Information Value & Separable Utility

• Information Value is the measure of how decision makers enhance expected utility by expanding feasible utility acts via Minkowski addition.
• The framework uses convex analysis to compare decision makers through the convexity differences of their belief-value functions.
• The study establishes that higher information value is equivalent to representing utility as additively separable through fusion rather than mere union.

Searching arXiv for the target paper and closely related work to ground the article in current arXiv records. “More valuable information” in decision theory can be studied by representing a decision maker not primarily by primitive actions, but by a feasible set of utility acts—state-indexed utility vectors—paired with beliefs by expected utility. In this framework, the paper “Increasing Value of Information Implies Separable Utility” (Lara, 13 Oct 2025) proves a structural equivalence: one decision maker values information more than another if and only if the former’s feasible utility acts are obtained from the latter’s by Minkowski addition with another c-utility act set. In the standard action–utility representation, this is exactly the existence of an expanded product decision problem with additively separable utility. The paper also organizes these constructions through a dioid structure whose two operations correspond to adding options by union and combining options by fusion, and identifies fusion—not union—as the exact structural counterpart of greater value of information (Lara, 13 Oct 2025).

1. Formal environment and representation of decision makers

The analysis is set on a finite state space $\mathcal{K}$, with $|\mathcal{K}|=K$, and identifies $\mathbb{R}^K$ with the space of utility acts $v=(v_k)_{k\in\mathcal{K}}$, one coordinate for each state of nature (Lara, 13 Oct 2025). Beliefs are probability distributions over $\mathcal{K}$, collected in the simplex

$$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$

and utility acts and beliefs are paired by the bilinear duality

$\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$

which is interpreted as expected utility (Lara, 13 Oct 2025).

A central object is the c-utility act set. A subset $G\subset\mathbb{R}^K$ is a c-utility act set when it is closed, convex, and comprehensive, with comprehensiveness given by

$\mathbb{R}^K_-+G\subset G,$

equivalently $\mathbb{R}^K_-+G=G$, so that if a utility act is feasible then any componentwise worse act is feasible as well (Lara, 13 Oct 2025). The set must also satisfy a continuity condition through its support function: $|\mathcal{K}|=K$0 with $|\mathcal{K}|=K$1 required to be a continuous, bounded, convex function on $|\mathcal{K}|=K$2 (Lara, 13 Oct 2025).

Within this representation, an abstract decision maker is simply a c-utility act set $|\mathcal{K}|=K$3, where $|\mathcal{K}|=K$4 denotes the class of all such sets. The decision maker’s value function over beliefs is

$|\mathcal{K}|=K$5

so the decision maker is an expected-utility maximizer over feasible utility acts (Lara, 13 Oct 2025). This formulation is equivalent to the classical one with a decision set $|\mathcal{K}|=K$6 and utility $|\mathcal{K}|=K$7: if each action $|\mathcal{K}|=K$8 induces a utility act $|\mathcal{K}|=K$9, then the associated c-utility act set is

$\mathbb{R}^K$0

and $\mathbb{R}^K$1 reproduces maximal expected utility under belief $\mathbb{R}^K$2 (Lara, 13 Oct 2025).

This representation matters because it shifts the analysis of informational responsiveness from individual actions to the geometry of feasible utility-act menus. That convex-analytic shift is the basis of the paper’s characterization theorem (Lara, 13 Oct 2025).

2. Information structures and the value of information

Information is modeled as a random posterior belief. Formally, on a probability space $\mathbb{R}^K$3, an information structure is a random variable

$\mathbb{R}^K$4

whose expectation $\mathbb{R}^K$5 is the associated prior (Lara, 13 Oct 2025). This is the Marschak–Miyasawa / Artstein–Wets style formulation in which an experiment is identified with a distribution of posteriors.

For a decision maker $\mathbb{R}^K$6, the value of information of $\mathbb{R}^K$7 is

$\mathbb{R}^K$8

It is the expected maximal expected utility with information minus maximal expected utility under the prior (Lara, 13 Oct 2025). Nonnegativity follows from Jensen’s inequality because $\mathbb{R}^K$9 is convex on $v=(v_k)_{k\in\mathcal{K}}$0 (Lara, 13 Oct 2025).

The paper also uses a martingale notion of informational refinement. Given two information structures $v=(v_k)_{k\in\mathcal{K}}$1 and $v=(v_k)_{k\in\mathcal{K}}$2, $v=(v_k)_{k\in\mathcal{K}}$3 is a garbling of $v=(v_k)_{k\in\mathcal{K}}$4, written $v=(v_k)_{k\in\mathcal{K}}$5, if there exists a sub-$v=(v_k)_{k\in\mathcal{K}}$6-algebra $v=(v_k)_{k\in\mathcal{K}}$7 such that

$v=(v_k)_{k\in\mathcal{K}}$8

The corresponding relative value of information is

$v=(v_k)_{k\in\mathcal{K}}$9

Moreover,

$\mathcal{K}$0

(Lara, 13 Oct 2025).

This notion belongs to the broader theory of value of information in which information is evaluated by the improvement in optimized performance. In classical expected-utility environments, analogous definitions appear in work on Shannon- or resource-constrained information, though there the constraint variable is often an information budget rather than a distribution of posteriors (Belavkin, 2014). The present paper remains within a pure expected-utility and posterior-belief setting (Lara, 13 Oct 2025).

3. Comparing decision makers by how much they value information

The paper studies an interpersonal comparison between decision makers $\mathcal{K}$1. It defines:

$\mathcal{K}$2

and, more strongly,

$\mathcal{K}$3

for all refinements $\mathcal{K}$4 (Lara, 13 Oct 2025).

A key intermediate result is that the weak and strong notions coincide, and both are equivalent to a simple functional property: $\mathcal{K}$5 is convex on $\mathcal{K}$6 (Lara, 13 Oct 2025). Thus the question “who values information more?” becomes a question about the convexity of the difference of the two belief-value functions.

This convex-difference characterization links the paper to earlier literature. The relation between more valuable information and convexity of value-function differences is described as already implicit in Jones–Ostroy and Whitmeyer, and the paper positions its contribution as a structural strengthening of that line: not merely a property of the value function, but a necessary-and-sufficient property of the feasible decision problem itself (Lara, 13 Oct 2025).

The comparison is very strong. It is not prior-specific, experiment-specific, or task-specific. It requires one decision maker to obtain at least as much incremental expected payoff from every information structure, equivalently from every Blackwell refinement. This suggests an interpretation of informational sensitivity that is invariant across information technologies rather than tailored to a particular experiment (Lara, 13 Oct 2025).

4. Minkowski addition and the structural theorem

The paper’s central theorem identifies the exact geometric structure corresponding to this ordering. For $\mathcal{K}$7, the following are equivalent:

1. $\mathcal{K}$8 values information strongly more than $\mathcal{K}$9.
2. $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$0 values information weakly more than $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$1.
3. $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$2 is convex on $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$3.
4. There exists a c-utility act set $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$4 such that

$$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$5

where $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$6 denotes Minkowski addition of sets of utility acts (Lara, 13 Oct 2025).

Minkowski addition is

$$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$7

Since c-utility act sets are closed, convex, and comprehensive, $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$8 remains in the same class (Lara, 13 Oct 2025).

This theorem converts an order over values of information into a decomposition of feasible utility acts. If $$\Delta=\Bigl\{p\in\mathbb{R}_+^K:\sum_{k\in\mathcal{K}}p_k=1\Bigr\},$$9 values information more than $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$0, then every feasible act of $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$1 can be expressed as the sum of an act feasible for $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$2 and an act from some additional c-utility act set $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$3 (Lara, 13 Oct 2025). Conversely, such a decomposition is sufficient to guarantee that $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$4 values information more than $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$5.

The same result is also expressed using the star-difference $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$6: the characterization can be written as

$\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$7

which internalizes the additional flexibility in the space $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$8 itself (Lara, 13 Oct 2025).

This structural theorem is the paper’s main novelty. It does not merely say that a more information-responsive decision maker has a more convex value function. It says that the entire feasible utility-act set must be obtainable by a geometric additive operation from the benchmark set (Lara, 13 Oct 2025).

5. Separable utility in the classical action–utility representation

The geometric theorem becomes especially transparent when translated back into the classical representation with actions and utility functions. Suppose decision maker $\langle p,v\rangle=\sum_{k\in\mathcal{K}}p_k v_k,$9 has decision set $G\subset\mathbb{R}^K$0 and utility $G\subset\mathbb{R}^K$1, while another decision problem $G\subset\mathbb{R}^K$2 has decision set $G\subset\mathbb{R}^K$3 and utility $G\subset\mathbb{R}^K$4. Consider the product decision set $G\subset\mathbb{R}^K$5 and define

$G\subset\mathbb{R}^K$6

Then the induced c-utility act set of $G\subset\mathbb{R}^K$7 is

$G\subset\mathbb{R}^K$8

a Minkowski sum, and the corresponding value of information decomposes additively: $G\subset\mathbb{R}^K$9 (Lara, 13 Oct 2025).

This establishes sufficiency: multiplying decisions and adding utility makes information more valuable. The paper then proves the converse in the classical setting. If two classical decision makers $\mathbb{R}^K_-+G\subset G,$0 and $\mathbb{R}^K_-+G\subset G,$1 satisfy the value-of-information ordering, then there exists $\mathbb{R}^K_-+G\subset G,$2 such that $\mathbb{R}^K_-+G\subset G,$3 is representable as an expanded product decision problem with utility

$\mathbb{R}^K_-+G\subset G,$4

which is additively separable (Lara, 13 Oct 2025).

This is the exact content of the title “Increasing Value of Information Implies Separable Utility” (Lara, 13 Oct 2025). The implication is not that every separable utility function yields the same informational behavior, but that systematic dominance in value of information over another decision maker is equivalent to representability as a separable extension of that decision maker’s problem.

A common misunderstanding would be to read the result as a statement about preference separability in general equilibrium or consumption theory. The paper’s separability claim is narrower and more precise: it concerns separability across components of an expanded product decision problem, with utilities added statewise, and is derived from the geometry of utility-act sets rather than from axioms on preferences over commodity bundles (Lara, 13 Oct 2025).

6. Dioid structure, union versus fusion, and limits of “adding options”

The paper introduces a dioid structure to organize two operations on decision makers. On c-value functions $\mathbb{R}^K_-+G\subset G,$5, the operations are

$\mathbb{R}^K_-+G\subset G,$6

On c-utility act sets $\mathbb{R}^K_-+G\subset G,$7, the corresponding operations are

$\mathbb{R}^K_-+G\subset G,$8

The map $\mathbb{R}^K_-+G\subset G,$9 is an isomorphism between these two dioids (Lara, 13 Oct 2025).

Economically, $\mathbb{R}^K_-+G=G$0 is union: adding options from one menu or another, with convexification and closure. By contrast, $\mathbb{R}^K_-+G=G$1 is fusion: choosing a pair of utility acts and receiving their sum, which in the classical formulation means multiplying decisions and adding utilities (Lara, 13 Oct 2025).

This distinction yields two notions of comparative flexibility:

• $\mathbb{R}^K_-+G=G$2 is more flexible by union than $\mathbb{R}^K_-+G=G$3 if $\mathbb{R}^K_-+G=G$4 for some $\mathbb{R}^K_-+G=G$5.
• $\mathbb{R}^K_-+G=G$6 is more flexible by fusion than $\mathbb{R}^K_-+G=G$7 if $\mathbb{R}^K_-+G=G$8 for some $\mathbb{R}^K_-+G=G$9 (Lara, 13 Oct 2025).

The main theorem can then be restated succinctly: $|\mathcal{K}|=K$00 So greater value of information is exactly flexibility by fusion, not generic flexibility by option expansion (Lara, 13 Oct 2025).

The paper also studies when union can increase information value. For $|\mathcal{K}|=K$01, the union $|\mathcal{K}|=K$02 values information more than $|\mathcal{K}|=K$03 if and only if

$|\mathcal{K}|=K$04

is convex on $|\mathcal{K}|=K$05 (Lara, 13 Oct 2025). This imposes restrictive geometric conditions; the set where $|\mathcal{K}|=K$06 must be convex, and the cell structure of $|\mathcal{K}|=K$07 must refine that of $|\mathcal{K}|=K$08 (Lara, 13 Oct 2025). Sufficient conditions are given under which union is effectively representable as a disguised fusion, for example when $|\mathcal{K}|=K$09 for some $|\mathcal{K}|=K$10 (Lara, 13 Oct 2025).

This analysis clarifies an important limitation. Simply adding more options does not in general produce a robust ordering of value of information. A stable ordering emerges only when the added flexibility has the stronger fusion form, i.e., when options combine multiplicatively while utility combines additively (Lara, 13 Oct 2025).

7. Conceptual significance and connections

The paper situates its contribution against several related lines of research. The refinement order on information structures is Blackwell’s order, but the paper addresses a different question: not which experiment is more informative, but which decision maker benefits more from any information structure (Lara, 13 Oct 2025). Earlier work had already linked this to convexity of value-function differences; the present contribution identifies the exact feasible-set structure behind that property (Lara, 13 Oct 2025).

This places the result alongside broader research programs that treat value of information as a structural object rather than a purely numerical one. In some work, information value is analyzed through constrained optimization or information budgets, yielding concave gain frontiers and even $|\mathcal{K}|=K$11-shaped value patterns under general information resources (Belavkin, 2014). In control and communication settings, value of information is defined as the marginal reduction in future cost induced by a packet or a measurement, often with event-triggered transmission rules of the form “send if and only if VoI is nonnegative” (Soleymani et al., 2024). By contrast, (Lara, 13 Oct 2025) works entirely within abstract expected-utility maximization over utility-act sets and asks for a universal comparison across all information structures.

Its main conceptual contribution is therefore a precise equivalence:

• More valuable information is a comparative statement over all experiments.
• Convex difference of belief-value functions is the functional signature of that comparison.
• Minkowski addition / fusion is the geometric structure generating it.
• Additively separable utility over product decisions is the classical decision-theoretic interpretation (Lara, 13 Oct 2025).

A plausible implication is that informational responsiveness is not merely a matter of curvature in a value function over beliefs; it is encoded in how feasible utility acts decompose. In this sense, the paper turns a comparative statement about the usefulness of information into a theorem about the algebra and geometry of decision problems themselves (Lara, 13 Oct 2025).