Gambling and Information Theory

Claude Shannon is recognised as the creator of the field of Information Theory. Information theory is a way of quantifying information so as to make the best decision in the face of imperfect information. It could be considered a formal expression of gambling theory, which means it also has applications to sport and other games of chance. With Edward O. Thorp, Shannon applied the same theory to the stock market with even better results than he had experienced with sports. Since the 1960s information theory has become mainstream investment theory followed by the likes of Warren Buffett, who first met Thorp in 1968.

Optimal Gambling

Discovered by John Larry Kelly, Jr. proportional betting, or Kelly betting, is an application of information theory to investing and gambling. A gambler maximises the expectation of the logarithm of his capital, which is additive in sequential bets, and "to which the law of large numbers applies." The additive nature of this measure is useful when weighing alternatives.

Surprisal and evidence in bits (the amount of entropy in an event with two possible outcomes and even odds) are logarithmic measures of probability and odds respectively. The logarithmic probability measures self-information or surprisal, whose average is information uncertainty and whose average difference is KL-divisals which reduce minuscule probabilities to numbers of manageable size, and add whenever probabilities multiply.

Gambling and Side Information

The Kelly criterion, considered an optimum betting strategy, can make money grow exponentially with whatever side information is obtained. The value of side/illicit information is measured as mutual information relative to the outcome of the event on which the gamble is being made. Side information needs to be evaluated over time, but once the information is assessed to be reliable bets can be precisely calculated according to the Kelly criterion. Even if the informant is lying, it is possible to still profit from their lies if some reverse correlation between the tips and the actual race results can be found.

How information will affect the outcome of a sporting event will raise differing opinions. Yet advanced statistical analysis now allows games to be quantified in ways not previously seen. On the stock exchange it is possible to explain and predict market behaviour to show how market inefficiency can occur through the new statistic known as DVOA, though many have questioned its usefulness.