Fuzzy Random Bimatrix Games Based on Possibility and Necessity Measures

In this study, we formulate bimatrix games with fuzzy random payoffs, and introduce equilibrium solution concepts based on possibility and necessity measures. It is assumed that each player has linear fuzzy goals for his/her payoff. To obtain equilibrium solutions based on the possibility and necessity measures, we propose two algorithms in which quadratic programming problems are solved repeatedly until equilibrium conditions are satisfied.

Stochastic Superiority Equilibrium in Game Theory

The definition of best response for a player in the Nash equilibrium is based on maximizing the expected utility given the strategy of the rest of the players in a game. In this work, we consider stochastic games, that is, games with random payoffs, in which a finite number of players engage only once or at most a limited number of times. In such games, players may choose to deviate from maximizing their expected utility. This is because maximizing expected utility strategy does not address the uncertainty in payoffs. We instead define a new notion of a stochastic superiority best response. This notion of best response results in a stochastic superiority equilibrium in which players choose to play the strategy that maximizes the probability of them being rewarded the most in a single round of the game rather than maximizing the expected received reward, subject to the actions of other players. We prove the stochastic superiority equilibrium to exist in all finite games, that is, games with a finite number of players and actions, and numerically compare its performance to Nash equilibrium in finite-time stochastic games. In certain cases, we show the payoff under the stochastic superiority equilibrium is 70% likely to be higher than the payoff under Nash equilibrium.

CONVEXITY IN STOCHASTIC COOPERATIVE SITUATIONS

This paper introduces a new model concerning cooperative situations in which the payoffs are modeled by random variables. We analyze these situations by means of cooperative games with random payoffs. Special attention is paid to three types of convexity, namely coalitional-merge, individual-merge and marginal convexity. The relations between these types are studied and in particular, as opposed to their deterministic counterparts for TU games, we show that these three types of convexity are not equivalent. However, all types imply that the core of the game is nonempty. Sufficient conditions on the preferences are derived such that the Shapley value, defined as the average of the marginal vectors, is an element of the core of a convex game.