Posted by Pete Moss on July 12, 1997 at 09:59:07: In Reply to: What is the Kelly's theorem? posted by Bisser on July 11, 1997 at 14:19:12:J.L. Kelly Jr.'s paper was about information transfer, and was published in the Bell System Technical Journal in 1956. He used betting on horse races to illustrate some principles of "information rate" and "Shannon entropy", measures of precision of probability functions.
The importance of the paper to us is not so much in the specific results he published as it is in the criterion he used for judging a betting-decision mechanism. If you have a favorable proposition available to you, by what measure do you decide how much you should bet? The greedy criterion is to maximize your "expected" return, the average amount you could expect to win from a large number of identical wagers. Doing that is simple: Bet everything you've got. The problem is, in order to make a large number of wagers, you must always have money available to wager. The greedy criterion will eventually lose, and you will be busted. Even if you do not go to that extreme and bet everything, you can still bet so much that your bankroll will over the long run decline rather than increase, because the losing plays deplete the capital excessively, rendering winning plays ineffectual due to under-funding. On the other end of the spectrum are conservative criterions in which you bet very small amounts, and choose to make only plays that are relatively certain. The bankroll grows at a steady but unnecessarily slow pace.
The "Kelly Criterion" is to wager so that the expected rate of increase of your total bankroll is maximized. You neither bet so little that you gain little on average per play, nor bet so much that the losing plays can be expected to deplete the bankroll excessively for future exploitation. The Kelly Criterion calls for striking the perfect balance between short term greed and miserly conservatism. It is a criterion for deciding how much of your seed corn you should plant.
In practice, calculating the Kelly Criterion decision invariably involves the use of logarithms, a recurring theme in information theory.
One application of the Kelly Criterion which has recently been discussed here is forming a plan for when to double down on a hand of blackjack. Many published strategy indexes use the "greedy" criterion of doubling whenever there is an advantage from doing so, regardless of how small that advantage might be.
Pete