  In decision problems under incomplete information, actions (identified to
payoff vectors indexed by states of nature) and beliefs are naturally paired by
bilinear duality. We exploit this duality to analyze the value of information,
using concepts and tools from convex analysis. We define the value function as
the support function of the set of available actions: the subdifferential at a
belief is the set of optimal actions at this belief; the set of beliefs at
which an action is optimal is the normal cone of the set of available actions
at this point. Our main results are 1) a necessary and sufficient condition for
positive value of information 2) global estimates of the value of information
of any information structure from local properties of the value function and of
the set of optimal actions taken at the prior belief only. We apply our results
to the marginal value of information at the null, that is, when the agent is
close to receiving no information at all, and we provide conditions under which
the marginal value of information is infinite, null, or positive and finite.
