A variation in the Brown method of solving games.

Abstract

The brown method of solving zero sum two person games by a method of successive approximations was programmed for the NCR-102A Digital Computer. Game matricies up to order 8x8 were investigated, although the program could easily be extended to order 16 x 16 without leaving the magnetic drum, or to arbitrarily higher order games by also using magnetic tape. The problem of obtaining all strategies of a convex set of optimal strategies was solved in a number of cases and the concept of a complementary game was developed. The Brown method was found to converge too slowly in most cases so that a modification of the method was used. In the Brown method, successive approximate strategies are developed for each player until a stage is reached in which the opposing strategies give equal values to the game (or give values sufficiently close), at which time (approximate) optimal strategies have been obtained. Julia Robinson has proved the convergence of the Brown method. The modification consists in comparing a maximum value of the game for Player I with a minimum value of the game for Player II at different stages of the iteration until these values are equal or sufficiently close to each other. The convergence of the new method follows from the convergence of the brown method and the new method was found to be generally much more rapid.