Statement of The Paradox
If a bet is equally likely to win or lose, and pays b times the stake for a win, the Kelly bet is:
times wealth. For example, if a 50/50 bet pays 2 to 1, Kelly says to bet 25% of wealth. If a 50/50 bet pays 5 to 1, Kelly says to bet 40% of wealth.
Now suppose a gambler is offered 2 to 1 payout and bets 25%. What should he do if the payout on new bets changes to 5 to 1? He should choose f* to maximize:
because if he wins he will have 1.5 (the 0.5 from winning the 25% bet at 2 to 1 odds) plus 5f*; and if he loses he must pay 0.25 from the first bet, and f* from the second. Taking the derivative with respect to f* and setting it to zero gives:
which can be rewritten:
So f* = 0.225.
The paradox is that the total bet, 0.25 + 0.225 = 0.475, is larger than the 0.4 Kelly bet if the 5 to 1 odds are offered from the beginning. It is counterintuitive that you bet more when some of the bet is at unfavorable odds. Todd Proebsting emailed Ed Thorp asking about this.
Ed Thorp realized the idea could be extended to give the Kelly bettor a nonzero probability of being ruined. He showed that if a gambler is offered 2 to 1 odds, then 4 to 1, then 8 to 1 and so on (2n to 1 for n = 1 to infinity) Kelly says to bet:
each time. The sum of all these bets is 1. So a Kelly gambler has a 50% chance of losing his entire wealth.
In general, if a bettor makes the Kelly bet on a 50/50 proposition with a payout of b1, and then is offered b2, he will bet a total of:
The first term is what the bettor would bet if offered b2 initially. The second term is positive if f2 > f1, meaning that if the payout improves, the Kelly bettor will bet more than he would if just offered the second payout, while if the payout gets worse he will bet less than he would if offered only the second payout.
Read more about this topic: Proebsting's Paradox
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