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 bjmath.com.
P. Grunwald (2004) "Maximum Entropy and the Glasses You are Looking Through."
E. De Giorgi (2002) "Evolutionary Portfolio Selection with Liquidity Shocks"