The idea of studying a two-person zero sum differential game in which the two players make their moves alternately, so that neither player has an information advantage throughout the game, is due to Danskin [l]. In this paper, by adapting the methods of [4], we relate the values of two differential games of this kind to the solutions of two nonlinear parabolic equations. It is then immediate that the values of the games obtained by considering finer and finer partitions of the time interval exist. In one game the players play constant controls at each step; in the other, measurable functions. It is of interest that in general the values are not equal. In the case of alternate play with piecewise constant controls (which is the only case discussed in [l]), the idea of using partitions in MN subintervals rather than just partitions in M subintervals is due to Danskin. However, the technical details here are significantly different. Thus, in particular, Danskin does not use the theory of parabolic partial differential equations. Two interesting “intermediate values” 4, and & , 0 ,< (I < 1, of a function 4(y, Z) of two variables arise. d,, occurs when piecewise constant controls arc introduced, and this intermediate value is implicit in Danskin’s paper. 4, lies between