This is precise the argument at the penultimate time-step in the dynamic programming solution of the multi-round case. The other interesting aspect is that the

This is precise the argument at the penultimate time-step in the dynamic programming solution of the multi-round case. The other interesting aspect is that the expected returns are logarithmic, i.e, with y = (2p -1) x p log(x+y) + (1-p) log(x-y) = log(x) + C where C is the Shannon capacity of the binary symmetric channel with cross-over probability p. By the same argument, the expected wealth after T rounds will be log(x) + T C So, in addition to the optimal strategy, we have also derived the rate of growth of wealth. This is also in tune with the motivation of Kelly's paper where he was showing a relationship between Shannon capacity and optimal gambling (without using a dynamic programming argument)