Abstract
We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.
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Appendix: Proof of Theorem 3.2
Appendix: Proof of Theorem 3.2
Proof
Fix \(\alpha \in [0.5,1[^2\). To show that \((x^*,y^*)\) defined by (16) is a Nash equilibrium, it is sufficient to show that there exists a vector \((\lambda _1^*, \lambda _2^*,v_1^*,v_2^*)\) which together with \((x^*,y^*)\) is a feasible point of [MP] with objective function value zero (see Theorem 3.1). By using i.i.d. property, we have \(\varSigma _1^{1/2}=\sigma I_{mn\times mn}\), \(\varSigma _2^{1/2}=\bar{\sigma }I_{mn\times mn}\). Take,
where \(\mathbf 1 _k\) denotes a \(k\times 1\) vector of ones. It is easy to check that \(\zeta ^*=(\lambda _1^*, \lambda _2^*,v_1^*,v_2^*,x^*,y^*)\) is a feasible point of [MP] and \(\psi (\zeta ^*)=0\). Hence, \((x^*,y^*)\) defined by (16) is a Nash equilibrium of a CCG. \(\square \)
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Singh, V.V., Lisser, A. A Characterization of Nash Equilibrium for the Games with Random Payoffs. J Optim Theory Appl 178, 998–1013 (2018). https://doi.org/10.1007/s10957-018-1343-0
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DOI: https://doi.org/10.1007/s10957-018-1343-0
Keywords
- Chance-constrained games
- Nash equilibrium
- Elliptically symmetric distribution
- Cauchy distribution
- Mathematical program
- Quadratic program