Private Private Information

TL;DR

This work introduces private private information structures where multiple signals about a state are mutually independent yet collectively inform the state. It establishes Blackwell-Pareto optimality as the benchmark for privacy-preserving informativeness and shows that, in the binary-state two-signal case, optimality is exactly characterized by conjugate belief distributions via $F_1(x)=1-F_2^{-1}(1-x)$. The authors connect optimal privacy structures to sets of uniqueness from tomography, yielding a general frontier for $n$ signals—partitions of uniqueness on $[0,1]^n$—and show how this structures privacy-constrained information design, with broad applications to privacy-preserving recommendations, multi-receiver design, causal strength, reduced-form feasibility, and zero-sum persuasion. They derive fundamental information-theoretic bounds on how much informativeness can be allocated across independent signals under privacy constraints, and provide constructive implications for designing dominant privacy-preserving recommendations and feasible reduced forms. Overall, the paper offers a unifying, information-theoretic and tomographic framework for understanding privacy in signaling, with practical impact on recommendation systems, mechanism design, and causal analysis.

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.

Figures (9)

• Figure 1: The pair of signals $(s_1,s_2)$ is uniformly distributed on the unit square, with $\omega=1$ in the black area and $\omega=0$ in the white area. The induced posteriors $p(s_1),p(s_2)$ coincide with the signals.
• Figure 2: The pair of signals $(s_1,s_2)$ is uniformly distributed on the unit square, with $\omega=1$ in the black area and $\omega=0$ in the white area. The induced posteriors $p(s_1),p(s_2)$ are binary, and equally likely to be either 1/4 or $3/4$. The posterior $p(s_2)$ is equal to $3/4$ on the top half on the square and to $1/4$ on the bottom half. Hence, $p(s_2)$ has the same distribution even conditioned on $s_1$, and so the induced second-order and higher-order beliefs are trivial.
• Figure 3: An example of a cumulative distribution function $F$ and its conjugate $\hat{F}$. The shapes under the curves are congruent: the transformation that maps one to the other is reflection around the anti-diagonal. Qualitatively, $F$ corresponds to the belief distribution of a more informative signal, and $\hat{F}$ corresponds to that of a less informative signal (because the former assigns less mass to posterior beliefs near 0.5).
• Figure 4: A private private information structure, where the beliefs $p(s_1)$ and $p(s_2)$ are distributed according to the pair of conjugate distributions $F$ and $\hat{F}$ from Figure \ref{['fig:conjugate']}: the signals are uniform on $[0,1]^2$, and $\omega=1$ if and only if $s_2\geq \hat{F}(1-s_1)$ (black region).
• Figure 5: The conjugate of a discrete distribution $F$ with three atoms at $0.1$, $0.4$, and $0.6$. Each atom becomes an interval of zero measure with the length equal to the atom's weight, and vice versa. Since $F$ does not have atoms at the endpoints of $[0,1]$, the number of intervals of zero measure exceeds the number of atoms by one, so its conjugate $\hat{F}$ has four atoms at $0$, $0.5$, $0.8$, and $1$.

Theorems & Definitions (79)

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