Showing papers in "International Journal of Game Theory in 1971"
TL;DR: In this article, it was shown that the optimal assignment game is a dual problem of a linear programming problem dual to optimal assignment, and that these outcomes correspond exactly to the price lists that competitively balance supply and demand.
Abstract: The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in thecore of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.
1,751 citations
TL;DR: In this article, it was shown that the core of a convex game is not empty and that it has an especially regular structure, and that certain other cooperative solution concepts are related in a simple way to the core.
Abstract: The core of ann-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one that is based on a convex set function. In this paper it is shown that the core of a convex game is not empty and that it has an especially regular structure. It is further shown that certain other cooperative solution concepts are related in a simple way to the core: The value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core.
1,634 citations
TL;DR: It was shown in this paper that for convex games, the bargaining set of the grand coalition of a convex game coincides with the core of the game, and that the kernel of a game with a grand coalition consists of a unique point which coincides with its nucleolus.
Abstract: It is shown that for convex games the bargaining setℳ1(i) (for the grand coalition) coincides with the core. Moreover, it is proved that the kernel (for the grand coalition) of convex games consists of a unique point which coincides with the nucleolus of the game.
260 citations
TL;DR: In this paper, repeated two-person zero-sum games with partial information about a chance move are considered and it is shown that the game with n repetitions has no simultaneous solutions.
Abstract: We consider repeated two-person zero-sum games in which each player has only partial information about a chance move that takes place at the beginning of the game. Under some conditions on the information pattern it is proved that\(\mathop {\lim }\limits_{n \to \infty } v_n\) exists,vn being the value of the game withn repetitions. Two functional equations are given for which\(\mathop {\lim }\limits_{n \to \infty } v_n\) is the only simultaneous solutions. We also find the least upper bound for the error term\(\left| {v_n - \mathop {\lim }\limits_{n \to \infty } v_n } \right|\).
170 citations
TL;DR: In this paper, a value for n-person games without side payments is given which coincides with the Shapley value for games with side payments, and with the Nash value for two-player games.
Abstract: A value forn-person games without side payments is given which coincides with theShapley value for games with side payments, and with theNash value for two-person games.
36 citations
TL;DR: For a class of repeated two-person zero-sum games with incomplete information, it was proved by Aumann and Maschler as mentioned in this paper that δn is of a lower order of magnitude.
Abstract: For a class of repeated two-person zero-sum games with incomplete information it was proved byAumann andMaschler that\(\mathop {\lim }\limits_{n \to \infty } v_n\) exists,Νn being the value of the game withn repetitions. As for the speed of convergenceAumann andMaschler showed that the error termδn=¦Νn−limΝn¦ is bounded from above byc/√n for some positive constantc. Both results have been generalized byMertens andZamir. It is shown in this paper that the above mentioned theorem about the speed of convergence is sharp in the sense that there are games in whichδn≥c′/√n for some positive constantc′. However there are games for which δn is of a lower order of magnitude, for instancec′(logn)/n≤δn≤c (logn)/n orc′/n≤δn≤c/n. Sufficient conditions are given here for games to belong to one of these categories as well as examples of games from each category.
36 citations
TL;DR: In this paper, the authors considered the simultaneous truel, a three-person game where each player is concerned only with her own survival and indifferent to the fate of her opponents.
Abstract: The simultaneous truel is a three-person game which is a generalization of the simple duel. The players' positions are fixed and their firing is simultaneous. Each player's only decision is which of his opponents will be his target. The (simultaneous) firing continues until there is at most one survivor or until all survivors have fired a specified number of times. Each player is assumed to be concerned only with his own survival; he is indifferent to the fate of his opponents. These games (parametrized by the maximum possible number of shots by each player) are examined for equilibrium points. It is found that, in many cases, the truel has a unique equilibrium point at which the player who is the poorest marksman has the greatest chance of survival.
28 citations
TL;DR: In this paper, it was shown that a finite two-person zero-sum game with non-Archimedean utilities may have no equilibrium-point solution, and either one or both players may have a "minimax" strategy.
Abstract: Contrary to what appears to have become an accepted part of the folklore of game theory, a finite two-person zero-sum game with non-Archimedean utilities may have no equilibrium-point solution, and either one or both players may have no “minimax” strategy. Even when both players have “minimax” strategies, such a game may lack an equilibrium point.
24 citations
TL;DR: Some theorems containing new results on the nucleolus and the central game are proven in this article, which is the case for all the results in this paper. But these results are based on the same model.
Abstract: Some theorems containing new results on the nucleolus and the central game are proven.
23 citations
TL;DR: Fiat money is a type paper or symbol with which any individual may buy most things by law and has virtually no intrinsic value but immediately assumes a trading value when its shortage (i.e., when it is no longer a slack variable to everyone in the appropriate set of simultaneous programs) can prevent trades that would have been deemed profitable in a nonmonetary competitive equilibrium system.
Abstract: Fiat money is a type paper or symbol with which any individual may buy most things by law. It has virtually no intrinsic value but immediately assumes a trading value when its shortage (i.e., when it is no longer a slack variable to everyone in the appropriate set of simultaneous programs) can prevent trades that would have been deemed profitable in a nonmonetary competitive equilibrium system.
22 citations
TL;DR: In this article, the authors extend the results of J. F. Mertens and S.Zamir, The Value of Two-Person Zero-Sum Repeated Games with Lack of Information on Both Sides, to the case where both players are not necessarily informed of each other's pure strategy choices at each stage.
Abstract: The purpose of this article is to extend the results of J. F.Mertens and S.Zamir, The Value of Two-Person Zero-Sum Repeated Games with Lack of Information on Both Sides (Intern. Journal of Game Theory,1, 39–64, 1971) to the case where both players are not necessarily informed of each other's pure strategy choices at each stage.
TL;DR: In this article, axioms for values of games with denumerably many players are introduced and, on a certain space of games, a value is defined as a limit of values of finite games.
Abstract: In this paper axioms for values of games with denumerably many players are introduced and, on a certain space of games, a value is defined as a limit of values of finite games. Further, some relationships between the value that the topology on the space of games of bounded variation are investigated. It is also shown and the regular weighted majority games are members of the space on which the value is defined.
TL;DR: In this paper, a bi-matrix threat game is defined as a triple (A,B,S) where A and B arem×n payoff matrices, and S is a closed convex subset of the plane, with (aij,Bij) eS for eachi,j. Given (threat) mixed strategiesx andy,Nash's model suggests that the eventual outcome will be that point (u, v), eS which maximizes the product (u −xAyt) (v −xByt) subject tou ≥x
Abstract: A bi-matrix threat game is defined as a triple (A,B,S) whereA andB arem×n payoff matrices, andS is a closed convex subset of the plane, with (aij,Bij) eS for eachi,j. Given (threat) mixed strategiesx andy,Nash's model suggests that the eventual outcome will be that point (u, v) eS which maximizes the product (u −xAyt) (v −xByt) subject tou ≥xAyt,v ≥xByt. Optimality of the threat strategies is then defined in the obvious way.
TL;DR: In this paper, the supergame of a non-cooperative game is defined as an infinite sequence of plays of the original game, and the notions of strong Pareto equilibrium point (s.p.) and essential core are introduced.
Abstract: Noncooperative games in normal form and in characteristic function form are considered. The supergame of the noncooperative game is defined as an infinite sequence of plays of the original game. The notions of strong Pareto equilibrium point (s.p.e.p.) and essential core are introduced. A relationship between the essential core of a noncooperative game and the set of s.p.e.p. of its supergame is asserted. This result is similar to that ofAumann for cooperative games without side payments.
TL;DR: In this paper, it was shown that anm-quota games are (m − 1)-stable if and only if there is no weak player, and some relationships between k-stable pair and anm -quota are shown.
Abstract: We consider thek-stability ofm-quota games ofn players. We prove that anm-quota game (N, v), which satisfies the conditionv(S)=0 for allS, ¦S¦ ≤m −1, is (m −1)-stable if and only if there is no weak player. Further, some relationships between ak-stable pair and anm-quota are shown. Some ofLuce's results [1955] on Shapley quota games are generalized tom-quota games.
TL;DR: In this paper, the authors discuss new methods of simplifying games in extensive form that should be useful for solving actual games, and they show that these methods are powerful enough to eliminate all dominations in the strategy matrix, where domination in three senses.
Abstract: Thevon Neumann-Morgenstern normal form of a game is conceptually and theoretically useful, but in practice leads to enormous matrix games. We discuss new methods of simplifying games in extensive form that should be useful for solving actual games. The first method is that of partially normalizing the game at an information set and, if dominations are found, making local “negative” decisions not to choose certain alternatives at the information set. Coupled with this idea is the reduction operation which eliminates parts of the game tree. These methods are shown to be powerful enough to eliminate all dominations in the strategy matrix, where we consider domination in three senses.
TL;DR: Starting from the solution of the bargaining game, symmetry, dynamics, stability and cooperability of the game are defined and investigated and may be taken as a possible structure of the conflict as it may develop at the official European Security Conference.
Abstract: A group of 31 West German political scientists has played a European Security conference. Eight nations were represented, FRG, GDR, USA, USSR, France, UK, Poland, CSSR. In the course of the play 26 statements were valued under the aspect of incorporating them into a conferencecommunique to be accepted by the conference. In the article, these values are processed into a gametheoretical analysis of the conflict played. Starting from the solution of the bargaining game, symmetry, dynamics, stability and cooperability of the game are defined and investigated. The result may be taken as a possible structure of the conflict as it may develop at the official European Security Conference.