General finite approximation of non-cooperative games played in staircase-function continuous spaces

A method of general finite approximation of N-person games 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 method consists in irregularly sampling the player's pure strategy value set, finding the best equilibria in 'short' N-dimensional-matrix games, each defined on a subinterval where the pure strategy value is constant, and stacking the equilibrium situations if they are consistent. As opposed to straightforwardly solving the sampled staircase game, which is intractable, stacking the subinterval equilibria extremely reduces the computation time. The stack of the 'short' (subinterval) N-dimensional-matrix game equilibria is an approximate equilibrium in the initial staircase game. The (weak) consistency of the approximate equilibrium is studied by how much the payoff and equilibrium situation change as the sampling density minimally increases. The consistency is decomposed into the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. It is practically reasonable to consider a relaxed payoff consistency. An example of a 4-person staircase game is presented to show how the approximation is fulfilled for a case of when every subinterval 4-dimensional-matrix (quadmatrix) game has pure strategy equilibria.