Showing papers on "Stochastic game published in 1987"
TL;DR: In this article, the authors study the Markov perfect equilibrium of an alternating move, infinite horizon duopoly model where the strategic variable is quantity and demonstrate that the MPE corresponding to the solution is the limit of the finite horizon equilibrium as the horizon tends to infinity.
252 citations
TL;DR: For games with discontinuous payoffs, this paper proved an equilibrium existence result for a class of games with an infinite number of strategies, and applied this result to obtain new existence results for the multi-firm, i-dimensional version of Hotellings's location game.
Abstract: We prove an equilibrium existence result for a class of games with an infinite number of strategies. Our theorem generalises an earlier result by Dasgupta and Maskin. We also identify conditions under which the limit of pure-strategy equilibria of a sequence of finite games is an equilibrium for the limit game. We apply this result to obtain new existence results for the multi-firm, i-dimensional version of Hotellings's location game. The techniques used suggest a technique for computing such equilibria. This paper studies games with discontinuous payoffs. Any game with a compact set of strategies can be approximated by a sequence of finite games. We study the correspondence mapping each approximation to the limit game to its mixed-strategy equilibria and identify conditions under which the limit of such equilibria is an equilibrium for the limit game. When payoffs are discontinuous, this property will not hold in general. An equilibrium for the limit game may not exist or, if it does exist, the equilibrium correspondence may not be upper hemi-continuous. The topic is of concern to economists for two, rather different, reasons. The existence question is clearly important in its own right. In many games that interest economists, payoff functions fail to be continuous or even semi-continuous. Standard existence theorems (Debreu (1952), Glicksberg (1950), (1952), etc.), therefore, are inapplicable. Two classical examples are Bertrand's (1883) model of price competition and Hotelling's (1929) model of spatial competition. Recent work on auction theory (see Milgrom-Weber (1982)) and dynamic oligopoly models (surveyed in Fudenberg-Tirole (1983)) has generated many more instances. The second reason for interest in the equilibrium correspondence is pragmatic. Finite games are generally much more tractable than infinite games. When the equilibrium correspondence is upper hemi-continuous, the easiest way to find an equilibrium for an infinite game may be to calculate the equilibria of a sequence of finite games and take limits.1 The seminal paper on games with discontinuous payoffs is Dasgupta-Maskin (DM) (1986a). This paper proves an existence theorem under assumptions that are strictly weaker than DM's. The key difference between our approach and DM's is as follows. DM identify conditions under which the limit of equilibria of any sequence of approximating finite games is an equilibrium for the limit game. These conditions are quite restrictive. They may be violated even though dominant strategy equilibria exist for the limit game. Our approach to the existence problem is different. We consider a given sequence of approximating games and ask when the limit of equilibria for this particular sequence will be an equilibrium for the limit game.2
115 citations
TL;DR: It is argued that it is important to distinguish between two meanings of “frequency dependent”, concerned with how the payoff to a given action in an animal's repertoire depends on the frequency with which the animal uses the action.
114 citations
TL;DR: In this article, the authors constructed optimal penal codes for a broad class of symmetric Bertrand supergames with capacity constraints and showed that these penal codes always drive punished firms to their security levels.
Abstract: This paper constructs optimal (pure strategy) penal codes for a broad class of symmetric Bertrand supergames with capacity constraints. The results should simplify the analysis of Bertrand supergames because the punishments comprising the penal codes constructed below have a very simple form and the associated payoffs are often easy to calculate. It is also shown that for a commonly employed subclass of the games considered the admission of mixed strategies does not allow severer punishments. The analysis draws heavily on the pathbreaking work of Abreu (1984). He demonstrated that in the search for optimal penal codes attention can be restricted to "simple penal codes". Simple penal codes are characterized by a vector of outcome paths, one for each player. The ith outcome path is interpreted as the punishment to be imposed on the ith player if he deviates from the equilibrium path or from any punishment path. Section 2 explains the basic framework of analysis. The class of Bertrand games studied is very general. No particular rationing rule is imposed; some mild restrictions are placed on the firms' sales functions instead. (The commonly used models of Levitan and Shubik (1972) and Beckman (1965) are special cases of the games analysed below.) It is assumed throughout that there is a price firms can be induced to charge if the punishment for charging a different price gives the offending firm a payoff equal to its security level for the supergame. (Thus either there must exist a static Nash equilibrium in pure strategies or else the discount factor must not be too close to zero.) In Section 3 optimal penal codes are constructed and it is established that the punishments comprising them always drive punished firms to their security levels. The arguments employed make clear that optimal penal codes are not generally unique. This issue is addressed in Section 4, where it is also shown that the Levitan-Shubik rationing rule exhibits more types of optimal penal codes than do other rules. In particular, if the Levitan-Shubik rule is employed then punishments that are analogous to Abreu's (1986) symmetric two-phase punishments for Cournot games can be used to construct optimal penal codes in Bertrand games. This property does not hold generally, however.
91 citations
TL;DR: In this article, the authors examined the asymptotic behavior of the set of equilibrium payoffs in a repeated game when there are bounds on the complexity of the strategies players may select.
90 citations
01 Jan 1987
TL;DR: It is proved that the team can obtain the optimal classifier to an arbitrary approximation when posed as a game with common payoff played by a team of mutually cooperating learning automata.
Abstract: The problem of learning correct decision rules to minimize the probability of misclassification is a long-standing problem of supervised learning in pattern recognition. The problem of learning such optimal discriminant functions is considered for the class of problems where the statistical properties of the pattern classes are completely unknown. The problem is posed as a game with common payoff played by a team of mutually cooperating learning automata. This essentially results in a probabilistic search through the space of classifiers. The approach is inherently capable of learning discriminant functions that are nonlinear in their parameters also. A learning algorithm is presented for the team and convergence is established. It is proved that the team can obtain the optimal classifier to an arbitrary approximation. Simulation results with a few examples are presented where the team learns the optimal classifier.
71 citations
TL;DR: In this article, the existence of subgame-perfect equilibria in infinite-action games of perfect information with finitely or countably many players was proved. And they proved that the subgame that is played from date t on depends on the history up to t only as this history affects some vector of state variables.
57 citations
TL;DR: In this paper, the authors present a Nash dynamic game model of interactions among oil inventory and tariff policies for oil importing countries, where each player in the game chooses its individual policies to maximize expected social welfare.
Abstract: In this paper we present a Nash dynamic game model of interactions among oil inventory and tariff policies for oil importing countries. The inventory policies consist of the rates for building up and drawing down strategic oil reserves, while the tariff policies consist of setting tariffs as a function of the state of the world oil market. The model represents a stochastic game in which countries are grouped into two players, where each player in the game chooses its individual policies to maximize expected social welfare. We use the model to present empirical results about the sizes of the stockpiles as a function of disruption probabilities, tariff policies and cooperation versus noncooperation between nations. In addition to results regarding the build-up and draw down rates as a function of the amount of disruption or probability of future disruption our central conclusion is that build-up and drawdown policies are essentially the same with or without international cooperation, due to the current inventory limits on U.S. and Foreign Strategic Petroleum Reserves.
47 citations
TL;DR: A new criterion for evaluating streams of payoffs for two person zero sum stochastic games is introduced, in which one of the players can play-optimal only by using history dependent strategies.
Abstract: For two person zero sum stochastic games we introduce a new criterion for evaluating streams of payoffs. When the players use this criterion we call such games total reward stochastic games. It is unknown whether total reward stochastic games, with the property that the average value is zero for each initial state, always have a value. We examine an example of such a total reward stochastic game in which one of the players can playź-optimal only by using history dependent strategies.
Fur stochastische Zwei-PersonenNull-Summen-Spiele wird ein neues Kriterium zur Bewertung der Auszahlungsstrome eingefuhrt, das Gesamt-Gewinn-Kriterium. Es ist bisher unbekannt, ob stochastische Spiele, deren Wert bezuglich des Durchschnittsgewinn-Kriteriums gleich Null ist, bezuglich des Gesamt-Gewinn-Kriteriums einen "Wert" besitzen. Es wird ein Beispiel untersucht, in dem ein Spieler nure-optimal spielen kann, wenn er von der Vorgeschichte abhangige Strategien benutzt.
31 citations
TL;DR: In this article, a non-cooperative multi-stage multi-person game is considered where each player selects one variable different from those selected by others at each stage of the game, a player who has not selected a variable before is allowed to select the presented one if and only if all players with lower order numbers rejected it.
Abstract: The paper is concerned with a nonco operative multi stage multi person game A finite sequence of random variables is observed sequentially by k ordered players Each player selects one variable different from those selected by others At each stage of the game a player who has not selected a variable before is allowed to select the presented one if and only if all players with lower order numbers rejected it Payoff functions depend on a sample and location of all selections For game strategies of the sequential behavioral type an equilibrium solution is found by the dynamic programming method Numerical results related to the secretary type problems are presented
28 citations
TL;DR: In this paper, a discounted non-cooperative stochastic game with an abstract measurable state space, compact metric action spaces of players, and additive transition and reward structure was studied.
Abstract: In this paper, we study a discounted noncooperative stochastic game with an abstract measurable state space, compact metric action spaces of players, and additive transition and reward structure in the sense of Himmelberget al. (Ref. 1) and Parthasarathy (Ref. 2). We also assume that the transition law of the game is absolutely continuous with respect to some probability distributionp of the initial state and together with the reward functions of players satisfies certain continuity conditions. We prove that such a game has an equilibrium stationary point, which extends a result of Parthasarathy from Ref. 2, where the action spaces of players are assumed to be finite sets. Moreover, we show that our game has a nonrandomized (e-\(\bar p\))-equilibrium stationary point for each e>0, provided that the probability distributionp is nonatomic. The latter result is a new existence theorem.
TL;DR: In this article, the authors show that senior executives who take the time to look closely at maintenance expenses and equipment downtime can pay a big payoff for taking the time and effort to look at maintenance and downtime.
TL;DR: In this article, an alternative finite algorithm is furnished similar to value iteration in a switching control stochastic game, where the law of motion is controlled by player I alone when being in a certain subset of states and by player II alone in the other states.
Abstract: In a Switching Control Stochastic Game the law of motion is controlled by player I alone when being in a certain subset of states and by player II alone when being in the other states. For the discounted case Vrieze [6] has given an algorithm which consists in solving a finite number of linear programs and is related to policy iteration. In this paper an alternative finite algorithm is furnished similar to value iteration.
In einem ź'stochastischen Spiel mit wechselnder Kontrolle" wird das Bewegungsgesetz in einem Teil der Zustande von Spieler I und in den anderen Zustanden von Spieler II kontrolliert. Fur den diskontierten Fall hat Vrieze [6] einen Politikiterations-Algo-rithmus angegeben, bei dem endlich viele lineare Programme gelost werden mussen. In dieser Arbeit wird ein alternativer endlicher Algorithmus vorgestellt, der eher der Wertiteration entspricht.
TL;DR: One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers, which provide the basis for the bilinear programming results.
Abstract: One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers. Solutions to such classes of games under the average reward criterion can be derived from optimal solutions to appropriate bilinear programs. The validity of using bilinear programming as a solution method follows from two preliminary theorems, the first of which is a complete classification of undiscounted stochastic games with optimal stationary strategies. The second of these preliminary theorems relaxes the conditions of the classification theorem for certain classes of stochastic games and provides the basis for the bilinear programming results. Analogous results hold for the discounted stochastic games with the above special structures.
TL;DR: In this article, game theory is used to gain insight into interfirm competition and the marketing decision-making process and the meaning of rational behavior in less-than-perfect conditions.
Abstract: Game theory may be used to gain insight into interfirm competition and the marketing decision-making process. This paper is aimed at the reader unfamiliar or only vaguely familiar with game theory and begins with a review of game theory terminology useful to marketing applications. Attention then focuses on the meaning of rational behavior in less-than-perfect conditions, which is a critical issue in the application of game theory to practical business situations (what may appear to be irrational behavior under conditions of complete information may be reasonable given the complex competitive environment typical of marketing decision-making situations). The paper concludes with an appraisal of the use of game theory as a decision-making model for managers.
TL;DR: In this article, the authors considered a zero-sum average cost stochastic game with the unbounded lower semi-continuous cost function, and by using the contraction property ([6, 7]) for the average case, they gave sufficient conditions for which there exists a minimax stationary strategy.
Abstract: We consider a zero-sum average cost stochastic game with the unbounded lower semi-continuous cost function, and by using the contraction property ([6, 7]) for the average case we give sufficient conditions for which there exists a minimax stationary strategy. Also, we formulate a rninimax inventory model as a stochastic game and show that for any € > 0 there exists an €-minimax .random (s, S) ordering policy, which is a modification of (s, S) ordering policy, under some weak conditions. 1. Introduction and Notation A zero-sum stochastic game has been investigated by many authors and the existence of equilibrium strategies has been discussed. For example, see [9,11] for the discounted case and [2,13] for the average case. In this paper we consider an average cost stochastic game with the un bounded lower semi-continuous cost function, ·and by using the contraction property ([6,7]) for the average case we give sufficient conditions for which there exists a minimax stationary strategy. Also, we apply these results to the inventory model with an unknown demand distribution and show that for any £ > 0 there exists an £-minimax random (s,S) ordering policy, which is a modi fication of (s,s) ordering policy, under some weak conditions. By a Borel set we mean a Borel subset of some complete separable metric space. For a Borel set X, ~ X denotes the Borel subsets of X. If X is a non+ + empty Borel set, B (X) [ B (X) ] denotes the set of all non-negative real s valued Borel measurable [lower semi-continuous] functions on X. The product of the sets D 1 ,D 2 , .•• will be denoted by D 1 D 2 •..• A zero-sum stochastic game is specified by five objects: S, {A(x),XES},
01 Jan 1987
TL;DR: Careful examination of this literature indicates that two major frameworks are being used in dynamic interaction analysis, and there are researchers who claim that only one of the two frameworks is appropriate while the other does not truly present the dynamic aspects of games.
Abstract: In recent years there has been a growing interest among economists in the multi-period dynamic interaction between players. Careful examination of this literature indicates that two major frameworks are being used in such analysis. The first discusses the dynamic interaction as a repeated game while the second uses the differential (or difference) game setting. Surprisingly, there is very little interaction in the literature between these two ‘branches’ of research, partly because differential games are described in a continuous time framework while supergames are discussed in a discrete time framework. Moreover, as was evident in this conference, there are researchers who claim that only one of the two frameworks is appropriate while the other does not truly present the dynamic aspects of games.
01 Jan 1987
TL;DR: In this article, an overview of the applications of the theory of games to defence analysis is given, and the important distinction is made between those models of conflict which can be adequately modelled as two-person constant-sum games and those which model as non-constant sum games with two or more players.
Abstract: An overview of the applications of the theory of games to defence analysis is given. The important distinction is made between those models of conflict which can be adequately modelled as two-person constant-sum games and those which model as non-constant-sum games with two or more players.
TL;DR: In this article, the uniqueness of Nash equilibria for a class of stochastic differential games where the dynamic constraints are linear in the control variables was shown for an oligopoly.
Abstract: The uniqueness of Nash equilibria is shown for a class of stochastic differential games where the dynamic constraints are linear in the control variables. The result is applied to an oligopoly.
TL;DR: In this paper, the authors provide closed-loop feedback solutions to their game in a finite-time horizon with deterministic evolution dynamics, using the Kushner test for the infinite-horizon problem.
Abstract: Recently, Clemhout and Wan (Ref. 1) provided solutions to a class of stochastic differential game models concerning common property resources. However, they only established closed-loop strategies for the infinite-horizon problem using the Kushner test. In this paper, we provide closed-loop feedback solutions to their game in a finite-time horizon with deterministic evolution dynamics.
TL;DR: In this article, the authors considered a non-cooperative n-persons game with mixed strategies and proved the existence of a fixed point for a closed and convex fuzzy point-to-set mapping on the product set of strategies.
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TL;DR: This study shows that the imprecision can arise from the measurement error that commonly exists in the data used to represent the dependent variable in cross-section parameter estimates in studies using longitudinal data.
Abstract: A large and growing line of research has used longitudinal data to eliminate unobservable individual effects that may bias cross-section parameter estimates. The resulting estimates, though unbiased, are generally quite imprecise. This study shows that the imprecision can arise from the measurement error that commonly exists in the data used to represent the dependent variable in these studies. The example of economists' salaries, which are administrative data free of measurement error, demonstrates that estimates based on changes in longitudinal data can be precise. The results indicate the importance of improving the measurement of the variables to which the increasingly high-powered techniques designed to analyze panel data are applied. The estimates also indicate that the payoff to citations to scholarly work is not an artifact of unmeasured individual effects that could be biasing previous estimates of the determinants of academic salaries.
TL;DR: In this paper, two classes of scalar stochastic differential games with hard constraints on controls are considered, and the solutions are found to be bang-bang, by extending a technique developed earlier for scalar optimal control problems.
Abstract: We consider two classes of scalar stochastic differential games with hard constraints on controls. The solutions are found to be bang-bang, by extending a technique developed earlier for stochastic optimal control problems.
TL;DR: In this paper, the results of a laboratory experiment investigating side-payment games represented in normal form are reported, and the question arises as to which one provides the most accurate basis for prediction of payoffs.
Abstract: This paper reports the results of a laboratory experiment investigating sidepayment games represented in normal form. Attempts to predict payoff allocations via the application of solution concepts (such as the Shapley value or the nucleolus) encounter a problem in games of this form, because the game must first be transformed into some other form. Commonly, this other form is a set function defined over coalitions, such as the von Neumann-Morgenstern characteristic function. Because there are numerous possible transformations, the question arises as to which one provides the most accurate basis for prediction of payoffs.
TL;DR: In this article, an N-person non-cooperative discounted vector-valued game with a metric space is investigated and a D-convex equilibrium point is proved in this game system.
Abstract: AnN-person noncooperative discounted vector-valued game with a metric space is investigated. We define aD-convex equilibrium point and prove that aD-convex equilibrium point exists in this game system. A modified version of the original game system is formulated so that an equilibrium point of the modified version is aD-convex equilibrium point of the original game system. Under appropriate conditions, the converse holds in the sense that aD-convex equilibrium point is also an equilibrium point of the modified game system.
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TL;DR: In this article, a class of games in normal form, suitable for a game-theoretic analysis of models of economic type, is studied and a Pareto-optimal equilibrium situation in any game of this class is shown.
Abstract: A class of games in normal form, suitable for a game-theoretic analysis of models of economic type, is studied. There is shown to be a Pareto-optimal equilibrium situation in any game of this class. When certain regularity conditions hold, there is an outcome which is stable in any reasonable sense.
TL;DR: In this paper, the authors investigated theories of payoff allocation in cooperative non-sidepayment games, that is, games where players can form coalitions and coordinate strategies but cannot redistribute payoffs via sidepayments.
TL;DR: In this article, many-player nonzero-sum games in which the dynamics is described by Ito stochastic differential equations are considered and some negative properties of the Nash-equilibrium strategies are studied.
Abstract: Many-player nonzero-sum games in which the dynamics is described by Ito stochastic differential equations are considered. Some of the negative properties of the Nash-equilibrium strategies are studied. Examples, illustrating these properties are presented. Notions of solutions of games improving the Nash-equilibrium are introduced.
TL;DR: In this article, the authors characterize sufficient conditions for reciprocative strategies to induce cooperation in (in)finitely repeated bimatrix games, assuming agents attempt to maximize either their long-run average payoffs or their discounted expected payoff streams.