Probability of action lower bounded when signals are separate
Persuasion by Parts
Computer Science and Game Theory
Summary
This paper looks at how much a decision maker can be convinced to act when information comes from multiple independent sources, each seeing only part of the full picture. The traditional way assumes signals can depend on the whole situation, but here the authors study more limited 'separate' signals from independent parts. They find a mathematical relationship that bounds how well persuasion by these separate signals can do compared to joint signals. The results hold whether or not the designer knows how the decision maker decides.
What this means in practice
- •For marketing teams: Design signal structures from independent data sources to better predict customer actions under limited joint data access.
- •For data integration engineers: Assess limits on combining partially independent information signals for informed decision systems when joint conditioning is infeasible.
A theory result. No direct application yet.
Authors
Shuyuan Jia, Kangning Wang, Renzhe Xu
Abstract
Classical Bayesian persuasion assumes that an information designer can condition signals on the entire state. We propose persuasion by parts. In this model, the receiver's payoff from acting is $X = \sum_{i=1}^n X_i$ for independent coordinates $X_1,\ldots,X_n$ observed by separate sources. The designer commits in advance to a product experiment, in which each source $i$ can condition its signal only on $X_i$. Given the signals, the receiver computes her posterior mean $y$ of $X$ and acts with probability $g(y)$, where $g\colon \mathbb{R} \to [0,1]$ is a nondecreasing function. The designer aims to maximize the probability of action. Let $V_{\mathrm{prod}}(g)$ and $V_{\mathrm{joint}}(g)$ be the supremum probabilities of action under persuasion by parts and under joint persuasion, respectively. When the designer knows $g$, we show $$ V_{\mathrm{prod}}(g) \;\ge\; c\,V_{\mathrm{joint}}(g)^2 \qquad\text{for every nondecreasing } g\colon \mathbb{R} \to [0,1], $$ where $c > 0$ is an absolute constant independent of $n$ and the prior; the exponent $2$ is tight. When $g$ is unknown, we construct a product experiment $Π$ from the prior alone, whose probability of action $V_Π(g)$ satisfies $$ V_Π(g) \;\ge\; c_\varepsilon\,V_{\mathrm{joint}}(g)^{2 + \varepsilon} \qquad\text{for every } \varepsilon > 0 \text{ and every nondecreasing } g\colon \mathbb{R} \to [0,1], $$ where $c_\varepsilon > 0$ depends only on $\varepsilon$.