918 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G 



cation system in which errors are being made at a non-negligible rate, 

 i.e., Avhere optimum coding is not being used. In its most general formu- 

 lation this problem seems to have but one solution. A cost function must 

 be defined on pairs of symbols which tell how bad it is to receive a cer- 

 tain symbol when a specified signal is transmitted. Furthermore, this 

 cost function must be such that its expected value has significance, i.e., 

 a system must be preferable to another if its average cost is less. The 

 utility theoiy of Von Neumann shows us one way to obtain such a cost 

 function. Generally this cost function would depend on things external 

 to the system and not on the probabilities which describe the system, so 

 that its average value could not be identified with the rate as defined 

 by Shannon. 



The cost function approach is, of course, not limited to studies of com- 

 munication systems, but can actually be used to analyze nearly any 

 branch of human endeavor. The author believes that it is too general to 

 shed any light on the specific problems of communication theory. The 

 distinguishing feature of a communication system is that the ultimate 

 receiver (thought of here as a person) is in a position to profit from any 

 knowledge of the input symbols or even from a better estimate of their 

 probabilities. A cost function, if it is supposed to apply to a communica- 

 tion system, must somehow reflect this feature. The point here is that 

 an arbitrary combination of a statistical transducer (i.e., a channel) and 

 a cost function does not necessarily constitute a communication system. 

 In fact (not knowing the exact definition of a communication system 

 on which the above statements are tacitly based) the author would not 

 know how to test such an arbitrary combination to see if it were a com- 

 munication system. 



What can be done, however, is to take some real-life situation which 

 seems to possess the essential features of a communication problem, and 

 to analyze it without the introduction of an arbitrary cost function. 

 The situation which will be chosen here is one in which a gambler uses 

 knowledge of the received symbols of a communication channel in order 

 to make profitable bets on the transmitted symbols. 



THE GAMBLER WITH A PRIVATE WIRE 



Let us consider a communication channel which is used to transmit the 

 results of a chance situation before those results become common 

 knowledge, so that a gambler may still place bets at the original odds. 

 Consider first the case of a noiseless binary channel, which might be 



^ Von Neumann and Morgenstein, Theory of Games and Economic Behavior, 

 Princeton Univ. Press, 2nd Edition, 1947. 



