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Econ-305 · economics · Economics

General-cost strategic disclosure on continuous type spaces

LLM status Open unverified

Statement

GT-052 | Information design | R: nonlinear-policy characterization agenda.

Definitions and mathematical statement

Fix $\omega\sim\mathcal N(0,\Sigma_0)$ on $\mathbb R^d$, $\Sigma_0$ positive definite. A receiver observing signal $s$ minimizes the conditional expected cost $\|Q\omega+Ru\|_2^2$, where $R^{\mathsf T}R$ is invertible. Thus $u(s)=-\Lambda\mathbb E[\omega\mid s]$, with $\Lambda=(R^{\mathsf T}R)^{-1}R^{\mathsf T}Q$. Let $\mathcal K$ be all Borel probability kernels from $\mathbb R^d$ to a standard Borel signal space. Given a continuous convex sender cost $c(\omega,u)$ of at most quadratic growth and a specified lower-bounded measurable signaling-cost functional $C:\mathcal K\to\mathbb R\cup\{+\infty\}$, define \[ J^*=\inf_{K\in\mathcal K}\left\{ \mathbb E_K c(\omega,-\Lambda\mathbb E_K[\omega\mid s])+C(K)\right\}. \] Restrict to kernels for which the expectations exist and the objective is finite, and assume at least one exists. Characterize $J^*$ and near-optimal kernels; determine conditions under which an optimizer exists and under which restricting to linear Gaussian signals $s=L\omega+\xi$, with independent Gaussian $\xi$, preserves the infimum.

Short English statement

Identify when general continuous-state disclosure problems can be solved by linear signals and how to characterize optimal nonlinear disclosure otherwise.

Variants and scope boundaries

No universal attainment claim is plausible for arbitrary $C$. The class and topology used in an attainment theorem must be stated. Cost on the signaling rule differs from a cost on individual messages. Online learning of unknown receiver types is a separate target.

Source relationship and status

The source's Section IV restricts general convex-cost analysis to linear noiseless strategies and leaves unrestricted optimal strategies to future work. Its entropy-cost Gaussian optimality theorem is already solved. The functional $C$ and the near-optimal formulation are editorial extensions; the source does not pose a universal arbitrary-cost existence conjecture.

References

[S1] Raj Kiriti Velicheti, Melih Bastopcu, S. Rasoul Etesami, and Tamer Başar (2024). Learning How to Strategically Disclose Information. arXiv:2403.08741v1. Verified locator: Section II-A and Lemma 2; Section IV, footnote on linear strategies and future investigation of optimal strategies. https://arxiv.org/html/2403.08741v1

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-305-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-305

General-cost strategic disclosure on continuous type spaces: research attempt

Target and scope

The state has law $\mu_0=N(0,\Sigma_0)$ on ${\mathbb{R}}^d$, with $\Sigma_0$ positive definite. After a signal $S$, the receiver chooses $u=-\Lambda{\mathbb{E}}[\omega\mid S]$. The target minimizes the expectation of a continuous convex cost $c(\omega,u)$ with at most quadratic growth, plus a lower-bounded measurable cost $C(K)$ on the entire signal kernel. The aim is a characterization, attainment conditions, and conditions for Gaussian linear signals to lose no value. These are three different questions: arbitrary measurable $C$ can encode discontinuities or distinguish payoff-equivalent signals.

The inspected primary paper treats quadratic costs through posterior covariance and discusses a restricted linear analysis for general convex costs. Its covariance and entropy results are established benchmarks. The argument below gives a direct compactness treatment for a specified subclass of the editorial target; it does not claim that those benchmarks are new.

Posterior means as the decision-relevant state

For a signal kernel put $M={\mathbb{E}}[\omega\mid S]$ and let $\pi$ be the joint law of $(\omega,M)$. It satisfies \[ \pi_\omega=\mu_0,\qquad \int(\omega-m)h(m)\,d\pi(\omega,m)=0 \quad(h\text{ bounded Borel}). \] Call this set of probability measures $\Pi$. The second identity says ${\mathbb{E}}[\omega\mid M]=M$. Conversely, disintegrate any $\pi\in\Pi$ with respect to its first marginal and send $S=M$. The resulting Borel kernel implements exactly $\pi$ and the required receiver response. Standard Borel spaces make this disintegration legitimate. Thus posterior means characterize the feasible payoff-relevant joint laws, without assuming that $M$ is Gaussian.

An exact reduced problem is available if \[ C(K)=\mathcal C(\mathcal L_K(\omega,{\mathbb{E}}[\omega\mid S])) \] for a functional $\mathcal C$. In that case \[ J^*=\inf_{\pi\in\Pi}\left\{\int c(\omega,-\Lambda m)\,d\pi+\mathcal C(\pi)\right\}. \] This assumption is substantive. Sending $(M,Z)$ with independent random $Z$ preserves the joint law above but can change an arbitrary signaling cost. Therefore replacing arbitrary $C$ by a posterior-mean cost without such invariance would be an incorrect reduction.

Proposition: The set $\Pi$ is weakly compact. If $\mathcal C$ is lower semicontinuous for weak convergence, is bounded below, and is finite at some feasible coupling, then the reduced problem attains its infimum.
Proof: Conditional Jensen gives ${\mathbb{E}}_\pi\|M\|^4\leq{\mathbb{E}}_{\mu_0}\|\omega\|^4=:B<\infty$ uniformly over $\Pi$. Hence the joint laws are tight and their squared norms are uniformly integrable. Let $\pi_n$ converge weakly along a subsequence. The first marginal remains $\mu_0$. For bounded continuous $h$, truncate the factor $\omega-m$ in the martingale identity. Weak convergence passes the bounded continuous truncations to the limit; the uniform moment bound makes the truncation error vanish. The limiting signed measure $A\mapsto\int 1_A(m)(\omega-m)d\pi$ is zero because its integrals against all bounded continuous functions are zero. This proves the conditional-mean constraint in the limit. Therefore $\Pi$ is closed and tight, hence compact.

The assumed quadratic growth and the uniform fourth-moment bound imply that the expectation of $c(\omega,-\Lambda m)$ is continuous along these weakly convergent couplings: truncate outside a large ball, use weak convergence inside, and bound the tails uniformly. Adding the lower semicontinuous functional preserves lower semicontinuity. Its sum is bounded below because the cost's absolute expectation is uniformly bounded, and it is finite somewhere by hypothesis. A minimizing sequence has a convergent subsequence attaining the infimum. ∎

This proof supplies an implementable kernel from the optimizer by disintegration, but not a finite-dimensional algorithm for finding it. Finite partitions of the posterior-mean signal do approximate the expected continuous cost: conditional means on refining finite partitions converge in $L^2$ to $M$, and the Gaussian fourth-moment bound controls the cost tails. This does not imply convergence of an arbitrary lower semicontinuous $\mathcal C$ from above. A finite-signal approximation theorem needs continuity or an explicit approximation property for that functional.

Precisely where covariance reduction works

Let \[ c(\omega,u)=\|A\omega+Bu\|^2, \quad D=B\Lambda,\quad T={\mathbb{E}}[MM^{\mathsf T}]. \] Because ${\mathbb{E}}[\omega M^{\mathsf T}]=T$, \[ {\mathbb{E}} c=\operatorname{tr}(A^{\mathsf T}A\Sigma_0) +\operatorname{tr}\left((D^{\mathsf T}D-D^{\mathsf T}A-A^{\mathsf T}D)T\right). \] Also $0\preceq T\preceq\Sigma_0$, by the conditional-variance decomposition. Every such $T$ is realized by a linear Gaussian signal, allowing degenerate independent Gaussian noise and omission of unobserved coordinates. To see this, whiten $\omega$ and diagonalize $\Sigma_0^{-1/2}T\Sigma_0^{-1/2}=U\operatorname{diag}(t_i)U^{\mathsf T}$. Observe each whitened coordinate with Gaussian noise variance $(1-t_i)/t_i$ when $0<t_i<1$, observe it exactly when $t_i=1$, and omit it when $t_i=0$. Its posterior-mean variance is $t_i$. Rotating back gives $T$.

Consequently a signaling cost of the form $\mathcal C(\pi)=g(T)$ gives the exact finite-dimensional infimum over the matrix interval, for any specified lower-bounded $g$. If $g$ is lower semicontinuous and finite somewhere, the minimum exists. Convexity of $g$ makes this a convex optimization problem, since the expected quadratic cost is affine in $T$. General nonquadratic convex $c$ does not obey the displayed trace identity and is not reduced by matching covariance alone.

A strict Gaussian gap allowed by the original cost class

The breadth of $C$ prevents universal Gaussian optimality even when the sender's running cost is convex. Set $d=1$, $\omega\sim N(0,1)$, and $c=0$. Let $a=\sqrt{2/\pi}$, and let $\pi_*$ be the law of $(\omega,a\operatorname{sign}\omega)$ induced by revealing the sign. Define the posterior-mean cost \[ \mathcal C(\pi)=\begin{cases}0&\pi=\pi_*,\\1&\pi\ne\pi_* .\end{cases} \] This function is lower semicontinuous: its sublevel sets below one are either empty or the closed singleton $\{\pi_*\}$. Thus the preceding attainment result applies. The sign signal achieves zero. A linear Gaussian signal has a Gaussian posterior mean, either degenerate or nondegenerate; it cannot have the two-point distribution $\{-a,a\}$ with both masses positive. Every linear Gaussian signal therefore costs one. This is a deliberately simple counterexample about permitted signaling costs, not a claim about economically usual entropy costs.

Checks and unresolved part

The no-information coupling $M=0$ and full-information coupling $M=\omega$ satisfy the constraints and give the covariance endpoints. The proof uses the fourth moment of the Gaussian prior, not compact support falsely attributed to it. Signal relabeling and extra independent messages are harmless only under the explicit cost invariance. The general problem with arbitrary kernel-dependent measurable $C$, and a substantive classification of nonquadratic running costs admitting Gaussian optimality, remain unresolved. No complete solution of the catalogue target is claimed.

Completion Estimate

61%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt characterizes posterior-mean invariant disclosure through feasible couplings, proves attainment under lower semicontinuity, and establishes Gaussian sufficiency for covariance costs plus a strict counterexample. Arbitrary kernel-dependent signaling costs and a substantive classification of nonquadratic convex costs preserving Gaussian optimality remain unresolved.

Source checked

Raj Kiriti Velicheti, Melih Bastopcu, S. Rasoul Etesami, and Tamer Başar, Learning How to Strategically Disclose Information, arXiv:2403.08741v1 (2024), Sections II-A, II-B and IV. The inspected text supports the covariance and entropy benchmarks and the restricted scope of its general-cost analysis. https://arxiv.org/html/2403.08741v1.

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