Bargaining in a non-stationary environment

We study an alternating offers bargaining model in which the set of possible utility pairs evolves through time in a non-stationary, but smooth manner. In general, there exists a multiplicity of subgame perfect equilibria. However, we show that in the limit as the time interval between two consecutive offers becomes arbitrarily small, there exists a unique subgame perfect equilibrium. Furthermore, we derive a powerful characterization of the unique (limiting) subgame perfect equilibrium payoffs. We then explore the circumstances under which Nash's bargaining solution implements this bargaining equilibrium. Finally, we extend our results to the case when the players have time-varying inside options.     

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.     

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.     

Voting with abstention

Most voting models in the literature neglect abstention, but is such a simplification justified? I investigate this question in a model with outside pressure on voters. For sequential voting (e.g., roll call votes), with and without an abstention option, there is a unique subgame perfect equilibrium, which implies that true majorities always succeed. Abstention can be an equilibrium strategy for some voters, in particular under complex decision rules (e.g., weighted voting, double majorities). Simultaneous voting often has a unique pure strategy equilibrium but also a plethora of mixed and pure/mixed strategy equilibria. Therefore, only with equilibrium selection, can we evaluate the consequences of neglecting abstention. For equal weight voting, equilibria selected by the procedure of Harsanyi and Selten change completely with an abstention option, even if abstention itself is not or rarely used. With small enough outside pressure, however, the selected equilibrium honors true majorities in both cases.     

Rational conjectures and evolutionary beliefs in public goods games

This paper applies conjectural variations (CVs) to a model of public good provision and shows that CVs are superior to Nash beliefs. In addition to imposing consistency, as Bresnahan, I show that consistent conjectures (CCs) are obtained from individual payoff maximization. CCs emerge as the unique subgame perfect Nash equilibrium (NE) in a two-stage game in which beliefs are chosen in Stage 1 and quantities in Stage 2. There is an individual payoff advantage to non-Nash behavior, generating a Prisoner's Dilemma in conjectures in addition to the usual free-rider problem associated with public goods. The correct and payoff maximizing conjecture is the unique equilibrium in an evolutionary framework against a player with Nash conjectures. The consistent conjecture equilibrium is the unique evolutionary equilibrium when both players conjectures evolve. Hence, the NE prediction is too optimistic when players have rational conjectures.     

Subgame Perfect Equilibria in Stage Games

It is well known that a stage game with infinite choice-sets, unless it contains a public coordination-device in each stage, may have no subgame perfect equilibria. We show that if a game with public coordination-devices has a subgame perfect equilibrium in which two players in each stage use non-atomic strategies, then the game without coordination devices also has a subgame perfect equilibrium. Journal of Economic Literature Classification Numbers: C6, C7, D8.     

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.     

A strategic implementation of the Average Tree solution for cycle-free graph games

In this note we provide a strategic implementation of the Average Tree solution for zero-monotonic cycle-free graph games. That is, we propose a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the average hierarchical outcome of the game. This mechanism takes into account that a player is only able to communicate with other players (i.e., to make proposals about a division of the surplus of cooperation) when they are connected in the graph.     

Efficient delays in a stochastic model of bargaining

Summary. We consider a k-player sequential bargaining model in which both the cake size and the identity of the proposer are determined by a stochastic process. For the case where the cake is a simplex (of random size) and the players share a common discount factor, we establish the existence of a unique stationary subgame perfect payoff which is efficient and characterize the conditions under which agreement is delayed. We also investigate how the equilibrium payoffs depend on the order in which the players move and on the correlation between the identity of the proposer and the cake size.Received: November 5, 1996; revised version: December 31, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Exploitable actions of believers in the “law of small numbers” in repeated constant-sum games

In repeated fixed-pair constant-sum games with unique equilibria in mixed strategies, such as matching pennies, the subgame perfect equilibrium is repeating the stage-game mixed-strategy equilibrium action. In such games rational players avoid strategies that are exploitable, in that current actions either deviate systematically from the equilibrium action probabilities or fail to be serially independent of past actions. I revisit classic experiments and find that subjects’ actions are sometimes exploitable because they are serially dependent. Subjects have difficulty in producing serially independent actions and in recognizing serially dependent sequences due to a bias called local representativeness.     

Union–firm bargaining: Order of play and efficiency

This paper shows that a modified alternating offers Rubinstein model can provide a Pareto superior outcome in the context of the right-to-manage union–firm bargaining. Two examples of bargaining protocols that yield a superior outcome are provided. In the first example, the parties engage in a game in which the order of play is determined as part of the bargaining. We show that the game has a unique subgame perfect equilibrium in which the firm always moves first in the wage bargaining game. The equilibrium wage is, therefore, unique. In the second example, we examine a two-part-tariff alternating offers bargaining protocol, where the parties bargain over the wage and transfer payments. We show that this bargaining protocol has a Pareto efficient, unique subgame perfect equilibrium. Thus, although the parties do not bargain over the level of employment, the outcome under this protocol is, nevertheless, socially optimal.     

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.     

Stochastic evolutionary stability in extensive form games of perfect information

Nöldeke and Samuelson [Games Econ. Behav. 5 (1993) 425] investigate a stochastic evolutionary model for extensive form games and show that even for games of perfect information with a unique subgame perfect equilibrium, non-subgame perfect equilibrium-strategies may well survive in the long run even when mutation rates tend to zero. In a different model of evolution in the agent normal form of these games Hart [Games Econ. Behav. 41 (2002) 227] shows that under suitable limit-taking, where small mutation rates are accompanied by large population sizes in a particular way, the unique prediction is again the subgame perfect equilibrium. This paper provides a proof of a similar result for the model of Nöldeke and Samuelson.     

Bargaining with history-dependent preferences

Two impatient players bargain over a pie of size one according to the infinite alternating-offers procedure. Players’ payoffs depend not only on the outcome but also on the process of the bargaining. Specifically, they prefer impasse to any agreement that gives them lower discounted utility than would have been derived from accepting earlier offers. We characterize the essentially unique subgame perfect equilibrium path, which consists of gradual concessions. The more patient players are, the longer it takes them to reach an agreement. When players become infinitely patient, the efficiency loss is substantial, yet the equilibrium division converges to the Nash solution.     

Majority Rule in a Stochastic Model of Bargaining

In this paper we consider multilateral stochastic bargaining models with general agreement rules. For n-player games where in each period a player is randomly selected to allocate a stochastic level of surplus and q?n players have to agree on a proposal to induce its acceptance, we characterize the set of stationary subgame perfect equilibrium payoffs and establish their existence. We show that for agreement rules other than the unanimity rule, the equilibrium payoffs need not be unique. Furthermore, even when the equilibrium is unique, it need not be efficient. Journal of Economic Literature Classification Numbers: C73, C78, D70.     

Bargaining foundation for ratio equilibrium in public‐good economies

We provide a bargaining foundation for the concept of ratio equilibrium in public‐good economies. We define a bargaining game of alternating offers, in which players bargain to determine their cost shares of public‐good production and a level of public good. We study the stationary subgame perfect equilibrium (SSPE) without delay of the bargaining game. We demonstrate that when the players are perfectly patient, they are indifferent between the equilibrium offers of all players. We also show that every SSPE without delay in which the ratios offered by all players are the same leads to a ratio equilibrium. In addition, we demonstrate that all equilibrium ratios are offered by the players at some SSPE without delay. We use these results to discuss the case when the assumption of perfectly patient players is relaxed and the cost of delay vanishes.     

Perfect Information and Potential Games

Games with perfect information giving rise to potential normal forms are described. For agent normal forms, a potential is defined such that a strategy profile is a maximizer for the potential if and only if it is a subgame perfect equilibrium. In the normal form of any game with perfect information, every better reply path where the players do not change their choices at irrelevant nodes leads to an equilibrium. Journal of Economic Literature Classification Number: C72.     

Serial dictatorship and Pareto optimality

In a deterministic allocation problem in which each agent is entitled to receive exactly one object, an allocation is Pareto optimal if and only if it is the outcome of a serial dictatorship. We extend the definition of serial dictatorship to settings in which some agents may be entitled to receive more than one object, and study the efficiency and uniqueness properties of the equilibrium allocations. We prove that subgame perfect equilibrium allocations of serial dictatorship games are not necessarily Pareto optimal; and generally not all Pareto optima can be implemented as subgame perfect equilibrium allocations of serial dictatorship games, except in the 2-agent separable preference case. Moreover, serial dictatorship games do not necessarily have unique subgame perfect equilibrium allocations, except in the 2-agent case, hence their outcomes are indeterminate and manipulable.     

A note on renegotiation in repeated Bertrand duopolies

Weak Renegotiation-Proofness (WRP) singles out marginal cost pricing as a unique pure-strategy equilibrium of the infinitely repeated Bertrand duopoly. We show that, with a discrete strategy space, WRP does not eliminate any relevant subgame perfect equilibrium outcome.