Undominated Nash Implementation in Bounded Mechanisms

We study implementation in undominated Nash equilibrium by bounded mechanisms. (An undominated Nash equilibrium is a Nash equilibrium in which no agent uses a weakly dominated strategy. A mechanism is bounded if every dominated strategy is dominated by some undominated strategy.) We identify necessary conditions and sufficient conditions for such implementation. These conditions are satisfied in virtually all economic environments, and are also satisfied by interesting correspondences from the social choice literature. For economic settings, we provide a particularly simple implementing mechanism for which the undominated equilibrium outcomes coincide with those obtained from the iterative elimination of weakly dominated strategies. Journal of Economic Literature Classification Numbers: 022, 025, 026.     

Implementation and Renegotiation

The paper characterizes the choice rules that can be implemented when agents are unable to commit themselves not to renegotiate the mechanism.     

Collusion and Renegotiation in a Principal–Supervisor–Agent Relationship

We describe a principal–supervisor–agent relationship in which agent and supervisor may collude. To prevent collusion, the principal may contract on a noisy signal which is correlated with the occurrence of collusion. When the signal is informative enough, the principal uses it and no collusion occurs in equilibrium. These contracts, however, are ex post inefficient and are only optimal if the principal can commit not to renegotiate. With renegotiation it is never optimal for the principal to prevent collusion and, at the same time, condition contracts on the signal. In fact, when the signal is informative enough collusion occurs in equilibrium.     

Moral Hazard and Renegotiation with Multiple Agents

We investigate the effects of contract renegotiation in multi-agentsituations where risk-averse agents negotiate a contract offer to theprincipal after the agents observe a common, unverifiable perfect signalabout their actions. We show that renegotiation with multiple agentsreduces the cost of implementing any implementable action profile downto the first-best level, even though the principal cannot observe theagents' actions. Moreover, it is sufficient for the principal to use a"simple" initial contract, in the sense that it consists of nomore than a single sharing scheme for each agent and the total payments tothe agents are the same regardless of the realised state. An importantimplication is that decentralization, in the sense of delegated negotiationand proposals from the agents, can be as effective as centralized schemesthat utilize revelation mechanisms in unrestricted ways.     

Renegotiation in Repeated Games with Side-Payments

We consider repeated games with side-payments: players have an endowment of wealth in each period in which transfers can be made. We show that if endowments are large enough and the common discount factor high enough, then a strongly renegotiation–proof equilibrium (SRP) in the sense of Farrell and Maskin exists. As the discount factor goes to 1, the set of SRP payoffs converges to the set of efficient, individually rational payoffs. These results provide a justification for the efficiency principle when agreements are not enforceable. Journal of Economic Literature Classification Numbers: C73, D23, L14.     

The debt hangover: Renegotiation with noncontractible investment

This paper analyses the incentives of the equityholders of a levered company to undertake noncontractible investments. This noncontrability is shown to seriously impede the efficiency of any renegotiation process in the debt overhang problems. Conditions for obtaining a fully efficient level of investment choice are derived.     

Collusion and group lending with adverse selection

In an environment with correlated returns, this paper characterizes optimal lending contracts when the bank faces adverse selection and borrowers have limited liability. Group lending contracts are shown to be dominated by revelation mechanisms which do not use the ex post observability of the partners' performances. However, when collusion between borrowers under complete information is allowed, group lending contracts are optimal in the class of simple revelation mechanisms (which elicit only the borrower's own private information) and remain useful with extended revelation mechanisms.     

Collusion and Information Exchange

Antitrust authorities view that exchange of individual firms’ sales data is more anti‐competitive than that of aggregate sales data. In this paper, I survey antitrust implications of such inter‐firm information exchange. I argue that both types of information exchange are anti‐competitive under some circumstances. More precisely, I compare profits when each type of information exchange is allowed to that when firms can only observe their own sales (Stigler’s secret price‐cutting model), and the former is bigger than the latter. I also provide a general method to bound the equilibrium profits without such information exchange.     

Collusion and biased tournaments

Tournaments are vulnerable to collusion. This paper finds that biased tournaments can be more effective at preventing collusion than unbiased ones. When agents can collude to exert low effort, introducing some bias into tournaments generates opposite effects on favored and disfavored agents׳ respective incentives to exert high effort and provides strong incentives for the favored agent to deviate from collusion. Introducing an adequate degree of bias reduces the principal׳s incentive cost for preventing collusion; however, granting excessive bias instead increases the incentive cost. We show that the optimal level of bias can be endogenously determined.     

Collusion and Price Rigidity

We consider an infinitely repeated Bertrand game, in which prices are publicly observed and each firm receives a privately observed, i.i.d. cost shock in each period. We focus on symmetric perfect public equilibria, wherein any "punishments" are borne equally by all firms. We identify a tradeoff that is associated with collusive pricing schemes in which the price to be charged by each firm is strictly increasing in its cost level: such "fully sorting" schemes offer efficiency benefits, as they ensure that the lowest-cost firm makes the current sale, but they also imply an informational cost (distorted pricing and/or equilibrium-path price wars), since a higher-cost firm must be deterred from mimicking a lower-cost firm by charging a lower price. A rigid-pricing scheme, where a firm's collusive price is independent of its current cost position, sacrifices efficiency benefits but also diminishes the informational cost. For a wide range of settings, the optimal symmetric collusive scheme requires (i) the absence of equilibrium-path price wars and (ii) a rigid price. If firms are sufficiently impatient, however, the rigid-pricing scheme cannot be enforced, and the collusive price of lower-cost firms may be distorted downward in order to diminish the incentive to cheat. When the model is modified to include i.i.d. public demand shocks, the downward pricing distortion that accompanies a firm's lower-cost realization may occur only when current demand is high.     

Collusion and discrimination in organizations

In a principal-multi-agent setting we investigate how optimal contracts should be modified under relative performance evaluation when agents collude. Agents may write side-contracts, which are not contingent on their effort choices but indirectly control them through side-transfers. We show that the optimal collusion-proof contract is to introduce a “discriminatory policy” in the sense that the wage schemes offered to agents depend on their identities even if they are identical with respect to productive abilities. Such discriminatory wage schemes explain the organizational strategy of “divide and conquer” as an optimal response to collusion.     

Merger,Partial Collusion and Relocation

We set up a three-firm model of spatial competition to analyse how a merger affects the incentives for relocation, and conversely, how the possibility of relocation affects the profitability of the merger, particularly for the non-participating firm. We also consider the cases of partial collusion in either prices or locations. Under the assumption of mill pricing, we find that a merger will generally induce the merger participants to relocate, but the direction of relocation is ambiguous, and dependent on the degree of convexity in the consumers transportation cost function. Furthermore, we identify a set of parameter values for which the free-rider effect of a merger vanishes, implying that the possibility of relocation could solve the merger paradox.Acknowledgement We thank Jonathan Cave, Frode Meland, participants at the Royal Economic Society Conference 2003 and an anonymous referee for valuable comments.     

Complexity and Renegotiation: A Foundation for Incomplete Contracts

The paper considers a hold-up model where only one of n future trading opportunities will prove to be efficient, and where ex post renegotiation of the ex ante contract cannot be prevented. As the environment becomes more complex ( n →∞), the outcome under any message-contingent long-term contract converges to that of the incomplete contracting model where trade is contractible ex post , but not ex ante . When trades are costly to describe, both ex ante and ex post , the incomplete contracting result is extended to a broader class of environments.     

Nash implementation with a private good

Summary. I construct a general model of social planning problems, including mixed production economies and regulatory problems with negative externalities as special cases, and I give simple mechanisms for Nash implementation under three increasingly general sets of assumptions. I first construct a continuous mechanism to implement the (constrained) Lindahl allocations of an economy, and I then extend this to arbitrary social choice rules based on prices. I end with a mechani sm to implement any monotonic social choice rule, assuming only the existence of a private (not necessarily transferable) good. In that general case, each agent simply reports an upper contour set, an outcome, and I need two agents to make binary numerical announcements. I do not require the usual no-veto-power condition. Received: February 19, 1998; revised version: January 30, 2002     

Collusion in homogeneous and heterogeneous tournaments

In tournaments, only the relative performance of agents is important. Therefore, the agents have an incentive to collude by jointly deviating to low effort levels. Previous papers on collusion have proposed to make the tournament asymmetric to prevent agents from colluding. However, by allowing side payments between the agents and focusing on implicit enforcement of collusion, the current study demonstrates that this is not true. Often, the principal prefers to hire homogeneous agents to make collusions less stable.     

Monotonicity and Nash implementation in matching markets with contracts

We consider general two-sided matching markets, so-called matching with contracts markets as introduced by Hatfield and Milgrom (in A Econ Rev, 95(4), 913–935, 2005), and analyze (Maskin) monotonic and Nash implementable solutions. We show that for matching with contracts markets the stable correspondence is monotonic and implementable. Furthermore, any solution that is Pareto efficient, individually rational, and monotonic is a supersolution of the stable correspondence. In other words, the stable correspondence is the minimal solution that is Pareto efficient, individually rational, and implementable.     

Collusion in Dynamic Bertrand Oligopoly with Correlated Private Signals and Communication

This paper studies collusion in repeated Bertrand oligopoly when stochastic demand levels for the product of each firm are their private information and are positively correlated. It derives general sufficient conditions for efficient collusion through communication and a simple grim-trigger strategy. This analysis is then applied to a model where the demand signal has multiple random components which respond differently to price deviations. In this model, it is shown that the above sufficient conditions hold if idiosyncratic noise terms are sufficiently small. Journal of Economic Literature Classification Numbers: C72, D82.     

Double implementation in Nash and undominated Nash equilibria in social choice environments

Summary Jackson [1] and Yamato [12] constructed game forms which implement social choice correspondences in Nash and undominated Nash equilibria simultaneously in exchange economies. In this paper, I deal with social choice environments and construct a game form which implements social choice correspondences satisfying monotonicity, no veto power and having at least three agents in Nash and undominated Nash equilibria under the existence of an alternative called a holocaust. The game form constructed in this paper includes an integer game but satisfies the boundedness condition introduced by Jackson [1].I am grateful to Professors Tatsuyoshi Saijo and Stephen Turnbull, and to an anonymous referee for their helpful comments and suggestions. However all remaining errors are mine. An earlier version of this paper was presented at the 1991 Annual Meeting of the Japan Association of Economics and Econometrics at Hokkaido University.After this paper was submitted toEconomic Theory I became aware of the work of Jackson, Palfrey, and Srivastava [2] who have obtained a similar result to mine as a by-product of their main result.     

Nash equilibrium in games with incomplete preferences

Summary. This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.Received: 22 September 2003, Revised: 24 June 2004, JEL Classification Numbers: D11, C72, D43.I would like to thank Jean-Pierre Benôit, Juan Dubra, Alejandrio Jofre, Debraj Ray, Kim-Sau Chung and the seminar participants at NYU and at the Universidad de Chile for their comments. I am most grateful to Efe Ok, for his comments, criticism, suggestions and questions.