Bell System Technical Journal · 1956 · 1.6K citations · 0 references
EngineeringGame TheoryChannel CodingCommunicationChannel Capacity EstimationTransmission RateCommunication ChannelCoding TheoryStatisticsInformation TheoryInformation RateInformation AsymmetryProbability TheoryInformation ManagementComputer ScienceEntropyInformation EconomicsBinary DigitsGame-theoretic Probability
When channel outputs represent outcomes of a chance event with fair odds, a gambler can exploit the received symbols to grow capital exponentially, a property previously linked only to Shannon's theorem on error‑free transmission. The result is extended to arbitrary odds. The maximum exponential growth rate equals the channel's information transmission rate, showing that this rate matters even without coding.
If the input symbols to a communication channel represent the outcomes of a chance event on which bets are available at odds consistent with their probabilities (i.e., “fair” odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially. The maximum exponential rate of growth of the gambler's capital is equal to the rate of transmission of information over the channel. This result is generalized to include the case of arbitrary odds. Thus we find a situation in which the transmission rate is significant even though no coding is contemplated. Previously this quantity was given significance only by a theorem of Shannon's which asserted that, with suitable encoding, binary digits could be transmitted over the channel at this rate with an arbitrarily small probability of error.