Showing papers on "Combinatorial game theory published in 1975"
TL;DR: Noncooperative and Dominant Player Solutions in Discrete Dynamic Games Author(s): Finn Kydland
Abstract: Noncooperative and Dominant Player Solutions in Discrete Dynamic Games Author(s): Finn Kydland Source: International Economic Review, Vol. 16, No. 2 (Jun., 1975), pp. 321-335 Published by: Wiley for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research -Osaka University Stable URL: http://www.jstor.org/stable/2525814 . Accessed: 07/04/2014 14:17
148 citations
TL;DR: In this paper, it was shown that the condition of having a nonempty core is sufficient for an ann-person simple majority game with ordinal preferences to have a non-empty core.
Abstract: Michael Dummett andRobin Farquharson [1961] provided a sufficient condition for ann-person simple majority game with ordinal preferences to have a nonempty core. In the present paper we generalize this result to an arbitrary proper simple game. It is proved that their condition is also sufficient for this game to have a nonempty core. Our proof of this theorem is much simpler than the proof given byDummett andFarquharson. Finally some applications of the theorem are presented.
46 citations
05 May 1975
TL;DR: It is shown that determining who wins such a game if each player plays perfectly is very hard; in fact, it is as hard as carrying out any polynomial-space-bounded computation.
Abstract: We consider a generalization, which we call the Shannon switching game on vertices, of a familiar board game called HEX. We show that determining who wins such a game if each player plays perfectly is very hard; in fact, it is as hard as carrying out any polynomial-space-bounded computation. This result suggests that the theory of combinatorial games is difficult.
45 citations
TL;DR: In this article, it is shown that appropriate union structures will generate a non-empty core for any game which is monotonic in zero-normalised form (such as a superadditive game).
19 citations
TL;DR: In this article, the authors generalized the concept of perfect information to games in which the players, while moving sequentially, remain uncertain about the actual payoff of the game because of an initial chance move.
Abstract: The present paper generalizes the concept of perfect information to games in which the players, while moving sequentially, remain uncertain about the actual payoff of the game because of an initial chance move. It is proved that the value of such games with “almost” perfect information can still be computed using backward induction in the game tree. The optimal behavioral strategies obtained by a dynamic procedure may, however, require randomization. A typical illustration of such games is poker.
12 citations
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TL;DR: In this article, the authors generalized the concept of perfect information to games in which the players, while moving sequentially, remain uncertain about the actual payoff of the game because of an initial chance move.
Abstract: The present paper generalizes the concept of perfect information to games in which the players, while moving sequentially, remain uncertain about the actual payoff of the game because of an initial chance move. It is proved that the value of such games with "almost" perfect information can still be computed using backward induction in the game tree. The optimal behavioral strategies obtained by a dynamic procedure may, however, require randomization. A typical illustration of such games is poker.
11 citations
TL;DR: In this article, it was shown that in a differential game, where termination is required and a playability requirement is imposed on decision pairs, some properties of classical zero-sum games need not hold.
Abstract: In a classical two-person zero-sum game, the decision pairs are members of a product space of admissible decision sets. Such a game possesses a number of desirable properties. In a differential game in which termination is required and a playability requirement is imposed on decision pairs, some properties of classical games need not hold. This is demonstrated by means of an example.
10 citations
9 citations
TL;DR: In this paper, it was shown that the information revealed by the informed player in an infinitely repeated game with simultaneous moves is essentially the same as that revealed by him in a one-stage game in which he must move first.
Abstract: Zero-sum two-person games with incomplete information on one side are considered. It is shown that the information revealed by the informed player in an infinitely-repeated game with simultaneous moves is essentially the same as the information revealed by him in a one-stage game in which he must move first.
8 citations
TL;DR: Zero-sum games with incomplete information are formulated as linear programs in which the players' behavioral strategies appear as primal and dual variables known properties for these games may then be derived from duality theory.
Abstract: Zero-sum games with incomplete information are formulated as linear programs in which the players' behavioral strategies appear as primal and dual variables Known properties for these games may then be derived from duality theory
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TL;DR: In this article, a semi-infinite game matrix has been defined, i.e., the game is specified by a sequence of vectors {Pj} ∈ Rm. The optimal strategies for both players and the value of the game can be obtained by solving a dual pair of linear programming problems.
Abstract: The ordinary finite, two-person, zero-sum game is completely defined by specifying an m × n game matrix A. The optimal strategies for both players, and the value of the game, can be obtained by solving a dual pair of linear programming problems. In this paper a semi-infinite game is defined; a semi-infinite game matrix has an infinite number of columns, i.e., the game is specified by a sequence of vectors {Pj} ∈ Rm. Optimal strategies and game values are shown to exist for the semi-infinite game by exploiting the relationship between these games and linear programming over cones.
01 Jan 1975
TL;DR: In this article, the authors discuss several discrete time versions of differential games involving resource allocation, and the games were developed over fifteen years ago to study the optimal employment of tactical air forces, and they are fairly realistic and complex.
Abstract: The interest in differential games is motivated to a large extent by the potential applications of the theory. We shall discuss several discrete time versions of differential games involving resource allocation. The games were developed over fifteen years ago to study the optimal employment of tactical air forces, and they are fairly realistic and complex. Complete mathematical solutions were then obtained and were of interest in the area of application. A significant feature of the solution in the most important model was that optimal play required the use of mixed strategies.
TL;DR: In this paper an algorithm for machine learning is defined and justified heuristically and empirically, and principles of perfect evaluation functions are derived and it is shown that these principles are sufficient to derive a perfect evaluation function and so arrive at a winning behavioral strategy.
Abstract: In this paper an algorithm for machine learning is defined and justified heuristically and empirically. “Consistency” properties of perfect evaluation functions are derived and these are used to select the best of a family of evaluation functions, that is, that evaluation function that is most consistent. Methods peculiar to a particular game such as rote learning, looking ahead are generally eschewed; “learning” consists of finding the evaluation which is most consistent. In the game of Nim, in which the winning strategy is well-known, we show that these principles are sufficient to derive a perfect evaluation function (under appropriate conditions) and so arrive at a winning behavioral strategy. In the Mod(6) game, for which a winning strategy is also known, we use the algorithm to deduce an evaluation function and evaluate its effectiveness. Finally, in the game of Hex we match the algorithm with a random player and observe its success.
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TL;DR: The interdependence of firms in oligopolistic markets and the inherent uncertainty about competitors’ reactions to any course of action adopted by a firm cannot be analysed effectively by the traditional tools of economic theory.
Abstract: The interdependence of firms in oligopolistic markets and the inherent uncertainty about competitors’ reactions to any course of action adopted by a firm cannot be analysed effectively by the traditional tools of economic theory.