A NEW INTERPRETATION OF INFORMATION RATE 921 



Only the case of independent transmitted symbols and noise will be 

 considered. We will consider first the case of "fair" odds, i.e., 



1 



Ois = 



p{s) 



In any sort of parimutual betting there is a tendency for the odds to be 

 fair (ignoring the "track take"). To see this first note that if there is no 

 "track take" 



Ei = i 



since all the money collected is paid out to the winner. Next note that if 



p(s) 



for some s a bettor could insure a profit by making repeated bets on the 

 s* outcome. The extra betting which would result would lower a., . 

 The same feedback mechanism probably takes place in more compli- 

 cated betting situations, such as stock market speculation. 

 There is no loss in generality in assuming that 



Z a(s/r) = 1 



s 



i.e., the gambler bets his total capital regardless of the received symbol. 

 Since 



he can effectively hold back money by placing canceling bets. Now 



r,s 



where Wsr is the number of times that the transmitted symbol is s and 

 the received, symbol is r. 



Log ^ = X) ^V^r log oisa{s/r) 



vl " "^ 



G = Urn ^ log -f/ = X) P(^> ^) log oLsa{s/r) 



N^x I\ Vq ts 



with probability one. Since 



1 



oil = 



' Pis) 



