On the implementation of the L-Nash bargaining solution in two-person bargaining games

The “Nash program” initiated by Nash (Econometrica 21:128–140, 1953) is a research agenda aiming at representing every axiomatically determined cooperative solution to a game as a Nash outcome of a reasonable noncooperative bargaining game. The L-Nash solution first defined by Forgó (Interactive Decisions. Lecture Notes in Economics and Mathematical Systems, vol 229. Springer, Berlin, pp 1–15, 1983) is obtained as the limiting point of the Nash bargaining solution when the disagreement point goes to negative infinity in a fixed direction. In Forgó and Szidarovszky (Eur J Oper Res 147:108–116, 2003), the L-Nash solution was related to the solution of multiciteria decision making and two different axiomatizations of the L-Nash solution were also given in this context. In this paper, finite bounds are established for the penalty of disagreement in certain special two-person bargaining problems, making it possible to apply all the implementation models designed for Nash bargaining problems with a finite disagreement point to obtain the L-Nash solution as well. For another set of problems where this method does not work, a version of Rubinstein’s alternative offer game (Econometrica 50:97–109, 1982) is shown to asymptotically implement the L-Nash solution. If penalty is internalized as a decision variable of one of the players, then a modification of Howard’s game (J Econ Theory 56:142–159, 1992) also implements the L-Nash solution.     

Analysis of a family of two-person bargaining games with incomplete information

This paper computes the Harsanyi-Selten solution for a family of two-person bargaining games with incomplete information where one player hastwo possible types while the other player has onlyone possible type. The actual computation procedure is also outlined.     

Relevant aspects in two-person games

The present paper is concerned with characterizing in a nonusual form the equilibrium points for the mixed extension of a two-person game. We study interesting properties about such equilibrium points which are concerned with different pairs of them. Finally, we introduce an elimination procedure for pure strategies and relate in a general way the complete set of equilibrium points.This work has been partially supported by the Consejo Nacional de Investigaciones Cientificas y Tecnicas, Buenos Aires, Argentina.     

Non-cooperative two-person games in biology: A classification

This article compares evolutionary equilibrium notions with solution concepts in rational game theory. Both static and dynamic evolutionary game theory are treated. The methods employed by dynamic theory, so-called “game dynamics”, could be discovered to be relevant for rational game theory also.     

Sequential Stackelberg equilibria in two-person games

The concept of sequential Stackelberg equilibrium is introduced in the general framework of dynamic, two-person games defined in the Denardo contracting operator formalism. A relationship between this solution concept and the sequential Nash equilibrium for an associated extended game is established. This correspondence result, which can be related to previous results obtained by Baar and Haurie (1984), is then used for studying the existence of such solutions in a class of sequential games. For the zero-sum case, the sequential Stackelberg equilibrium corresponds to a sequential maxmin equilibrium. An algorithm is proposed for the computation of this particular case of equilibrium.This research was supported by SSHRC Grant No. 410-83-1012, NSERC Grant No. A4952, and FCAR Grants Nos. 86-CE-130 and EQ-0428.The authors thank T. R. Bielecki and J. A. Filar, who pointed out some mistakes and helped improving the paper.At the time of this research, this author was with GERMA, Ecole Mohammedia d'Ingénieurs, Rabat, Morocco.     

Solution concepts in two-person multicriteria games

In this paper, we propose new solution concepts for multicriteria games and compare them with existing ones. The general setting is that of two-person finite games in normal form (matrix games) with pure and mixed strategy sets for the players. The notions of efficiency (Pareto optimality), security levels, and response strategies have all been used in defining solutions ranging from equilibrium points to Pareto saddle points. Methods for obtaining strategies that yield Pareto security levels to the players or Pareto saddle points to the game, when they exist, are presented. Finally, we study games with more than two qualitative outcomes such as combat games. Using the notion of guaranteed outcomes, we obtain saddle-point solutions in mixed strategies for a number of cases. Examples illustrating the concepts, methods, and solutions are included.     

Nonzero-sum differential games with bargaining solutions

Nash's bargaining solution for finite games is extended to differential games with nonzero-sum integral payoffs. Sufficient conditions for the optimality of a strategy pair are established. An example is given.     

Stochastic dominance equilibria in two-person noncooperative games

Two-person noncooperative games with finitely many pure strategies are considered, in which the players have linear orderings over sure outcomes but incomplete preferences over probability distributions resulting from mixed strategies. These probability distributions are evaluated according to t-degree stochastic dominance. A t-best reply is a strategy that induces a t-degree stochastically undominated distribution, and a t-equilibrium is a pair of t-best replies. The paper provides a characterization and an existence proof of t-equilibria in terms of representing utility functions, and shows that for large t behavior converges to a form of max–min play. Specifically, increased aversion to bad outcomes makes each player put all weight on a strategy that maximizes the worst outcome for the opponent, within the supports of the strategies in the limiting sequence of t-equilibria.The paper has benefitted from the comments of four referees and an associate editor.     

Correlated equilibria in some classes of two-person games

A correlated equilibrium in a two-person game is “good” if for everyNash equilibrium there is a player who prefers the correlated equilibrium to theNash equilibrium. If a game is “best-response equivalent” to a two-person zero-sum game, then it has no good correlated equilibria. But games which are “almost strictly competitive” or “order equivalent” to a two-person zero-sum game may have good correlated equilibria.     

Pure Nash equilibria in finite two-person non-zero-sum games

In this paper we study bimatrix games. The payoff matrices have properties closely related to concavity of functions. For such games we find sufficient conditions for the existence of pure Nash equilibria.     

Threat bargaining games with a variable population

In this paper we establish links between desirable properties satisfied by familiar solutions to bargaining games with a variable population and the Nash equilibrium concept for threat bargaining games. We introduce three new concepts for equilibrium threat strategies called strategic stability, strategic monotonicity with respect to changes in the number of agents and strategic constancy. Our primary objective in this paper is to show that familiar assumptions satisfied by bargaining games with a variable population yield equilibrium threat strategies which satisfies in a very natural way the concepts we have introduced.     

Perfect information two-person zero-sum markov games with imprecise transition probabilities

Based on an extension of the controlled Markov set-chain model by Kurano et al. (in J Appl Prob 35:293–302, 1998) into competitive two-player game setting, we provide a model of perfect information two-person zero-sum Markov games with imprecise transition probabilities. We define an equilibrium value for the games formulated with the model in terms of a partial order and then establish the existence of an equilibrium policy pair that achieves the equilibrium value. We further analyze finite-approximation error bounds obtained from a value iteration-type algorithm and discuss some applications of the model.     

Computational complexity of winning strategies in two-person polynomial games

One considers two-person games, with players called I and II below. In order, they choose natural numbers, for example, for length 4, I chooses x₁, II chooses x₂. I chooses x₃, II chooses x₄. Then I wins if P(x₁,x₂,x₃,x₄)=0.Here P is a polynomial with integer coefficients. An old theorem of von Neumann and Zermelo shows that such a game is determined, i.e., there exists a winning strategy for one player or the other but not necessarily a computable winning strategy or one computable in polynomial time. It will be shown that there exists a game of polynomial type of length 4 for which there do not exist winning strategies for either player which are computable in polynomial time.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 69–73, 1991.     

Generalized correlated equilibrium for two-person games in extensive form with perfect information

A correlation scheme (leading to a special equilibrium called “soft” correlated equilibrium) is applied for two-person finite games in extensive form with perfect information. Randomization by an umpire takes place over the leaves of the game tree. At every decision point players have the choice either to follow the recommendation of the umpire blindly or freely choose any other action except the one suggested. This scheme can lead to Pareto-improved outcomes of other correlated equilibria. Computational issues of maximizing a linear function over the set of soft correlated equilibria are considered and a linear-time algorithm in terms of the number of edges in the game tree is given for a special procedure called “subgame perfect optimization”.     

Approximate calculation of Nash equilibria for two-person games

An approximate method for calculating Nash equilibrium points in a two-person game is developed.     

A note on two-person zero-sum communicating stochastic games

For undiscounted two-person zero-sum communicating stochastic games with finite state and action spaces, a solution procedure is proposed that exploits the communication property, i.e., working with irreducible games over restricted strategy spaces. The proposed procedure gives the value of the communicating game with an arbitrarily small error when the value is independent of the initial state.     

Solutions for some bargaining games under the Harsanyi-Selten solution theory,part II: Analysis of specific bargaining games

Part II of the paper (for Part I see Harsanyi (1982)) describes the actual solutions the Harsanyi-Selten solution theory provides for some important classes of bargaining games, such as unanimity games; trade between one seller and several potential buyers; and two-person bargaining games with incomplete information on one side or on both sides. It also discusses some concepts and theorems useful in computing the solution; and explains how our concept of risk dominance enables us to analyze game situations in terms of some intuitively very compelling probabilistic (subjective-probability) considerations disallowed by classical game theory.     

Time inconsistency and learning in bargaining games

This paper studies an alternating-offers bargaining game between possibly time-inconsistent players. The time inconsistency is modeled by quasi-hyperbolic discounting and the “naive backwards induction” solution concept is used in order to obtain the results. Both naive agents who remain naive and those who learn about their own preferences are considered. Offers of the players who are naive or partially naive are never accepted by any type of player in either no learning or gradual learning cases. The game between a naive or partially naive player who never learns and a time-consistent agent ends in an immediate agreement if the time-consistent agent is the proposer. A one period delay occurs if the time-consistent agent is the responder. The more naive the player is, the higher the share received. In addition, two naive agents who never learn disagree perpetually. When naive and partially naive agents play against exponential or sophisticated agents and they are able to learn their types over time, there exists a critical date before which there is no agreement. Therefore, the existence of time-inconsistent players who can learn their types as they play the game can be a new explanation for delays in bargaining. The relationship among the degree of naivete, impatience level and bargaining delay is also characterized. Specifically, for sufficiently high discount factors, agreement is always delayed. On the other hand, if the naive agent has sufficiently firm initial beliefs (slow learning or high degree of naivete), agents agree immediately.