Two-player Knock 'em Down

We analyze the two-player game of Knock 'em Down, asymptotically as the number of tokens to be knocked down becomes large. Optimal play requires mixed strategies with deviations of order sqrt(n) from the naive law-of-large numbers allocation. Upon rescaling by sqrt(n) and sending n to infinity, we show that optimal play's random deviations always have bounded support and have marginal distributions that are absolutely continuous with respect to Lebesgue measure.