Smithian Value of Information (SVI)

• Smithian Value of Information (SVI) is a formal metric that quantifies decision gains by linking updated beliefs with flexible utility structures.
• It leverages bilinear duality and Minkowski addition to precisely measure the added value of new information in decision-making.
• SVI drives pragmatic inference in communicative models and computational experiments, demonstrating significant benefits in economic and AI settings.

The Smithian Value of Information (SVI) formalizes the value that new information provides to a decision-maker, grounded in the duality between beliefs (probability distributions over states of nature) and utility acts (state-contingent payoff vectors). SVI extends classical value of information concepts by capturing the structure of decision-maker (DM) preferences and their flexibility under information, and by precisely distinguishing between adding alternative options and introducing separable/augmenting decision possibilities. SVI is also operative within models of communicative action, such as pragmatic pointing, where relevance is quantified as the improvement in expected utility resulting from new information, as predicted and evaluated separately by communicative agents. SVI has been foundational in formal economic theory and in computational models of social reasoning, providing a lens for understanding when information is valuable and what structural properties of choices govern that value (Lara, 13 Oct 2025, Jiang et al., 2021).

1. Mathematical Foundations: Bilinear Duality and Value of Information

Let $\Omega = \{1,\ldots,K\}$ be a finite set of possible states. The set of utility-acts is $V = \mathbb{R}^K$, and beliefs are represented by the probability simplex $$\Delta = \{p \in \mathbb{R}^K_+ : \sum_{k} p_k = 1\}$$. For any $v \in V$ and $p \in \Delta$, the expected utility is computed via the bilinear pairing $\langle v, p \rangle = \sum_{k} v_k p_k$.

Given a closed, bounded set $G \subset V$ (interpreted as a "c-utility-act set" characterizing a DM), the support function is defined as

$$\sigma_G(p) := \sup \{ \langle v, p \rangle : v \in G \}, \quad p \in \Delta.$$

For a random information structure $\tilde{q}$ taking values in $\Delta$ (with prior $E[\tilde{q}] \in \Delta$), the Value of Information (VoI) for $G$ is

$$\mathrm{VoI}_G(\tilde{q}) := \mathbb{E}[\sigma_G(\tilde{q})] - \sigma_G(\mathbb{E}[\tilde{q}]) \geq 0.$$

This structure formalizes the quantifiable gain from information, independently of the specific realization of the state.

2. Structural Comparison: Separable Utility and Minkowski Addition

Each DM is specified up to expected utility equivalence by its c-utility-act set $G \subset \mathbb{R}^K$, where $G$ is closed, convex, comprehensive ($G + \mathbb{R}^K_{-} = G$), and $\sigma_G$ is continuous.

Comparing two DMs $M$ and $L$ with respective c-utility-act sets $G_M$, $G_L$, "M values information more than L" is defined as $$\mathrm{VoI}_{G_M}(\tilde{q}) \geq \mathrm{VoI}_{G_L}(\tilde{q})$$ for all $\tilde{q}$. Equivalently, the support function difference $\sigma_{G_M}(p) - \sigma_{G_L}(p)$ is convex on $\Delta$.

The main SVI characterization theorem establishes the following equivalence:

• $M$ values information more than $L$ if and only if there exists $T \in \mathcal{A}$ such that

$G_M = G_L \otimes T := \overline{G_L + T}$

where $+$ denotes Minkowski addition. Thus, SVI is fundamentally tied to the existence of a separable (additively extendable) utility structure, captured algebraically via the Minkowski sum (Lara, 13 Oct 2025).

3. Algebraic Operations: The Dioid Structure of Decision Makers

The collection $\mathcal{A}$ of c-utility-act sets forms a commutative dioid under two operations:

• Union $\oplus$: $G \oplus H = \overline{\mathrm{conv}}(G \cup H)$ (adds new options in parallel).
• Fusion $\otimes$: $G \otimes H = \overline{G + H}$ (adjoins options via Minkowski sum; compound decisions with summing utilities).

The dioid $(\mathcal{A} \cup \{\emptyset\}, \oplus, \otimes)$ has zero $\emptyset$, unit $\mathbb{R}^K_-$, idempotent union, and fusion distributive over union. This algebraic structure enables a precise classification of DMs' flexibility and their comparative value of information.

Economic interpretation:

• Fusion allows the DM to execute joint decisions with aggregate payoff (multiplicative flexibility).
• Union provides additional exclusive alternatives but does not guarantee increased informational value unless special convexity or refinement conditions are satisfied (Lara, 13 Oct 2025).

4. Flexibility: Fusion, Union, and Increases in Value of Information

Flexibility is stratified as follows:

• Fusion-flexibility: $M$ is more flexible by fusion than $L$ if $M = L \otimes T$ for some $T \in \mathcal{A}$.
• Union-flexibility: $M$ is more flexible by union than $L$ if $M = L \oplus T$ for some $T \in \mathcal{A}$.

The decisive result is that $G_M$ values information more than $G_L$ if and only if $G_M$ is a fusion-flexible extension of $G_L$, i.e., $G_M = G_L \otimes T$ for some $T$ (Lara, 13 Oct 2025). Merely expanding the set of alternatives by union does not generically increase VoI; necessary and sufficient conditions involve convexity of $\max\{0, \sigma_G - \sigma_L\}$ on $\Delta$ and structural refinement of normal cone lattices.

A plausible implication is that genuine informational gains (i.e., guaranteed strictly higher VoI across all information structures) require augmenting the DM's space of decisions via separable (Minkowski-sum) extensions, not just proliferating alternatives.

5. SVI in Communication: Relevance and Pragmatic Pointing

In multi-agent settings, SVI formalizes relevance in communicative action. For a signaler (guide) with belief $b_{sig}$ and receiver with belief $b_{rec}$ (possibly before and after receiving a communicative act $u$), the Smithian utility from the signaler's perspective is

$$U_{\mathrm{Smith}}(b_{rec} \mid b_{sig}) = \sum_{a_{rec}} P(a_{rec} \mid b_{rec}) \cdot \mathrm{EU}_{\mathrm{Smith}}(a_{rec} \mid b_{sig}),$$

$$\mathrm{EU}_{\mathrm{Smith}}(a_{rec} \mid b_{sig}) = \sum_{s'} U_{sig}(s') \sum_{s} P(s' \mid s, a_{rec}) b_{sig}(s).$$

The Smithian Value of Information for $u$ is defined as

$$\mathrm{SVI}(u \mid b_{sig}) = U_{\mathrm{Smith}}(b_{rec}' \mid b_{sig}) - U_{\mathrm{Smith}}(b_{rec} \mid b_{sig}),$$

where $b_{rec}'$ is the receiver's posterior after $u$ (Jiang et al., 2021).

SVI is thereby operationalized as the improvement in the signaler’s estimate of the receiver’s expected utility, predicting actions under the receiver’s posterior and evaluating them with the signaler’s epistemic stance. This construction directly instantiates Adam Smith’s principle of non-Humean ("paternalistic") empathy, contrasting with models in which prediction and evaluation are both based on the receiver’s perspective.

6. Integration in Computational Models and Empirical Results

SVI provides the utility metric for choosing among communicative acts in Rational Speech Act models, where the production probability of $u$ is

$$P_{sig}(u|b_{sig}) \propto \exp\{ \alpha \cdot [\mathrm{SVI}(u|b_{sig}) - c(u)] \}$$

with $\alpha$ a rationality parameter and $c(u)$ a pointing cost. This model supports pragmatic inference, allowing agents to recover task-relevant states from observed communicative actions (Jiang et al., 2021).

In the Wumpus-world experiment, employing a Smithian pointing model—where a guide’s pointing act is interpreted through SVI—demonstrated statistically significant improvement in cumulative reward over both conventional POMDP baselines and alternative observation-based approaches. The benefit was sensitive to the cost of action, vanishing when tasks were trivially easy or too costly for information to alter optimal policy choices (e.g., at extreme move costs, $c = -9$ or $-1$). At moderate difficulty, Smithian agents focused exploration and optimized performance, substantiating the practical impact of SVI in AI and social computation contexts (Jiang et al., 2021).

7. Economic and Theoretical Significance

SVI establishes a rigorous ordering on DMs: $G_M \succeq_{\mathrm{VoI}} G_L$ if and only if $G_M = G_L \otimes T$ for some $T$, formalizing the principle that increases in the value of information necessarily stem from separable utility extensions (i.e., fusion-flexibility). Adding exclusive alternatives (union-flexibility) does not generically yield higher VoI, except under restrictive convexity and refinement conditions.

The SVI framework unifies classical economic analysis of information, contemporary decision theory, and computational models of social and communicative behavior, providing a robust mathematical machinery for quantifying and explaining the roots of informational value in complex decision environments (Lara, 13 Oct 2025, Jiang et al., 2021).