A bargaining approach to the Owen value and the Nash solution with coalition structure

The mechanism by Hart and Mas-Colell (1996) for non-transferable utility (NTU) games is generalized so that a coalition structure among players is taken into account. The new mechanism yields the Owen value for transferable utility (TU) games with coalition structure as well as the consistent value (Maschler and Owen 1989, 1992) for NTU games with trivial coalition structure. Furthermore, we obtain a solution for pure bargaining problems with coalition structure which generalizes the Nash (1950) bargaining solution.     

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.     

Bargaining in Uncertain Environments: A Systematic Distortion of Perfect Equilibrium Demands

We present data supporting the notion that subjects distort monotonically subgame perfect equilibrium demands in two-person discount bargaining games, where, in addition to the traditional offer counter-offer sequence of moves, a player who rejects an offer may either be forced out of the bargaining or be given the option to opt out. Subjects also exhibit behavioral adaptivity—learning from past demands in a search for the highest acceptable demand. These reuslts are consistent with previous experimental evidence for these types of games, but stand in sharp contrast to those obtained in discount bargaining where side options are not available.Journal of Economic LiteratureClassification Numbers: C72, C78.     

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).     

Uniqueness of equilibrium for smooth multistage concave games

Smooth, noncooperative, multistage, concave games are formulated so that a new uniqueness condition—based on the Poincaré-Hopf theorem—can be applied. The new condition is the weakest to appear in the uniqueness literature. The uniqueness subgame perfect equilibrium is obtained and examples are given.     

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.     

Multilateral Bargaining Problems

In many situations in economics and political science there are gains from forming coalitions but conflict over which coalition to form and how to distribute the gains. This paper presents an approach to suchmultilateral bargaining problems. Asolutionto a multilateral bargaining problem specifies an agreement for each coalition that is consistent with the bargaining process in every coalition. We establish the existence of such solutions, show that they are determined by reservation prices, and characterize these reservation prices as the payoffs ofsubgame perfect equilibrium outcomesof a non-cooperative bargaining model.Journal of Economic LiteratureClassification Numbers: C71, C72, C78.     

A Backward Induction Experiment

This paper reports experiments with one-stage and two-stage alternating-offers bargaining games. Payoff-interdependent preferences have been suggested as an explanation for experimental results that are commonly inconsistent with players' maximizing their monetary payoffs and performing backward induction calculations. We examine whether, given payoff-interdependent preferences, players respect backward induction. To do this, we break backward induction into its components, subgame consistency and truncation consistency. We examine each by comparing the outcomes of two-stage bargaining games with one-stage games with varying rejection payoffs. We find and characterize systematic violations of both subgame and truncation consistency. Journal of Economic Literature Classification Numbers: C70, C78.     

Convergence of the Aumann–Davis–Maschler and Geanakoplos bargaining sets

Summary. Geanakoplos [17] defined a notion of bargaining set, and proved that his bargaining set is approximately competitive in large finite transferable utility (TU) exchange economies with smooth preferences. Shapley and Shubik [26] showed that the Aumann–Davis–Maschler bargaining set is approximately competitive in replica sequences of TU exchange economies with smooth preferences. We extend Geanakoplos result to nontransferable utility (NTU) exchange economies without smooth preferences, and we extend the Shapley and Shubik result to non-replica sequences of NTU exchange economies with smooth preferences.Received: November 11, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Average Monotonic Cooperative Games

A subclass of monotonic transferable utility (T.U.) games is studied: average monotonic games. These games are totally balanced. We prove that the core coincides with both the bargaining set à la Davis and Maschler and the bargaining set à la Mas-Colell. To obtain this result a technique based on reduced games is used. Journal of Economic Literature Classification Number: C71     

Noncooperative foundations of bargaining power in committees and the Shapley–Shubik index

In this paper we explore the noncooperative foundations of the bargaining power that a voting rule gives to each member of a committee that bargains in search of consensus over a set of feasible agreements under a voting rule. Assuming complete information, we model a variety of bargaining protocols and investigate their stationary subgame perfect equilibria. We show how the Shapley–Shubik index and other power indices can be interpreted as measures of ‘bargaining power’ that appear in this light as limit cases.     

Bargaining with Imperfect Commitment

We study the effect of imperfect commitment in noncooperative two-person bargaining games. By establishing the reputation for being stubborn, a player sometimes commits to her initial demand, becoming unable to change her demands or to accept an inferior offer from her opponent. When the probability of being stubborn is small, the set of equilibria is shown to be small and agreement may be reached immediately despite the possibility of stubbornness. A player has greater bargaining power when she is more patient and/or is more likely to be stubborn. Journal of Economic Literature Classification Number: C78.     

A Closed Form Characterization of the Stationary Outcome in Multilateral Bargaining

In this paper we consider infinite horizon multilateral bargaining with alternate offers. We prove that there exists only one stationary subgame perfect equilibrium outcome and it corresponds to the unique invariantmeasure of a column stochastic matrix. We characterize this stationary subgame perfect equilibrium outcome in a closed form, and also extend the approach to the multilateral bargaining with random moves.     

Cooperative equilibria in finite horizon noncooperative supergames

It is well known that cooperative outcomes can be supported by noncooperative equilibrium strategy combinations in games that are infinitely repeated; however, the example of the repeated prisoner's dilemma has led people to conclude that such cooperative outcomes cannot be sustained if the game is repeated finitely often. In this paper it is proved that the belief about finitely repeated games is false. Conditions are given that allow the existence of cooperative outcomes in finitely repeated games that are supported by subgame perfect noncooperative equilibrium strategies.     

Subgame Monotonicity in Extensive Form Evolutionary Games

A theory of subgame monotonicity is developed for finite extensive form games to model evolutionary processes that respect the subgame structure of the game. It is shown how these evolutionary processes can be decomposed into their evolution on smaller games and also recombined to simplify the analysis of the original game. When applied to the replicator dynamic, a generalization of the Wright manifold from population genetics is a natural consequence of monotonicity. Subgame monotonicity and the Wright manifold are also shown to arise naturally in imitation models based on the extensive form. Journal of Economic Literature Classification Numbers: C70, C72.     

N-PERSON NASH BNARGAINING WITH VARIABEL THREATS*

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.     

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.     

On Renegotiation-Proof Equilibria in Finitely Repeated Games

This paper studies renegotiation-proof equilibria of Beno?̂t and Krishna (1993) in finitely repeated games with more than two players. We provide a simple characterization of the set of average payoffs from renegotiation-proof equilibria: the limiting set of average equilibrium payoffs is either efficient or dimensionally small as the horizon of the repeated game goes to infinity. Two sufficient conditions for the former to occur are provided. An example shows that those conditions cannot be weakened.Journal of Economic LiteratureClassification Numbers: C72, C73.     

Renegotiation-proof contract in repeated agency

Renegotiation-proof contracts are studied in infinitely repeated principal–agent problem. Contracts satisfying a weaker notion of renegotiation-proofness always exist. The renegotiation-proof value function has a simple characterization: It is the principal's optimal value function when an appropriate lower bound is placed on the agent's expected utility. Sufficient conditions are provided for renegotiation-proof value function in finite horizon to converge to renegotiation-proof value function in infinite horizon as time goes to infinity.