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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: This paper provides the first polynomial time algorithms constructing ε-SuppNE for normalized or win lose bimatrix games, for any nontrivial constant 0≤ε<1, bounded away from 1.
Abstract: In view of the apparent intractability of constructing Nash Equilibria (NE in short) in polynomial time, even for bimatrix games, understanding the limitations of the approximability of the problem is an important challenge.
In this work we study the tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short). Roughly speaking, while the typical notion of approximate NE demands that each player gets a payoff at least an additive term less than the best possible payoff, in a SuppNE each player is assumed to adopt with positive probability only approximate pure best responses to the opponent’s strategy.
As a first step, we demonstrate the existence of SuppNE with small supports and at the same time good quality. This is a simple corollary of Althofer’s Approximation Lemma, and implies a subexponential time algorithm for constructing SuppNE of arbitrary (constant) precision.
We then propose algorithms for constructing SuppNE in win lose and normalized bimatrix games (i.e., whose payoff matrices take values from {0,1} and [0,1] respectively). Our methodology for attacking the problem is based on the solvability of zero sum bimatrix games (via its connection to linear programming) and provides a 0.5-SuppNE for win lose games and a 0.667-SuppNE for normalized games.
To our knowledge, this paper provides the first polynomial time algorithms constructing e-SuppNE for normalized or win lose bimatrix games, for any nontrivial constant 0≤e<1, bounded away from 1.
64 citations
TL;DR: In this article, the authors give an explicit representation of the lowest cost strategy (or "cost-efficient" strategy) to achieve a given payoff distribution, and highlight the connections between cost-efficiency and dependence (copulas).
Abstract: In this paper, we give an explicit representation of the lowest cost strategy (or "cost-efficient" strategy) to achieve a given payoff distribution. For any inefficient strategy, we are able to construct financial derivatives which dominate in the sense of first-order or second-order stochastic dominance. We highlight the connections between cost-efficiency and dependence (copulas). This allows us to extend the theory to deal with state-dependent constraints to better reflect real world preferences. We show in particular that path-dependent strategies (although inefficient in the Black Scholes setting) may become optimal in the presence of state-dependent constraints.
64 citations
TL;DR: In this paper, it was shown that continuous-time fictitious play converges uniformly at ratet − 1 in any finite two-person zero-sum game with payoff and strategy.
64 citations
TL;DR: In this paper, the authors consider the notion of optimal payoff as that maximizing the terminal position for a chosen preference functional and investigate the relationship between both concepts, optimal and efficient payoffs, as well as the behavior of the efficient payoff under different market dynamics.
Abstract: In 1988 Dybvig introduced the payoff distribution pricing model (PDPM) as an alternative to the capital asset pricing model (CAPM). Under this new paradigm agents preferences depend on the probability distribution of the payoff and for the same distribution agents prefer the payoff that requires less investment. In this context he gave the notion of efficient payoff. Both approaches run parallel to the theory of choice of von Neumann-Morgenstern (1947), known as the Expected Utility Theory and posterior axiomatic alternatives. In this paper we consider the notion of optimal payoff as that maximizing the terminal position for a chosen preference functional and we investigate the relationship between both concepts, optimal and efficient payoffs, as well as the behavior of the efficient payoffs under different market dynamics. We also show that path-dependent options can be efficient in some simple models.
63 citations
TL;DR: This paper introduces a concept of uncertain bimatrix game within the framework of uncertainty theory, and three solution concepts of uncertain equilibrium strategies as well as their existence theorem are proposed.
Abstract: In real-world games, the players are often lack of the information about the other players' (or even his own) payoffs. Assuming that all entries of payoff matrices are uncertain variables, this paper introduces a concept of uncertain bimatrix game. Within the framework of uncertainty theory, three solution concepts of uncertain equilibrium strategies as well as their existence theorem are proposed. Furthermore, a sufficient and necessary condition is presented for finding the uncertain equilibrium strategies. Finally, an example is provided for illustrating the usefulness of the theory developed in this paper.
63 citations