Consistency of equilibrium stacks in finite uniform approximation of a noncooperative game played with staircase-function strategies

Vadim Romanuke

doi:10.3934/naco.2022027

Article Contents

2024, Volume 14, Issue 2: 298-321. Doi: 10.3934/naco.2022027

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Consistency of equilibrium stacks in finite uniform approximation of a noncooperative game played with staircase-function strategies

Vadim Romanuke

Faculty of Mechanical and Electrical Engineering, Polish Naval Academy, Gdynia, Poland

Received: February 2022

Revised: August 2022

Early access: September 19, 2022

Published online: September 19, 2022

The author is technically supported by the Faculty of Mechanical and Electrical Engineering, Polish Naval Academy.

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Abstract

A method of finite uniform approximation of an $ N $-person noncooperative game played with staircase-function strategies is presented. A continuous staircase $ N $-person game is approximated to a staircase $ N $-dimensional-matrix game by sampling the player's pure strategy value set. The set is sampled uniformly so that the resulting staircase $ N $-dimensional-matrix game is hypercubic. An equilibrium of the staircase $ N $-dimensional-matrix game is obtained by stacking the equilibria of the subinterval $ N $-dimensional-matrix games, each defined on a subinterval where the pure strategy value is constant. The stack is an approximate solution to the initial staircase game. The (weak) consistency of the approximate solution is studied by how much the players' payoff and equilibrium strategy change as the sampling density minimally increases. The consistency is equivalent to the approximate solution acceptability. An example of a 4-person noncooperative game is presented to show how the approximation is fulfilled for a case of when every subinterval quadmatrix game has pure strategy equilibria.

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Mathematics Subject Classification: Primary: 91A10, 90C59; Secondary: 62C99.

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Figure 1. The players' pure-strategy equilibrium stacks in quadmatrix games (67) with (68)–(75) which approximate the initial staircase-function game with (53)–(62) by $ S = \overline {2,\,\,32} $

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Figure 2. The players' best pure-strategy equilibrium stacks by $ S = \overline {2,\,\,32} $ and circle-highlighted equilibrium stacks by $ S = \overline {28,\,\,32} $

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Figure 3. The player's payoff (from equilibria in Figure 2 by which the sum of the players' payoffs is maximal) versus $ S = \overline {2,\,\,32} $ in the 4-person staircase game (approximated by the sampling)

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