Perfect and Proper Equilibria of Large Games

This paper deals with the existence and other related issues of perfect and proper equilibria of games with a continuum of players. A sufficient condition for the existence of a perfect (proper) equilibrium as an almost everywhere limit of a sequence of ε-perfect (ε-proper) equilibria is given. An example shows that almost everywhere convergence need not obtain if the condition is violated. Extension to the case where the set of actions available to the players can differ is discussed.Journal of Economic LiteratureClassification Number: C79.     

Perfect Bayesian implementation

Summary We study Social Choice Sets (SCS) implementable as perfect Bayesian equilibria of some incomplete information extensive form game. We provide a necessary condition which we callcondition . The condition is analogous tocondition C that Moore and Repullo [1988] show to be necessary for subgame perfect implementation in games of complete information, and it is weaker than the Bayesian Monotonicity condition stated in Jackson [1991]. Our first theorem establishes that Incentive Compatibility, Closure and Condition are necessary for implementation.Our second theorem establishes sufficient conditions. We show that any SCS which satisfies Incentive Compatibility, Closure and a condition called Sequential Monotonicity No Veto (SMNV) is implementable. SMNV is similar in spirit but weaker than the Monotonicity No Veto condition stated in Jackson [1991]. It is also similar to a combination of condition and No Veto Power, which Abreau and Sen show to be sufficient for implementation in subgame perfect equilibrium.This paper is a revised version of a chapter of my dissertation at Stanford University, Graduate School of Business. I would like to thank, without implicating, Faruk Gul, Bob Wilson and especially John Roberts for their constant advice and encouragement. I would also like to thank participants at the 1993 Summer Meeting of the Econometric Society and a referee for comments leading to substantial improvement in the paper. Financial support from Bocconi University is gratefully acknowledged.     

Nash implementing non-monotonic social choice rules by awards

Summary. By a slight generalization of the definition of implementation (called implementation by awards), Maskin monotonicity is no more needed for Nash implementation. In fact, a weaker condition, to which we refer as almost monotonicity is both necessary and sufficient for social choice correspondences to be Nash implementable by awards. Hence our framework paves the way to the Nash implementation of social choice rules which otherwise fail to be Nash implementable. In particular, the Pareto social choice rule, the majority rule and the strong core are almost monotonic (hence Nash implementable by awards) while they are not Maskin monotonic (hence fail to be Nash implementable in the standard framework).Received: 12 April 2004, Revised: 18 March 2005, JEL Classification Numbers: D 79.I thank François Maniquet, Efe Ok and Ipek Ozkal-Sanver for their valuable contributions. The paper is part of a project on Electoral System Design supported by Bilgi University Research Fund.     

Cooperative Outcomes through Noncooperative Games

This paper shows the equivalence between the stable solution set of any cooperative game in characteristic form (G1) and the subgame perfect Nash equilibria in pure strategies of a certain noncooperative game (G2). Players of G1 are named "agents." G2 is played by different players ("principals") who compete in wages to attract agents. The equivalence result holds when there are enough principals (if the game is superadditive, two principals suffice). Finally, another related cooperative game (G3) is constructed with both principals and agents as players. For G2 and G3 the same result is then proven, for any number of principals. Journal of Economic Literature Classification Numbers: C71 and C72.     

A general impossibility result on strategy-proof social choice hyperfunctions

A social choice hyperfunction picks a non-empty set of alternatives at each admissible preference profile over sets of alternatives. We analyze the manipulability of social choice hyperfunctions. We identify a domain D^λ of lexicographic orderings which exhibits an impossibility of the Gibbard–Satterthwaite type. Moreover, this impossibility is inherited by all well-known superdomains of D^λ. As most of the standard extension axioms induce superdomains of D^λ while social choice correspondences are particular social choice hyperfunctions, we are able to generalize many impossibility results in the literature.     

Bargaining under a deadline: evidence from the reverse ultimatum game

We study a “reverse” ultimatum game, in which proposers have multiple chances to offer responders a division of some fixed pie. The game ends if the responder accepts an offer, or if, following a rejection, the proposer decides not to make a better offer. The unique subgame perfect equilibrium gives the proposer the minimum possible payoff. Nevertheless, the experimental results are not too different from those of the standard ultimatum game, although proposers generally receive slightly less than half of the surplus.We use the reverse ultimatum game to study deadlines experimentally. With a deadline, the subgame perfect equilibrium prediction is that the proposer gets the entire surplus.Deadlines are used strategically to influence the outcome, and agreements are reached near the deadline. Strategic considerations are evident in the differences in observed behavior between the deadline and no deadline conditions, even though agreements are substantially less extreme than predicted by perfect equilibrium.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

WHEN AND HOW THE PUNISHMENT MUST FIT THE CRIME

In repeated normal‐form (simultaneous‐move) games, simple penal codes (Abreu, Journal of Economic Theory 39(1) (1986), 191–225; and Econometrica 56(2) (1988), 383–96) permit an elegant characterization of the set of subgame‐perfect outcomes. We show that in repeated extensive‐form games such a characterization no longer obtains. By means of examples, we identify two types of settings in which a subgame‐perfect outcome may be supported only by a profile with the property that the continuation play after a deviation is tailored not only to the identity of the deviator but also to the nature of the deviation.     

Robust implementation in general mechanisms

A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation.Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.     

Subgame perfect implementation: A full characterization

Moore and Repullo [Subgame perfect implementation, Econometrica 56 (1988) 1191-1220], and Abreu and Sen [Subgame perfect implementation: a necessary and almost sufficient condition, J. Econ. Theory 50 (1990) 285-299] introduce distinct necessary and sufficient conditions for SPE implementation, when the number of players is at least three. This paper closes the gap between the conditions—a complete characterization of the SPE implementable choice rules is provided. The characterization consists of α^*, which strengthens α of Abreu-Sen by adding it a restricted veto-power condition, and the unanimity condition. Under strict preferences α^* is equal to α.     

Demand bargaining and proportional payoffs in majority games

We study a majoritarian bargaining model in which players make payoff demands in decreasing order of voting weight. The unique subgame perfect equilibrium outcome is such that the minimal winning coalition of the players that move first forms with payoffs proportional to the voting weights. This result advances previous analysis in terms of one or more of the following: a) the simplicity of the extensive form (finite horizon with a predetermined order of moves); b) the range of the majority games covered; c) the equilibrium concept (subgame perfect equilibrium is sufficient for a unique prediction).     

Existence of equilibrium in common agency games with adverse selection

We establish the existence of subgame perfect equilibria in general menu games, known to be sufficient to analyze common agency problems. Our main result states that every menu game satisfying enough continuity properties has a subgame perfect equilibrium. Despite the continuity assumptions that we make, discontinuities naturally arise due to the absence, in general, of continuous optimal choices for the agent. Our approach, then, is based on (and generalizes) the existence theorem of [Simon, L., Zame, W., 1990. Discontinuous games and endogenous sharing rules. Econometrica 58 (4), 861–872] designed for discontinuous games.     

Finite Horizon Bargaining and the Consistent Field

We explore the relationships between noncooperative bargaining games and the consistent value for nontransferable utility (NTU) cooperative games. A dynamic approach to the consistent value is introduced: the consistent vector field. The main contribution is to establish that theconsistent fieldis intimately related to the concept ofsubgame perfectionfor finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. We then show that in the general NTU case—unlike the transferable utility and pure bargaining cases—the finite horizon subgame perfect equilibria need not approach the consistent value.Journal of Economic LiteratureClassification Numbers: C71, C72.     

An explicit solution to an optimal tax problem

Under specific but perhaps not over-restrictive assumptions on social welfare and consumer preferences, an explicit closed-form solution for an optimal linear income tax is derived. Specifically, given linear income supply functions and a rank-order social welfare function, the optimal tax rate and benefit level are characterized by four parameters: I, a measure of pre-tax inequality in the ability (wage) distribution; r, the fraction of potential total income required for (non-redistributed) government revenue; σ, the fraction of potential total income required for consumer subsistence expenditures; and a disincentive parameter, δ, the marginal propensity to spend on leisure or the amount by which earned income is reduced in response to a unit increase in unearned benefit. Defining , the ratio I/(1 - σ - r), the optimal tax rate τ is given by: The formula is used to fully characterize τ in terms of the parameters. Results include the following: τ = 0 if I = 0; τ = 1 if δ = 0; τ is increasing in I, σ and r; τ may be increasing or decreasing in δ depending on the value of ; when disincentive effects are large, τ becomes close to so that, in such economies, if σ and r are small, the optimal tax rate is equal to the measure of pre-tax inequality. Formulae for the deadweight loss associated with the tax are derived and some observations are offered on the empirical issues associated with the model.     

Punishment paths and cartel size

Using a ‘Dominant-group-fringe’ model, we show that the choice of a subgame perfect Nash equilibrium (SPNE) punishment path in the event that unilateral defection occurs depends upon the common discount parameter, and that when the discount parameter is given the punishment path may depend upon the size of the dominant group. The choice of punishment path is thus endogenous. A striking result is that only for sufficiently small dominant groups in sufficiently small industries are there cases where costs affect these choices. Consequently, costs are irrelevant to the choice of punishment in industries with a large number of firms.     

Finitely repeated games with social preferences

A well-known result from the theory of finitely repeated games states that if the stage game has a unique equilibrium, then there is a unique subgame perfect equilibrium in the finitely repeated game in which the equilibrium of the stage game is being played in every period. Here I show that this result does in general not hold anymore if players have social preferences of the form frequently assumed in the recent literature, for example in the inequity aversion models of Fehr and Schmidt (Quartely Journal of Economics 114:817–868, 1999) or Bolton and Ockenfels (American Economic Review 100:166–193, 2000). In fact, repeating the unique stage game equilibrium may not be a subgame perfect equilibrium at all. This finding should have relevance for all experiments with repeated interaction, whether with fixed, random or perfect stranger matching.     

Selective Acceptance and Inefficiency in a Two-Issue Complete Information Bargaining Game

This paper explores the effects of a “selective acceptance” rule on the outcome of two-issue negotiations. The alternating-offer game introduced here allows for the possibility that settlement may be reached on one issue while negotiation continues about the other. This model captures features of laws that are generally believed to increase efficiency. The analysis shows that if one issue is indivisible, there are inefficient subgame perfect equilibria with no Pareto-improving alternative equilibria. With opposing valuations, rapid communication guarantees inefficiency. These are unique examples of this strong form of inefficiency in an alternating-offer bargaining game with complete (and perfect) information. Journal of Economic Literature Classification Numbers: C72, J30.     

A note on bargaining over a finite number of feasible agreements

Summary In this note we show that theuniqueness of the subgame perfect equilibrium of Rubinstein's (1982) bargaining theory does not hold if the number of feasible agreements isfinite. It will be shown thatany Pareto-efficient agreement (belonging to thefinite set of feasible agreements) can be supported as a subgame perfect equilibrium of the Rubinstein alternating-offers bargaining game, provided the length of a single bargaining period is sufficiently small.     

Ex post implementation

We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and—in economic environments—sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is sufficient. Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints.We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity. The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. We further describe a Pareto correspondence that fails ex post monotonicity but satisfies Maskin monotonicity.     

Repeated games with one-memory

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1-memory strategies. We establish the following in games with perfect (rich) action spaces: First, when the players are sufficiently patient, the subgame perfect Folk Theorem holds with 1-memory. Second, for arbitrary level of discounting, all strictly enforceable subgame perfect equilibrium payoffs can be approximately supported with 1-memory if the number of players exceeds two. Furthermore, in this case all subgame perfect equilibrium payoffs can be approximately supported by an ε-equilibrium with 1-memory. In two-player games, the same set of results hold if an additional restriction is assumed: Players must have common punishments. Finally, to illustrate the role of our assumptions, we present robust examples of games in which there is a subgame perfect equilibrium payoff profile that cannot be obtained with 1-memory. Thus, our results are the best that can be hoped for.