The fundamental problem in gambling is to find positive expectation betting opportunities. The analogous problem in investing is to find investments with excess risk-adjusted expected rates of return. Once these favorable opportunities have been identified, the gambler or investor must decide how much of his capital to bet. This is the problem which we consider here. It has been of interest at least since the eighteenth century discussion of the St. Petersburg Paradox (Feller, 1966) by Daniel Bernoulli.
One approach is to choose a goal, such as to minimize the probability of total loss within a specified number of trials, N. Another example would be to maximize the probability of reaching a fixed goal on or before N trials (Browne, 1996).
A different approach, much studied by economists and others, is to value money using a utility function. These are typically defined for all non-negative real numbers, have extended real number values, and are non-decreasing (more money is at least as good as less money). Some examples are U(x) = x a, and , where means , and . Once a utility function is specified, the object is to maximize the expected value of the utility of wealth.
Daniel Bernoulli used the utility function to ``solve'' the St. Petersburg Paradox. (But his solution doesn't eliminate the paradox because every utility function which is unbounded above, including log, has a modified version of the St. Petersburg Paradox.) The utility function was revisited by J.L. Kelly (1956) where he showed that it had some remarkable properties. These were elaborated and generalized in an important paper by Brieman (1961). Markowitz (1959) illustrates the application to securities. I was introduced to the Kelly paper by Claude Shannon at M.I.T. in 1960, shortly after I had created the mathematical theory of card counting at casino blackjack. Kelly's criterion was a bet on each trial so as to maximize , the expected value of the logarithm of the (random variable) capital X. I used it in actual play and introduced it to the gambling community in the first edition of Beat the Dealer, Thorp, (1962). If all blackjack bets paid even money, had positive expectation and were independent, the resulting Kelly betting recipe when playing one hand at a time would be extremely simple: bet a fraction of your current capital equal to your expectation. This is modified somewhat in practice (generally down) to allow for having to make some negative expectation ``waiting bets'', for the higher variance due to the occurrence of payoffs greater than one to one, and when more than one hand is played at a time.
Here are the properties that made the Kelly criterion so appealing. For ease of understanding, we illustrate using the simplest case, coin tossing, but the concepts and conclusions generalize greatly.