  This paper is part of an ongoing investigation of "pragmatic information",
defined in Weinberger (2002) as "the amount of information actually used in
making a decision". Because a study of information rates led to the Noiseless
and Noisy Coding Theorems, two of the most important results of Shannon's
theory, we begin the paper by defining a pragmatic information rate, showing
that all of the relevant limits make sense, and interpreting them as the
improvement in compression obtained from using the correct distribution of
transmitted symbols.
  The first of two applications of the theory extends the information theoretic
analysis of the Kelly Criterion, and its generalization, the horse race, to a
series of races where the stochastic process of winning horses, payoffs, and
strategies depend on some stationary process, including, but not limited to the
history of previous races. If the bettor is receiving messages (side
information) about the probability distribution of winners, the doubling rate
of the bettor's winnings is bounded by the pragmatic information of the
messages.
  A second application is to the question of market efficiency. An efficient
market is, by definition, a market in which the pragmatic information of the
"tradable past" with respect to current prices is zero. Under this definition,
markets whose returns are characterized by a GARCH(1,1) process cannot be
efficient.
  Finally, a pragmatic informational analogue to Shannon's Noisy Coding Theorem
suggests that a cause of market inefficiency is that the underlying
fundamentals are changing so fast that the price discovery mechanism simply
cannot keep up. This may happen most readily in the run-up to a financial
bubble, where investors' willful ignorance degrade the information processing
capabilities of the market.
