A tight sufficient condition for Radner–Stiglitz nonconcavity in the value of information

This paper deals with the existence of a nonconcavity in the value of information, as was first explained by Radner and Stiglitz [A nonconcavity in the value of information, in: M. Boyer, R.E. Kihlstrom (Eds.), Bayesian Models in Economic Theory, Elsevier Science Publishers, Amsterdam, 1984, pp. 33–52 (Chapter 3)]. After defining infinitesimal information distance variation  IIDVIIDV, we find that IIDV=0IIDV=0 is sufficient for a zero marginal value of information at the null. This is a condition only on the information structure and in particular is independent of the decision maker's preferences. This condition is tight: when IIDV>0IIDV>0, there exists a payoff function for which the marginal value of information at the null is positive under general assumptions.