Showing papers on "Stochastic game published in 1990"

Rational Learning Leads to Nash Equilibrium

Abstract: Two players are about to play a discounted infinitely repeated bimatrix game. Each player knows his own payoff matrix and chooses a strategy which is a best response to some private beliefs over strategies chosen by his opponent. If both players' beliefs contain a grain of truth (each assigns some positive probability to the strategy chosen by the opponent), then they will eventually (a) accurately predict the future play of the game and (b) play a Nash equilibrium of the repeated game. An immediate corollary is that in playing a Harsanyi-Nash equilibrium of a discounted repeated game of incomplete information about opponents' payoffs, the players will eventually play an equilibrium of the real game as if they had complete information.

Venture Theory: A Model of Decision Weights

Abstract: Several theories suggest that people replace probabilities by decision weights when evaluating risky outcomes. This paper proposes a model, called venture theory, of how people assess decision weights. It is assumed that people first anchor on a stated probability and then adjust this by mentally simulating other possible values. The amount of mental simulation is affected by the absolute size of payoffs, the extent to which the anchor deviates from the extremes of 0 and 1, and the level of perceived ambiguity concerning the relevant probability. The net effect of the adjustment i.e., up or down vis-i-vis the anchor reflects the relative weight given in imagination to values above as opposed to below the anchor. This, in turn, is taken to be a function of both individual and situational variables, and in particular, the sign and size of payoffs. Cognitive and motivational factors therefore both play important roles in determining decision weights. Assuming that people evaluate outcomes by a prospect theory value function Kahneman and Tversky 1979 and are cautious in the face of risk, fourteen predictions are derived concerning attitudes toward risk and ambiguity as functions of different levels of payoffs and probabilities. The results of three experiments are reported. Whereas only a subset of the model's predictions can be tested in Experiment 1, all fourteen are tested in Experiments 2 and 3 using hypothetical and real payoffs, respectively. Several of the model's predictions are not supported in Experiment 2 but almost all are validated in Experiments 1 and 3. The failures relate to the exact nature of probability × payoff interactions in attitudes toward risk and ambiguity for losses. The theory and results are discussed in relation to other experimental evidence, future tests of the theory, alternative models of risky choice, and implications of venture theory for explaining further phenomena.

Repeated Games with Long-run and Short-run Players

Abstract: This paper studies the set of equilibrium payoffs in repeated games with long- and short-run players and little discounting. Because the short-run players are unconcerned about the future, each equilibrium outcome is constrained to lie on their static reaction (best-response) curves. The natural extension of the folk theorem to games of this sort would simply include this constraint in the definitions of the feasible payoffs and minmax values. In fact, this extension does obtain under the assumption that each player's choice of a mixed strategy for the stage game is publicly observable but, in contrast to standard repeated games, the set of equilibrium payoffs is different if players can observe only their opponents' realized actions.

Cyclic games and an algorithm to find minimax cycle means in directed graphs

Abstract: An algorithm is described that finds optimal stationary strategies in dynamic two-person conflicts with perfect information, deterministic transitions, finite sets of positions, and time-averaged limiting integral payoff.

A paradox of congestion in a queuing network

Abstract: In an uncongested transportation network, adding routes and capacity to an existing network must decrease, or at worst not change, the average time individuals require to travel through the network from a source to a destination. Braess (1968) discovered that the same is not true in congested networks. Here we give an example of a queuing network in which added capacity leads to an increase in the mean transit time for everyone. Self-seeking individuals are unable to refrain from using the additional capacity, even though using it leads to deterioration in the mean transit time. This example appears to be the first queuing network to demonstrate the general principle that in non-co-operative games with smooth payoff functions, user-determined equilibria generically deviate from system-optimal equilibria.

Equilibrium Behavior in Crisis Bargaining Games

Abstract: This paper analyzes a general model of two-player bargaining in the shadow of war, where one player possesses private information concerning the expected benefits of war. I derive conclusions about equilibrium behavior by examining incentive compatibility constraints, where these constraints hold regardless of the game form; hence, the qualitative results are "game-free." I show that the higher the informed player's payoff from war, the higher is his or her equilibrium payoff from settling the dispute short of war, and the higher is the equilibrium probability of war. The latter result rationalizes the monotonicity assumption prevalent in numerous expected utility models of war. I then provide a general result concerning the equilibrium relationship between settlement payoffs and the probability of war.

Bounded Rationality and Strategic Complexity in Repeated Games

Abstract: Publisher Summary This chapter discusses three important areas in modern decision theory, which are bounded rationality, artificial decision making, and management information systems. It discusses some new methodologies and results within the area of repeated games and with complexity measures that use notions of automata. The proposal to apply the notion of automaton to describe a player in a repeated game comes from a survey of repeated games. This notion is suggested as a way to distinguish between simple and complicated strategies based on the number of states of automata describing them. It studies the interactive behavior of bounded players by studying a game with appropriately restricted sets of strategies. The chapter also discusses the effect of complexity costs on the outcome of the game. The players are restricted to use automata of any finite size to play the game. However, their final payoffs decrease as they use automata of bigger sizes. Thus, tension in a player is created between high overall utility and increasing complexity. In the modified version of the game, the equilibrium outcomes have a nice simple structure and the set of equilibrium payoffs is dramatically reduced. This is even the case as the complexity costs approach zero and thus, their model points out a fundamental discontinuity regarding complexity costs.

Overview of Different Approaches for Solving Stochastic Programming Problems with Multiple Objective Functions

Abstract: Stochastic programming is one of the most exciting and challenging developments of mathematical programming. It aims to treat uncertainty within decision oriented models in a coherent and systematic way. Lack of such an approach is one of the objections raised to deterministic mathematical programming modelling. The requirement for a single objective or payoff functions is another objection; it can be argued that most decision makers usually have several decision criteria, and multi-objective programming aims to reflect this. Also simple examples show (similar to the Endorsed paradox and Arrow impossibility theorems) that there are, in general, no good ways of aggregating several criteria into one objective function. But maybe sometimes there are. Worse, even when there is a natural objective function, but stochastic elements come into play maximizing the expectation will often involve unacceptable large variances. In this way, a new interdisciplinary science is about to be born-stochastic programming with several objective functions.

Nash equilibria of n -player repeated games with semi-standard information

Abstract: The folk theorem is extended here to the case where after each stage of the repeated game each player is informed only about the equivalence classes of the pure actions which were used by the other players. The sets of upper equilibrium payoffs and of lower equilibrium payoffs are characterized here, and they are found to be different.

Differential game of optimal approach of two inertial pursers to a noninertial

Abstract: In this paper, the game of the optimal approach of two identical inertial pursuers to a noninertial evader is investigated. The duration of the game is fixed. The payoff functional is the distance between the evader and the closest pursuer when the game terminates. The value function is constructed for all possible positions of the game. The regions where the pursuit is one-to-one and the regions where it is essentially collective are described algorithmically. Some analogies between this game and the linear differential game with elliptical vectograms are indicated. It is noted that the focal surface and the dispersal surface are in proximity of one another.

A zero-sum differential game in a finite duration with switching strategies

Abstract: A zero-sum differential game of finite horizon with both players using switching controls is studied. Positive switching costs are associated with each player. Under some suitable conditions, it is proved that the Elliot–Kalton upper and lower value functions of the game are the unique viscosity solution of the same Isaacs’ equation, which turns out to be a system of evolutionary quasi-variational inequalities with bilateral obstacles. The existence of the Elliot–Kalton value of the game then follows. Some limiting cases are also discussed.

The search game on a network with immobile hider

Abstract: In this paper we discuss a search game on a network, Q, with two players, a searcher and a hider. The hider chooses a point on Q at which to hide, while the searcher starts at a given point on Q and moves with continuous trajectory subject to a maximal speed. This is a zero-sum game with payoff given by the time elapsed until the searcher reaches the point occupied by the hider. The solution to this game is known for certain types of network. Here by formulating the problem as an infinite-dimensional linear program, we derive an algorithm for its solution. We discuss the performance of the algorithm on three examples.

The organization as an adaptive network

Abstract: Organizations can be viewed as communication networks whose information-processing units are individual human agents. The agents in such systems can be modeled either as forwardlooking optimizers who know the true structure of the social system in which they participate, or as poorly-informed learners who revise their actions in light of past payoff experiences. Accordingly, an organizational structure can be regarded either as a vector of equilibrium strategies in a non-cooperative game, or as a pattern of interaction among individual learning mechanisms. I show that a system of interacting learning mechanisms (an adaptive network) converges to a unique pseudo-stationary organizational structure.

Stable bargained equilibria for assignment games without side payments

Abstract: We consider NTU assignment games, which are generalizations of two-sided markets. Matched pairs bargain over feasible allocations; the disagreement outcome is endogenuously determined, taking in account outside options which are based on the current payoff of other players. An allocation is in equilibrium if and only if each pair is in equilibrium (no player wishes to rebargain). The set of equilibria is not empty and it naturally generalizes the intersection of the core and prekernel of TU assignment games. A set with similar properties does not exist for general NTU games. The main source of technical difficulties is the relatively complicated structure of the core in NTU games. We make a strong use of reduced games and consistency requirements. We generalize also the results obtained by Rochford (1984) for TU assignment games.

Sequential games with random priority

Abstract: The paper deals with a class of two-person zero-sum sequential games related to a partial observation of random variables X1,X2,...,XN by the players. Each player, based on some indirect information, selects a moment t, l≤t≤N. In this way he communicates that he would like to accept an unknown realization xt of Xt. When only one player selects the moment t then he obtains the required realization. If both players select the same moment t, then there is a lottery choosing either Player 1 or Player 2. The player chosen by the lottery obtains the realization xt and the player thus deprived of the acceptance of xt at t

Cooperative equilibria in discounted stochastic sequential games

Abstract: This paper addresses the problem of computation of cooperative equilibria in discounted stochastic sequential games. The proposed approach contains as a special case the method of Green and Porter (developed originally for repeated oligopoly games), but it is more general than the latter in the sense that it generates nontrivial equilibrium solutions for a much larger class of dynamic games. This fact is demonstrated on two examples, one concerned with duopolistic economics and the other with fishery management.

The gold-mine game

Abstract: The paper considers the following two-person zero-sum game. The minimizing player chooses to hide his gold and a mine in two distinct boxes from an infinite number of boxes labelled 1, 2, 3,.... The maximizing player now chooses to open the boxes in some order, and if he finds the gold before the mine the payoff to him is 1; otherwise, the payoff is zero. The game is solved in the sense of Kindler.

A modification of the folk theorem to apply to time-dependent supergames

Abstract: label pertain to the attainment of cooperative outcomes (i.e., payoff vectors) in repeated games that are supported by noncooperative equilibrium strategy combinations. An example is a folk theorem based on trigger strategies. A repeated game consists of the repeated play of a particular game. For example, imagine a specific prisoner's dilemma game. A repeated game based on this would involve an unchanging pair of players who play this prisoner's dilemma game over and over again. The prisoner's dilemma itself is called the constituent game or the single-shot game or the stage game. In a repeated game, the individual repetitions are structurally independent. That is, associated with each repetition are payoffs that depend only on the actions taken in that period. Some would add a further condition to the definition of folk theorem: The set of constituent game outcomes that can be supported as noncooperative equilibrium outcomes should include virtually every (attainable) payoff vector affording to each player a higher payoff than the minimum to which he can be held by the remaining players when the latter act in concert to hold down his payoff. That is, a player i can be held to zi if there is a strategy (n - 1)-tuple available to the other players such that, no matter what strategy is selected by player i, his payoff will be no more than zi. Such a payoff, zi, is called the minimax level of the player. Game theorists generally use this narrower definition. Minimax payoffs are sometimes called individually rational payoffs. The minimax payoff is conceptually distinct from, and is no smaller than, the payoff a player can assure himself, no matter what others do (the maximin payoff). In much of the literature related to the folk theorem the constituent game is finite (i.e., each player has a finite number of pure strategies) and mixed strategies are allowed. In such games minimax and maximin payoffs coincide; however, the games in the present paper are not finite, so that the coincidence of these two concepts is not assured. A repeated game is a special case of a supergame. To distinguish repeated

Communication, Computability And Common Interest Games

Abstract: This paper provides a theory of equilibrium selection for one-shot two- player finite-action strategic-form common interest games. A single round of costless unlimited pre-play communication is allowed. Players are restricted to use strategies which are computable in the sense of Church's thesis. The equilibrium notion used involves perturbations which are themselves computable. The only equilibrium payoff vector which survives these strategic restrictions and the computable perturbations is the unique Pareto-efficient one.

On weakly completely mixed bimatrix games

Abstract: Weakly completely mixed bimatrix games are defined to be games with a completely mixed Nash component. For these games this component turns out to consist of only one point, which is isolated. Special classes of these games are completely mixed matrix and bimatrix games, the first introduced by Kaplansky, the latter by Raghavan. We give a characterization of these games, which can be used for completely mixed matrix games also. Given a completely mixed strategy pair, we are able to construct a (weakly) completely mixed bimatrix game having this pair as an equilibrium. We derive interesting results for the case where the payoff matrices have a nonnegative and irreducible inverse.