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An Adjustment Process-based Algorithm with Error Bounds for Approximating a Nash Equilibrium

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Regarding the approximation of Nash equilibria in games where the players have a continuum of strategies, there exist various algorithms based on adjustment processes: from one step to the next, one player updates his strategy solving an optimization problem where the strategies of the other players come from the previous steps. These iterative schemes generate sequences of strategy profiles which are constructed by using continuous optimization techniques and which converge to a Nash equilibrium of the game. In this paper, we propose an adjustment process based on continuous optimization which guarantee the convergence to a Nash equilibrium in two-player non zero-sum games when the best response functions are not linear, both compositions of the best response functions are not contractions, and the strategy sets are Hilbert spaces. Firstly, we address the issue of uniqueness of the Nash equilibrium extending to a more general class the result obtained in F. Caruso, M.C. Ceparano, and J. Morgan (J Math Anal Appl 2018) for weighted potential games. Secondly, we describe a theoretical adjustment process involving the best response functions which converges to the unique Nash equilibrium of the game. Finally, in order to ap- proximate the unique Nash equilibrium of the game, we present a discretization scheme and a numerical adjustment process based on continuous optimization, and we compute error bounds.

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  • Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2018. "An Adjustment Process-based Algorithm with Error Bounds for Approximating a Nash Equilibrium," CSEF Working Papers 502, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
  • Handle: RePEc:sef:csefwp:502
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    Keywords

    Non-cooperative game; Approximation of Nash equilibrium; Uniqueness of Nash equilibrium; Fixed point; Adjustment process; Best response; Error bound.;

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