Proebsting's paradox

In probability theory, Proebsting's paradox is an argument that appears to show that the Kelly criterion can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first discussed by Edward O. Thorp in 2008. The paradox was named for Todd Proebsting, its creator.

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.

Many bets have the feature that payoffs and probabilities can change before the outcome is determined. In sports betting for example, the line may change several times before the event is held, and news may come out (such as an injury or weather forecast) that changes the probability of an outcome. In investing, a stock originally bought at $20 per share might be available now at $10 or $30 or any other price. Some sports bettors try to make income from anticipating line changes rather than predicting event outcomes. Some traders concentrate on possible short-term price movements of a security rather than its long-term fundamental prospects.

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