ −˜σ)|σ>˜σ   ≡MG(α)  The marginal cost of increasing α,MC(α), is the marginal decrease in expected utility in the probabilistic scenario that pooling is not optimal (σ<˜σ), while the marginal gain of increasing α,MG(α), is the marginal increase in expected utility in the probabilistic scenario that pooling is optimal (σ>˜σ). Concavity of implies that the marginal cost is increasing in α, whereas the marginal gain is decreasing in α. This implies that there is a unique αthat maximizes the buyer’s expected payoff. The conditions for an interior solution are MC(0)<MG(0)and MC(1)>MG(1) The following proposition summarizes this result. Proposition 4.1. The optimal menu of contracts for a buyer with smooth ambiguity aversion is {(1α∗ch+(1−α∗)cl) (α∗α∗ch)}with α∗=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0if MC(0)≥MG(0) 1if MC(1)≤MG(1) such that MCα∗=MGα∗otherwise. Proposition 4.1 is the smooth counterpart of Proposition 2.1 in the introductory example. To see the connection to the optimal mechanism under maxmin expected utility, assume that absolute ambiguity aversion is constant, i.e., −(x) (x) =γ. Assume further that the support of μis [σσ]and that σ < ˜σ<σ. Under maxmin preferences, the buyer’s optimal mechanism, characterized by αMEU ≡vl−cl vh−cl, makes her payoff unambiguous. Under smooth ambiguity aversion, the buyer compromises between maximizing the second-order expectation of her payoff, EμEσ[πb], and limiting her exposure to ambiguity. The expected payoff EμEσ[πb]is maximized by offering a posted price (equal Theoretical Economics 13 (2018) Robust contracting 193 to either clor ch), whereas ambiguity is eliminated by offering the menu characterized by αMEU. The optimal mechanism under smooth ambiguity aversion is a convex combination of the two. If offering the pooling price maximizes EμEσ[πb], then the optimal contracting parameter α∗lies in the interval [αMEU1]; otherwise α∗lies in the interval [0αMEU]. The more ambiguity averse the buyer is, the closer is α∗to αMEU.Thisis summarized in Proposition 4.2. Proposition 4.2. Assume −(x) (x) =γ. •If Eμ[σ]<˜σ,thenα∗∈[0αMEU]and dα∗ dγ≥0. •If Eμ[σ]>˜σ,thenα∗∈[αMEU1]and dα∗ dγ≤0. 5. Conclusion This paper considers the contracting problem between a seller and a buyer, who demands robustness with regard to the distribution of the seller’s private information, in an environment with interdependent values. The analysis shows that if the buyer’s valuation increases in the seller’s valuation, the nature of the optimal mechanism crucially depends on the buyer’s confidence in the underlying type distribution. The maximally robust contract in this situation is a menu that separates all seller types and thereby hedges against the ambiguity perceived by the buyer. As a result, the larger the buyer’s demand for robustness is, the more separation the optimal mechanism displays. This stands in contrast to the case where buyer and seller have independent private values and a posted price is optimal, no matter what the degree of the buyer’s ambiguity aversion is. One question not addressed in this paper is how the presence of uncertainty in the form of ambiguity affects the efficiency of the equilibrium allocation. In the environment considered, gains from trade are strictly positive with each type of seller and are thus maximized when the buyer offers the pooling price. As the analysis demonstrates, if the buyer knows the underlying type distribution, she offers a posted price potentially smaller than the pooling price, whereas if the buyer perceives and dislikes ambiguity, she proposes a separating menu. How these two mechanisms compare in terms of the social surplus they generate, and, more generally, how the presence of ambiguity affects the welfare of trading parties in environments with asymmetric information might be an interesting question for future research. Appendix A A.1 Proof of Proposition 2.1 The buyer maximizes infσ∈[σσ]Eσ[πb]. To identify the minimizing prior, consider ∂Eσ[πb] ∂σ =α(vh−cl)−(vl−cl) 194 Sarah Auster Theoretical Economics 13 (2018) The buyer’s payoff Eσ[πb]is decreasing in σfor all α≤vl−cl vh−cland increasing in σ for all α≥vl−cl vh−cl.Wethushaveσ∈arginfσ∈[σσ]Eσ[πb]if α≤vl−cl vh−clholds and σ∈ arginfσ∈[σσ]Eσ[πb(α)]if α≥vl−cl vh−clholds. The buyer consequently maximizes the step function =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ σα(vh−ch)+(1−σ)vl−αch+(1−α)clif α≤vl−cl vh−cl  σα(vh−ch)+(1−σ)vl−αch+(1−α)clif α> vl−cl vh−cl  By definition of ˜σ, the buyer’s payoff Eσ[πb]is decreasing in αif σ≤˜σand increasing in αif σ≥˜σ. Consequently, if σ≤˜σ, both parts of the step function are decreasing in αand is maximized at α=0. Similarly, if σ≥˜σ, both parts of the step function are increasing in αand is maximized at α=1.Ifσ<˜σ<σ,is strictly increasing in αon the interval [0vl−cl vh−cl]and strictly decreasing in αon the interval [vl−cl vh−cl1], and therefore is maximized at vl−cl vh−cl. A.2 Proof of Proposition 3.1 Suppose that the solution of problem (II)issuchthatπb(c) is constant across c.This implies that x(c) is differentiable in c. Indeed, notice that, given πb(c) =Cfor some C∈R,wehavex(c)(c) =πs(c) +Cfor all c∈[01].Asπsis absolutely continuous, a standard consequence of the seller’s incentive constraints, x(c) is continuous in c.Butas πs(c) =1 cx(u)duand (c) is differentiable and bounded away from zero, we get from the fundamental theorem of calculus that x(c) is differentiable in c. We then have π b(c) =x(c)(c) +x(c)v(c), so the optimal mechanism solves the differential equation x(c)(c) +x(c)v(c) =0(1) The solution to (1)isgivenbyx(c) =Dexp[− c 0 v(t) (t) dt],D∈R. Together with the condition x(0)=1, this yields D=1and hence x∗(c) as characterized in Proposition 3.1.Let the associated (constant) payoff be denoted by ¯πmax. We then need to show that any optimal mechanism indeed equalizes the buyer’s payoff across c. Consider a nonincreasing function ˜ x:[01]→[01]that satisfies ˜ x(c)(c)− 1 c˜ x(u)du≥¯πmax. We can first show that ˜ x(c) ≥x∗(c) for all c.Foreachc∈[01],let ˜ xc:[01]→Rbe an auxiliary function, defined by ˜ xc(u) =Dcexp[−u 0 v(t) (t) dt]with Dc such that ˜ xc(c) =˜ x(c). Since x∗(u) =exp[−u 0 v(t) (t) dt], for all csuch that ˜ x(c) < x∗(c),we have Dc<1and hence ˜ xc(u) < x∗(u) ∀u∈[01]. Toward a contradiction, suppose now there exists some c∈[01]such that ˜ x(c) < x∗(c). Notice that since the payoff with the highest type, ˜ x(1)(1),mustbeweakly greater than ¯πmax =x∗(1)(1),wehave ˜ x(1)≥x∗(1)and hence D1≥1. Now we know that Dc<1since by assumption ˜ x(c) < x∗(c). We therefore have ˜ x(1)> ˜ xc(1). Since ˜ xis nonincreasing and ˜ xcis continuous, we then have ˜ x(u) > ˜ xc(u) on a left neighborhood of 1. Let cbethesupremumoftheset{u∈[01]: ˜ x(u) ≤˜ xc(u)}, which is well defined Theoretical Economics 13 (2018) Robust contracting 195 as ˜ x(c) =˜ xc(c). Since ˜ xis nonincreasing, it must then hold that ˜ x(c)≤˜ xc(c)<x ∗(c). Considering the buyer’s payoff with type c,weobtain ˜ xc−1 c ˜ x(u)du≤˜ xcc−1 c ˜ xc(u)du=˜ xc(1)(1)<x ∗(1)(1)=¯πmax The first inequality follows from the facts that ˜ x(c)=˜ xc(c)(by definition of ˜ xc)and ˜ x(u) > ˜ xc(u) for all u>c . The following equality follows comes from the property that the buyer’s payoff is constant in cunder ˜ xc. The last inequality follows from ˜ x(c)< x∗(c), which implies Dc<1and hence ˜ xc(u) < x∗(u) for all u∈[01]. Together, this contradicts our initial assumption ˜ x(c)(c) −1 c˜ x(u)du≥¯πmax for all cand therefore implies ˜ x(c) ≥x∗(c) for all c. Finally, consider the buyer’s payoff with type c=0. Given that the minimum payoff associated to ˜ xmust be weakly greater than ¯πmax,itmustholdthat ˜ x(0)(0)−1 0 ˜ x(u)du≥(0)−1 0 x∗(u)du=¯πmax Since ˜ x(0)(0)≤(0)and ˜ x(c) ≥x∗(c) for all c, this inequality can only be satisfied if ˜ x(c) =x∗(c) for all c. A.3 Proof of Proposition 3.2 To characterize the solution of problem (III), consider the auxiliary optimization problem, where the endogenous minimum payoff is treated as an exogenous parameter ¯π∈R: max x∈1 0 H(c)x(c)dc(III) subject to the pointwise constraints x(c)(c) −1 c x(u)du≥¯π∀c∈[01](2) Evidently, any solution of problem (III) must also be a solution of (III)forsome ¯π, as otherwise the buyer could increase his expected payoff evaluated at Fwhile maintaining the same minimum payoff infc∈[01]{x(c)(c) −1 cx(u)du}.Wecanrestrictour attention to values of ¯πthat are weakly greater than the minimum payoff under the mechanism that maximizes 1 0H(c)x(c)dc, i.e., that solves problem (I). Recall that this payoff is given by ¯πmin =max{(0)−c∗0}. Moreover, the minimum payoff ¯πhas to be weakly smaller than the maximum value of infc∈[01]{x(c)(c) −1 cx(u)du}, attained at the mechanism that solves problem (II), as otherwise the feasible set is empty. This payoff is given by ¯πmax =(1)exp(−1 0 v(t) (t) dt). It will be useful to derive the function xmin ¯π:[01]→[01]under which the buyer’s payoff πb(c) is equal to ¯πif ¯π>0and equal to 0 if ¯π≤0for all c∈[01].Thisrequires that the buyer’s payoff is constant across c∈[01]and therefore that xmin ¯π(c) = Dexp[−c 0 v(t) (t) dt]for some D∈R(see Appendix A.2). The associated (constant) payoff 196 Sarah Auster Theoretical Economics 13 (2018) is equal to max{¯π0}if xmin ¯π(1)(1)=max{¯π0}, i.e., if D=max{¯π0} (1)exp(1 0 v(t) (t) dt).We thus have xmin ¯π(c) =max{¯π0} (1)exp1 c v(t) (t) dt We can first show that the pointwise constraints (2) are satisfied only if x(c) ≥xmin ¯π(c) for all c∈[01].First,if¯π≤0so that xmin ¯π(c) =0for all c∈[01],x(c) ≥xmin ¯π(c) must be trivially satisfied. Second, if ¯π>0and x(c) < xmin ¯π(c) for some c∈[01],byan analogous argument to that in Appendix A.2,thereexistssomecand some function xc(u) =Dcexp[−u 0 v(t) (t) dt]with Dcsuch that xc(c) =x(c) such that xc−1 c x(u)du≤xcc−1 c xc(u)du=xc(1)(1)<x min ¯π(1)(1)=¯π thus violating (2). Suppose now a solution to (III) exists (which we are going to show later) and let it be denoted by x¯π. We can then demonstrate that for each c∈[01],eitherx¯π(c) =1or x¯π(c) =xmin ¯π(c). To see this, let C≡{c∈[01]:x¯π(c) ∈(xmin ¯π(c) 1)}and consider the function ˆ x:[01]→[01],definedby ˆ x(c) =1if c≤ˆ c xmin ¯π(c) if c>ˆ c where ˆ c∈(01)is such that 1 0ˆ x(u)du=1 0x¯π(u)du. We can first verify that ˆ xsatisfies the pointwise constraints (2). For c≤ˆ c,whereˆ x(c) =1, notice that the buyer’s payoff, given by (c) −1 cˆ x(u)du=(c) −(ˆ c−c)−1 ˆ cxmin ¯π(u)du, strictly increases in c,sothat the only relevant constraint is (0)−1 0ˆ x(u)du≥¯π. Since 1 0ˆ x(u)du=1 0x¯π(u)du,this constraint is indeed satisfied. For c>ˆ c, constraints (2) are satisfied by construction. Define next the functions ˆ X(c) ≡c 0ˆ x(u)duand X¯π(c) ≡c 0x¯π(u)du, and consider their difference ˆ X(c) −X¯π(c). Since ˆ x(c) ≥x¯π(c) for all c≤ˆ c,ˆ X(c) −X¯π(c) is increasing on [0ˆ c], strictly so on the interior of C.3Similarly, since ˆ x(c) ≤x¯π(c) for all c≥ˆ c, ˆ X(c) −X¯π(c) is decreasing on [ˆ c1], again strictly on the interior of C. This together with ˆ X(0)=X¯π(0)and ˆ X(1)=X¯π(1)implies that ˆ X(c)−X¯π(c) ≥0for all c∈[01]and ˆ X(c)−X¯π(c) > 0for all c∈int C. Consider then the difference in the buyer’s expected payoff evaluated at Fassociated to ˆ xand x¯π. After an integration by parts, we obtain 1 0 H(c)ˆ x(c)dc−1 0 H(c)x¯π(c)dc =H(c)ˆ X(c)−X¯π(c)1 0−1 0 H(c)ˆ X(c)−X¯π(c)dc The first term on the right-hand side equals 0.GivenH(c) < 0and ˆ X(c) ≥X¯π(c) for all c∈[01]and ˆ X(c) > X¯π(c) for all c∈int C,theterm1 0H(c)( ˆ X(c) −X¯π(c)) dc 3Notice that dˆ X(c)−X¯π(c) dc=ˆ x(c) −x¯π(c) almost everywhere. Theoretical Economics 13 (2018) Robust contracting 197 is nonnegative, and hence 1 0H(c)ˆ x(c)dc≤1 0H(c)x¯π(c)dc,onlyifthesetint Cis empty. The previous argument, together with the monotonicity constraint on x¯πimplies that there exists a threshold ˆ csuch that x¯π(c) =1for all c≤ˆ cand x¯π(c) =xmin ¯π(c) for all c>ˆ c. Notice that, in principle, the function x¯πcan take infinitely many values at ˆ c. This is the only degree of flexibility, so setting x=1at ˆ cis without loss of generality. The threshold ˆ cmaximizes the buyer’s expected payoff evaluated at F, ˆ c 0 H(c)dc+1 ˆ c H(c)xmin ¯π(c) dc (3) subject to the pointwise constraints (2). Given the structure of x¯π, the only relevant of those constraints is that for c=0, as argued above. The first derivative of (3)withrespect to ˆ c,givenbyH(ˆ c)(1−xmin ¯π(ˆ c)),showsthat(3) is strictly increasing in ˆ con [0c∗]and strictly decreasing in ˆ con [c∗0]. The optimal mechanism is therefore characterized by the maximal value of ˆ con [0c∗]such that the buyer’s payoff with type c=0,givenby (0)−ˆ c−1 ˆ cxmin ¯π(u)du, is weakly greater than ¯π. If this condition is satisfied at ˆ c=c∗, i.e., if (0)−c∗−1 c∗ max{0¯π} (1)exp1 c v(t) (t) dtdc≥¯π (4) the optimal mechanism is characterized by the threshold c∗. Otherwise the optimal threshold is (uniquely) determined by the condition (0)−ˆ c−1 ˆ c max{0¯π} (1)exp1 c v(t) (t) dtdc=¯π (5) Problem (III) is thus uniquely solved by the function x¯π(c) =⎧ ⎪ ⎨ ⎪ ⎩ 1if c≤ˆ c max{0¯π} (1)exp1 c v(t) (t) dtif c>ˆ c (6) where ˆ c=c∗if (4) is satisfied and ˆ cis such that (5) holds otherwise. A.4 Proof of Proposition 3.3 Since a solution of problem (III) must also be a solution of (III)forsome ¯π∈[¯πmin ¯πmax], to solve the problem (III), we can consider the simpler problem where the buyer chooses from the collection of functions x¯πas defined in (6)with ¯π∈[¯πmin¯πmax].The buyer’s optimization problem can thus be written as max x¯π (1−ε)1 0 H(c)x¯π(c)dc+εinf c∈[01]x¯π(c)(c) −1 c x¯π(u)du(7) Define F(¯π) =1 0H(c)x¯π(c)dcand notice that infc∈[01]{x¯π(c)(c) −1 cx¯π(u)du}= ¯π. Problem (7) is then analogous to the problem of maximizing εF+(1−ε) ¯πsubject to 198 Sarah Auster Theoretical Economics 13 (2018) F=F(¯π),whereF(¯π) is the maximal expected payoff evaluated at Fas a function of the minimum payoff ¯π, analogous to a conventional utility-possibility frontier. The optimal mechanism is then determined by the tangency points of the buyer’s indifference curves of εF+(1−ε) ¯πand the possibility constraint F(¯π). The indifference curves are straight lines with a slope equal to −ε 1−ε. So as to derive the tangency points, it will be useful to distinguish the two cases where ¯πmin is positive or negative. Suppose first ¯πmin ≥0and let ˆπdenote the value of ¯π such that (4) is satisfied with equality. Below ˆπthe threshold ˆ cis equal to c∗, while above ˆπthe value of ˆ cis determined by (5). Since this condition depends on ¯π,toanalyze F(¯π), we need to find the derivative of ˆ cwith respect to ¯π. Taking the total differential of (5), we obtain dˆ c d¯π=− 1+1 (1)1 ˆ c exp1 c v(t) (t) dc 1−¯π (1)exp1 ˆ c v(t) (t) dt With this, we have  F(¯π) =⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 (1)1 c∗exp1 c v(t) (t) dtH(c)dcif ¯π∈¯πminˆπ −H(ˆ c) −1 (1)1 ˆ c exp1 c v(t) (t) dtH(ˆ c) −H(c)dcif ¯π∈(ˆπ ¯πmax] (8)  F(¯π) =⎧ ⎪ ⎨ ⎪ ⎩ 0if ¯π∈¯πminˆπ −H(ˆ c)1+1 (1)1 ˆ c exp1 c v(t) (t) dtdcdˆ c d¯πif ¯π∈(ˆπ ¯πmax] It can be verified that the derivative of F(¯π) at ¯π=ˆπexists and that  F(¯π) is strictly negative and weakly decreasing in ¯π. The function F(¯π) is thus differentiable, strictly decreasing, and weakly concave on its domain. More specifically, F(¯π) is linear on [0ˆπ]and strictly concave on [ˆπ ¯πmax]. This implies that there exists only one parameter point at which the buyer’s problem has multiple solutions, namely when εis such that −ε 1−ε= F(¯π),¯π∈[0ˆπ]. We can then derive the thresholds εand ε. The optimal mechanism is characterized by ¯π=0if  F(0)≤− ε 1−ε, while it is characterized by ¯π=¯πmax if  F(¯πmax)≥− ε 1−ε.We thus have ε=− F(0) 1− F(0)and ε=− F(¯πmax) 1− F(¯πmax). Since  F(0)and  F(πmax)take finite, nonzero values, we have 0<ε<ε<1. Consider next the case ¯πmin <0. Under this specification, ˆπ=¯πmin. The threshold ˆ c is therefore always determined by condition (5). For weakly negative values of ¯π,x¯π(c) is a function with a single step at (0)−ˆ cand so F(¯π) =(0)−¯π 0H(c)dcfor all ¯π≤0. The first and second derivatives of F(¯π) in this case are  F(¯π) =−H(0)−¯π<0and  F(¯π) =H(0)−¯π<0 On [¯πmin0],F(¯π) is thus strictly decreasing and strictly concave. The properties of F(¯π) for ¯π>0have been described above. At ¯π=0, the function F(π) has a corner.