Chance-constrained games with mixture distributions

In this paper, we consider an n-player non-cooperative game where the random payoff function of each player is defined by its expected value and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We consider the case when the probability distribution of each random constraint vector belongs to a subset of elliptical distributions as well as the case when it is a finite mixture of the probability distributions from the subset. We propose a convex reformulation for the joint chance constraint of each player and derive the bounds for players’ confidence levels and the weights used in the mixture distributions. Under mild conditions on the players’ payoff functions, we show that there exists a Nash equilibrium of the game when the players’ confidence levels and the weights used in the mixture distributions are within the derived bounds. As an application of these games, we consider the competition between two investment firms on the same set of portfolios. We use a best response algorithm to compute a Nash equilibrium of the randomly generated games of different sizes. Shen Peng Optimization and Systems Theory, Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden E-mail: shenp@kth.se Navnit Yadav Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India. E-mail: navnitydv@gmail.com Abdel Lisser L2S, CentraleSupélec Bât Breguet A4.22 3 rue Joliot Curie 91190 Gif-sur-Yvette, France E-mail: abdel.lisser@l2s.centralesupelec.fr Vikas Vikram Singh Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India. E-mail: vikassingh@iitd.ac.in

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