[trixtrix]

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So anyway, here's a quick and dirty back-of-the-envelope type proof of the proposition I made above. (Specifically, In the face of a positive EV multi-way wager set spanning every possible outcome of an event, the optimal Kelly allocation will be to invest that fraction of bankroll on each outcome equal to the probability of that outcome occurring.) I'm just going to prove it for the two-outcome case. (If memory serves correctly, Kelly's original paper does prove it for the n-outcome case).

Given two events, occurring with probability p and 1-p, and paying out at decimal odds of n and o (where 1/n + 1/o < 1 -- the "arbitrage condition", meaning a risk-free arbitrage exists), with bet allocations (as percent of bankroll) of x and y.

Maximize wrt x,y:
E(U(x,y)) = p * ln(1 + (n-1)*x - y) + (1-p) * ln(1 - x + (o-1)*y)

Subject to:
x >= 0, y >= 0, x+y <= 1

There's no loss of generality in assuming the third constraint binds as the bettor may always place cancelling bets by wagering in n/(o+n) on event 1 o/(o+n) on event 2, yielding a guaranteed return of no/(n+o) - 1. This return will be strictly positive due the arbitrage condition, 1/n + 1/o < 1. It should also be readily apparent from the arbitrage condition that the non-negativity constraints won't bind, so for the sake of brevity we ignore them here.

So, after introducing the Lagrangean, L, on the third constraint:

Maximize wrt x,y,L:
F = p * ln(1 + (n-1)*x - y) + (1-p) * ln(1 - x + (o-1)*y) - L * (y + x - 1)

Yielding (the d's should actually be partial derivatives, but I can't get the special character to appear on UBB):
dF/dx = -L + ((-1 + n)*p)/(1 + (-1 + n)*x - y) - (1 - p)/(1 - x + (-1 + o)*y)
dF/dy = -L - p/(1 + (-1 + n)*x - y) + ((-1 + o)*(1 - p))/(1 - x + (-1 + o)*y)
dF/dL = y + x - 1

Setting to zero and solving gives us:
L = 1 - 1/o - 1/n
x = p
y = 1-p


QED

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years of using fincad and matlab instead of paper+ pen has deterioted my already meager math skills

mega reps to ganchrow for providing the proof and reminding me what i had already forgotten..

i realize now that the goal of kelly is to maximize the log-normal return of your br so that your overall br size will grow at an exponential rate.

the fact that appropriate bet-size of each stake is = to the probability of occurrence is merely a by-product of the solution, not the intended goal...

vnh sir