Gambling and information theory

From Wikipedia, the free encyclopedia

Kelly gambling or proportional gambling is an application of information theory to gambling and (with some ethical and legal reservations) investing.

In gambling and investment, the goal is to maximize the rate of growth. Doubling rate in gambling on a horse race is

$W(b,p) = \mathbb E[\log_2 S(X)] = \sum_{i=1}^m p_i \log_2 b_i o_i$

where there are $m$ horses, the probability of the $i$ th horse winning being $p i$ , the proportion of wealth bet on the horse being $b i$ , and the odds (payoff) being $o i$ (e.g., $o i = 2$ if the $i$ th horse winning pays double the amount bet). This quantity is maximized by proportional (Kelly) gambling:

$b = p \,$

for which

$\max_b W(b,p) = \sum_i p_i \log_2 o_i - H(p) \,$

where $H (p)$ is information entropy.

An important but simple relation exists between the amount of side information a gambler obtains and the expected exponential growth of his capital (Kelly). The so-called equation of ill-gotten gains can be expressed in logarithmic form as

$\mathbb E \log K_t = \log K_0 + \sum_{i=1}^t H_i$

for an optimal betting strategy, where $K 0$ is the initial capital, $K t$ is the capital after the tth bet, and $H i$ is the amount of side information obtained concerning the ith bet (in particular, the mutual information relative to the outcome of each betable event). This equation applies in the absence of any transaction costs or minimum bets. When these constraints apply (as they invariably do in real life), another important gambling concept comes into play: the gambler (or unscrupulous investor) must face a certain probability of ultimate ruin, which is known as the gambler's ruin scenario. Note that even food, clothing, and shelter can be considered fixed transaction costs and thus contribute to the gambler's probability of ultimate ruin.

This equation was the first application of Shannon's theory of information outside its prevailing paradigm of data communications (Pierce).