The weak core of simple games with ordinal preferences: implementation in Nash equilibrium

In a simple game, coalitions belonging to a given class are “absolutely powerful” while others have no power. We attempt to make this distinction operational. Toward this end, we propose two axioms on social choice correspondences, Strong Non-Discrimination and Exclusion. Strong Non-Discrimination describes circumstances under which certain coalitions, the losing coalitions, have no influence over social choice. Exclusion requires that there are situations in which certain coalitions, the winning coalitions, can exercise their power. We show that the weak core correspondence is the minimal correspondence satisfying Maskin Monotonicity and Strong Non-Discrimination. We also show that the weak core is the unique correspondence satisfying Nash implementability, Strong Non-Discrimination, and Exclusion.     

Belief-free equilibria in games with incomplete information: Characterization and existence

We generalize the results of Hörner and Lovo (2009) [15] to N-player games with arbitrary information structure. First, we characterize the set of belief-free equilibrium payoffs under low discounting as the set of feasible payoffs that are individually rational, jointly rational, and incentive compatible. Second, we provide necessary and sufficient conditions on the information structure for this set to be non-empty.     

Transferable utility games with uncertainty

We introduce the concept of a TUU-game, a transferable utility game with uncertainty. In a TUU-game there is uncertainty regarding the payoffs of coalitions. One out of a finite number of states of nature materializes and conditional on the state, the players are involved in a particular transferable utility game. We consider the case without ex ante commitment possibilities and propose the Weak Sequential Core as a solution concept. We characterize the Weak Sequential Core and show that it is non-empty if all ex post TU-games are convex.     

Uniqueness of stationary equilibria in bargaining one-dimensional policies under (super) majority rules

We consider negotiations selecting one-dimensional policies. Individuals have instantaneous preferences represented by continuous, concave and single-peaked utility functions, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. We provide sufficient conditions that guarantee the existence of a unique stationary subgame perfect equilibrium, and we provide its explicit characterization. The uniqueness of the equilibrium permits an analysis of the set of Pareto optimal voting rules. For symmetric distributions of peaks and uniform recognition probabilities unanimity is the unanimously preferred majority rule.     

Nonminimal product differentiation as a bargaining outcome

It is well known that in a duopoly model of product choice with quadratic transportation cost, the firms locate at the extreme endpoints of the market. Jehiel (1992, Int. J. Ind. Organ, 10, 633–641) has examined this model in an infinite horizon setting where in the initial period the firms choose location and in subsequent periods charge the Nash bargaining solution prices. Then, in the unique equilibrium both firms locate at the center of the market. This paper examines the case when the firms instead charge the prices determined by either the egalitarian bargaining solution or the Kalai–Smorodinski bargaining solution. It is shown that central agglomeration is an equilibrium. Furthermore, there is a continuum of symmetric equilibria in addition where the firms locate apart from each other. So the products are not necessarily minimally differentiated. Thus different bargaining solutions provide quite different outcomes.     

Advice and behavior in intergenerational ultimatum games: An experimental approach

In the real world, when people play a game, they often receive advice from those that have played it before them. Such advice can facilitate the creation of a convention of behavior. This paper studies the impact of advice on the behavior subjects who engage in a non-overlapping generational Ultimatum game where after a subject plays he is replaced by another subject to whom he can offer advice.Our results document the fact that allowing advice fosters the creation of a convention of behavior in Ultimatum games. In addition, by reading the advice offered we conclude that arguments of fairness are rarely used to justify the offers of Senders but are relied upon to justify rejections by Receivers.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

Exchange of indivisible goods and indifferences: The Top Trading Absorbing Sets mechanisms

There is a wide range of economic problems that involve the exchange of indivisible goods with no monetary transfers, starting from the housing market model of the seminal paper by Shapley and Scarf (1974) to problems such as the kidney exchange or the school choice problem. The classical solution to many of these models is to apply a mechanism called Top Trading Cycles, attributed to David Gale, which satisfies good properties for the case of strict preferences. In this paper, we propose a family of mechanisms, called Top Trading Absorbing Sets mechanisms, which generalize the Top Trading Cycles to the general case in which individuals are allowed to report indifferences, while preserving a maximal possible set of its desirable properties.     

Learning from private information in noisy repeated games

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games where players observe imperfect public signals of the actions played, and both the payoff functions and the map from actions to signal distributions depend on an unknown state. The PTXE payoffs when players are patient are determined by the solutions to a family of linear programming problems. Using this characterization, we develop conditions under which play can be as if the players have learned the state. We provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

An implementation of the Owen value

We present an implementation of the Owen value (Owen, 1977, Essays in Mathematical Economics and Game Theory, pp. 76–88), inspired by the bidding mechanism introduced by Pérez-Castrillo and Wettstein (2001, J. Econ. Theory 100, 274–294). The idea is the following: first players of each coalition play the bidding mechanism trying to obtain the resources of the coalition. Later, players who obtained the resources play the bidding mechanism in order to share the surplus.     

Core-stable rings in auctions with independent private values

We propose a semi-cooperative game theoretic approach to check whether a given coalition is stable in a Bayesian game with independent private values. The ex ante expected utilities of coalitions, at an incentive compatible (noncooperative) coalitional equilibrium, describe a (cooperative) partition form game. A coalition is core-stable if the core of a suitable characteristic function, derived from the partition form game, is not empty. As an application, we study collusion in auctions in which the bidders? final utility possibly depends on the winner?s identity. We show that such direct externalities offer a possible explanation for cartels? structures (not) observed in practice.     

Coalitional stability under perfect foresight

Summary. We analyze strategic social environments where coalitions can form through binding or nonbinding agreements and actions of a coalition may impose externalities upon the welfare of the rest of the players. We define a solution concept that (1) captures the perfect foresight of the players that has been overlooked in the literature (e.g., Harsanyi [10] and Chwe [6]) and (2) identifies the coalitions that are likely to form and the “stable” outcomes that will not be replaced by any coalition of rational (and hence farsighted) players. The proposed solution concept thereby offers a notion of agreements and coalition formation in complex social environments. Received: February 12, 1996; revised version: March 3, 1997     

The least core, kernel and bargaining sets of large games

Summary. We study the least core, the kernel and bargaining sets of coalitional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countably additive measures. Then we show that in such games the intersection of the prekernel and the least core is non-empty. Finally, we show that the Aumann-Maschler and the Mas-Colell bargaining sets contain the set of all countably additive payoff measures in the prekernel. Received: June 6, 1996; revised version: March 1, 1997     

On the folk theorem with one-dimensional payoffs and different discount factors

Proving the folk theorem in a game with three or more players usually requires imposing restrictions on the dimensionality of the stage-game payoffs. Fudenberg and Maskin (1986) assume full dimensionality of payoffs, while Abreu et al. (1994) assume the weaker NEU condition (“nonequivalent utilities”). In this note, we consider a class of n-player games where each player receives the same stage-game payoff, either zero or one. The stage-game payoffs therefore constitute a one-dimensional set, violating NEU. We show that if all players have different discount factors, then for discount factors sufficiently close to one, any strictly individually rational payoff profile can be obtained as the outcome of a subgame-perfect equilibrium with public correlation.     

Voting blocs, party discipline and party formation

I study the strategic incentives to coordinate votes in an assembly. Coalitions form voting blocs, acting as single players and affecting the policy outcome. In an assembly with two exogenous parties I show how the incentives to accept party discipline depend on the types of the agents, the sizes of the parties, and the rules the blocs use to aggregate preferences. In a game of fully endogenous party formation, I find sufficient conditions for the existence of equilibria with one bloc, two blocs, and multiple blocs.     

Quantal response equilibria with heterogeneous agents

This paper introduces a model of quantal response equilibrium with heterogeneous agents and demonstrates the existence of a representative agent for such populations. Except in rare cases, the representative agent?s noise terms cannot be independently and identically distributed across the set of actions, even if that is assumed for the individual agents. This result demonstrates a fundamental difference between a representative agent and truly heterogeneous quantal responders and suggests that when fitting quantal response specifications to aggregate data from a population of subjects, the noise terms should be allowed to be jointly dependent across actions. Even though this introduces additional degrees of freedom, it makes the model well specified. The representative agent inherits a regular quantal response function from the actual agents, so this model does impose falsifiable restrictions.     

A new necessary condition for implementation in iteratively undominated strategies

We uncover a new necessary condition for implementation in iteratively undominated strategies by mechanisms that satisfy the “best element property” where for each agent, there exists a strategy profile that gives him the highest payoff in the mechanism. This class includes finite and regular mechanisms. We conclude that either the quasilinearity-like assumptions of available sufficiency results cannot be completely dispensed with or some mechanisms that do not satisfy the best element property must be employed. We term the condition “restricted deception-proofness.” It requires that, in environments with identical preferences, the social choice function be immune to all deceptions, making it then stronger than incentive compatibility. In some environments the conditions for (exact or approximate) implementation by mechanisms satisfying the best element property are more restrictive than previously thought.     

Non-cooperative solutions for estate division problems

In an estate division problem an estate has to be divided among several players whose total entitlement to the estate exceeds its size. This paper extends the non-cooperative approach through a claim game, as initiated by O?Neill (1982), by allowing players to put multiple claims on the same part of the estate, and by considering the case where individual entitlements may exceed the estate. A full characterization of the set of Nash equilibria of the claim game is obtained both for restricted estate division problems, where individual entitlements do not exceed the estate, and for the general case. Variations on the claim game are considered, which result in proportional division in equilibrium.