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arxiv: 2112.14356 · v5 · submitted 2021-12-29 · 💰 econ.TH · cs.GT· math.PR

Private Private Information

Kevin He , Fedor Sandomirskiy , Omer Tamuz This is my paper

Pith reviewed 2026-05-24 12:37 UTC · model grok-4.3

classification 💰 econ.TH cs.GTmath.PR
keywords private signalsprivacyinformation designrecommendation systemscausal inferencemechanism design
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0 comments X

The pith

Private private signals inform about a state while carrying no information about each other, and the paper characterizes the versions that maximize informativeness under this constraint.

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

The paper defines private private signals as noisy observations about an unknown state whose joint distribution makes each signal independent of the others. It studies the maximum informativeness achievable under this mutual privacy requirement and gives a characterization of the signals that attain it. A sympathetic reader would care because many settings need to deliver useful information without letting one observation reveal details about another. The work shows the precise tradeoff between signal quality and the privacy condition. It then applies the idea to several economic contexts where such signals arise naturally.

Core claim

Private private signals contain information about the state but not about other signals. The paper characterizes those that are optimal in the sense that they cannot be made more informative without violating privacy.

What carries the argument

Private private signals, defined by mutual independence in their joint distribution while each depends on the unknown state; the characterization of those that maximize informativeness subject to this condition.

If this is right

  • Recommendation systems can issue signals that inform users about a state without the signals informing each other.
  • Information design problems admit solutions that use optimal private private signals to respect privacy while delivering information.
  • Causal inference can employ private private signals as independent observations of the underlying state.
  • Mechanism design can incorporate private private signals to provide or elicit information without cross-signal leakage.

Where Pith is reading between the lines

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

  • The characterization could be used to design experiments that generate independent yet state-dependent observations.
  • Real-world data sets could be checked to see whether observed signals satisfy the private private property.
  • The same construction might extend to settings with repeated interactions or multiple related states.

Load-bearing premise

It is possible to construct signals whose joint distribution satisfies the privacy condition while still depending on the unknown state.

What would settle it

A concrete family of mutually independent signals, each depending on the state, whose informativeness about the state strictly exceeds that of the paper's characterized optimal family.

Figures

Figures reproduced from arXiv: 2112.14356 by Fedor Sandomirskiy, Kevin He, Omer Tamuz.

Figure 1
Figure 1. Figure 1: The pair of signals ps1, s2q is uniformly distributed on the unit square, with ω “ 1 in the black area and ω “ 0 in the white area. The induced posteriors pps1q, pps2q coincide with the signals. This notion captures a strong sense in which s contains more information about ω than sˆ does: in any decision problem, an agent maximizing expected utility performs better when observing s than when observing sˆ. … view at source ↗
Figure 2
Figure 2. Figure 2: The pair of signals ps1, s2q is uniformly distributed on the unit square, with ω “ 1 in the black area and ω “ 0 in the white area. The induced posteriors pps1q, pps2q are binary, and equally likely to be either 1/4 or 3{4. We use the concept of dominance to define Blackwell-Pareto optimality: which private private information structures provide a maximal amount of information to the agents, so that more i… view at source ↗
Figure 3
Figure 3. Figure 3: An example of a cumulative distribution function [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A private private information structure, where th [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The conjugate of a discrete distribution [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimal privacy-preserving recommendation when [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A social welfare-maximizing private private info [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: In the private private information structure asso [PITH_FULL_IMAGE:figures/full_fig_p067_8.png] view at source ↗
read the original abstract

Private signals model noisy information about an unknown state. Although these signals are called "private," they may still carry information about each other. Our paper introduces the concept of private private signals, which contain information about the state but not about other signals. To achieve privacy, signal quality may need to be sacrificed. We study the informativeness of private private signals and characterize those that are optimal in the sense that they cannot be made more informative without violating privacy. We discuss implications for privacy in recommendation systems, information design, causal inference, and mechanism design.

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

2 major / 0 minor

Summary. The paper introduces 'private private signals' as signals that convey information about an unknown state but are unconditionally independent of each other (no information about other signals). It studies the informativeness of such signals and characterizes those that are optimal in the sense that they cannot be made more informative without violating the privacy condition. The paper discusses implications for privacy in recommendation systems, information design, causal inference, and mechanism design.

Significance. If non-trivial examples exist and the characterization is valid, the results could inform settings where signals must remain independent while still depending on a common state, with potential applications in mechanism design and privacy-preserving information structures. However, the significance is limited by the unresolved question of existence for standard (scalar) state spaces.

major comments (2)
  1. [Abstract] Abstract: the central claim of characterizing 'optimal' private private signals that cannot be made more informative without violating privacy presupposes the existence of non-trivial signals satisfying both I(S_i; Θ) > 0 and unconditional independence I(S_i; S_j) = 0. For a scalar state Θ this is typically impossible, since E[S1 S2] = E[E[S1|Θ] E[S2|Θ]] induces correlation unless the conditional expectations are constant (rendering signals uninformative). The characterization therefore rests on either a non-standard privacy definition or an implicit multi-dimensional state; neither is secured in the provided text.
  2. [Abstract] The weakest assumption (that signals can be constructed to satisfy the privacy condition while depending on the state) is load-bearing for all subsequent results on optimality and informativeness. Without an explicit construction, example, or state-space assumption (e.g., independent components of Θ), the optimality characterization cannot be evaluated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed reading and constructive comments. We address the concerns regarding existence of non-trivial private private signals and the need for explicit state-space assumptions and constructions below. These points will be clarified in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of characterizing 'optimal' private private signals that cannot be made more informative without violating privacy presupposes the existence of non-trivial signals satisfying both I(S_i; Θ) > 0 and unconditional independence I(S_i; S_j) = 0. For a scalar state Θ this is typically impossible, since E[S1 S2] = E[E[S1|Θ] E[S2|Θ]] induces correlation unless the conditional expectations are constant (rendering signals uninformative). The characterization therefore rests on either a non-standard privacy definition or an implicit multi-dimensional state; neither is secured in the provided text.

    Authors: We agree that the covariance argument shows non-trivial examples are typically impossible for a scalar state under many common distributions. Our framework is formulated for a general state space Θ, which includes the multi-dimensional case with independent components. In this setting, non-trivial private private signals exist (e.g., each signal can be informative about a distinct component). We will revise the abstract and add an explicit statement of the state-space assumption in Section 2 to make this clear. revision: yes

  2. Referee: [Abstract] The weakest assumption (that signals can be constructed to satisfy the privacy condition while depending on the state) is load-bearing for all subsequent results on optimality and informativeness. Without an explicit construction, example, or state-space assumption (e.g., independent components of Θ), the optimality characterization cannot be evaluated.

    Authors: We acknowledge that an explicit construction and clearer state-space assumption would strengthen the paper and allow readers to evaluate the results more readily. In the revision we will add a simple example (two-dimensional Θ with independent components and binary signals) demonstrating existence, along with a dedicated paragraph in the introduction stating the maintained assumptions on Θ. revision: yes

Circularity Check

0 steps flagged

No circularity; characterization rests on external definitions of privacy and informativeness

full rationale

The provided abstract and description introduce the concept of private private signals and state that the paper characterizes optimal ones. No equations, self-citations, fitted parameters, or derivation steps are visible that reduce a claimed result to its own inputs by construction. The skeptic concern about existence for scalar states is a potential correctness or modeling issue, not a circularity in the derivation chain. The paper appears self-contained against external benchmarks of information theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5609 in / 942 out tokens · 19424 ms · 2026-05-24T12:37:53.590381+00:00 · methodology

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