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Stochastic game
About: Stochastic game is a research topic. Over the lifetime, 9493 publications have been published within this topic receiving 202664 citations.
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TL;DR: P Peng's BSDE method is extended from the framework of stochastic control theory into that of Stochastic differential games and is shown to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way.
Abstract: In this paper we study zero-sum two-player stochastic differential games with the help of the theory of backward stochastic differential equations (BSDEs). More precisely, we generalize the results of the pioneering work of Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293-314] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game. This extension of the class of admissible control processes has the consequence that the cost functionals become random variables. However, by making use of a Girsanov transformation argument, which is new in this context, we prove that the upper and the lower value functions of the game remain deterministic. Apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information, its combination with BSDE methods, in particular that of the notion of stochastic “backward semigroups" introduced by Peng [BSDE and stochastic optimizations, in Topics in Stochastic Analysis, Science Press, Beijing, 1997], allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way. The upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method is extended from the framework of stochastic control theory into that of stochastic differential games.
268 citations
TL;DR: The Stackelberg strategy in nonzero-sum games is a reasonable solution concept for games where, either due to lack of information on the part of one player about the performance function of the other, or due to different speeds in computing the strategies, or because of differences in size or strength, one player dominates the entire game by imposing a solution which is favorable to himself as mentioned in this paper.
Abstract: The Stackelberg strategy in nonzero-sum games is a reasonable solution concept for games where, either due to lack of information on the part of one player about the performance function of the other, or due to different speeds in computing the strategies, or due to differences in size or strength, one player dominates the entire game by imposing a solution which is favorable to himself. This paper discusses some properties of this solution concept when the players use controls that are functions of the state variables of the game in addition to time. The difficulties in determining such controls are also pointed out. A simple two-stage finite state discrete game is used to illustrate these properties.
267 citations
TL;DR: In this article, the existence of a saddle point in the bounded case is obtained if the Isaacs' condition holds, and this technique is also a very simple approach for finding an optimal strategy in the case of controlled diffusions.
266 citations
TL;DR: It is shown that in reasonably large populations, so-called zero-determinant strategies can act as catalysts for the evolution of cooperation, similar to tit-for-tat, but that they are not the stable outcome of natural selection.
Abstract: Iterated games are a fundamental component of economic and evolutionary game theory. They describe situations where two players interact repeatedly and have the ability to use conditional strategies that depend on the outcome of previous interactions, thus allowing for reciprocation. Recently, a new class of strategies has been proposed, so-called “zero-determinant” strategies. These strategies enforce a fixed linear relationship between one’s own payoff and that of the other player. A subset of those strategies allows “extortioners” to ensure that any increase in one player’s own payoff exceeds that of the other player by a fixed percentage. Here, we analyze the evolutionary performance of this new class of strategies. We show that in reasonably large populations, they can act as catalysts for the evolution of cooperation, similar to tit-for-tat, but that they are not the stable outcome of natural selection. In very small populations, however, extortioners hold their ground. Extortion strategies do particularly well in coevolutionary arms races between two distinct populations. Significantly, they benefit the population that evolves at the slower rate, an example of the so-called “Red King” effect. This may affect the evolution of interactions between host species and their endosymbionts.
265 citations
TL;DR: This article found that salient labels yield frequent coordination in symmetric games, but when the payoff is asymmetric, labels lose much of their effectiveness and miscoordination abounds, which raises questions about the extent to which the effectiveness of focal points based on label salience persists beyond the special case of symmetric game.
Abstract: Since Schelling, it has often been assumed that players make use of salient decision labels to achieve coordination. Consistent with previous work, we find that given equal payoffs, salient labels yield frequent coordination. However, given even minutely asymmetric payoffs, labels lose much of their effectiveness and miscoordination abounds. This raises questions about the extent to which the effectiveness of focal points based on label salience persists beyond the special case of symmetric games. The patterns of miscoordination we observe vary with the magnitude of payoff differences in intricate ways that suggest nonequilibrium accounts based on "level-k" thinking and "team reasoning." (JEL C12, C92)
259 citations