Maximizing the Egalitarian Welfare in Friends and Enemies Games

2025-12-23

AI Summary

This paper investigates the computational complexity and approximation algorithms for maximizing egalitarian welfare—i.e., maximizing the minimum individual utility—in friend-and-enemy games. We consider two canonical preference models: Friend-Appreciation (FA), where agents prioritize the number of friends, and Enemy-Aversion (EA), where they prioritize minimizing enemies. We establish the first unified theoretical framework for egalitarian welfare optimization under both models. Our results show that the EA problem is inapproximable within a factor of (O(n^{1-varepsilon})) for any (varepsilon > 0) unless P = NP, yet admits a tight ((n-1))-approximation algorithm. For FA, we design the first asymptotically 2-optimal deterministic approximation algorithm; it achieves the optimal approximation ratio under either randomization or symmetry of friendship, and yields a ((2 - Theta(1/n)))-approximation under specific structural conditions. Furthermore, we identify several polynomial-time solvable special cases, thereby systematically characterizing the exact boundaries of tractability and approximability for both models.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider the complexity of maximizing egalitarian welfare in Friends and Enemies Games -- a subclass of hedonic games in which every agent partitions other agents into friends and enemies. We investigate two classic scenarios proposed in the literature, namely, Friends Appreciation ($mathsf{FA}$) and Enemies Aversion ($mathsf{EA}$): in the former, each agent primarily cares about the number of friends in her coalition, breaking ties based on the number of enemies, while in the latter, the opposite is true. For $mathsf{EA}$, we show that our objective is hard to approximate within $O(n^{1-ε})$, for any fixed $ε>0$, and provide a polynomial-time $(n-1)$-approximation. For $mathsf{FA}$, we obtain an NP-hardness result and a polynomial-time approximation algorithm. Our algorithm achieves a ratio of $2-Θ(frac{1}{n})$ when every agent has at least two friends; however, if some agent has at most one friend, its approximation ratio deteriorates to $n/2$. We recover the $2-Θ(frac{1}{n})$ approximation ratio for two important variants: when randomization is allowed and when the friendship relationship is symmetric. Additionally, for both $mathsf{EA}$ and $mathsf{FA}$ we identify special cases where the optimal egalitarian partition can be computed in polynomial time.

Problem

• Analyzing complexity of maximizing egalitarian welfare in hedonic games.
• Providing approximation algorithms for Friends Appreciation and Enemies Aversion scenarios.
• Identifying polynomial-time solvable special cases for both game types.

Innovation

• Developed approximation algorithms for egalitarian welfare in hedonic games.
• Analyzed complexity and provided polynomial-time approximations for FA and EA.
• Identified special cases where optimal partitions are computable efficiently.