A NEW INTERPRETATION OF INFORMATION RATE 919 



used, for example, to transmit the results of a series of baseball games 

 between two equally matched teams. The gambler could obtain even 

 money bets even though he already knew the result of each game. The 

 amount of money he could make \\'ould depend only on how much he 

 chose to bet. How much would he bet? Probably all he had since he 

 would win with certainty. In this case his capital would grow expo- 

 nentially and after N bets he would have 2^ times his original bankroll. 

 This exponential growth of capital is not uncommon in economics. In 

 fact, if the binary digits in the above channel were arriving at the rate 

 of one per week, the sequence of bets would have the value of an invest- 

 ment paying 100 per cent interest per week compounded weekly. We 

 will make use of a quantity G called the exponential rate of growth of 

 the gambler's capital, where 



G = Urn -^ log ^ 



iVH.00 iV V 



where Vn is the gambler's capital after A'' bets, Vo is his starting capital, 

 and the logarithm is to the base two. In the above example (j = 1. 



Consider the case now of a noisy binary channel, where each trans- 

 mitted symbol has probability, p, or error and q of correct transmission. 

 Now the gambler could still bet his entire capital each time, and, in 

 fact, this would maximize the expected value of his capital, (Fjv), 

 which in this case would be given by 



(F;v) = (2qfVo 



This would 1)6 little comfort, however, since when A^ was large he would 

 probably be broke and, in fact, would be broke with probability one if 

 he continued indefinitely. Let us, instead, assume that he bets a frac- 

 tion, (, of his capital each time. Then 



v^ = (1 + (y\i -^fvo 



where W aiid L are the number of wins and losses in the N bets. Then 



G = Lim 



^logd -f o+^iog(i -i) 



= g log (1 -f /') -1- p log (1 — i) with probability one 



Let us maximize G with respect to /. The maximum value with respect 

 to the Yi of a quantity of the form Z = ^ Xi log Yi , subject to the 

 constraint ^ Yi = Y, is obtained by putting 



Y 

 Yi = j^ Xi , 



