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.
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.
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.
[3] A classic investing example is a trader who has exposure limits, say he is not allowed to have more than $1 million at risk in any one stock.
What the paradox says, essentially, is that if a Kelly bettor has incorrect beliefs about what future bets may be offered, he can make suboptimal choices, and even go broke.
[2] More light on the issues was shed by an independent consideration of the problem by Aaron Brown, also communicated to Ed Thorp by email.
The second formulation makes clear that the change in behavior results from the mark-to-market loss the investor experiences when the new payout is offered.
In this interpretation, the infinite series of doubling payouts does not ruin the Kelly bettor by enticing him to overbet, it extracts all his wealth through changes beyond his control.