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Learn how to calculate bet amounts using Kelly's strategy for coin tosses, wins, and losses. Understand the exponential rate of growth per trial and the zone of positive or ruinous over-betting. Compare Kelly strategy with other betting methods.
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Using a little math, here’s Kelly’s answer for how much to bet in the case of the coin. Let n, W and L be the number of tosses, wins and losses, respectively. The win and loss probabilities are p and q = 1 – p. We start with bankroll X0 and bet a fixed fraction f at each trial. After n tosses we have Xn=X0 (1+f )W(1-f )L. With a little rearranging we have Xn/X0=exp{n log([Xn/X0]1/n) = exp{nGn(f)}}where Gn(f)=log ([Xn/X0]1/n) = (W/n) log (1+f) + (L/n) log (1-f) measures the exponential rate of growth per trial. The expected value of Gn(f) is g(f) = p log(1+f) + q log(1-f) which has a maximum at f* = p – q. In this instance f* happens to be the same as the bettor’s edge, namely the expected value of one unit bet.
If g(f) > 0 then limn→∞ Xn = ∞, almost surely, i.e. one’s fortune tends to infinity with probability one: the zone of positive growth. If g(f) < 0 then limn→∞ Xn= 0, almost surely, i.e. one’s fortune tends to zero with probability one: the zone of ruinous over betting. If g(f) = 0 then Xnoscillates wildly from 0 to ∞. With the Kelly strategy Φ*verus any “essentially different” strategy Φ, the ratio Xn(Φ*)/Xn(Φ) tends to infinity with probability one. The expected time to reach any specified goal tends to be least with Kelly.
