Kelly-Breiman and "How Much Should I Bet?"

Given an unfavorable game, you should not bet a nickel. That much is obvious.

Now, suppose you have the opportunity to play a favorable game. It makes sense to bet something, but it seldom makes sense to bet the ranch. Thus, one faces a quantitative question with a fine mixture of real-world charm, probability theory, and economic modeling.

In due course, we will survey much of the literature on this topic. For the moment, I just want to ice down a few links. We'll talk about it later.


Breiman, L., "Optimal Gambling Systems For Favorable Games," Jerzy Neyman, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1, 65-78, 1961.

Gottleib, G., "An Optimal Betting Strategy For Repeated Games," Journal of Applied Probability ( 22), 787-795, 1985.

Kelly, J.L. Jr.. "A New Interpretation of Information Rate," Bell Systems Technical Journal, 35, (1956), 917-926.


You can find several further resources relegated to bet sizing at the "Featured Articles" page of

Still, more --- and more technical

P. Grunwald (2004) "Maximum Entropy and the Glasses You are Looking Through."

E. De Giorgi (2002) "Evolutionary Portfolio Selection with Liquidity Shocks"