Payoffs-beliefs duality and the value of information
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.
SIAM Journal on Optimization, 30(1), 464-489, 2020.
Citation
@article{de_lara2020,
author = {de Lara, Michel and Gossner, Olivier},
title = {Payoffs-Beliefs Duality and the Value of Information},
journal = {SIAM Journal on Optimization},
date = {2020},
url = {https://gossner.me/papers/payoffs-beliefs-duality-and-the-value-of-information.html},
langid = {en},
abstract = {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.}
}