Entropy and the Value of Information for Investors

Abstract

Consider any investor who fears ruin when facing any set of investments that satisfy no-arbitrage. Before investing, he can purchase information about the state of nature in the form of an information structure. Given his prior, information structure α is more informative than information structure β if, whenever he is willing to buy β at some price, he is also willing to buy α at that price. We show that this informativeness ordering is complete and is represented by the decrease in entropy of his beliefs, regardless of his preferences, initial wealth, or investment problem. We also show that no prior-independent informativeness ordering based on similar premises exists.

Payoffs-Beliefs Duality and the Value of Information

“…A). We prove the upper inequality CAE q −p ≥ VoIA(q) in (12). By definition (24) of a support function, we have that σA(·) ≤ A × · , where A = sup{ a , a ∈ A} < +∞.…”

“…By definition (24) of a support function, we have that σA(·) ≤ A × · , where A = sup{ a , a ∈ A} < +∞. Thus CA = A in the left hand side inequality in (12). B).…”

“…More precisely, we assume that around the prior, optimal actions smoothly depend on a 1-1 way on the belief. This assumption is met when, for instance, the decision problem faced by the agent is a scoring rule [11], or an investment problem [1,12].…”

“…For instance, [26], [31] and [2] restrict attention to families of decision problems that generate monotone decision rules. Focusing on investment decision problems, [12] obtains and characterizes a complete ranking of information sources based on a uniform criterion; [13] use a duality approach to characterize the value of an information purchase that consists of an information structure with a price attached to it.…”

Bayesian general equilibrium

“…In this paper I clarify the relation between maximum entropy and Bayesian inference, which seems to be still relatively unknown in the economics literature. Since informationtheoretic techniques are recently more and more applied to economics (Krebs, 1997;Sims, 2003;Veldkamp, 2011;Cabrales et al, 2013), the results on maximum entropy and Bayesian inference presented in this paper might be of interest to economists. With this clarification, Foley's statistical equilibrium is precisely an interim equilibrium that I define in this paper.…”