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Increasing Value of Information Implies Separable Utility

Michel de Lara

TL;DR

This paper studies how information becomes more valuable in decision problems with incomplete state information by framing decisions as closed convex comprehensive sets of payoff acts and exploiting a duality with beliefs. It introduces c-utility act sets and a Minkowski-addition structure to capture how information value rises when decisions are multiplied and utilities are added (additively separable utility). The main result shows that an information-valuable decision-maker M relative to L exists if and only if there is a c-utility act set T such that M = L ⊗ T (fusion), equivalently the difference of their value functions is convex. The authors formalize this via dioids, defining union and fusion operators on decision-makers, and demonstrate that fusion is the precise form of flexibility that guarantees more valuable information, with connections to and extensions of prior work (Whitmeyer 2024, Denti 2022, Yoder 2022). The framework provides a robust, generalized lens for assessing VoI and offers a bridge between abstract convex analysis and classic decision problems.

Abstract

We consider decision-making under incomplete information about an unknown state of nature. Utility acts (that is, utility vectors indexed by states of nature) and beliefs (probability distributions over the states of nature) are naturally paired by bilinear duality, giving the expected utility. With this pairing, an expected utility maximizer (DM) is characterized by a continuous closed convex comprehensive set of utility acts (c-utility act set). We show that DM M values information more than DM L if and only if the c-utility act set of DM M is obtained by Minkowski addition from the cutility act set of DM L. In the classic setting of decision theory, this is interpreted as the equivalence between more valuable information, on the one hand, and multiplying decisions and adding utility, on the other hand (additively separable utility). We also introduce the algebraic structure of dioid to describe two operations between DMs: union (adding options) and fusion (multiplying options and adding utilities). We say that DM M is more exible by union (resp. by fusion) than DM L if DM M is obtained by union (resp. by fusion) from DM L. Our main result is that DM M values information more than DM L if and only if DM M is more exible by fusion than DM L. We also study when exibility by union can lead to more valuable information.

Increasing Value of Information Implies Separable Utility

TL;DR

This paper studies how information becomes more valuable in decision problems with incomplete state information by framing decisions as closed convex comprehensive sets of payoff acts and exploiting a duality with beliefs. It introduces c-utility act sets and a Minkowski-addition structure to capture how information value rises when decisions are multiplied and utilities are added (additively separable utility). The main result shows that an information-valuable decision-maker M relative to L exists if and only if there is a c-utility act set T such that M = L ⊗ T (fusion), equivalently the difference of their value functions is convex. The authors formalize this via dioids, defining union and fusion operators on decision-makers, and demonstrate that fusion is the precise form of flexibility that guarantees more valuable information, with connections to and extensions of prior work (Whitmeyer 2024, Denti 2022, Yoder 2022). The framework provides a robust, generalized lens for assessing VoI and offers a bridge between abstract convex analysis and classic decision problems.

Abstract

We consider decision-making under incomplete information about an unknown state of nature. Utility acts (that is, utility vectors indexed by states of nature) and beliefs (probability distributions over the states of nature) are naturally paired by bilinear duality, giving the expected utility. With this pairing, an expected utility maximizer (DM) is characterized by a continuous closed convex comprehensive set of utility acts (c-utility act set). We show that DM M values information more than DM L if and only if the c-utility act set of DM M is obtained by Minkowski addition from the cutility act set of DM L. In the classic setting of decision theory, this is interpreted as the equivalence between more valuable information, on the one hand, and multiplying decisions and adding utility, on the other hand (additively separable utility). We also introduce the algebraic structure of dioid to describe two operations between DMs: union (adding options) and fusion (multiplying options and adding utilities). We say that DM M is more exible by union (resp. by fusion) than DM L if DM M is obtained by union (resp. by fusion) from DM L. Our main result is that DM M values information more than DM L if and only if DM M is more exible by fusion than DM L. We also study when exibility by union can lead to more valuable information.

Paper Structure

This paper contains 49 sections, 13 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Abstract decision-maker
  3. Duality between utility acts and beliefs
  4. Classic decision problems under imperfect information
  5. Abstract decision problems under imperfect information
  6. Definition of c-utility act set s.
  7. Abstract decision problem.
  8. From abstract to classic decision problem.
  9. More valuable information
  10. Information structures and value of information
  11. Comparison of information structures and relative value of information
  12. Characterization of more valuable information
  13. Characterization of more valuable information in the classic setting
  14. Flexibility and more valuable information
  15. Dioids of expected utility maximizer s
  16. ...and 34 more sections

Key Result

Proposition 7

Consider a decision-maker with c-utility act set ${G} \in \mathcal{A}_{c}( {\cal K} )$, as in Definition de:4C_action_set, All terms in eq:relative_VoI (hence in eq:VoI) are well defined, and the relative value of information $\mathrm{V\!o\!I}_{{G}}( \mathbf{q^{\sharp}}\slash\mathbf{q^{\flat}} )$ in

Figures (3)

  • Figure 1: Example of polyhedral c-utility act set in the case of ${\lvert {\cal K} \rvert}=2$ states of nature and, in the classic decision problem setting, of a finite set of three actions (corresponding to the points $B$, $C$ and $D$)
  • Figure 2: Example of (nonpolyhedral) c-utility act set in the case of ${\lvert {\cal K} \rvert}=2$ states of nature and, in the classic decision problem setting, of an infinite set of actions which is the union of four actions (corresponding to the points $B$, $C$, $D$ and $E$) and of a continuum of actions (between $C$ and $D$)
  • Figure 3: If ${L}$ denotes the hashed polyhedral utility act set (see Figure \ref{['fig:polyhedralActionPayoffSet']}), the polyhedral utility act set to the left of the polyline $\infty$-B-C-D-E-$\infty$ is obtained either as ${L} \oplus \{ E \} = \overline{\mathrm{co}}( {L}\cup\{ E \})$ or as ${L} \otimes [0,E-D] = {L} + [0,E-D]$

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 7
  • Definition 8
  • Theorem 9
  • Proposition 10
  • ...and 15 more