Games and Sub-games1

where B is a real matrix with m rows and n columns, x ranges over the set of row vectors with m components, all non-negative and adding up to one, y ranges over the corresponding set of «-component column vectors, and the pay-off, (x2?, y), indicates the inner product of the two vectors xB and y. One device which may simplify a game computation is that of "dominance" or "majorization" [vNM, p. 174] by which the solution of a game is reduced to the solution of a smaller game, that is, one with a smaller number of pure strategies. There is another device which, when conditions are right, may simplify the solution of a game by reducing it to the solution of smaller games. This device, presented here, gives either the value or a bound for it, depending on the information available about the sub-games. It also gives an optimal strategy or a strategy sufficient to insure an outcome not worse than that predicted by the aforementioned bound. It is particularly effective when there are rows (or columns) in B, which are constant or have large constant segments. Let fibea game matrix (rows maximizing) decomposed into