In the last post I gave an introduction of Kelly’s paper where he describes optimal gambling strategies based on information received over a potentially noisy channel. Here I’ll talk about the general case, where the channel has several inputs symbols, each with a given probability of being transmitted, and which represent the outcome of some chance event. First, we need to set up some notation:
– probability that the transmitted symbol is s.
– the conditional probability that the received symbols is r given that the transmitted symbol is s.
– the joint probability that the received symbol is r and the transmitted symbol is s.
– probability that the received symbol is r.
– the conditional probability that the transmitted symbol is s given that the received symbols is r.
– the odds paid on the occurrence of s, i.e. the number of dollars returned on a one-dollar bet on s.
– the fraction of capital that the gambler decides to bet on the occurrence of s after receiving r.
First, suppose that . Note that if , then the gambler can capitalize on making repeated bets on s, and this extra betting would in turn lower in most natural settings (this is known as “arbitrage” in the market world). Also, lets suppose that there is no “track take,” i.e. , and that the gambler bets his entire capital regardless of the symbol received, i.e. (note: he can withhold money by placing canceling bets so this is a reasonable assumption).
The amount of capital after N bets is now , where is the number of times the transmitted symbol is s and the received symbol is r. The log difference of the N’th and starting bet is
and we have
with probability 1.
where H(X) is Shannon’s source rate. We can maximize the firs term by putting and what we get is which is the Shannon’s rate of transmission as talked about by Matt here.
Now if , we have
and G is still maximized by setting . Hence,
Notice that to maximize G, the gambler has ignored the odds, i.e. it is only the information rate that matters in his strategy. In a future post I’ll discuss further implication of these results, as well as the “track take” scenario.