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Payoffs-Beliefs Duality and the Value of Information

Author

Listed:
  • Michel de Lara

    (CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTech)

  • Olivier Gossner

    (X - École polytechnique - IP Paris - Institut Polytechnique de Paris)

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.

Suggested Citation

  • Michel de Lara & Olivier Gossner, 2020. "Payoffs-Beliefs Duality and the Value of Information," Post-Print hal-01941006, HAL.
    • Handle: RePEc:hal:journl:hal-01941006
    DOI: 10.1137/18M1230049
    Note: View the original document on HAL open archive server: https://hal.science/hal-01941006v3
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    References listed on IDEAS

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    3. Daniele Pennesi, 2020. "Identity and information acquisition," Carlo Alberto Notebooks 610, Collegio Carlo Alberto, revised 2021.

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    Keywords

    AMS classification: 46N10; payoffs-beliefs duality; value of information; convex analysis; AMS classification: 91B06;
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