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arxiv: 2202.10883 · v6 · submitted 2022-02-22 · 💰 econ.TH

Information Design in Smooth Games

Alex Smolin , Takuro Yamashita This is my paper

Pith reviewed 2026-05-24 12:11 UTC · model grok-4.3

classification 💰 econ.TH
keywords information designsmooth gamesprincipal-agent contractinglinear-quadratic gamestargeted disclosurelinear disclosurecommon valuesinterdependent values
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The pith

In smooth games an information structure is optimal exactly when its induced equilibrium can be implemented by a principal-agent contracting problem.

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

The paper proves that in games where players select from a continuum of actions with continuously differentiable payoffs, the best information structure is the one whose equilibrium outcome can also arise from a principal designing contracts for agents. This equivalence turns the problem of choosing signals into a more tractable contracting exercise. The authors then apply the result to symmetric linear-quadratic games and obtain explicit solutions: targeted disclosure is optimal for any prior when values are common, while linear disclosure is uniquely optimal when values are interdependent and normally distributed. The same logic yields concrete prescriptions for venture capital, polarization, and price competition.

Core claim

An information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, the paper characterizes optimal information structures in symmetric linear-quadratic games: targeted disclosure is robustly optimal across all priors when values are common, and linear disclosure is uniquely optimal when values are interdependent and normally distributed.

What carries the argument

The equivalence between an information structure being optimal and the equilibrium it induces being implementable in a principal-agent contracting problem.

If this is right

  • In symmetric linear-quadratic games with common values, targeted disclosure is robustly optimal across all priors.
  • With interdependent and normally distributed values, linear disclosure is uniquely optimal.
  • The same characterizations apply directly to venture capital financing, Bayesian polarization, and price competition.
  • Optimal information design reduces to finding a contractible action profile that matches the desired equilibrium.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The contracting equivalence could be used as an approximation tool in games with nearly continuous action spaces even if actions are formally discrete.
  • Similar reductions might simplify information design in dynamic or repeated settings where direct optimization over signals is intractable.
  • Laboratory experiments could test whether subjects respond to information policies that match contracting patterns more readily than other policies.

Load-bearing premise

The result relies on players choosing from a continuum of actions with continuously differentiable payoffs.

What would settle it

A counterexample in a game with discrete actions or non-differentiable payoffs where an optimal information structure cannot be implemented via contracting would falsify the characterization.

Figures

Figures reproduced from arXiv: 2202.10883 by Alex Smolin, Takuro Yamashita.

Figure 1
Figure 1. Figure 1: Price responsiveness to own demand shock (left) and opponent’s demand shock [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Optimal price volatility (left) and price correlation (right). Calculated at [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
read the original abstract

We study information design in games where players choose from a continuum of actions and have continuously differentiable payoffs. We show that an information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, we characterize optimal information structures in symmetric linear-quadratic games. With common values, targeted disclosure is robustly optimal across all priors. With interdependent and normally distributed values, linear disclosure is uniquely optimal. We illustrate our findings with applications in venture capital, Bayesian polarization, and price competition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies information design in smooth games (continuum of actions, continuously differentiable payoffs). It establishes that an information structure is optimal precisely when the equilibrium it induces can also be implemented as the outcome of a principal-agent contracting problem. Building on this equivalence, it derives characterizations of optimal information structures in symmetric linear-quadratic games: targeted disclosure is robustly optimal under common values, while linear disclosure is uniquely optimal under interdependent normally distributed values. The results are illustrated with applications to venture capital, Bayesian polarization, and price competition.

Significance. If the contracting equivalence holds, the paper supplies a practical bridge between information design and mechanism-design techniques, allowing researchers to import contracting tools to identify optimal signals in smooth environments. The explicit characterizations for linear-quadratic games are directly usable in applications and generate testable predictions about disclosure policies. The scoping to C1 payoffs and continuum actions is stated upfront, so the result is internally consistent within its domain.

minor comments (3)
  1. [Abstract / Introduction] The abstract and introduction should clarify whether the contracting equivalence is if-and-only-if or one-directional; the current phrasing leaves the logical direction slightly ambiguous for readers who have not yet reached the formal statement.
  2. [Section 3] Notation for the signal space and the induced posterior beliefs could be standardized earlier; several applications reuse similar symbols without an explicit cross-reference table.
  3. [Section 4.1] The robustness claim for targeted disclosure would benefit from a short remark on whether the result survives small perturbations to the prior or to the payoff functions outside the exact LQ class.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The summary accurately reflects the paper's contributions and the recommendation for minor revision is noted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: characterization result is self-contained

full rationale

The paper's central result is a characterization theorem: an information structure is optimal precisely when the induced equilibrium is implementable via principal-agent contracting, explicitly scoped to continuum-action games with C1 payoffs. This equivalence is derived as a mathematical statement within the stated domain and does not reduce any claimed prediction or optimality criterion to a fitted parameter, self-definition, or load-bearing self-citation. The abstract and setting description contain no ansatz smuggling, renaming of known results, or uniqueness claims imported from prior author work. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such items are unknown.

pith-pipeline@v0.9.0 · 5597 in / 939 out tokens · 17327 ms · 2026-05-24T12:11:38.809448+00:00 · methodology

discussion (0)

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Reference graph

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