IRRELEVANCY OF THE PRINCIPAL’S PRIVATE INFORMATION IN INDEPENDENT PRIVATE VALUES ENVIRONMENTS TYMOFIY MYLOVANOV AND THOMAS TROEGER Abstract. The existing mechanism design literature on linear independent private values offers conflicting answers to the question whether the assumption that the principal does not have private information is innocuous in the sense that the mechanism implemented by a principal whose valuation is uncertain to the agents coincides with the mechanism that would be optimal if the principal’s valuation were publicly known. In this paper, we show that the assumption about the privacy of the principal’s information is irrelevant if the players’ payoffs are monotonic in their types. This condition is satisfied in standard linear private values environments such as, e.g., bilateral bargaining, single and multi-unit auctions, and public good provision, where the default outcome is allocating the good to one of the agents (the seller) or providing no public good. Furthermore, this condition is necessary in regular bilateral bargaining environments (Mylovanov and Tr¨ oger 2012).

The optimal mechanism design in environments with asymmetric information is a fundamental field of economics. Under the assumption that all participants have quasi-linear preferences over market outcomes, a rich theory has emerged (see, e.g., the books by Krishna (2002) and Milgrom (2004)). A caveat in much of this theory is that the mechanism proposer (the principal) is assumed to have no private information although, in many applications, she is one of the market participants and, as such, should have private information. As an example, the designer of an auction is often in fact the seller of the auctioned good and is privately informed about her opportunity cost of selling.

Date: June 11, 2012. Mylovanov: University of Pennsylvania, Department of Economics, 3718 Locust Walk, Philadelphia, PA 19104, USA, [email protected]. Troeger: University of Mannheim, Department of Economics, L7, 3-5 68131 Mannheim, Germany, [email protected] We are deeply grateful to the editor and two anonymous referees for their very helpful comments. We gratefully acknowledge financial support from the National Science Foundation, grant 1024683, and from the German Science Foundation (DFG) through SFB/TR 15 “Governance and the Efficiency of Economic Systems.” 1

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In this paper, we focus on linear independent private values environments with single dimensional private information. This class is prominent in the literature on mechanism design (see, e.g., Ledyard and Palfrey (2007)) and includes bilateral bargaining, single and multi-unit auctions, and public good provision, among many other examples. Using the revelation principle, we focus on Bayesian incentive compatible and individually rational allocations. We take the viewpoint that one of the privately informed players has the right to propose an allocation, that is, she becomes the principal. For example, in a bilateral trade environment, the seller may be the principal. A few papers have studied linear and quasilinear independent private value environments. The recurrent theme in the literature is that in some of these environments the allocation implemented by the principal is not affected by whether the principal’s information is private or commonly known. In particular, Maskin and Tirole (1990) show a privately informed principal will offer the same mechanism as when her information is commonly known in their environment.1 Similar results have subsequently been obtained for other private value environments with transferable utility (Tan 1996, Yilankaya 1999, Balestrieri 2008, Skreta 2009). There are several linear and quasilinear independent private value environments in which this irrelevance result does not hold. Fleckinger (2007) modifies the environment of Maskin and Tirole’s to allow for countervailing incentives and shows in the new environment that a privately informed principal can extract the entire surplus while this not feasible for the principal if her valuation is commonly known the agents. The leading example in Mylovanov and Tr¨oger (forthcoming) has the same property; another example is offered in Mylovanov and Tr¨oger (2008). Finally, in the bilateral exchange environment with type-dependent outside option of the agent considered in Mylovanov and Tr¨oger (2012), the privately informed principal is almost surely strictly better off than when her information is commonly known. Intuitively, the agents’ uncertainty about the principal’s valuation might allow the principal to extract a higher surplus from the agents through relaxing their individual rationality constraints.2 For example, imagine that the principal has a choice between two lotteries over the outcomes, which generate the payoff of 1 and −1, respectively, to the sole agent. There are two equally likely 1

The main focus of Maskin and Tirole (1990) is on non-linear environments, in which case the result is different. 2 Manelli and Vincent (2010) and Gershkov et al. (forthcoming) show that in linear independent private values environments Bayesian incentive compatible allocations can be implemented in dominant strategies; this implies that the uncertainty about the principal’s valuation cannot benefit her through the channel of relaxing the incentive compatibility constraints.

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types of the principal who differ in their preference over the lotteries. The outside options are 0 for both players. If the principal’s type is unknown to the agent, the principal can implement her most preferred lottery by delaying the choice of the lottery until after the agent commits to accept the outcome of the lottery (i.e., the principal will offer a mechanism in which, after the mechanism is accepted, the principal chooses the lottery and the agent takes no action). Clearly, this allocation is not individually rational conditional on the type of the principal who prefers the lottery with the negative payoff for the agent. In this paper, we provide a condition under which the privacy of the principal’s information is irrelevant for the choice of the mechanism and the principal selects the same allocation she would select if her information were publicly known. To do this, we apply the characterization of a solution to the informed principal problem for independent private value environments obtained in Mylovanov and Tr¨oger (forthcoming, 2012), which implies that the privacy of the principal’s information is irrelevant if and only if the allocation that would be implemented if the principal’s information were commonly known, called a best separable allocation, is also ex-ante optimal. We show that in any linear environment with monotonic payoffs, any best separable allocation is ex-ante optimal, where monotonicity means that each agent’s payoff is monotonic in the same direction in his type for any outcome. This result generalizes the existing results in the literature on the irrelevance of the privacy of the principal’s information in linear independent private value environments. The monotonicity condition is tight in the sense that in some environments it becomes necessary. In Mylovanov and Tr¨oger (2012), we show that the monotonicity condition is necessary for the irrelevance result in the bilateral bargaining environment. We expect that this condition is necessary in many other linear independent private value environments. Much of the intuition for the monotonicity condition can be gained from the analysis of a bilateral trade environment in which a single unit is traded (Myerson and Satterthwaite, 1983). The optimal mechanism for the seller (principal) if her type (=cost) is commonly known is a posted price. Hence, the best separable allocation arises from a collection of these optimal posted prices. Under a regularity condition on the distribution of the agent’s valuation, a best separable allocation arises from pointwise maximization of the virtual surplus function. Any ex-ante optimal allocation is a best separable allocation because it still arises from pointwise maximization of the virtual surplus function (e.g., Ledyard and Palfrey, 2007). The reason this works is that, whatever type the principal has, the agent’s equilibrium payoff is always increasing in the agent’s type. Hence, it is always the same lowest-valuation type of the agent for whom the participation constraint is binding. In other words, any best separable

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allocation is ex-ante optimal because the agent’s payoff is monotonic in her type. We generalize this idea and show that in any linear environment with monotonic payoffs, any best separable allocation is ex-ante optimal and hence will be selected by the informed principal. The irrelevance of the principal’s private information in linear environments with monotonic payoffs shows that the informed principal problem can indeed by ignored in many applications including private as well as public good problems. The remainder of the paper is organized as follows. In Section 1, we present the model. The main results are in Section 2. 1. Model We consider the interaction of a principal (player 0) and n agents (players i ∈ N = {1, . . . , n}) in linear independent private values environments (Ledyard and Palfrey 2007). The players must collectively choose from a space of basic outcomes Z = A × IRn , where A represents a finite set of verifiable collective actions, and IRn is the set of vectors of agents’ payments. For example, A = {0, 1, . . . , n} may represent an environment where the collective action is the allocation of a single unit of a private good among the principal and the agents. Every player i has a payoff type ti ∈ Ti = [ti , ti ] that captures her private information. The product of players’ type spaces is denoted T = T0 × · · · × Tn . The types t0 , . . . , tn are realizations of stochastically independent Borel probability measures with cdf functions F0 , . . . , Fn with supp(Fi ) = Ti for all i; the prior beliefs F0 , . . . , Fn are common knowledge. Player i’s payoff function is denoted ui : Z × Ti → IR. Players have linear payoff functions, u0 (a, x, t0 ) = sa0 t0 + ca0 + x1 + · · · + xn , ui (a, x, ti ) = sai ti + cai − xi , i ∈ N, where x = (x1 , . . . , xn ) and sai and cai are some constants. An environment is regular if, for all agents i (not necessarily the principal), the type distribution Fi has a strictly positive continuous density fi , and the virtual valuation function ψi (ti ) = ti − (1 − Fi (ti ))/fi (ti ) is strictly increasing (here, Fi denotes the cumulative distribution function for Fi ).

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The players’ interaction results in an outcome that is a probability measure on the set of basic outcomes; the set of outcomes is denoted Z = A × IRn , where A denotes the set of probability measures on A, and IRn is the vector of the agents’ expected payments. If the players cannot agree on an outcome, some exogenously given disagreement outcome z obtains. The disagreement outcome z = (α, 0, . . . , 0) for some (possibly random) collective action α ∈ A. We normalize cai such that each player’s valuation from the disagreement outcome equals 0, that is, R a expected a (s t + ci )dα(a) = 0 for all i and ti . A i i A player’s (expected) payoff from any outcome ζ = (α, x) ∈ Z is denoted Z Z def a a ui (ζ, ti ) = ti si dα(a) + ci − xi = ui (z, ti )dζ(z), A

Z

where x0 = −x1 − · · · − xn . An allocation is a complete type-dependent description of the result of the players’ interaction; it is described by a map ρ(·) = (α(·), x(·)) : T → Z such that payments are uniformly bounded (that is, supt∈T ||x(t)|| < ∞, to guarantee integrability) and such that the appropriate measurability restrictions are satisfied (that is, for any measurable set B ⊆ A, the map T → IR, t 7→ α(t)(B) is Borel measurable, and x(·) is Borel measurable). Thus, an allocation rule describes the outcome of the players’ interaction as a function of the type profile. Alternatively, an allocation rule ρ can be interpreted as a direct mechanism, where the players i = 0, . . . , n simultaneously announce types tˆi ∈ Ti and the outcome ρ(tˆ0 , . . . , tˆn ) is implemented. Let G0 denote a cdf of a Borel probability measure on T0 that represents the agents’ belief about the principal’s type. Given an allocation ρ and a belief G0 , the expected payoff of type ti of player i if she announces type tˆi and all other players announce their types truthfully is denoted Z ρ,G0 ˆ Ui (ti , ti ) = ui (ρ(tˆi , t−i ), ti )dG−i (t−i ), T−i

where G−i denotes the product measure obtained from deleting dimension i of G0 , F1 , . . . , Fn . The expected payoff of type ti of player i from allocation ρ is Uiρ,G0 (ti ) = Uiρ,G0 (ti , ti ). We will use the shortcut U0ρ (t0 ) = U0ρ,G0 (t0 ), which is justified by the fact that the principal’s expected payoff is independent of G0 .

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An allocation ρ is called G0 -feasible if, for all players i, the G0 -incentive constraints (1) and the G0 -participation constraints (2) are satisfied, (1)

∀ti , tˆi ∈ Ti :

(2)

∀ti ∈ Ti :

Uiρ,G0 (ti ) ≥ Uiρ,G0 (tˆi , ti ), Uiρ,G0 (ti ) ≥ 0.

Given allocations ρ and ρ0 and a belief G0 , we say that ρ is G0 -dominated by ρ0 if ρ0 is G0 -feasible and 0

∀t0 ∈ supp(G0 ) :

U0ρ (t0 ) ≥ U0ρ (t0 ),

∃B ⊆ supp(G0 ), PrG0 (B) > 0 ∀t0 ∈ B :

U0ρ (t0 ) > U0ρ (t0 ).

0

Definition 1. An allocation ρ is strongly neologism-proof if (i) ρ is F0 -feasible and (ii) ρ is not G0 -dominated for any belief G0 that is absolutely continuous relative to F0 . In finite-type-space environments, any strong neologism-proofness is a perfectBayesian equilibrium in a mechanism-selection game in which any finite game form with perfect recall may be proposed as a mechanism (Mylovanov and Tr¨oger forthcoming).3 In the environments in which the type of the principal is commonly known to the agents, the principal will select a best separable4 allocation. Equivalently, a best separable allocation will be selected if the principal is restricted to offer a mechanism in which she is not a player herself.5 Definition 2. An allocation ρ is best separable if, for all point beliefs G0 , ρ is G0 -feasible and ρ is not G0 -dominated. Observe that the concept of a best separable allocation is entirely independent of the agent’s prior belief F0 . In independent private value environments, a best separable allocation which is not F0 -dominated is a strong solution (Myerson 1983). Definition 3. An allocation ρ is strong solution if, for all point beliefs G0 , ρ is G0 -feasible and ρ is not F0 -dominated. 3In

environments with infinite type spaces, there is no “natural” set of feasible mechanisms, nor is there an obvious choice for the definition of equilibrium. Any definition would have to deal with exempting deviations to mechanisms such that in the resulting continuation games no equilibrium exists, or in which the belief-equilibrium correspondence does not have the requisite continuity properties (c.f., Zheng (2002)). On the other hand, the idea behind our arguments in the earlier paper (Mylovanov and Tr¨oger, forthcoming, proof of Proposition 1) is simple and appears robust. 4Maskin and Tirole (1990) use the term full-information optimal allocation instead. 5Zheng (2002) calls such mechanisms “transparent”.

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Strong solutions yield a unique payoff prediction for the principal: if there are multiple strong solutions, each type of the principal obtains the same payoff in any of these.6 By the definition of strong neologism proofness, a strongly neologism-proof allocation cannot be dominated by a best separable allocation, while a best separable allocation ρ is a strong solution if it is strongly neologism-proof. Moreover, a best separable allocation is the only candidate for a strong solution.7 Hence, Remark 1. If both a strongly neologism proof allocation and a strong solution exist, then they lead to the same principal-type payoffs vector. Mylovanov and Tr¨oger (forthcoming) show that a strongly neologism proof allocation exists in independent private values environments that satisfy a separability condition. In these environments, a best separable allocation is a strong solution if and only if it is strongly neologism proof. Myerson (1983, Theorem 2) proves that in any environment with finite type spaces and a finite outcome space, a strong solution is a perfect Bayesian equilibrium outcome of an informed-principal game where any finite simultaneousmove game form is a feasible mechanism. Furthermore, in the environments with transferable utility, independent private values, and one agent, any equilibrium outcome is a strong solution, if the former exists. 2. Results For any belief G0 , the problem of maximizing the principal’s G0 -ex-ante expected payoff across all allocations that are G0 -feasible is Z (3) U0ρ (t0 )dG0 (t0 ). max ρ G0 -feasible

T0

Definition 4. An allocation ρ is ex-ante optimal if it solves problem (3) with G0 = F0 . In order to demonstrate that best separable allocations are strongly neologismproof, we use a result from Mylovanov and Tr¨oger (2012). 6For

any two strong solutions ρ1 and ρ2 , one can construct a third strong solution ρ3 by choosing for each type of the principal the better of the two allocation rules (because ρ1 and ρ2 are safe, ρ3 is safe as well). If there was a type t∗ that is better off in ρ3 compared to ρ1 or ρ2 , then ρ3 would dominate ρ1 or ρ2 , a contradiction. 7Assume that there exists an allocation ρ that is strong solution and is not best separable. Then, there exists t00 ∈ T0 with the payoff strictly less than in any best separable allocation. By incentive compatibility of ρ and any best separable allocation, there exists a set of types of positive measure containing t00 with the payoffs strictly less than in any best separable allocation. Thus, ρ is F0 -dominated by any best separable allocation.

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Corollary 1. (Mylovanov and Tr¨oger 2012) A best separable allocation is strongly neologism-proof if and only if it solves problem (3) for all G0 that are absolutely continuous relative to F0 . Our results will use the following monotonicity condition. Definition 5. The players’ payoffs are monotonic in types if8 sai ≥ 0

(4)

for all i, a.

It is well-known (see, e.g., Ledyard and Palfrey (2007)) that in any regular linear environment with monotonic payoffs, a best separable allocation can be computed by point-wise maximization of the virtual surplus function ! Z n X sa0 t0 + ca0 + (sai ψi (ti ) + cai ) dα(a) (α ∈ A, t ∈ T), V (α, t) = A

i=1

and choosing a payment scheme such that the resulting allocation is incentive feasible, and such that type ti of each player i obtains her disagreement payoff 0. A straightforward extension of the corresponding arguments shows that an ex-ante optimal allocation is also given by point-wise maximization of the virtual surplus function. Hence, we have the following result. Proposition 1. Consider a linear regular environment with monotonic payoffs. Then any best separable allocation is ex-ante optimal. In any such allocation ρ(·) = (α(·), x(·)), α(t) ∈ arg max V (α, t). α∈A

Hence, using Corollary 1, we have the following result. Proposition 2. Consider a linear regular environment with monotonic payoffs. Then any best separable allocation is strongly neologism-proof. The interpretation of Proposition 2 is simple: if payoffs are monotonic, then the privacy of the principal’s private information is irrelevant for mechanism design—she will offer the same mechanism as when her type is commonly known, and she does not gain from being a player in her own mechanism. Next, we extend the result to the irregular environments in which the pointwise maximizer of the virtual surplus function is non-monotone. Proposition 3. Consider a linear environment with monotonic payoffs. Then any best separable allocation is ex-ante optimal and strongly neologism-proof. 8Assuming

weakly increasing payoff functions is without loss of generality. If the payoff of some player i is weakly decreasing for all actions, we can redefine her types as tˆi = −ti and obtain a weakly increasing payoff function.

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Proof. For any i ∈ N , ti ∈ Ti , and α : T → A, define Z α ˆ i (t) = sai dα(t), ZA α ˆ i (ti ) = α ˆ (ti , t−i )f−i (t−i )dt−i . T−i

A straightforward extension of Myerson’s Lemma 3 (1981, p. 64) to linear independent private values environments yields the following result: Suppose that α solves Z max (5) V (α(t), t)f (t)dt α:T→A

(6)

s.t.

T

ti 7→ α ˆ i (ti )

is weakly increasing on Ti for all i ∈ N .

Suppose also that (α, x) satisfies Z (7)

cai dα(t)

xi (t) = α ˆ i (t)ti +

Z

ti

−

α ˆ i (t−i , t˜i )dt˜i .

ti

A

and the principal incentive and participation constraints (1) and (2), i = 0, hold. Then (α, x) is ex-ante optimal. Define functions Hi , Gi , and ci (i ∈ N ) as in Myerson (1981, p. 68). Analogous to an argument by Myerson (1981, (6.10)), the objective in (5) can be rewritten as ! Z Z X sa0 t0 + ca0 + (sai ci (ti ) + cai ) dα(t)f (t)dt T

(8)

A

−

i∈N

XZ i∈N

(Hi (Fi (ti )) − Gi (Fi (ti )))dˆ αi (ti ) . | {z } Ti

=:∆α (ti )

Analogously to Myerson (1981, p. 69-70), one argues that the constraint (6) implies ∆α (ti ) ≥ 0. allocation. Then, ∆α (ti ) = 0 and α = Let (α, x) beR any best separable
P α is such that A sa0 t0 + ca0 + i∈N (sai ci (ti ) + cai ) dα(t) is maximal for each type profile t. Hence, (8) is maximized at α = α, subject to the constraint (6). Because (α, x) is best separable and the environment has private values, the principal incentive and participation constraints are satisfied. Note that the argument applies for any F0 and therefore (α, x) is ex-ante optimal and strongly neologism-proof.  The irrelevance result extends to the irregular case because the virtual surplus function is additively separable in types of different players and multiplicatively separable in the agent’s virtual utility ψi (ti ) and the marginal value

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of action sai . Consequently, the objectives in (5) as (8) are additively separable in the agents’ and the principal’s types and the type-wise maximizer of (8) is not affected by the beliefs about the principal’s type. References Balestrieri, Filippo, “A modified English auction for an informed buyer,” January 2008. mimeo. Fleckinger, Pierre, “Informed principal and countervailing incentives,” Economics Letters, February 2007, 94 (2), 240–244. Gershkov, Alex, Jacob K. Goeree, Alexey Kushnir, Benny Moldovanu, and Xianwen Shi, “On the equivalence of Bayesian and dominant strategy implementation,” Econometrica, forthcoming. Krishna, Vijay, Auction Theory, Academic Press, 2002. Ledyard, John O. and Thomas R. Palfrey, “A general characterization of interim efficient mechanisms for independent linear environments,” Journal of Economic Theory, March 2007, 133 (1), 441–466. Manelli, Alejandro M. and Daniel R. Vincent, “Bayesian and Dominant Strategy Implementation in the Independent Private Values Model,” Econometrica, November 2010, 78 (6), 1905–1938. Maskin, Eric and Jean Tirole, “The principal-agent relationship with an informed principal: The case of private values,” Econometrica, 1990, 58 (2), 379–409. Milgrom, Paul, Putting Auction Theory to Work, Cambridge University Press, 2004. Myerson, Roger B., “Mechanism design by an informed principal,” Econometrica, 1983, 51 (6), 1767–1798. Mylovanov, Tymofiy and Thomas Tr¨ oger, “Optimal Auction Design and Irrelevance of Privacy of Information,” mimeo, 2008. and , “Mechanism design by an informed principal: the quasi-linear private-values case,” mimeo, 2012. and , “Informed-principal problems in environments with generalized private values,” Theoretical Economics, forthcoming. Skreta, Vasiliki, “On the informed seller problem: optimal information disclosure,” Review of Economic Design, November 2009, 15 (1), 1–36. Tan, Guofu, “Optimal Procurement Mechanisms for an Informed Buyer,” Canadian Journal of Economics, August 1996, 29 (3), 699–716. Yilankaya, Okan, “A note on the seller’s optimal mechanism in bilateral trade with twosided incomplete information,” Journal of Economic Theory, July 1999, 87 (1), 125– 143. Zheng, Charles Z., “Optimal auction with resale,” Econometrica, November 2002, 70 (6), 2197–2224.