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The logarithmic utility function makes the Kelly criterion optimal even when there is only one remaining bet. For example say you have $x, the probability of wi
by OscarCunningham 5y ago
The logarithmic utility function makes the Kelly criterion optimal even when there is only one remaining bet. For example say you have $x, the probability of winning is p, and you bet $y. Then you want to maximise
plog(x+y) + (1-p)log(x-y).
Setting the derivative with respect to y equal to 0 yields
p/(x+y) - (1-p)/(x-y) = 0.
This rearranges to give
y = (2p - 1)x,
which is precisely the Kelly criterion.
- adiM 5y agoThis 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)