Multi-Agent and Common Agency Games with Complete Information: A Survey.∗ Alberto Galasso† July 18, 2004

Abstract This paper provides a general overview of the literature on multi-principal multi-agent games with complete information. The analysis compares standard models with more recent results and reviews possible applications of these games.

1

Introduction

The classical principal-agent model considers cases in which asymmetries of information exist between players at the time of contracting. The absence of imperfect information renders the problem trivial: the agent receives just his reservation utility and the principal implements an efficient outcome and appropriates of the difference between the total social surplus and the agent’s reservation utility. This triviality can be removed enlarging the number of players in the game. Introducing a multiplicity either of agents or of principals and keeping the complete information assumption there can arise inefficiencies due to externalities among players. In this survey we are going to study these types of games and to compare their outcomes. To focus on the complete information setting can appear too superficial but, as Laussel and Le Breton (2001), we stress two reasons for focusing on the complete information case. First, as stated by Bernheim and Whinston (1986a), “although a fully satisfactory theory of menu actions would certainly allow for incomplete information, we shall soon see that significant complexities arise even when there is no private information.” Second, ruling out informational considerations permits us to isolate ∗

I am grateful to Clara Graziano for useful discussions and suggestions. Departement of Economics at the LSE, Houghton Street, London WC2A 2AR. E-mail: [email protected] †

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the effect on the outcome of the game of the competition between the principals from the effect of the existence of some private information. The paper is organized as follows. In Section 2, we present a general common agency model, in Section 3 we describe the structure of the equilibrium payoffs and in Sections 4 and 5 we present extensions of the standard model. In Section 6 we analyze the multi-agent model and in Section 7 and Section 8 we describe extensions of this game. In Section 9 we present an original game to study the effect of bargaining power on equilibrium outcome. Section 10 is devoted to the analysis a model displaying multiplicity both of agents and of principals. Section 11 concludes.

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The Common Agency Game

Bernheim and Whinston (1986a) present a game in which an agent (the auctioneer) selects an action affecting the well being of M principals (the bidders) each of whom offers a menu of payments contingent on the action chosen. This common agent chooses the action from a set of possible choices X and the payoffs of the agent and principal m are described by:

(1)

u(x) +

M X

ti (x)

i=1

(2)

fm (x) − tm (x).

The intuition is the following. If the agent chooses the action x ∈ X he receives ti (x) from each principal i = 1, ..., M and experiences an utility (or disutility) of u(x). Similarly principal m receives a gross monetary payoff described by the function fm (x) to which it has to be subtracted the monetary reward paid to the agent. As an illustrative example we can consider auctions in which bidders name a menu of offers for the various possible actions available to the auctioneer (e.g. allocations of the components of a construction project)1 .

2.1

Nash Equilibria of the Game

The M principals simultaneously offer contingent transfers to the agent who subsequently chooses the action that maximizes his total payoff. In a Nash equilibrium the agent chooses an action in order to maximize (1) given the transfers of the principals and each principal chooses a transfer that is a best reply to the transfers of the other M − 1 principals. 1

Peters (2003) shows that the results obtained offering a single menu are robust to the possibility that principals offer sets of menus from which final contract is negotiated.

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There is a large number of Equilibria and only some of them select an efficient action, namely an action maximizing total surplus or, more formally : ∗

x ∈ arg max u(x) + x

M X

f m (x).

i=1

Inefficient Equilibria are not eliminated by standard refinements as trembling hand perfectness. The fundamental question that Bernheim and Whinston(1986a) want to answer is the following one: are all of these Equilibria equally plausible? They argue that they are not. They analyze a subclass of equilibrium that are considered to be focal especially in situations in which no communication occurs between principals. In these Equilibria each principal plays a truthful strategy that can be illustrated with the following definition: Definition 1 tm () is said to be a truthful strategy relative to x0 if and only if for all x∈X , £ ¤ tm (x) = max 0, f m (x) + tm (x0 ) − f m (x0 ) .

³ ´ m M 0 {t }m=1 , x is said to be a Truthful Nash Equilibrium if and only if it is a Nash

0 Equilibrium and {f m }M m=1 are truthful strategies relative to x .

Note that in any Truthful Equilibrium each bidder offers a reward for action x that exactly refers his net willingness to pay for x as opposed to the equilibrium outcome x0 . We can describe various properties of Truthful Nash Equilibria that renders this set an appealing refinement of the Nash set: 1. For any set of offers by his opponents {tj }j6=m principal m’s best response correspondence contains a truthful strategy. 2. In all Truthful Nash Equilibria the agent selects an efficient action. 3. Net payoffs of the principals are easily characterized: for each efficient action there exists a (nonempty) set of net payoffs for the principals such that the payoffs of the corresponding Truthful Nash Equilibrium is on the Pareto Frontier of this set (see Section 3 for computational details). 4. If we restrict the game to two principals and the agent has no inherent preferences over his action set we have uniqueness (in Section 3 we analyze other contributions on uniqueness).

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2.2

A Simple Example

To better understand the concepts introduced above let us describe a simple example. Consider a game where there are two principals and one agent. The agent can choose one out of four different actions: the payoffs of the principals are monetary, and are determined by the action chosen by the agent. The payoff of the agent depends only on transfers of the principals, and he is otherwise indifferent among the actions. As a tiebreak rule it is assumed that the agent resolves his indifference in favor of the allocation with the highest social payoff. The payoffs of the two principals are: u1 = (8, 0, 6, 5) and u2 = (0, 7, 6, 5). Notice that the efficient allocation is the third one (the sum of the payoffs is twelve). The game has a multiplicity of Equilibria. Following Prat and Rustichini (1998) we classify them in three types: 1. Those in which each agent bids on his most favorite action and the agent chooses the first action as: t1 = (7, 0, 0, 0) ; t2 = (0, 7, 0, 0). 2. Those in which principals coordinate on a “wrong action” and the agent still chooses the first action as: t1 = (6, 0, 0, 3) ; t2 = (0, 6, 0, 3). 3. Those in which the action chosen is the efficient one. transfers supporting this equilibrium is the following:

A particular case of

t1 = (3, 0, 1, 0) ; t2 = (0, 3, 2, 1). In this equilibrium the difference between the payoff from an action and the transfer for that action is the same for each principal, when the transfer is positive: this is a Truthful Nash Equilibrium.

2.3

Coalitions

For environments in which bidders can communicate with each other, a strong additional justification for focusing the attention upon Truthful Nash Equilibria is available. Indeed, every Truthful Nash Equilibrium is Coalition Proof : no coalition of principals can arrange a stable and mutually preferred joint deviation. The concept of Coalition Proof Nash Equilibrium is developed in Bernheim et alt. (1987a,1987b). They analyze environments in which players can freely discuss their strategies but cannot make binding commitments. In such circumstances, agreements among players are meaningless unless they are self enforcing. Moreover the Nash best response property is certainly a requirement for self enforceability but it is not

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generally sufficient: it may be possible for coalitions of players to arrange plausible and mutually beneficial deviations from Nash Equilibria. To properly understand the concept of Coalition Proof Nash Equilibrium it is necessary to consider another refinement: the Strong Nash Equilibrium proposed by Aumann(1959). This equilibrium concept requires stability against deviations by every conceivable coalition: an equilibrium is strong if no coalition, taking the actions of its complement as given, can cooperatively deviate in a way that benefits all its members. The analysis of Bernheim et alt.(1987a, 1987b) points out the limits of the Strong Nash Equilibrium. Indeed, this concept is actually “too strong”. In particular coalitions are allowed to much freedom in choosing their joint deviations: it is not required that the agreements must be immune to deviations by subcoalitions. This inconsistency in the Strong Nash Equilibrium concept is evident in the stringent requirement that a Strong Nash Equilibrium must be Pareto efficient. As a result of this requirement, Strong Nash Equilibria almost never exist. To overcome these limitations Bernheim et alt.(1987a) propose the concept of Coalition-Proof Nash equilibrium. An agreement is coalition-proof if and only if it is Pareto efficient within the class of self enforcing agreements. In turn, an agreement is self-enforcing if and only if no coalition of players can plan to deviate in a way that makes all of its members better off (taking the actions of its complement as fixed). The big difference with the Strong Equilibrium Concept is that a valid deviation by a coalition must be self-enforcing, in the sense that no proper subcoalitions can reach a mutually beneficial agreement to deviate from the deviation. Likewise, any potential deviation by a sub-coalition must be judged by the same criterion and so on. To better understand the difference between Strong Nash Equilibrium and Coalition Proof Nash Equilibrium, consider the classical Prisoner’s Dilemma Game as described in Table 1.

Alfred

Barbara Don’t Confess Confess Don’t Confess 3,3 0,4 Confess 4,0 1,1 Table1 : Prisoners’ Dilemma

The Unique Nash Equilibrium strategy profile is to play Confess for both players. However this Nash Equilibrium is not a Strong Equilibrium: Alfred and Barbara can form a coalition and decide not to confess. Moreover, this agreement is not self enforcing: both players have an incentive to deviate if the other confesses. Therefore we cannot consider this deviation for the Coalition Proofness Criterion and we can say that the unique equilibrium of the Prisoner’s Dilemma is not a Strong Equilibrium but it is a Coalition Proof Nash Equilibrium.

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3

The Structure of Equilibrium Payoffs

Laussel and Le Breton (2001) investigate how the fundamentals of the common agency setting affect the equilibrium payoff of the game. In doing this, they highlight a precise link between common agency theory and cooperative game theory. Given a common agency game they describe a corresponding transferable utility cooperative game, denoted by W , with the set of players being the set of principals. In particular, for each coalition of principals S they calculate the highest joint payoff W (S) of the agent and principals in S. In this new environment a vector u is a vector of equilibrium payoffs for the principals if and only if uPis a Pareto optimum of the polyhedron defined by the set of linear inequalities: i∈S ui ≤ W (N) − W (N\S) for all S. This system of inequalities is necessary to have that the total payoff of group S can never exceed the contribution of group S to the total surplus. The main objective of Laussel and Le Breton (2001) is to know the magnitude of the rent obtained by the agent as a result of the competition between principals. This rent is an important benchmark because it is the minimum of the amounts the agent can obtain in other games in which he has either some private information or more bargaining power. To do so they identify conditions for which, in the common agency game, the agent gets no rent in all Equilibria (what they call the no-rent property). They show that this property is conceptually related to notions introduced by cooperative game theorists in their investigation of the core. Indeed it is proved that balancedness of W is sufficient for the no rent property if there are two principals; total balancedness is sufficient if there are three principals or the game is symmetric and that convexity of W guarantees no rent property regardless the number of principals. The analysis of Laussel and Le Breton on the structure of equilibrium payoff has been continued by Bergemann and Valimaki (2003) showing that the truthful equilibrium payoff is unique if and only if the marginal contributions of the principals to the value of the grand coalition are weakly superadditive. They show that in such equilibria, all principals receive their marginal contribution as payoffs and therefore they call equilibria satisfying this property marginal contribution equilibria.2

4

Sequential and Dynamic Common Agency

While Bernheim and Whinston’s common agency game assumes that principals make their offers simultaneously, Prat and Rustichini (1998) consider the case in which the game is played in a sequential manner. Each principal makes an offer to the agent 2

Laussel and Le Breton (2001) give only a sufficient condition for uniqueness while Bergemann and Valimaki (2003) find necessary conditions.

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following a pre-specified order. Moreover each offer is public, so that the principals who have not yet made their offer can condition their strategy on the offers already made. After having observed the offers of every principal, the agent makes his choice. In this new environment the efficiency property of common agency still holds. Moreover it does not depend on the refinement chosen: for any given ordering of principals, in all Subgame Perfect Equilibria the agent selects an efficient action. These Subgame Perfect Equilibria can be supported by various transfer profiles. There always exists a SPE transfer profile which corresponds to one of the truthful equilibria of the simultaneous game. There also exist another transfer profile, that Prat and Rustichini call Thrifty Nash Equilibrium Transfer, in which the total amount of transfers on the equilibrium action is lower than in the truthful case. They show that if there exist a Coalition-Proof Equilibrium, it must entail thrifty transfers. Hence, the connection between truthfulness and coalition-proofness does not carry over to the sequential case. Considering the example described in Section 2.2 and assuming that principal 1 moves first the Thrifty Equilibrium transfers are: t1 = (0, 0, 1, 0) ; t2 = (0, 0, 0, 0). Bergemann and Valimaki (2003) analyze another dynamic framework. They consider an agenda setting game: a two period game where in the first period the agent chooses the available actions for the second stage and in the second period the common agency game is played with the set of actions as determined in the first stage. As described in Laussel and Le Breton (2001) the equilibrium payoff of a common agency game depends on the degree of competition between the principals. Since the competitiveness in turn depends on the set of available actions, Bergemann and Valimaki (2003) show that the agent is sometimes able to increase her second period rent by excluding the efficient action. This leads to an inefficiency in the overall game unless we allow the principals to lobby the first period choices of the agent as well. With lobbying in both periods, the overall efficiency is restored, but the payoff to the agent is higher in the two stage game than in the static game where the first period choice of action is ignored. Moreover, Bergemann and Valimaki (2003) describe a general model of dynamic common agency proving the existence and the efficiency of truthful Markov equilibria in this general setting.

5

Contractual externalities and Common Agency Equilibria

Martimort and Stole (2003) study the equilibria of common agency games where the contracting variable of one principal directly affects the other principal’s payoff. The particular example discussed is that of two retailers who distribute the output of a common manufacturer and compete on a final market. Both principals (the retailers) independently and non-cooperatively contract with the common agent (the 7

manufacturer) and the production sold on the final market by one retailer affects the price received by the other. Principals are assumed to have all bargaining power in offering contracts to the retailer. Two variations of the common agency game are considered3 : the intrinsic common agency where the agent can either accept the whole set of contract offered by the principals or reject all of the contracts and the delegated common agency where the agent can also choose a strict subset of principals to whom he wants to sell. In the intrinsic common agency game if principals use non linear price-quantity schedule every output between the Cournot and the competitive outcomes can be sustained as an equilibrium and in all such equilibria the agent gets zero rent. Moreover, if we restrict attention to contracts which are singletons (obtaining a model which is the complementary of Segal (1999)) the unique equilibrium is the Cournot output. Also in the case of delegated common agency the game displays a multiplicity of equilibria if we allow principals to offer non linear price-quantity schedules. As in the previous case, every output between the Cournot quantity and the competitive one can be sustained as a Nash equilibrium. The important difference respect to the previous case is that in all such equilibria the agent gets a strictly positive rent. With singleton contracts there does not exist any pure strategy equilibrium in which the agent serves both principals.

6

Contracting with Externalities

Segal (1999) presents a game in which one principal makes simultaneous offers to N agents. The principal trade with each agent i is denoted by xi ∈ Xi ⊂ R+ with 0 ∈ Xi . Payoffs for agent i and the principal are given respectively by: ui (x1, x2 , ..., xN ) − ti N X f (x1, x2 , ..., xN ) + ti .

(1) (2)

i=1

Where (using the notation introduced in Section 2) ui (x) and f (x) denote the utilities for agent i and the principal and ti is the transfer from agent i to the principal. Externalities among agents arise because each agent’s utility depends not only on his own trade with the principal but also on other agents’ trades. To analyze this game Segal (1999) imposes some conditions on the welfare and the domain of the agents’ trade. Under those conditions the total surplus is a function of the aggregate trade only. Using techniques of monotone comparative statics (Milgrom and Shannon 1994, Topkins 1998) various interesting results are obtained. 3

This distinction has been introduced for the first time by Bernheim and Whinston (1986b).

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If the principal commits to a set of publicly observable bilateral contracts, the presence of externalities may imply that the outcome implemented is different from the efficient one. Indeed Segal (1999) shows that only if each agent’s reservation utility does not depend on other agents’ trades then the principal implements efficient outcomes. With positive externalities on nontraders (ui (0, x−i ) is nondecreasing in x−i ∈ X−i ) the principal will optimally reduce agents’ reservation utilities by trading too little from the social viewpoint. The converse occurs with negative externalities. If the principal does not have as much commitment power as assumed in the previous case and each agent only observes his own offer there may be inefficient outcomes even in the absence of externalities on nontraders. In this setting Segal (1999) restricts agents to hold so-called “passive beliefs”: even after observing an unexpected offer from the principal, an agent believes that other agents face their equilibrium offers. In this case when externalities are absent at an efficient trade profile (there exists an efficient allocation x∗ such that ui (x∗i , x−i ) does not depend on x−i ∈ X−i for all i) private contracting produces efficient outcomes, regardless of any externalities that might exist at other trade profiles. When externalities are present at all efficient trade profiles, they must distort the contracting outcome. Indeed, with positive (negative) externalities on efficient traders, the equilibrium trade is lower (higher) than socially optimal. To conclude Segal compares the outcomes of public and private contracting. Clearly if externalities on nontraders are equal to those on efficient traders the two contracting regimes yield the same outcomes. To study the general case consider the following definition: Definition 2 Externalities are increasing (decreasing) if for each agent i, all x−i ,x0−i ∈ X−i with x0−i ≥ x−i , ui (x, x0−i ) − ui (xi , x−i ) is nondecreasing in xi ∈ Xi . The intuition is that with increasing externalities, the externality imposed on agent i by increasing other agents’s trades is more positive when he trades more. If there is symmetry across agents and externalities are increasing the principal implements an higher outcome with public contracting than with private contracting. If externalities on nontraders and on efficient traders are both positive and increasing private contracting is less efficient than public contracting. Conversely when the externalities on nontraders are of the same sign, but of smaller magnitude, than those on nontraders, the principal’s inability to commit can raise the total surplus of contracting parties.

6.1

Sequential Contracting

Moller (2002) considers sequential contracting between one principal and two agents in the presence of externalities. The principal offer a contract to one agent first, and 9

after this agent has decided whether to accept or to reject he offers a contract to the second agent. The second agent observes the contract offered to the first agent and his acceptance or rejection and then decides whether to accept or to reject his own contract. The results obtained in this sequential setting differs from those in Segal(1999). To simplify call agent 1 the first agent the principal is going to contract with and agent 2 the second one. Sequential public contracting implements the efficient quantity only if externalities on agent 1 are absent at his efficient trade level and externalities on agent 2 are absent at zero. This implies that when externalities are absent at agents’ outside option but present at their efficient trade levels, sequential public contracting is inefficient whereas simultaneous contracting is efficient. Moreover, Moller shows that sequential contracting may arise naturally as a consequence of the externalities among agents if the principal can choose the contracting procedure. For example when externalities are positive at the agents’ outside option and increasing, then the principal will prefer to contract sequentially rather than simultaneously. This timing leads (with the externalities specified above) to less total surplus than the simultaneous thereby creating an additional welfare loss.

6.2

Applications

The models developed in Segal (1999) and Moller (2002) can be applied to study various contracting situations involving multilateral externalities. One of the most important is Vertical Contracting as described in Hart and Tirole (1990), O’Brien and Schaffer (1992), McAfee and Schwartz (1994), Rey and Tirole (1996) Katz and Shapiro (1986) and Kamien et alt. (1992). In these models a principal supplies an intermediate good to N downstream firms who then produce substitute consumer goods. While the first four papers assume that a downstream firm cannot produce without using principal’s good (ui (0, x−i ) = 0), Katz and Shapiro (1986) and Kamien et alt. (1992) analyse situations in which the firm has an inferior technology which does not use the principal’s input ( ui (0, x−i ) is positive and decreasing). Another interesting application is the analysis of Takeovers using models like those developed in Grossman and Hart (1980), Bagnoli and Lipman (1988), Holmstron and Nalebuff (1992), Burkart et alt (1998). The principal is a corporate rider, who makes a tender offer to N shareholders (agents). If the raider has a greater ability to enhance the firm’s value when she holds a larger stake in the firm (superior raider) a tendering shareholder has positive externalities on other shareholders. On the other hand if the raider is inferior a tendering shareholder imposes a negative externality on other shareholders. Finally let us consider models of Pure Public Good/Bad as presented in Bergstrom et alt. (1986) and Neeman (1997). The principal is a provider of a public good/bad, who can contract with N consumers of the good. Each consumer’s contribution to the public good (bad) has a positive (negative) externality on the other consumers. 10

7

Robust Predictions for Bilateral Contracting with Externalities

Segal and Whinston (2001) generalize the model of Segal (1999) considering the case where the principal does not offer a point contract but an entire menu of contracts to the agent. Only if the menu is accepted the principal chooses a bundle from it. Contracting is not publicly observed but there are not ad hoc restrictions on agents’ beliefs. In this way they consider a family of noncooperative games of contracting with externalities, which they call bilateral contracting games, and which include Segal (1999) and Bernheim and Whinston (1986a) as special cases. They search for properties of equilibrium outcome that are robust in the sense that they must be satisfied by all Equilibria of all bilateral contracting games. The main result of the paper is that the set of equilibrium outcomes is dramatically affected when the parties are allowed to offer each other menus from which the principal can then choose, rather than simple point contracts. From an agent’s viewpoint, a menu can separate the different “types ” of the principal corresponding to different trades with other agents. It is shown that there exists a menu giving all types of principals their maximum payoff among all the menus that the agent must accept regardless of his belief. This menu is called the Rothshild Stilglitz and Wilson (RSW) menu. The requirement that deviations in which principal offers a RSW menu be unprofitable imposes a significant bound on the set of equilibrium outcomes. To understand the idea consider a setting of vertical contracting where one manufacturer (the principal) sells her output to N ≥ 2 retailers (agents). The retailers then resell their purchases in the downstream market. In this example the RSW menu is the competitive menu: C = {(x, t) : x ∈ [0, X c ] , t = pc x}

where pc is the price observed in the downstream market when it is sold a quantity, X , for which the marginal cost of the manufacturer equals the price. Observe that each retailer is guaranteed a zero payoff if he accepts the offer, regardless of his belief about the aggregate quantity X−i sold to other retailers. Any equilibrium outcome must h be ³immune ´i to the manufacturer deviation using b P X b is an equilibrium outcome and consider such RSW menu. Suppose that X, the manufacturer’s deviation to retailer i with the competitive menu while keeping her offer to other agents unchanged. Note that manufacturer’s equilibrium revenue b from retailer i was at most i the revenue loss on existing sales due to h ³P (X)b ´ xi and c b −p x bi . On the other hand, the manufacturer’s the deviation is at most P X h i b . Finally he incurs the extra production cost revenue on new sales is pc X c − X b Adding up, we see that for the deviation not to be profitable we must c(X c ) − c(X). c

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have: h i i h ³ ´ b − P X b − pc x b pc X c − X bi ≤ c(X c ) − c(X)

or equivalently

i b h ³ ´i h c b c X b b . (3) [p X − c(X )] − p X − c X ≤ p(X) − p N This is a necessary condition for equilibrium. In many parametrized examples the b ∈ [X N , X c ]. Notice that as N → ∞ this solutions of the inequality have the form: X inequality implies competitive convergence. Note that the competitive menu is not the only menu that the manufacturer can offer that assures a retailer a non negative payoff. The degenerate menu offering only the null trade (0,0) is of course one such menu. For another consider the class of linear prices menus giving the manufacturer the right to sell to the retailer any quantity at price p. The retailer profit will then be nonnegative for any belief he could hold about X−i if and only if p ≤ pc .Therefore, the best linear-price menu for the manufacturer among those that are acceptable by the retailer regardless of his beliefs is the competitive menu considered above. Moreover, Segal and Whinston (2001), considering the special class of payoff function with externalities, ui (xi , x−i ) = α(X)xi and f (X) = −c(X), define necessary conditions on c() and α() to have competitive convergence. In general it is necessary that: c

c

c

1. pc X − c(X) is strictly decreasing for X ≥ X c 2. α(X) ≤ pc for all X ≥ X c . In environments in which these conditions do not hold the bound yields other asymptotic predictions. This result can be used to reanalyze the literature on General Equilibrium with externalities to see which models can be justified as a limit of outcomes of non cooperative contracting games with N agents as N → ∞ and which models cannot.

8

Coordination and Discrimination

In a subsequent paper Segal(2001) studies the effects of prohibiting principals from coordinating agents on her preferred equilibrium and making different contract available to different agents. He considers two stage contracting game of the following kind: in the first stage the principal offers each agent i a menu of contracts and in the second stage each agent chooses a point contract. Each agent observes the

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menu offered to all agents. This assumes a commitment on the principal’s part not to renegotiate contracts secretly. The first result consists in showing that with increasing externalities, offering a single menu to ex ante identical agents the principal can only implement symmetric outcomes. With decreasing externalities the principal can implement asymmetric equilibria even when offering the same menu to all agents. To characterize the possible implementation is difficult but it is possible to show that as the number of agents gets large, the set of aggregate trade without discrimination converges to the set on “nonpivotal” aggregate trades that obtain asymptotically with discrimination. Consider now again the general model where the principal can offer different menus to different agents. Segal shows that with increasing externalities, the game can have a multiplicity of Nash equilibria. Consider the setting of takeovers. If the raider is superior we have decreasing externalities and “all tender ” is a unique Nash equilibrium. If the raider is inferior we have increasing externalities there exists two Nash equilibria: one in which nobody tenders and one in which everybody tenders. Moreover the first equilibrium is preferred by all agents to the second. Let us assume there are just two agents. In this case the principal can offer at least one agent a price that induces him to trade even if he expects the other agent not to. Then she can offer the other agent a price that induces him to trade if he expect the first agent to trade. Iterating this round-robin procedure (Topkins (1998))we will converge to the game lowest Nash equilibrium. Notice that, given the increasing externalities, the first agent receives more transfers than the second agent also if they are ex-ante identical. For this reason Segal calls this strategy a “divide and conquer” strategy. In the previous case if the principal is not allowed to discriminate and must offer the same menu to all agents, he can no longer use this divide and conquer strategy. In order to uniquely implement trade with both agents, he must offer a price that induces an agent to trade even if he expects the other agent not to. It is possible to compare the outcome of the various regimes. Each contract setting is a source of distortions and inefficiencies and the externalities which are relevant for the distortion depend on the contract regime. It is interesting to note that inefficiencies are normally reduced by both prohibitions.

9

Efficiency and Bargaining Power

We introduce a simple original example reviewing the solution concepts defined above and illustrating how the bargaining power affects equilibrium outcomes. We consider a simple framework where a principal, the Airport Society, can trade with two agents, the Farmers, that we are going to call Farmer A and Farmer B. The gross payoff of the Airport if it concludes a contract with only one of the two farmers is equal to K

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and it is equal to 2K if it trades with both.4 There are negative externalities on nontraders [Segal (1999)] between the two farmers. More precisely, if they both refuse to trade each of them obtains a payoff of r2 . If just one of the two trades with the Airport the one not contracted obtains r1 . We assume r2 > K > r1 which implies that trade is an inefficient outcome: i.e. each farmer prefers having a farm nearby rather than having an airport. Moreover we assume: 2K − r1 − r2 > 0. Solutions to this game vary, depending on the contracting procedure chosen. Let us start with the offer game approach of Segal (1999) in which the Airport publicly commits to a vector of transfers (tA , tB ), where ti has to be interpreted as transfer to Farmer i, and each farmer either accepts or rejects the offer. The equilibrium he describes is one in which the Airport offers transfers that render each farmer indifferent between accepting and rejecting the contract as long as the other farmer trades. More specifically Segal considers the transfer vector t = (r1 , r1 ) and a game between farmers described by the following bimatrix : Farmer B Farmer A

Accept Accept tA = r1 , tB = r1 Reject r1 , tB = r1

Reject tA = r1 , r1 r2 , r2

(Accept, Accept) is indeed a Nash equilibrium of the proposed game. It is important to notice that because of increasing externalities on nontraders this equilibrium is not unique. Segal’s analysis focuses only on this equilibrium point because he assumes that the Airport can coordinate farmers on his preferred equilibrium. This is equivalent to impose coordination failure between the two farmers. (Reject, Reject) is another Nash Equilibrium of the game. A comparison of the two shows that only the second is coalition proof. Segal (2001) shows that to implement (Accept, Accept) as a unique Nash Equilibrium the Airport must use a “divide-and-conquer” strategy offering at least one farmer, say Farmer A, a transfer that induces him to trade even if he expects the other farmer not to. Simultaneously he can offer Farmer B a price that induces him to trade if he expects Farmer A to trade. In our model it implies the vector of transfers t = (r2 , r1 ). In this case we are going to have trade since we assumed 2K −r1 −r2 > 0. We now turn to the bidding game in which both farmers propose take it or leave it offers to the Airport. To analyze this case we follow Martimort and Stole (2003) and we consider a game in which each farmer can only observe and contract upon 4

The following game was inspired by Coase’s (1960) famous railroad-farmer model in which a railroad may have to get permission from each of a number of farmers to undertake a profitable project.

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the Airport’s action that he directly cares about. We assume that the contractual space of this game is exactly the same as that in the offer game described above: each farmer proposes a price in order to sell his own field. This bidding game has two types of Nash Equilibria. In one of these both farmers ask for K and the Airport trades with both of them. The other is the set of equilibria in which both farmers ask for a transfer larger than K and the Airport refuses to trade. This second set of Nash Equilibria is coalition proof. Thus we can say that in the bidding game, all coalition proof Nash Equilibria are efficient. Summing up, the Airport Game has a continuum of efficient coalition proof Nash Equilibria if it is solved using the bidding game approach whereas it has a unique inefficient coalition proof Nash Equilibrium with transfers t = (r2 , r1 ) if it is solved using the offer game procedure. It is easy to see that both outcomes do not differ if we consider the sequential version of the two games. There is therefore a striking difference between the outcomes of the two procedures: reassigning bargaining power from the Airport to the Farmers not only changes the share of surplus that each party obtains but also alters the size of the total surplus to split. It is interesting to notice that both the coordination failure and the coalition proof nash equilibria of the bidding games correspond to a particular specification of the weighted Nash bargaining solution. Let α and β denote the bargaining power of the two agents with respect to the principal where α, β ∈ [0, 1] and α + β ≤ 1. Higer values for α and β imply less bargaining power for the principal (given by (1-α − β)). The Nash bargaining solution to the problem solves max(2K − tA − tB )1−α−β (tA − dA )α (tB − dB )β tA ,tB

where dA ≥ 0 and dB ≥ 0 denote agent A and agent B disagreement payoffs if negotiations breakdown. Solving the problem we obtain: tA = dA + α [2K − dA − dB ] tB = dB + β [2K − dA − dB ] . It is easy to see now that if β = α = 0 and dA = dB = r1 the Nash bargaining solution corresponds to the coordination failure Nash equilibrium. Moreover if β = α = 0 and dA = r2 and dB = r1 it coincides with the divide and conquer strategy.5

10

Games Played through Agents

Prat and Rustichini (2000) generalize the model of Bernheim and Whinston (1986a) defining the concept of game played through agents (GPTA). A GPTA is a game 5

The issue of bargainig power is studied much deeper in Galasso (2004)

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where a set of players (the agents) take decisions that affect the payoff of another set of player (the principals) and the principals can, by means of monetary transfers, try to influence the decision of the agents. There is a set of M principals and a set of N agents. The payoffs are defined as: X un (xn ) + tm (4) n (xn ) m∈M

f m (x1 , ...xN ) −

X

tm n (xn ).

(5)

n∈N

Both in Bernheim and Whinston (1986a) and in Segal (1999)’s models the absence of direct externalities among agents always implies the existence of an efficient equilibrium. In a GPTA, even in the absence of direct externalities, an efficient equilibrium may not exist. The presence of multiple players on both sides creates a strategic externality that makes it impossible to achieve the efficient outcome. The main result of the paper is to provide a sufficient condition for a GPTA to have an efficient equilibrium. This condition relates to the cooperative concept of balancedness. In this contest balancedness is defined in terms of weighted deviations from the equilibrium outcome. Definition 3 In a GPTA, the vectors w and z with respective dimension MS and H are balanced weights with respect to x b if all their elements are nonnegative, and for every m ∈ M, n ∈ N, xn ∈ Xn /b xn : X wm (x) = zn (an ). (6) (x:xn =an )

A GPTA is balanced with respect to x b if and only if for every pair of vectors of balanced weights w and z we have: XX X X wm (x) (f m (b x) − f m (x)) + zn (xn ) (un (b xn ) − un (xn )) . (7) m∈M x∈X

n∈N xn ∈Xn

This definition of balancedness differs from the one used in cooperative game theory (see e.g. Osborne and Rubinstein (1994)) because here it is not the coalition that has a weight but each possible outcome is associated with a weight for each principal and a weight for each agent. Moreover, Prat and Rustichini (2000) adapted the definition of truthful equilibrium (Bernheim and Whinston (1986a)) to a GPTA with many agents: Definition 4 For principal m, tm is weakly truthful relative to x b if for every x ∈ X: X X x) − tm xn ) ≥ f m (x) − tm f m (b n (b n (xn ). n∈N

n∈N

A weakly truthful equilibrium with outcome x b is a pure strategy equilibrium with outcome x b in which the transfer of every principal is weakly truthful relative to sb. 16

A consequence of this definition is that, like truthful equilibria of common agency games, weakly truthful equilibria of GPTA’s are always efficient. We can now state the main theorem: Theorem 5 A GPTA has a weakly truthful equilibrium with outcome x b if and only if it is balanced with respect to x b.

Thus, since a GPTA has a weakly truthful equilibrium if and only if is balanced and a weakly truthful equilibrium is always efficient, balancedness is a sufficient condition for the existence of an equilibrium with an efficient outcome. Note that this is just a sufficient condition. There can be games that do not have a weakly truthful pure strategy equilibrium but have a non truthful pure strategy equilibrium supporting the efficient outcome. The connection with cooperative game theory and GPTA is deeper than the use of balacedness. GPTA may provide a non-cooperative foundation for the core. Every cooperative game with transferable utility (TUG) can be put in correspondence with a special type of GPTA in which two identical principals compete to hire agents (the action of the agent consists in selecting one of the principals). The value of a coalition in the TUG becomes the payoff of the principal who hires the agent in that coalition. Prat and Rustichini (2000) show that the core of a superadditive TUG is nonempty if and only if the corresponding GPTA has an efficient equilibrium.

10.1

Applications

Multi-principal multi-agent problems arises in various fields. An important application of the theory is lobbying. Dixit, Grossman and Helpman (1997), for instance, consider a setting where there are many lobbying and a single politician and the theory of GPTA can be useful to extend their result to a more realistic setting of multiple politicians. Another important case is the one of vertical restraints in which an industry with several firms produces goods that are used by several buyers who can be final consumers or intermediate producers. A contract offered to one seller may cover not only the relation between that supplier and the buyer but also the relation between the buyer and the other suppliers, such as an exclusive clause. In a common agency environment Bernheim and Whinston (1986a) show that the equilibrium contracts are efficient and this argument has been used by the Chicago School, e.g. Bork(1978), to deny the anticompetitive nature of vertical restraints. Extending the setting form common agency to GPTA allows to study more deeply the issue of efficiency and how it relates to the balancedness of the game chosen. From the analysis of Prat and Rusitichini (2003) we see that in classical production environment Bork’s claim is correct in the sense that there is an equilibrium in which firms sign vertical contracts that ensure to productive efficiency.

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11

Conclusions

In this survey we have analyzed literature on multi-agent and common agency games with complete information. We have seen how the presence of externalities among players is source of inefficiency in these games. We presented the classical common agency game introduced by Bernheim and Whinston (1986a) and we analyzed the properties of the equilibrium payoff. Moreover we described links existing between this model and cooperative game theory. We presented extensions of the model: the dynamic versions of Prat and Rustichini (1998) and Bergemann and Valimaki (2003) and the version with externalities modeled by Martimort and Stole (2003). We then studied the multi-agent game described in Segal (1999) with its problems of coordination and discrimination, the sequential version of Moller (2003), and we discussed the main applications of the model. Moreover we reported the generalized version of the game of Segal and Whinston (2001) in which they search for properties of equilibrium outcome that are robust in the sense that they must be satisfied by all equilibria of all bilateral contracting games. Furthermore we presented an original game (the Airport Game) to study the effect of bargaining power on equilibrium outcomes. Finally we analyzed games played through agents using the model of Prat and Rustichini (2000) and their applications to lobbying and vertical restraints.

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