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

Abstract

<p style='text-indent:20px;'>A method of finite uniform approximation of an <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-person noncooperative game played with staircase-function strategies is presented. A continuous staircase <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula>-person game is approximated to a staircase <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula>-dimensional-matrix game by sampling the player's pure strategy value set. The set is sampled uniformly so that the resulting staircase <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula>-dimensional-matrix game is hypercubic. An equilibrium of the staircase <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>-dimensional-matrix game is obtained by stacking the equilibria of the subinterval <inline-formula><tex-math id="M6">\begin{document}$ N $\end{document}</tex-math></inline-formula>-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.</p>