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Payoffs-Beliefs Duality and the Value of Information

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper makes the value of information a geometric fact: information is worthless exactly when posteriors stay in the set where prior-optimal actions remain optimal, and worth a measurable amount otherwise.

desk verdict A genuinely useful convex-analysis treatment of the value of information, with a fixable proof gap in Theorem 3 and a wrong running example; worth refereeing once repaired. read the letter →

arxiv 1908.01633 v2 pith:WSNFUHTH submitted 2019-08-05 math.OC

Michel de Lara (CERMICS) , Olivier Gossner (X) This is my paper · ORCID
classification math.OC MSC 46N1091B06
keywords valueofinformationconvexanalysispayoffs-beliefsdualitysupportfunctionconfidencesetmarginalstructuresdecisiontheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper gives a convex-analysis account of the value of information in decision problems. It treats each action as a payoff vector indexed by states of nature and each belief as a probability vector, pairing them by the scalar product that defines expected payoff; the value function is then the support function of the set of available actions. The central result characterizes zero information value: information is worthless exactly when, almost surely, the posterior belief falls in the confidence set of beliefs at which every action optimal at the prior remains optimal. When posteriors can leave that set with positive probability, information has positive value, and the paper supplies global upper and lower bounds on that value in terms of how far posteriors move. At the margin, near no information, the agent's local behavior at the prior, whether confident, undecided, or flexible, determines whether the marginal value of information is zero, positive and finite, or infinite.

What carries the argument

The load-bearing object is the support function $\sigma_A(s)=\sup_{a\in A}\langle s,a\rangle$ of the action set $A\subset\mathbb{R}^K$, restricted to beliefs; this is the value function. Convex analysis supplies the dictionary: the subdifferential of the value function at a belief is the exposed face of optimal actions, and the normal cone at an action collects the beliefs that reveal that action as optimal. From these, the paper defines the confidence set of a prior, the posteriors at which every prior-optimal action stays optimal, and the indifference kernel, the directions that break none of the prior's ties. The confidence set carries Proposition 2 and Theorem 3; the indifference kernel and the curvature of the boundary of the action set, via the spherical image and Weingarten maps, carry the undecided and flexible bounds of Theorems 5 and 7.

What would settle it

Compute the value of information in the paper's four-action example (Table 1) at prior $p=1/2$ with a signal that sends the posterior to $p=0$ or $p=1$ equiprobably: Proposition 2 predicts $\mathrm{VoI}=\frac12 v(0)+\frac12 v(1)-v(\tfrac12)=0.75>0$ because both posteriors lie outside the confidence set $[1/3,4/5]$. If such a direct calculation ever returned zero, or if a numerical search over compact convex polytope action sets produced a posterior distribution that escapes the confidence set with positive probability yet has zero expected gain, the central characterization would be refuted.

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Extended reading notes

Core claim

The discovery is that the economic question of how much a piece of information is worth is governed by the geometry of the set of available payoff vectors near the prior. Writing $A$ for the compact convex set of actions and $v_A(p)=\max_{a\in A}\langle p,a\rangle$ for the value function, the set of optimal actions at belief $p$ is the subdifferential $\partial v_A(p)$, and the set of beliefs at which an action $a$ is optimal is the normal cone $N_A(a)$ intersected with the simplex. Against this backdrop, Proposition 2 states that $\mathrm{VoI}_A(q)=\mathbb{E}[v_A(q)]-v_A(\bar p)$ vanishes if and only if the random posterior $q$ lies almost surely in the confidence set $\Delta^c_A(\bar p)=\bigcap_{a\in A^\star(\bar p)}\Delta^\star_A(a)$. Theorems 3, 5, and 7 then bound the value of any information structure by, respectively, the expected distance from the posterior to the confidence set, the expected seminorm distance from prior to posterior when the agent is undecided, and the expected squared distance when the agent is flexible. These local-to-global estimates turn the value of information into a quantity that can be read off from the shape of the action set at the prior alone.

Load-bearing premise

The whole argument depends on representing any decision problem as a compact convex set of payoff vectors and on expected-utility maximization $\max_{a\in A}\langle p,a\rangle$; if preferences are not expected utility, randomization is forbidden, or actions cannot be reduced to state-indexed payoffs, the duality and every theorem built on it collapse.

Editorial extensions

If this is right

  • If a signal never moves the posterior outside the confidence set of the prior, it has zero value, no matter how informative it is by other criteria.
  • If a signal moves the posterior outside that set with positive probability, the value is positive, with upper and lower bounds in terms of expected distance from the posterior to the confidence set and of the probability of leaving an epsilon-neighborhood of it.
  • At priors where several actions are optimal, a small signal that breaks a tie has first-order value: the value grows like the expected distance between prior and posterior, so the marginal value can be infinite for signals whose belief displacement is of order the square root of the information parameter.
  • At priors where the optimal action varies smoothly with belief, the value of information grows like the expected squared distance from prior to posterior, making small information second-order and giving diffusion-type signals a positive finite marginal value.
  • Near no information, the local regime of the value function at the prior, confident, undecided, or flexible, together with the speed at which posteriors spread, decides among three marginal values: zero, a positive finite number, or infinite.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The three regimes can be read as a local Taylor expansion of the value function, flat, kinked, or quadratically curved; this suggests that for any information structure, the asymptotic value is fixed by the lowest-order nonzero term in that expansion along the signal's belief displacements.
  • Because the confidence set is computed only from prior-optimal actions, the bounds offer a robustness tool: an analyst who knows only the local face of the action set at the prior can bound the worst-case value of any information structure without knowing the distribution of signals.
  • The same payoff-beliefs duality may carry over to infinite state spaces through support functions on dual pairs of locally convex spaces, but the normal-cone and curvature arguments would need functional-analytic reworking; testing the quadratic bound in a Gaussian belief model would be a natural first step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies the value of information in finite-state decision problems through convex duality. The decision problem is encoded by a compact convex set A of state-indexed payoff vectors; the value function v_A is the support function of A, optimal actions are exposed faces, and beliefs supporting an action form normal cones. An information structure is modeled as a random posterior q with expectation equal to the prior. The paper gives a necessary and sufficient condition for zero value of information (Proposition 2), global upper and lower bounds based on the confidence set, the indifference kernel, and the smoothness of the value function (Theorems 3, 5, and 7), and applies these bounds to the marginal value of information near no information in Section 5, including Brownian, Poisson, and binary-signal examples. The main technical tool is standard convex analysis, with proofs collected in an appendix.

Significance. The framework is elegant and potentially useful: representing the value function as a support function gives a transparent geometric interpretation of when information matters, and the paper separates conditions on the decision problem from conditions on the information structure in a way that the Radner-Stiglitz literature often does not. Proposition 2 is a clean characterization, and the asymptotic classification in Section 5 (zero, finite, or infinite marginal value) is a nice synthesis of existing results, including a comparison with De Lara and Gilotte (2007). The paper is also honest about the scope of its model: it assumes expected utility with randomization, so actions are identified with payoff vectors in a convex compact set. However, two substantive issues need attention before the paper can be accepted: the proof of the upper bound in Theorem 3 uses an invalid constant, and Section 3.2 misidentifies the undecided priors in Example 1. These are repairable, but they affect the reliability of the main results as written.

major comments (2)
  1. [Appendix A.3.1, Eq. (49), Theorem 3] The proof of the upper bound in Theorem 3 is incorrect as written. The proof fixes an arbitrary a in A and asserts φ_a(q) ≤ (sup_{a' in A} ||a−a'||) d(q, Δ^c_A(pbar)), using φ_a(p)=0 for p in the confidence set via (46b). But (46b) is only established for a in F_A(pbar)=A^*(pbar), not for arbitrary a in A. Consequently the derived constant C_A = inf_{a in A} sup_{a' in A} ||a−a'|| is not valid. This is not merely a cosmetic issue: in Example 1, with prior pbar=(1/2,1/2) and binary full information, VoI_A(q)=3/4, while the claimed constant gives C_A·E[d(q,Δ^c_A(pbar))]=√122/15≈0.736, so the displayed upper bound fails. The theorem is salvageable by taking C_A to be the diameter of A or by restricting the infimum to a in A^*(pbar), but the corrected proof is needed because this bound is used in Proposition 11 and in the Section 5 examples.
  2. [Section 3.2 and the remark after Theorem 5] The manuscript states that in Example 1 the agent is undecided at pbar=1/2 and pbar=3/4, with several optimal actions and a nondifferentiable value function. This is incorrect: at both of these beliefs the unique optimal action is (2,2) and the value function is differentiable. The actual kinks, where two pieces of the value function meet, are at p=1/3 (between (3,0) and (2,2)) and p=4/5 (between (2,2) and (0,5/2)). This misidentification appears twice, in Section 3.2 and in the subsequent remark on Theorem 5, and it undermines the illustration of the undecided case. The text should either use p=1/3 and p=4/5 for Example 1 or replace Example 1 with a genuinely indifferent example.
minor comments (3)
  1. [Example 14, Section 5.2] There is a missing closing parenthesis in the displayed expression for E[d(q_θ, Δ^c_A(pbar))]; it should read E[d(q_θ, Δ^c_A(pbar))]=0 for θ small enough.
  2. [Example 12, Section 5.2] The text says 'a standard Browian process'; this should be 'Brownian process'.
  3. [Proof of Proposition 2, Appendix A.3.1] In the density argument near the end of the proof, the notation for the closure of the set is hard to follow as rendered; consider rewriting this step with an explicit closure operation to improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the value-of-information bounds are proved from support-function convex analysis; the only self-citation is comparative, not load-bearing.

full rationale

Score 0: no circular reasoning detected. The derivation chain is self-contained: the value function is defined in (2) as expected payoff maximization, the value of information in (6) is the expected gain over the prior, and Theorems 3, 5, and 7 are proved in Appendix A.3 from the support-function identity (24)-(25), the subgradient inequality (27), and standard external convex-analysis facts from Rockafellar, Hiriart-Urruty-Lemarechal, and Schneider. Proposition 2's equivalence between zero value and posterior beliefs lying in the confidence set follows from the nonnegativity of phi_a in (46a)-(46b), not from the definition of the confidence set: the confidence set in (7) is defined through optimal actions, while VoI=0 is a condition on expected support values, so the equivalence is a convexity theorem rather than a tautology. The only self-citation, De Lara and Gilotte (2007), appears in Section 5.1 as a comparison: Proposition 11 is proved from Theorems 3 and 7 and is said to imply the earlier paper's main condition, so the earlier paper is not used as a premise. A separate issue outside the scope of circularity: the proof of Theorem 3's upper bound invokes (46b) to justify (49), but (46b) is stated only for a in FA(pbar), while the printed constant takes an infimum over all a in A; as the skeptic's Example 1 computation shows, the printed constant can fail. This is a correctness/proof-gap concern, not circularity, because the bound is not obtained by assuming its own conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters or invented physical entities are introduced. The confidence set, indifference kernel, and seminorm are mathematical definitions, not entities with external commitments. The load-bearing assumptions are the expected-utility payoff-vector representation, the posterior representation of information, and the smoothness assumptions in the flexible case.

assumptions (6)
  • domain assumption The agent's payoff under any belief p is max_{a in A} <p,a> for a compact convex action set A, the closed convex hull of payoffs (equations (1)-(2)).
    This is the expected utility plus randomization assumption that the whole paper is built on. It is stated in Section 2 and is the basis for the support-function representation.
  • domain assumption Any information structure is represented by a random posterior q with E[q]=p-bar (equation (5)).
    This is the Blackwell and posterior martingale representation of information; used to define VoI in equation (6) and throughout.
  • domain assumption The prior belief p-bar has full support (Section 2).
    Full support is used in the local diffeomorphism arguments of Proposition 6 and Theorem 7, via the map nu in equation (57).
  • domain assumption For the flexible case, the boundary of A is a C2 submanifold with positive curvature at the optimal action (Proposition 6).
    This regularity assumption drives Theorem 7's quadratic bounds and is not needed for the finite-choice results.
  • standard math Background finite-dimensional convex analysis: support functions are convex, subdifferentials equal exposed faces, normal cones, Weingarten map properties from Schneider (2014) and Hiriart-Urruty-Lemarechal.
    Used throughout the proofs in the appendix.
  • standard math For the Brownian and Poisson examples, the posterior processes follow the stated diffusions or Poisson updating (Examples 12 and 13).
    These are standard results cited to Bolton and Harris (1999) and Faingold and Sannikov (2011).

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Pith. "Pith review of Payoffs-Beliefs Duality and the Value of Information." pith.science (2026). https://pith.science/paper/WSNFUHTH

@misc{pith2026190801633,
  author       = {Pith},
  title        = {Pith review of: Payoffs-Beliefs Duality and the Value of Information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WSNFUHTH}},
  note         = {Machine review of arXiv:1908.01633}
}
read the original 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.

Figures

Figures reproduced from arXiv: 1908.01633 by the authors.

Figure 1
Figure 1. The set A of actions on the left, and the value function vA on the right. Each of the four arrows on the left represents an action a such that p = 4/5 belongs to the set ∆? A(a) of beliefs revealed by action a. On the right side, these four actions (each attached to an arrow) can be seen as four elements of the subdifferential of the value function vA at p = 4/5. The set ∆? A(3, 0) = [0, 1/3] can be visualized both … view at source ↗
Figure 2
Figure 2. The action set A on the left and the corresponding value function v = vA in (21) for the insurance example on the right. Parameter values are α = 0.08, f = 10, $ = 1000, R = 10. 5 The marginal value of information The question of the marginal value of information is studied in [32]. They provide joint conditions on a parameterized family of information structures together with a decision problem such that, when the … view at source ↗

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    We have that p∈ ∆c A(¯p) ⇐⇒ FA(¯p)⊂FA(p) (44a) ⇐⇒ σA(p)−σA(¯p)−⟨p− ¯p,a⟩ = 0, ∀a∈FA(¯p) (44b) ⇐⇒ σA(p)−σA(¯p) +σ−A⋆(p)(p− ¯p) = 0

    Let p∈ ∆. We have that p∈ ∆c A(¯p) ⇐⇒ FA(¯p)⊂FA(p) (44a) ⇐⇒ σA(p)−σA(¯p)−⟨p− ¯p,a⟩ = 0, ∀a∈FA(¯p) (44b) ⇐⇒ σA(p)−σA(¯p) +σ−A⋆(p)(p− ¯p) = 0. (44c) 19

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    The indifference kernel Σi A(¯p) of (11) is the vector subspace Σi A(¯p) = [FA(¯p)−FA(¯p)]⊥ = [A⋆(¯p)−A⋆(¯p)]⊥ = ⋂ a∈FA(¯p) NFA(¯p)(a). Proof

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    (a) Let p∈ ∆

    We prove the three equivalences in (44). (a) Let p∈ ∆. Using the property (31) that exposed face FA and normal cone NA are conjugate, we obtain: p∈ ∆c A(¯p) ⇐⇒p∈ ⋂ a∈FA(p) NA(a) by (43) ⇐⇒a∈FA(p), ∀a∈FA(¯p) by (31) ⇐⇒ FA(¯p)⊂FA(p). (b) Let p∈ ∆. We have that σA(p)−σA(¯p)−⟨p− ¯...

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    Then, use the definition of NFA(¯p)(a) in (29)

    Express (11) using (28). Then, use the definition of NFA(¯p)(a) in (29). This ends the proof. 2 A.3.1 Valuable information Proof.[Proof of Proposition 2] Let a∈FA(¯p) and q be an information structure as in (5). We have that V oIA(q) = 0 ⇐⇒ E [σA(q)−σA(¯p)] = 0 by (39) ⇐⇒ E [σA...

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