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# Approximation of Unit-Hypercubic Infinite Noncooperative Game Via Dimension-Dependent Samplings and Reshaping the Player’s Payoffs into Line Array for the Finite Game Simplification

Department of Applied Mathematics and Social Informatics, Khmelnitskiy National University, Khmelnitskiy, Ukraine

Computer Modeling in Engineering & Sciences 2015, 108(2), 113-134. https://doi.org/10.3970/cmes.2015.108.113

## Abstract

The problem of solving infinite noncooperative games approximately is considered. The game may either have solution or have no solution. The existing solution may be unknown as well. Therefore, an approach of obtaining the approximate solution of the infinite noncooperative game on the unit hypercube is suggested. The unit-hypercubic game isomorphism to compact infinite noncooperative games allows to disseminate the approximation approach on a pretty wide class of noncooperative games. The approximation intention is in converting the infinite game into a finite one, whose solution methods are easier rather than solving infinite games. The conversion starts with sampling the players’ payoff functions. Each dimension of the player’s pure strategies unit hypercube is sampled with its own sampling constant, being the number of equal-measure intervals between the selected points along the dimension. There are stated requirements for the sufficiently accurate sampling. Having got the finite game on hypercubic lattice after the sampling, every player’s payoff multidimensional matrix is reshaped to reduce number of its dimensions down to the number of players. Dimensionality reduction will commonly accelerate computations, connected with the approximate solution consistency. The introduced consistency mechanism rejects the finite game solution, pretended to being the initial game approximate solution, if the solution depends vastly on the sampling steps. If the solution is weakly consistent then, changing the sampling steps minimally, there are non-decreasing difference of the players’ payoffs and difference of the players’ equilibrium strategies and cardinalities of their supports. If the solution is consistent then the non-decreasing property holds stricter for cardinalities of the supports and their upper densities.

## Keywords

Romanuke, V. V. (2015). Approximation of Unit-Hypercubic Infinite Noncooperative Game Via Dimension-Dependent Samplings and Reshaping the Player’s Payoffs into Line Array for the Finite Game Simplification. CMES-Computer Modeling in Engineering & Sciences, 108(2), 113–134.

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