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Fair Allocation in Dynamic Mechanism Design

Alireza Fallah, Michael I. Jordan, Annie Ulichney

Abstract

We consider a dynamic mechanism design problem where an auctioneer sells an indivisible good to groups of buyers in every round, for a total of $T$ rounds. The auctioneer aims to maximize their discounted overall revenue while adhering to a fairness constraint that guarantees a minimum average allocation for each group. We begin by studying the static case ($T=1$) and establish that the optimal mechanism involves two types of subsidization: one that increases the overall probability of allocation to all buyers, and another that favors the groups which otherwise have a lower probability of winning the item. We then extend our results to the dynamic case by characterizing a set of recursive functions that determine the optimal allocation and payments in each round. Notably, our results establish that in the dynamic case, the seller, on the one hand, commits to a participation bonus to incentivize truth-telling, and on the other hand, charges an entry fee for every round. Moreover, the optimal allocation once more involves subsidization, which its extent depends on the difference in future utilities for both the seller and buyers when allocating the item to one group versus the others. Finally, we present an approximation scheme to solve the recursive equations and determine an approximately optimal and fair allocation efficiently.

Fair Allocation in Dynamic Mechanism Design

Abstract

We consider a dynamic mechanism design problem where an auctioneer sells an indivisible good to groups of buyers in every round, for a total of rounds. The auctioneer aims to maximize their discounted overall revenue while adhering to a fairness constraint that guarantees a minimum average allocation for each group. We begin by studying the static case () and establish that the optimal mechanism involves two types of subsidization: one that increases the overall probability of allocation to all buyers, and another that favors the groups which otherwise have a lower probability of winning the item. We then extend our results to the dynamic case by characterizing a set of recursive functions that determine the optimal allocation and payments in each round. Notably, our results establish that in the dynamic case, the seller, on the one hand, commits to a participation bonus to incentivize truth-telling, and on the other hand, charges an entry fee for every round. Moreover, the optimal allocation once more involves subsidization, which its extent depends on the difference in future utilities for both the seller and buyers when allocating the item to one group versus the others. Finally, we present an approximation scheme to solve the recursive equations and determine an approximately optimal and fair allocation efficiently.
Paper Structure (43 sections, 16 theorems, 123 equations, 6 figures)

This paper contains 43 sections, 16 theorems, 123 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notions of Fairness
  4. Truthful Fair Allocation Mechanisms
  5. Dynamic Fairness
  6. Fairness and Efficiency
  7. Empirical Perspectives
  8. Model
  9. Timeline of the auction:
  10. Utility functions:
  11. The fairness constraint:
  12. Direct truthful mechanisms:
  13. The Static Case
  14. Static fairness constraint:
  15. The Dynamic Case
  16. ...and 28 more sections

Key Result

Proposition 1

A mechanism $(\bm{x}, \bm{p})$ is EPIC if and only if for every $i$ and $k$ we have that with $c(\bm{v}_{-(i,k)})=0$ for IR mechanisms. Moreover, the seller's expected utility for any EPIC and IR mechanism is given by

Figures (6)

  • Figure 1: Optimal allocation in the static case.
  • Figure 2: Difference in utility relative to unconstrained optimal allocation for $T=2$
  • Figure 3: Difference in utility relative to unconstrained optimal allocation for $T=3$
  • Figure 4: Difference in utility relative to unconstrained optimal allocation for $T=4$
  • Figure 5: The case where $\eta_2 < \eta_1 + \gamma$.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Proposition 1: myerson1981optimal
  • Theorem 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Remark 1
  • Theorem 2
  • Remark 2: The ex post fairness constraint
  • ...and 22 more