A NEW INTERPRETATION OF INFORMATION RATE 923 



(b) Since the minimum value of H{a) subject to 



s as 

 obtains when 



a. = 



p(s) 



and H(X) = H(a), any deviation from fair odds helps the gambler. 



(c) Since the gambler's exponential gain would be H{a) — H(X) if 

 he had no inside information, we can interpret R = H{X) — H{X/Y) 

 as the increase of Gmax due to the communication channel. When there 

 is no channel, i.e., H{X/Y) = H{X), Gmax is minimized (at zero) by set- 

 ting 



1 



as = — 



Ps 



This gives further meaning to the concept "fair odds." 



WHEN THERE IS A "TRACK TAKE" 



In the case there is a "track take" the situation is more complicated. 

 It can no longer be assumed that ^s a{s/r) = 1. The gambler cannot 

 make canceling bets since he loses a percentage to the track. Let br = 

 1 — X)s ais/r), i.e., the fraction not bet when the received symbol is 

 the r one. Then the quantity to be maximized is 



G = 11 p(s, r) log [br + aMs/r)], (2) 



rs 



subject to the constraints 



br+ E«(sA) = 1. 



In maximizing (2) it is sufficient to maximize the terms involving a 

 particular value of r and to do this separately for each value of r smce 

 both in (2) and in the associated constraints, terms involving different 

 r's are independent. That is, we must maximize terms of the type 



Gr = q(r)^ q(s/r) log [6, + asa(s/r)] 



s 



subject to the constraint 



