[Paper Review] Fair Division with Interdependent Values

Georgios Birmpas, Tomer Ezra+2 more|arXiv (Cornell University)|May 23, 2023

Auction Theory and ApplicationsDecision Sciences3 citations

TL;DR

This paper introduces fair division mechanisms for indivisible goods under interdependent valuations, where agents' values depend on all agents' private signals. It proposes simple mechanisms that always admit pure Nash equilibria fair with respect to true signals, while proving that no mechanism can guarantee all equilibria are fair—highlighting a fundamental gap between independent and interdependent value settings.

ABSTRACT

We introduce the study of designing allocation mechanisms for fairly allocating indivisible goods in settings with interdependent valuation functions. In our setting, there is a set of goods that needs to be allocated to a set of agents (without disposal). Each agent is given a private signal, and his valuation function depends on the signals of all agents. Without the use of payments, there are strong impossibility results for designing strategyproof allocation mechanisms even in settings without interdependent values. Therefore, we turn to design mechanisms that always admit equilibria that are fair with respect to their true signals, despite their potentially distorted perception. To do so, we first extend the definitions of pure Nash equilibrium and well-studied fairness notions in literature to the interdependent setting. We devise simple allocation mechanisms that always admit a fair equilibrium with respect to the true signals. We complement this result by showing that, even for very simple cases with binary additive interdependent valuation functions, no allocation mechanism that always admits an equilibrium, can guarantee that all equilibria are fair with respect to the true signals.

Motivation & Objective

• To design allocation mechanisms that ensure at least one fair pure Nash equilibrium with respect to true signals in interdependent value settings.
• To extend fairness and equilibrium concepts from independent to interdependent valuation functions.
• To investigate whether mechanisms can guarantee fairness in all equilibria under interdependent values.
• To identify structural limitations in achieving universally fair equilibria despite the existence of fair ones.
• To explore the implications of interdependence for strategic fairness in mechanism design.

Proposed method

• Extends definitions of pure Nash equilibrium and fairness (e.g., MMS, EF1) to interdependent valuation settings.
• Designs simple allocation mechanisms that always admit at least one pure Nash equilibrium fair with respect to true signals.
• Uses signal-dependent valuation functions where each agent’s value depends on all agents’ private signals.
• Employs counterexample constructions to prove impossibility results: no mechanism can ensure all equilibria are fair under interdependent values.
• Analyzes binary additive interdependent valuations to demonstrate the separation between independent and interdependent settings.
• Applies game-theoretic analysis to show that truthful reporting can lead to fair outcomes even without monetary transfers.

Experimental results

Research questions

• RQ1 Can allocation mechanisms be designed to always admit a pure Nash equilibrium that is fair with respect to the true signals in interdependent value settings?
• RQ2 Is it possible to construct mechanisms where all pure Nash equilibria are fair with respect to true valuations under interdependent values?
• RQ3 How do fairness guarantees in interdependent value models compare to those in independent value models?
• RQ4 What structural properties do fair equilibria have in interdependent settings, and can they be simplified for practical deployment?
• RQ5 Are there restricted classes of interdependent valuations where all equilibria can be guaranteed to be fair?

Key findings

• The paper constructs mechanisms that always admit at least one pure Nash equilibrium that is fair with respect to the true signals, even under interdependent valuations.
• It proves that no mechanism can guarantee all pure Nash equilibria are fair with respect to true signals, even in the simplest case of binary additive interdependent valuations.
• The impossibility result holds for both MMS and EF1 fairness, demonstrating a fundamental separation between independent and interdependent value models.
• In the independent value setting, truthful mechanisms exist that are MMS and EF1 fair; this is not possible under interdependent values.
• The fair equilibria identified in the proposed mechanisms require agents to have significant knowledge of others’ true signals, suggesting practical limitations.
• The study reveals that while fair equilibria exist, achieving them universally across all equilibria is impossible under interdependent valuations.