92(i THE BELL SYSTEM TECHNICAL .lOlKXAL, JCLY 195G 



the value function Avhich he attached to his money, but merely with the 

 fact that it is the logarithm A\hic'h is additive in repeated bets and to 

 which the law of large numbers applies. Suppose the situation were 

 different; for example, suppose the gambler's wife allowed him to bet 

 one dollar each week but not to reinvest his winnings. He should then 

 maximize his expectation (expected value of capital) on each bet. He 

 would bet all his available capital (one dollar) on the event j-ielding the 

 highest expectation. With probability one he would get ahead of any- 

 one dividing his money differently. 



It should be noted that we have only shown that our gambler's capital 

 will surpass, with probability one, that of any gambler apportioning his 

 money different!}^ from ours but still m a fixed way for each received 

 sjanbol, independent of time or past events. Theorems remain to be 

 proved showing in what sense, if any, our strategy is superior to others 

 involving a{s/r) which are not constant. 



Although the model adopted here is draAvn from the real-life situation 

 of gambling it is possible that it could apph' to certain other economic 

 situations. The essential requirements for the validity of the theory are 

 the possibilit}' of reinvestment of profits and the abilit}^ to control or 

 vary the amount of money invested or bet in different categories. The 

 "channel" of the theory might correspond to a real communication 

 channel or simply to the totality of inside information available to 

 the investor. 



Let us summarize briefly the results of this paper. Tf a gambler places 

 bets on the input symbol to a comnumication channel and l)ets his money 

 in the same proportion each time a particular symbol is receiA'cd his, 

 capital will grow (or shrink) exponentially. If the odds are consistent 

 with the probabilities of occvu'rence of the transmitted symbols (i.e., 

 equal to their reciprocals), the maximum value of this exponential rate 

 of growth will be equal to the rate of transmission of information. If the 

 odds are not fair, i.e., not consistent with the transmitted symbol proba- 

 bilities but consistent with some other set of probabilities, the maximum 

 exponential rate of growth will be larger than it would have been with no 

 channel by an amount equal to the rate of transmission of information. 

 In case there is a "track take" similar results are obtained, but the 

 formulae involved are more complex and have less direct information 

 theoretic interpretations. 



ACNOWLEDGMENTS 



I am indebted to R. E. Graham and C. E. Shannon for their assist- 

 ance in the preparation of this paper. 



