An unresolved problem in Bayesian decision theory is how to value and price information. For instance, Blackwell's Theorem, the workhorse of information economics, is exquisitely unhelpful: Unless signal a is a garbling of signal b, there is some decision maker who will prefer a to b. On the demand front, Radner and Stiglitz showed that depending on how information units are defined, one can quite easily have a nonconcave value of information for any cost function.
This paper resolves both problems by assuming inexpensive information. Building on Large Deviation Theory, we produce a generically complete asymptotic order on samples of i.i.d. signals in finite-state, finite-action Bayesian models. That is, even though signal a was not a garbling of signal b, for generic signals a and b, eventually all decision makers agree on the answer to the simple question: Would you rather have a size n pool of conditionally iid a signals or b signals (where n is large)? We believe that this question well captures the essence of large demand, such as internet purchases of information services: Which database would you rather buy from: Lexus-Nexus or Roper? We show that the value of perfect information less the value of the n sample (the full-information gap, or FIG) is eventually exponentially falling, and is higher for lower quality signals. For the two-state model {L,H}, the FIG for n signals equals rn, where r equals the minimum of the state-H expectation of the moment generating function of the log likelihood ratio of L vs. H, This quantity is easily seen to be state symmetric, and turns out to be the intrinsic measure of the value of a signal. While statisticians have pursued analogues of this first question we posed, it was for non Bayesian environments, and with a weaker ordering.
We now entirely lose the trail of the related statistical work.
We extend the above order from the `total' to the `marginal value of information' --- i.e. the value of an additional signal.
By appealing to an additional term in a useful asymptotic series derived by Cramer (1938), we show that the marginal value too vanishes at the same exponential rate as did the full-information gap:
i.e. it is asymptotically proportional to rn.
We finally exploit this result to provide a precise formula for the information demand (how many signals n to purchase), valid at low prices.
It is an integer within one of
where the constant D depends on the underlying signal, and the decision maker's preferences.
(Conveniently, demand asymptotically does not depend on preferences.)
The proof of this result is remarkably brief.
[This formula is not merely valid for phantom unrealistic prices: Simulations excised from the paper show that this formula `kicks in' at surprisingly low demands, like 10 or 20, for typical example utilities and signal distributions.]
So demand is logarithmic, falling in the price, and falling in the signal quality for a given price.
While our derivations owe their simplicity to the two-state model, we conclude our paper by extending all of our theory to the multiple state model. Here we take inspiration from statistical results of Torgerson, although his proofs were only valid for 0-1 demands, and applied solely to the comparison of experiments.