Stochastic game

About: Stochastic game is a research topic. Over the lifetime, 9493 publications have been published within this topic receiving 202664 citations.

Games with Discontinuous Payoffs

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

Cooperation in an asymmetric volunteer’s dilemma game Theory and experimental evidence

Abstract: The symmetric Volunteer’s dilemma game (VOD) models a situation in which each of N actors faces the decision of either producing a step-level collective good (action “C”) or freeriding (“D”). One player’s cooperative action suffices for producing the collective good. Unilateral cooperation yields a payoff U for D-players and U - K for the cooperative player(s). However, if all actors decide for “freeriding”, each player’s payoff is zero (U > K > 0). In this article, an essential modification is discussed. In an asymmetric VOD, the interest in the collective good and/or the production costs (i.e. work) may vary between actors. The generalized asymmetric VOD is similar to market entry games. Alternative hypotheses abaout the behavior of subjects are derived from a game-theoretical analysis. They are investigated in an experimental setting. The application of the mixed Nash-equilibrium concept yields a rather counter-intuitive prediction which apparently contradicts the empirical data. The predictions of the Harsanyi-Selten-theory and Schelling’s “focal point theory” are in better accordance with the data. However, they do not account for the “diffusion-of-responsibility-effect” also observable in the context of an asymmetric VOD game.

The Paradox of Spontaneous Formation of Private Legal Systems

Abstract: ion is made by determining a ‘payoff’ to each player (i.e., benefit conferred on the player) based on both what that player did, and what others interacting with that player do. The payoff is measured in “utils”, a generic scale measuring benefits of any kind conferred on a player (e.g., money, other material benefits, spiritual elation, a sense of being loved, etc.). There might be a different payoff to the player for each combination of her and others’ actions; mirroring real life, the choice a player makes affects her welfare, but so do the choices others make. To reduce complication, several abstractions will be made in the games examined in this paper. First, it is assumed there are two players—a network member and the other network members (or a network member and the network governance institution). Second, each player is limited a choice between two actions. These actions will change from game to game depending on the illustrative story of the game, but generally they will be called ‘Cooperate’ ({C}) and ‘Default’ ({D}). To clarify the concept of a game type, consider the ‘function’ of providing a venue for scorn: Statler and Waldorf, the grumpy old men sitting on the balcony in “The Muppets’ Show”, love to express derision at anything and everything. When the Muppets Show is not on and they find no targets for their venom on stage, they must verbally attack each other. The best that can possibly happen from Statler’s point of view is when he says something nasty to Waldorf, and Waldorf does not reply. Second to this “oneupsmanship” is a situation in which they exchange gibes. Much less satisfying is a 69 To make the illustrative examples more intuitive, this paper will sometimes call the more socially beneficial of the two actions “cooperating”, while the other, less virtuous action will be called “defaulting”. But this is not always the case. There does not have to be anything morally or socially better in an action called “cooperating” over an action called “defaulting”. In some games, the two options would be equivalent morally and from a welfare-maximizing perspective. For example, in the Battle of the Sexes game, described supra, in Section III.3, the “cooperating” action is going to see a baseball game, while the “defaulting” action is going to see a movie. The tags of cooperation and default are used merely to make this game comparable to other games discussed in this section, and not to denote a positive or negative connotation to either action. The Paradox of Spontaneous Formation of Private Legal Systems/Amitai Aviram 29 situation in which both Statler and Waldorf are nice to each other, and their venom fails to find an outlet. Bad as that sounds, it can get worse—Statler might act nicely to Waldorf, who in return will mock Statler with a nasty jeer; suffering an unanswered jab is even worse than having everyone play nice. Waldorf has the same preferences as Statler (reversing the roles, of course). Abstracting these preferences into a table, the payoff structure will look like this: Waldorf acts nicely Waldorf mocks Statler acts nicely 1,1 0,3 Statler mocks 3,0 2,2 Placing the payoff information in a table helps us identify the likely outcome. Let’s put ourselves in Statler’s shoes. If he expects Waldorf to act nicely, Statler is better off mocking him (he will then get 3 “utils” (southwest box) instead of one util (northwest box)). And if Statler expects Waldorf to mock him, Statler will—once again, mock Waldorf (he will get 2 utils (southeast box) rather than zero utils (northeast box)). So Statler will mock Waldorf regardless of what he expects Waldorf to do. Since Waldorf has the same preferences, he will reach the same conclusion, and the two will end up teasing and insulting each other. That’s good news—this happens to be the welfaremaximizing solution, since they get two utils each, or 4 total—a larger total than in any of the other boxes. This game is known as the “Deadlock” game, because if acting nicely were considered to be “cooperating”, the parties would be deadlocked in refusal to cooperate. The Deadlock game is among the most costly to enforce mutual cooperation—not only do the parties tend to not cooperate, but the welfare maximizing situation for them is 70 To summarize, the set of preferences for each player of the Deadlock game is: {D,C}>{D,D}>{C,C}>{C,D}. 71 For the payoff set in each box, Statler’s payoff is noted first, then Waldorf’s payoff. 72 A commonly cited real world example of this game would be arms control negotiations between two countries who do not want to disarm (i.e., would prefer that both they and their enemy be armed than both they and their enemy be unarmed). The likely result is a failure of the arms control negotiations. See, e.g., Janet Chen, Su-I Lu & Dan Vekhter, Game Theory—Non Zero Sum Games—Other Games, available at: http://cse.stanford.edu/classes/sophomore-college/projects-98/game-theory/dilemma.html. The Paradox of Spontaneous Formation of Private Legal Systems/Amitai Aviram 30 mutual default, so if they could coordinate, they’d attempt to enforce mutual default rather than mutual cooperation. Imagine, for example, Statler & Waldorf’s response if Kermit tried to force them to act kindly to each other... The following subsections will examine other games, their illustrative stories, their payoff structure, the likely behavior of the players and the relative ease of enforcing cooperation in them. A. Harmony The Harmony game can be seen to be an inverse of the Deadlock game. It is the easiest game in which to enforce mutual cooperation. In fact, no enforcement at all is necessary. Alice and Bill, two very good friends, face a choice between the same two actions that Statler and Waldorf chose from in the Deadlock game: they can act nicely to the other or they could mock him/her. Unlike Statler and Waldorf, each of them prefers to be nice to the other, even if he himself is slighted by the other (after all, the other’s slight may have been merely a misperception, and at any rate, they care for each other so much that hurting the other would indirectly hurt them). Next worst possibility is that they themselves somehow failed and mocked the other. In that case, each hopes that the other would show restraint and not mock back (this would be worse than being mocked while acting nicely, since the shame of being rude to one’s friend in the former case outweighs the anger at being mocked in the latter case). The worst for these two would be slipping into mutual taunting. Putting these preferences into a payoff table yields this: Bill acts nicely Bill mocks Alice acts nicely 3,3 2,1

A Subexponential Randomized Algorithm for the Simple Stochastic Game Problem

Abstract: We describe a randomized algorithm for the simple stochastic game problem that requires 2O(?n) expected operations for games with n vertices. This is the first subexponential time algorithm for this problem.