Equilibrium stacks for a non-cooperative game defined on a product of staircase-function continuous and finite strategy spaces

Abstract

A method of finite uniform approximation of 3-person games played with staircase-function strategies is presented. A continuous staircase 3-person game is approximated to a staircase trimatrix game by sampling the player’s pure strategy value set. The set is sampled uniformly so that the resulting staircase trimatrix game is cubic. An equilibrium of the staircase trimatrix game is obtained by stacking the equilibria of the subinterval trimatrix games, each defined on an interval where the pure strategy value is constant. The stack is an approximate solution to the initial staircase game. The (weak) consistency, equivalent to the approximate solution acceptability, is studied by how much the players’ payoff and equilibrium strategy change as the sampling density minimally increases. The consistency includes the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. The most important parts are the payoff consistency and equilibrium strategy support cardinality (weak) consistency, which are checked in the quickest and easiest way. However, it is practically reasonable to consider a relaxed payoff consistency, by which the player’s payoff change in an appropriate approximation may grow at most by epsilon as the sampling density minimally increases. The weak consistency itself is a relaxation to the consistency, where the minimal decrement of the sampling density is ignored. An example is presented to show how the approximation is fulfilled for a case of when every subinterval trimatrix game has pure strategy equilibria.