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Variational inequality formulation for the games with random payoffs

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Abstract

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Vikas Vikram Singh

  2. Laboratoire de Recherche en Informatique, Université Paris Sud, 91405, Orsay, France

    Abdel Lisser

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  1. Vikas Vikram Singh
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  2. Abdel Lisser
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Correspondence to Vikas Vikram Singh.

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Singh, V.V., Lisser, A. Variational inequality formulation for the games with random payoffs. J Glob Optim 72, 743–760 (2018). https://doi.org/10.1007/s10898-018-0664-8

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  • DOI: https://doi.org/10.1007/s10898-018-0664-8

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