On the Informed Principal Model with Common Values

On the Informed Principal Model with Common Values Anastasios Dosis September 23, 2016 Abstract In the informed principal model with common values, I identify as key in the characterisation of equilibrium mechanisms of the standard three-stage game studied in Maskin and Tirole (1990, 1992), the set of Belief-Free Undominated (BFU) mechanisms. A BFU is simply a mechanism that is undominated within the set of incentive compatible and individually rational mechanisms relative to all possible beliefs. A BFU and every mechanism that dominates it relative to the prior beliefs, if at all, can be sustained as an equilibrium mechanism. This result holds even in environments without transfers and/or the usual sorting conditions. Furthermore, I specify an intuitive sufficient condition that allows for the complete characterisation of the set of equilibrium mechanisms. This condition states that there exists a sequence of strictly incentive compatible and strictly individually rational mechanisms relative to all possible beliefs with payoffs converging to these resulting from a BFU mechanism. If this condition is satisfied, then the lower bound in the payoff of any type is that from a BFU mechanism. KEYWORDS: Mechanism design, informed principal, belief-free mechanisms, perfect Bayesian equilibria JEL CLASSIFICATION: C72, D82 1 INTRODUCTION Conventional contract theory assumes that an uninformed principal wishes to enter into a contract agreement with an agent who either possesses private information (adverse selection) or/and can take unobservable actions that affect the payoff of both the principal Department of Economics - ESSEC Business School and THEMA, 3 Av. Bernard Hirsch, B.P. 50105, Cergy, 95021, France, Email: dosis@essec.com. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). 1

and the agent (moral hazard). The question then becomes one of characterising a mechanism (contract) that maximises the expected payoff of the principal subject to a set of constraints that induce the agent to be both obedient and truthful. Nonetheless, in many real-life situations, it is also the principal, along with the agent, who possesses private information about the state of the world or take non-observable actions (or both). For instance, consider a seller and a buyer who both have imperfect information about the true value of the object they wish to trade but each one of them receives a private signal that is statistically dependent on the true value. In that case, when one of the two parties proposes a mechanism to the other party, the proposal itself may reveal information about the value of the object which in its own turn may influence the behaviour of both the buyer and the seller in the mechanism. In this article, I analyse such general principal-agent relationships with an informed principal. In particular, I study the set of equilibrium mechanisms of the standard threestage game of Maskin and Tirole (1990, 1992). The principal offers a mechanism. The agent accepts or rejects. Upon acceptance, the mechanism is executed. Rejection leads to the termination of further negotiations and the reservation allocation is in force. I suppose that only the principal has private information and this information might affect the payoff of the agent, i.e. common values, hence the analysis is closer to Maskin and Tirole (1992) (henceforth MT) rather than Maskin and Tirole (1990). Key in the characterisation of the set of equilibrium mechanisms of the three-stage game is the specification of the set of Belief-Free Undominated (BFU) mechanisms. A BFU is a mechanism that is undominated within the set of incentive compatible and individually rational mechanisms relative to any beliefs. The main result I establish is that the BFU and every mechanism that dominates it relative to the prior beliefs, if at all, can be sustained as an equilibrium mechanism. Moreover, I provide a sufficient condition that fully characterises the set of equilibrium mechanisms. Intuitively, this condition states that the payoff from a BFU mechanism can be approached arbitrarily close by strictly incentive compatible and strictly individually rational mechanisms relative to all possible beliefs. If this condition is satisfied, then the lower bound in the payoff of any type is that from a BFU mechanism. Evidently, the article is closely related to MT and therefore it is vital to point out how it distinguishes from it. MT take the following approach: first, they characterise efficient allocations relative to any beliefs (not necessarily the prior ones) as well as the Rothschild- Stiglitz-Wilson (RSW) allocation, i.e. named after the seminal contributions of Rothschild and Stiglitz (1976) and Wilson (1977). Consequently, they establish that if the RSW allocation is efficient relative to some non-degenerate beliefs, then the equilibrium set consists of those incentive compatible and individually rational allocations that weakly dominate 2

the RSW allocation. They show that the sufficient condition, i.e. that the RSW allocation be efficient relative to some non-degenerate beliefs, holds in real two-dimensional environments, i.e. environments with transferable utility, in which preferences satisfy singlecrossing, monotonicity and that the agent s reservation utility is increasing in the type of the principal. In this article I take a different approach. I place the BFU mechanisms in a central position and I show that the BFU mechanisms are key in the characterisation of equilibrium mechanisms. This applies even to environments without transferable utility that might not satisfy any of the conditions described above. In fact, one can easily show that a RSW mechanism is indeed a BFU mechanism. Therefore, the article strengthens the results of MT rather than providing criticism. Undoubtedly, the seminal article in the literature of mechanism design by an informed principal is Myerson (1983). This article introduces the general collective Bayesian problem with multiple agents and interdependent values. It elegantly axiomatically characterises reasonable solutions for the selection of the mechanism by the informed principal. Even though the analysis is mostly cooperative, it touches upon the non-cooperative equilibria of a game similar to this in MT without intending to fully characterise the set of equilibrium mechanisms but rather showing that the mechanisms characterised axiomatically can be supported as non-cooperative equilibrium mechanisms. Unlike Myerson (1983), Maskin and Tirole (1990) consider the problem of mechanism design by an informed principal in a noncooperative environment under the assumption that the private information of the principal does not directly affect the payoff of the agent. With quasilinear utilities, they show that the principal neither gains nor loses from her private information, i.e. the ex ante, interim and ex post optimal mechanisms are equivalent. Similar results are obtained in Myerson (1985), Tan (1996), Yilankaya (1999) and Skreta (2011). Nonetheless, this equivalence result is challenged in Fleckinger (2007) and more recently in Mylovanov and Tröger (2014). In a principal-agent model with correlated types, Cella (2008) shows that the principal can extract a larger share of the surplus than when information is public. Skreta (2011) shows in an informed seller model that the optimal information disclosure crucially depends on whether values are private or common. Koessler and Skreta (2014) and Balestrieri and Izmalkov (2016) study optimal mechanisms for an informed seller with common values when the seller cares only about revenue. Dosis (2016) allows the seller to have a type-dependent valuation for the object and characterises the set of interim optimal mechanisms. He also shows that a slight modification of the three-stage game allows only interim optimal mechanisms to be supported as equilibrium mechanisms. Severinov (2008) analyses a general informed principal environment with interdependent 3

values and many agents and establishes conditions under which a variant of the threestage game has a perfect sequential equilibrium, i.e. Grossman and Perry (1986), that is ex post efficient. Balkenborg and Makris (2015) analyse an environment with common values and transferable utility. They identify a solution for the informed principal with appealing properties, i.e. the assured allocation, and they show that it is sustainable as an equilibrium in the three-stage game. The remainder of this article is organised as follows. In Section 2, I present the general informed principal model and I define, mechanisms, incentive compatibility, dominance and a BFU mechanism. I also provide a sufficient condition in the set of direct-revelation mechanisms. In Section 3, I characterise the set of equilibrium mechanisms of the threestage game. Section 4 concludes the article. 2 THE MODEL Actions and Payoffs. Consider two players i = 1, 2. For player i = 1, let the set of possible types be denoted by Θ (finite). The two players wish to select a contractible action x from a compact set X. Let x 0 denote the reservation action and assume that x 0 X. The reservation action corresponds to some action that takes place if the two players do not come to an agreement. It can be thought either as the outside option or as a prior contract that binds the two players which they wish to renegotiate. The preferences of all players and types over the set of actions admit an expected utility representation, with the (VNM) utility index of player i be denoted by u i (x θ), a continuous function for every i and θ. Let p(θ) denote the prior belief that player i = 2 assigns to player i = 1 being θ. The payoff of player i from the reservation action is allowed to be type-dependent, or u i (x 0 θ). Suppose that player i = 1 is the principal and player i = 2 is the agent. 1 This distinction, due to Myerson (1983), is only in terms of which party selects the mechanism to be played. Mechanisms. A mechanism is a mapping from the set of types Θ to the set of contractible actions X. Because only the principal has private information, there is no loss of generality in assuming that only she makes meaningful announcements in the mechanism. Due to the reservation principle, there is no loss of generality in concentrating on direct revelation mechanisms in which the principal simply announces a type (not necessarily the true one). A representative mechanism is denoted by µ : Θ X. Let q(θ) denote a belief held by the agent that the principal s type is θ, where θ Θ q(θ) = 1. q may be 1 For simplicity, the principal will be pronounced with a feminine (she or her) and the agent with a masculine (he or him). When I refer to any player (either the principal or the agent), I will use the feminine pronounce. 4

different than p as the agent might have updated his information about the type of the principal. The following definitions are by now standard: Definition 2.1. A mechanism is incentive compatible (IC) if and only if u 1 (µ(θ) θ) u 1 (µ(ˆθ) θ) for every θ, ˆθ θ. Definition 2.2. An incentive compatible mechanism is individually rational (IR) relative to beliefs q iff u 1 (µ(θ) θ) u 1 (x 0 θ) θ and θ q(θ)u 2(µ(θ) θ) θ q(θ)u 2(x 0 θ) Let M ICR (q) denote the set of IC and IR mechanisms relative to beliefs q. Key in the characterisation of the mechanism selection by the principal is the concept of mechanism domination. This is formally stated as follows: Definition 2.3. A mechanism µ dominates another mechanism µ relative to beliefs q iff µ, µ M ICR (q) and u 1 (µ (θ) θ) u 1 (µ(θ) θ) for every θ with the inequality being strict for at least one θ. It is convenient to denote by M D (µ q) the set of mechanisms that dominate mechanism µ relative to beliefs q. We are now ready to define the most fundamental set of mechanisms, the belief-free undominated mechanisms. Definition 2.4. µ is a Belief-Free Undominated (BFU) mechanism iff µ M ICR (q) for every q and q M D (µ q) =. In words, the set of BFU mechanisms includes all mechanisms that are IC and IR relative to all possible beliefs and there is no other mechanism with the same property that dominates them. Every mechanism in the set of BFU mechanisms is payoff equivalent and hence, without loss of generality, let us assume that there is a unique BFU mechanism and denote this as µ. It is only straightforward to verify that a Rothschild-Stiglitz-Wilson (RSW) mechanism, as this is defined in MT, is a BFU mechanism. Note also that given our assumption that x 0 X, the set of BFU mechanisms is non-empty. In environments with private values, it should be clear that a BFU mechanism coincides with the ex post optimal mechanism, i.e. the mechanism that could be optimal if the type of the principal was observable by the agent. For the complete characterisation of the set of equilibrium mechanisms, one needs to impose further structure. Two additional straightforward definitions are required. Definition 2.5. A mechanism is strictly incentive compatible iff u 1 (µ(θ) θ) > u 1 (µ(ˆθ) θ) for every θ, ˆθ θ Definition 2.6. An incentive compatible mechanism is strictly individually rational given beliefs q iff u 1 (µ(θ) θ) > u 1 (x 0 θ) for every θ and θ q(θ)u 2(µ(θ) θ) > θ q(θ)u 2(x 0 θ). 5

Let us denote by M SICR (q) the set of SIC and SIR mechanisms relative to beliefs q. The following condition is a sufficient condition for the complete characterisation of the set of equilibrium mechanisms. Condition 2.7. {µ n } n M SICR (q) q: {u 1 (µ n (θ) θ)} n u 1 (µ (θ) θ) θ. In words, Condition 2.7 states that the payoff from a BFU mechanism can be approached by any type infinitesimally close by SIC and SIR mechanisms for every possible beliefs. The value of Condition 2.7 will become apparent in the characterisation of the equilibrium mechanisms. The Game. As I already argued in the introduction, the goal of the paper is to characterise the set of mechanisms that can be supported as perfect Bayesian equilibria (PBE) in the extensive form game studied in MT. This extensive form has three stages. In the first stage the principal proposes a mechanism. In the second stage the agent accepts or rejects the mechanism. Acceptance of the proposal leads to the execution of the mechanism that has been selected, in the third stage, whereas rejection terminates further negotiations and the reservation action is selected. A strategy for the principal consists of a proposal of a mechanism in the first stage, for every possible type, and an announcement of a message in the third stage if the agent accepts the mechanism for every possible proposal of a mechanism. A strategy for the agent consists of an acceptance or rejection decision in the second stage for every possible proposal. A (posterior) belief for the agent is a probability distribution over the set of types of the principal for every possible proposal of a mechanism. A pure strategy perfect Bayesian equilibrium is a profile of strategies and beliefs such that: (i) The strategy of every player is optimal given the strategy of the other player and his beliefs at every node of the game tree (sequential rationality), (ii) Beliefs are updated by Bayes rule whenever a node is reached given the equilibrium strategies, and, (iii) Beliefs are arbitrarily determined in nodes that are not reached given the equilibrium strategies but are consistent with the equilibrium strategies. 3 EQUILIBRIUM MECHANISMS Myerson (1983) shows that for every possible equilibrium where different groups of types offer distinct mechanisms there is a payoff-equivalent equilibrium in which all types offer the same mechanism. This illuminating observation is known as the inscrutability principle and it bears a great value when one studies informed principal problems. Thanks to this principle, there is no loss of generality in focusing on equilibria in which all types offer 6

the same mechanism. The following theorem sheds more light on the set of equilibrium mechanisms of the three-stage game. Theorem 3.1. µ is an equilibrium mechanism. If M D (µ p), then every µ M D (µ p) is an equilibrium mechanism. Proof. Consider the following profile of strategies. Every type of the principal proposes the same mechanism µ in the first stage and truthfully reveals her type in the third stage. The agent accepts in the second stage. The posterior beliefs remain equal to the prior beliefs if the principal selects mechanism µ and the agent accepts the proposal, i.e. along the equilibrium path. The question now is whether we can find beliefs such that for every other possible mechanism that improves the payoff of at least one type of the principal over µ, the agent will reject the mechanism. Suppose to the contrary that there exists a mechanism µ M ICR (q) for every q and a type θ such that: u 1 ( µ( θ) θ) > u 1 (µ ( θ) θ) (3.1) and u 1 ( µ(θ) θ) < u 1 (µ (θ) θ) θ θ (3.2) Because µ M ICR (q) for every q, the following is true: u 2 ( µ( θ) θ) u 2 (x 0 θ) (3.3) Consider now mechanism ˆµ, where ˆµ( θ) = µ( θ) and ˆµ( θ) = µ (θ) for every θ θ. Due to Ineq. (3.1) and (3.2), ˆµ M ICR (q) for every q and ˆµ q M D (µ q) which contradicts that q M D (µ q) =. A similar argument establishes the second statement of the theorem. Notably, one of the central solutions identified in Myerson (1983) is the strong solution; a safe and undominated mechanism. It is not difficult to verify that a strong solution exists if and only if M D (µ p) =. Theorem 3.1 only partially characterises the set of equilibrium mechanisms. Even though all mechanisms that dominate the BFU mechanism relative to the prior beliefs can supported as PBE mechanisms, we do not know what happens for mechanisms that provide payoff strictly lower that this in a BFU mechanisms for some types. Condition 2.7 allows us to prove the following theorem: Theorem 3.2. If Condition 2.7 is satisfied, then every equilibrium mechanism µ is such that u 1 (µ(θ) θ) u 1 (µ (θ) θ) for every θ. 7

Proof. Consider a type θ and a hypothetical equilibrium mechanism µ such that u 1 (µ ( θ) θ) u 1 ( µ( θ) θ) = ɛ > 0. Due to the inscrutability principle, suppose that µ M ICR (p). Because of Condition 2.7, there exists a sequence of mechanisms {µ n } n M SICR (q) for every q and N ɛ such that u 1 (µ ( θ) θ) u 1 (µ Nɛ ( θ) θ) < ɛ or u 1 (µ Nɛ ( θ) θ) > u 1 ( µ( θ) θ) Suppose that type θ offers mechanism µ Nɛ. Because this mechanism is strictly incentive compatible and strictly individually rational for all possible beliefs, player i = 2 should accept, otherwise the equilibrium fails to be sequentially rational. Type θ can achieve higher payoff by deviating to mechanism µ Nɛ ; a contradiction. 4 CONCLUSION In this article, I studied a general informed principal model with common values. I showed that key in the characterisation of equilibrium mechanisms of the three-stage game of Maskin and Tirole (1990, 1992) is the notion of a belief-free undominated mechanism (BFU). A BFU and every mechanism that dominates it relative to the prior beliefs, if at all, can be sustained as an equilibrium mechanism. I also provided a sufficient condition that allows for the complete characterisation of the set of equilibrium mechanisms. REFERENCES Balestrieri, Filippo, and Sergei Izmalkov, Informed seller in a Hotelling market, Available at SSRN 2398258, (2016). Balkenborg, Dieter, and Miltiadis Makris, An undominated mechanism for a class of informed principal problems with common values, Journal of Economic Theory, 157 (2015), 918 958. Cella, Michela, Informed principal with correlation, Games and Economic Behavior, 64 (2008), 433 456. Dosis, Anastasios, Interim Optimal Trading for and Informed Seller, Available at SSRN http://ssrn.com/abstract=2828185, (2016). Fleckinger, Pierre, Informed principal and countervailing incentives, Economics Letters, 94 (2007), 240 244. 8

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