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Fairness and Incentive Compatibility via Percentage Fees

Shahar Dobzinski, Sigal Oren, Jan Vondrak

TL;DR

The paper addresses incentive-compatible mechanism design for maximizing the Nash Social Welfare under a novel percentage fee model, where payments are fractions of the chosen outcome's value. It shows NSW-maximizing allocations are implementable in this model under mild valuation assumptions and establishes a Roberts-type characterization via a log-transform correspondence to the traditional model. The authors then develop several computationally efficient NSW mechanisms, including deterministic and randomized maximal-in-range schemes, achieving strong approximation guarantees for XOS and subadditive valuations, along with fundamental hardness results for certain settings. Overall, the work demonstrates that percentage-fee payments can enable implementable fairness objectives that are difficult or impossible to realize under traditional fixed-fee mechanisms, offering both theoretical insights and practical mechanism designs.

Abstract

We study incentive-compatible mechanisms that maximize the Nash Social Welfare. Since traditional incentive-compatible mechanisms cannot maximize the Nash Social Welfare even approximately, we propose changing the traditional model. Inspired by a widely used charging method (e.g., royalties, a lawyer that charges some percentage of possible future compensation), we suggest charging the players some percentage of their value of the outcome. We call this model the \emph{percentage fee} model. We show that there is a mechanism that maximizes exactly the Nash Social Welfare in every setting with non-negative valuations. Moreover, we prove an analog of Roberts theorem that essentially says that if the valuations are non-negative, then the only implementable social choice functions are those that maximize weighted variants of the Nash Social Welfare. We develop polynomial time incentive compatible approximation algorithms for the Nash Social Welfare with subadditive valuations and prove some hardness results.

Fairness and Incentive Compatibility via Percentage Fees

TL;DR

The paper addresses incentive-compatible mechanism design for maximizing the Nash Social Welfare under a novel percentage fee model, where payments are fractions of the chosen outcome's value. It shows NSW-maximizing allocations are implementable in this model under mild valuation assumptions and establishes a Roberts-type characterization via a log-transform correspondence to the traditional model. The authors then develop several computationally efficient NSW mechanisms, including deterministic and randomized maximal-in-range schemes, achieving strong approximation guarantees for XOS and subadditive valuations, along with fundamental hardness results for certain settings. Overall, the work demonstrates that percentage-fee payments can enable implementable fairness objectives that are difficult or impossible to realize under traditional fixed-fee mechanisms, offering both theoretical insights and practical mechanism designs.

Abstract

We study incentive-compatible mechanisms that maximize the Nash Social Welfare. Since traditional incentive-compatible mechanisms cannot maximize the Nash Social Welfare even approximately, we propose changing the traditional model. Inspired by a widely used charging method (e.g., royalties, a lawyer that charges some percentage of possible future compensation), we suggest charging the players some percentage of their value of the outcome. We call this model the \emph{percentage fee} model. We show that there is a mechanism that maximizes exactly the Nash Social Welfare in every setting with non-negative valuations. Moreover, we prove an analog of Roberts theorem that essentially says that if the valuations are non-negative, then the only implementable social choice functions are those that maximize weighted variants of the Nash Social Welfare. We develop polynomial time incentive compatible approximation algorithms for the Nash Social Welfare with subadditive valuations and prove some hardness results.
Paper Structure (10 sections, 15 theorems, 19 equations)

This paper contains 10 sections, 15 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Implementability in the Percentage Fee Model
  3. Maximizing the Nash Social Welfare
  4. Characterizations: An Analog of Roberts Theorem
  5. Single Parameter Domains
  6. Computationally Efficient Approximation Mechanisms
  7. Approximations for a Constant Number of XOS / Subadditive Players
  8. A Polynomial Time $O(m^{\frac{2}{3}})$-Approximation Mechanism
  9. An $\tilde{O}(m^{\frac{1}{2}})$-Approximation Mechanism using $O(n+m)$ Value Queries
  10. An Impossibility Result for MIR Mechanisms with Additive Valuations

Key Result

Theorem 2.1

Consider a domain $\mathcal{V}$ where all valuations are positive or all valuations are non-negative and there is an alternative $null$ such that $v_i(null)=0$ for every player $i$. Let $f$ be a social choice function defined on $\mathcal{V}^n$ that selects an alternative that maximizes the Nash Soc

Theorems & Definitions (34)

  • Theorem 2.1: Nash Social Welfare Maximization
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4: an analog of Roberts' theorem
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • ...and 24 more