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Acceptable-and-attractive Approximate Solution of a Continuous Non-Cooperative Game on a Product of Sinusoidal Strategy Functional Spaces

Romanuke Vadim

Foundations of Computing and Decision Sciences

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2021

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Vol. 46, No. 2

173--197

EN

A problem of solving a continuous noncooperative game is considered, where the player’s pure strategies are sinusoidal functions of time. In order to reduce issues of practical computability, certainty, and realizability, a method of solving the game approximately is presented. The method is based on mapping the product of the functional spaces into a hyperparallelepiped of the players’ phase lags. The hyperparallelepiped is then substituted with a hypercubic grid due to a uniform sampling. Thus, the initial game is mapped into a finite one, in which the players’ payoff matrices are hypercubic. The approximation is an iterative procedure. The number of intervals along the player’s phase lag is gradually increased, and the respective finite games are solved until an acceptable solution of the finite game becomes sufficiently close to the same-type solutions at the preceding iterations. The sufficient closeness implies that the player’s strategies at the succeeding iterations should be not farther from each other than at the preceding iterations. In a more feasible form, it implies that the respective distance polylines are required to be decreasing on average once they are smoothed with respective polynomials of degree 2, where the parabolas must be having positive coefficients at the squared variable.

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Finite approximation of continuous noncooperative two-person games on a product of linear strategy functional spaces

Romanuke Vadim

Journal of Mathematics and Applications

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2020

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Vol. 43

123--138

EN

A method of the finite approximation of continuous non-cooperative two-person games is presented. The method is based on sampling the functional spaces, which serve as the sets of pure strategies of the players. The pure strategy is a linear function of time, in which the trend-defining coefficient is variable. The spaces of the players’ pure strategies are sampled uniformly so that the resulting finite game is a bimatrix game whose payoff matrices are square. The approximation procedure starts with not a great number of intervals. Then this number is gradually increased, and new, bigger, bimatrix games are solved until an acceptable solution of the bimatrix game becomes sufficiently close to the same-type solutions at the preceding iterations. The closeness is expressed as the absolute difference between the trend-defining coefficients of the strategies from the neighboring solutions. These distances should be decreasing once they are smoothed with respective polynomials of degree 2.

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Interval Methods for Computing Strong Nash Equilibria of Continuous Games

Kubica B. J., Woźniak A.

Decision Making in Manufacturing and Services

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2015

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Vol. 9, no. 1

63--78

EN

The problem of seeking strong Nash equilibria of a continuous game is considered. For some games, these points cannot be found analytically, only numerically. Interval methods provide us with an approach to rigorously verify the existence of equilibria in certain points. A proper algorithm is presented. We formulate and prove propositions, that give us features which have to be used by the algorithm (to the best knowledge of the authors, these propositions and properties are original). Parallelization of the algorithm is also considered, and numerical results are presented. As a particular example, we consider the game of “misanthropic individuals”, a game, invented by the first author, that may have several strong Nash equilibria depending on the number of players. Our algorithm is able to localize and verify these equilibria.