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Majorized Bayesian Persuasion and Fair Selection

Siddhartha Banerjee, Kamesh Munagala, Yiheng Shen, Kangning Wang

Abstract

We address the fundamental problem of selection under uncertainty by modeling it from the perspective of Bayesian persuasion. In our model, a decision maker with imperfect information always selects the option with the highest expected value. We seek to achieve fairness among the options by revealing additional information to the decision maker and hence influencing its subsequent selection. To measure fairness, we adopt the notion of majorization, aiming at simultaneously approximately maximizing all symmetric, monotone, concave functions over the utilities of the options. As our main result, we design a novel information revelation policy that achieves a logarithmic-approximation to majorization in polynomial time. On the other hand, no policy, regardless of its running time, can achieve a constant-approximation to majorization. Our work is the first non-trivial majorization result in the Bayesian persuasion literature with multi-dimensional information sets.

Majorized Bayesian Persuasion and Fair Selection

Abstract

We address the fundamental problem of selection under uncertainty by modeling it from the perspective of Bayesian persuasion. In our model, a decision maker with imperfect information always selects the option with the highest expected value. We seek to achieve fairness among the options by revealing additional information to the decision maker and hence influencing its subsequent selection. To measure fairness, we adopt the notion of majorization, aiming at simultaneously approximately maximizing all symmetric, monotone, concave functions over the utilities of the options. As our main result, we design a novel information revelation policy that achieves a logarithmic-approximation to majorization in polynomial time. On the other hand, no policy, regardless of its running time, can achieve a constant-approximation to majorization. Our work is the first non-trivial majorization result in the Bayesian persuasion literature with multi-dimensional information sets.

Paper Structure

This paper contains 37 sections, 11 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fairness via Information Revelation.
  3. Summary of Results.
  4. Technical Highlight: Reduction to Majorized Network Flow.
  5. Related Work
  6. Bayesian Persuasion.
  7. Majorization.
  8. Fair Selection.
  9. Selection via Persuasion.
  10. Our Model and Results
  11. Signaling Model
  12. Independent Signaling Schemes.
  13. Signaling Policies.
  14. Fairness via Approximate Majorization
  15. Our Results
  16. ...and 22 more sections

Key Result

Proposition 2.3

The signaling policy $\Omega$ is $\alpha$-majorized if and only if for every symmetric and concave functionSuch a function will also be monotonically non-decreasing.$f \colon \mathbb{R}_{\geq 0}^n \to \mathbb{R}_{\geq 0}$ (called a welfare function or fairness function) and any other signaling polic

Figures (3)

  • Figure 1: The network flow instance for an example with $n = 2$ agents, with $D_1 = \mathtt{Bernoulli}(0.3)$ and $D_2 = \mathtt{Bernoulli}(0.6)$. The number in blue next to each directed edge indicates the capacity of the edge.
  • Figure 2: Illustration of the network flow instance. The text in blue denotes the capacity of each edge, while the text in black denotes the feasible flow value on each edge.
  • Figure 3: Illustration of the reduction process for two instances. In both instances, we have $s_i^1 = 2/7$, $s_i^2 = 2/7$, $s_i^3 = 3/7$; with $\mu_i(\sigma_i^1) = 2$ and $\mu_i(\sigma_i^2) = 4$. In the blue (left) instance, $\mu_i(\sigma_i^3) = 5$, so $\alpha_i = 1/3$. We absorb all the probability mass from $\sigma_i^3$ (i.e., $3/7$) together with $3/14$ probability of $\sigma_i^1$ into $\sigma_i^2$. In the red (right) instance, $\mu_i(\sigma_i^3) = 6$ thus $\alpha_i = 1/2$. We absorb all the probability mass of $\sigma_i^1$ (i.e., $2/7$) with the same amount of probability mass in $\sigma_i^3$ into $\sigma_i^2$.

Theorems & Definitions (25)

  • Definition 2.1: expected utility
  • Definition 2.2: $\alpha$-majorization
  • Proposition 2.3: adapted from goel2006simultaneous
  • Example 2.4
  • Theorem 2.5: Proved in \ref{['sec:network_flow']}
  • Theorem 2.6: Proved in \ref{['sec:full_maj']}
  • Example 2.7
  • Theorem 2.8: Proved in \ref{['sec:approx']}
  • Remark
  • Theorem 2.9: Proved in \ref{['sec:main_lb']}
  • ...and 15 more