A NEW INTERPRETATION OF INFORMATION RATE 925 



then X must consist of all s ^ i where t is a, positive integer or zero. 

 Consider how the fraction 



I — (Xt 



varies with t, where 



t t ^ 



Pt = ^ p(s), <Tt = ^ — ; Fo = 1 



I 1 as 



Now if p(l)ar < 1, Ft increases with i until at ^ 1. In this case t = 

 satisfies the desired conditions and X is empty. If p{l)ai > 1 Ft de- 

 creases with t until p(t + l)at+i < Ft or at ^ 1. If the former occurs, 

 i.e., p(t + l)oit+i < Ft , then i^^+i > Ft and the fraction increases until 

 cr< ^ 1. In any case the desired value of t is the one which gives Ft its 

 minimum positive value, or if there is more than one such value of /, 

 the smallest. The maximizing process may be summed up as follows: 



(a) Permute indices so that p(s)as ^ p(s + l)Qrg+i 



(b) Set h equal to the minimum positive value of 



-I t < - 



— where Pt = ILp (s), at = ^ — 



i- — (Tt 1 1 ffj 



(c) Set a(s) = p(s) — b/as or zero, whichever is larger. (The a(s) 

 will sum to 1 — h.) 



The desired maximum G will then be 



(rmax = Z) P(s) log p(s)as + (1 - Pt) log 



1 -Pt 



I - CTt 



where t is the smallest index which gives 



1 -Pt 

 1 - <rt 



its minimum positive value. 



It should be noted that if p{s)as < 1 for all s no bets are placed, but 

 if the largest p(s)as > 1 some bets might be made for which p(s)as < 1, 

 i.e., the expected gain is negative. This violates the criterion of the 

 classical gambler who never bets on such an event. 



CONCLUSION 



The gambler introduced here follows an essentially different criterion 

 from the classical gambler. At every bet he maximizes the expected 

 value of the logarithm of his capital. The reason has nothing to do with 



