Elsevier

Operations Research Letters

Volume 44, Issue 5, September 2016, Pages 640-644
Operations Research Letters

Existence of Nash equilibrium for chance-constrained games

https://doi.org/10.1016/j.orl.2016.07.013Get rights and content

Abstract

We consider an n-player strategic game with finite action sets and random payoffs. We formulate this as a chance-constrained game by considering that the payoff of each player is defined using a chance constraint. We consider that the components of the payoff vector of each player are independent normal/Cauchy random variables. We also consider the case where the payoff vector of each player follows a multivariate elliptically symmetric distribution. We show the existence of a Nash equilibrium in both cases.

Introduction

In 1928, John von Neumann  [19] showed that there exists a mixed strategy saddle point equilibrium for a two player zero sum matrix game. In 1950, John Nash  [18] showed that there always exists a mixed strategy Nash equilibrium for an n-player general sum game with finite number of actions for each player. In both [18], [19], it is considered that the players’ payoffs are deterministic. However, there can be practical cases where the players’ payoffs are better modeled by random variables following certain distributions and as a result players compete in a stochastic Nash game. The wholesale electricity markets are the good examples that capture this situation. The randomness in an electricity market is present due to various external factors, e.g., wind integration [17], and consumers’ random demand  [7].

One way to study stochastic Nash games is by using expected payoff criterion. Ravat and Shanbhag  [21] considered stochastic Nash games using expected payoff criterion. They showed the existence and uniqueness of Nash equilibrium, under certain conditions, in various cases. Xu and Zhang  [25] used a sample average approximation method to solve stochastic Nash equilibrium problems. Jadamba and Raciti  [10] used a variational inequality approach on probabilistic Lebesgue spaces to study stochastic Nash games. The stochastic Nash games under expected payoff criterion using stochastic variational inequalities are considered in[11], [15], [16], [26] and references therein. In these papers, the stochastic approximation based schemes to compute the Nash equilibria of stochastic Nash games have been given.

The expected payoff criterion is more appropriate for the cases where the decision makers are risk neutral. The risk averse stochastic Nash games arising from electricity market using risk measures as CVaR and variance are considered in  [14], [21] and  [6] respectively. A risk averse payoff criterion based on chance constraint programming  [3], [20] has also received some attention in electricity market  [7], [17]. These games are called chance-constrained games. In  [17], the randomness in payoffs is due to the installation of wind generators on the electricity market. The authors consider the case where the random variables that represent the amount of wind are independent normal random variables, and they also consider the case where the random vector follows a multivariate normal distribution. In [7], the consumers’ random demand is assumed to be normally distributed. A game theoretic situation in electricity market where the action sets are finite is considered in  [23]. Although the players’ payoffs are deterministic in  [23], the counterpart of the model, where the payoffs are random variables, in chance-constrained game setting can be considered. Only few theoretical results on zero sum chance-constrained games with finite action sets of the players are available in the literature so far  [1], [2], [4], [5], [24].

In this paper, we focus on the games where the payoffs of the players are random variables with known probability distributions. The case where probability distributions of the random payoffs are not known completely is considered in  [22]. The authors use distributionally robust approach to handle these games. To the best of our knowledge, there is no result on the existence of a Nash equilibrium for a chance-constrained game even when the action sets of all the players are finite. We consider an n-player strategic game where the action set of each player is finite and the payoff vector of each player is a random vector. We formulate this problem as a chance-constrained game by considering that the payoff of each player is defined using a chance constraint. We show the existence of a Nash equilibrium for a chance-constrained game in the following cases.

  • 1.

    If all the components of the payoff vector of each player are independent normal/Cauchy random variables, there exists a mixed strategy Nash equilibrium for a chance-constrained game. As a special case, if only one component of the payoff vector of each player is a random variable and all other components are deterministic, the Nash equilibrium existence result can be extended to all the continuous probability distributions whose quantile functions exist.

  • 2.

    If the payoff vector of each player follows a multivariate elliptically symmetric distribution, there always exists a mixed strategy Nash equilibrium for a chance-constrained game.

The structure of the rest of the paper is as follows: in Section  2 we give the definition of a chance-constrained game. Existence of a mixed strategy Nash equilibrium is then given in Section  3.

Section snippets

The model

We consider an n-player strategic game. Let I={1,2,,n} be a set of all players. For each iI, let Ai be a finite action set of player i and its generic element is denoted by ai. A vector a=(a1,a2,,an) denotes an action profile of the game. Let A=Ai be the set of all action profiles of the game. Denote, Ai=Aj, and aiAi is a vector of actions aj, ji. The action set Ai of player i is also called the set of pure strategies of player i. A mixed strategy of a player is represented by a

Existence of Nash equilibrium

We assume that the payoffs of each player are random variables following a certain distribution. We consider various cases and show the existence of a mixed strategy Nash equilibrium of chance-constrained game for different values of α.

Acknowledgment

This research was supported by Fondation DIGITEO, SUN Grant No. 2014-0822D.

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