Maximizing the Egalitarian Welfare in Friends and Enemies Games

Edith Elkind, Michele Flammini, Giovanna Varricchio

Keywords: Hedonic Games, Friends and Enemies Games, Egalitarian Welfare

TL;DR: We study the compexity of maximizing egalitarian welfare in friends and enemies hedonic games, and obtain hardness results and approximation algorithms.

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 the two classical scenarios proposed in the literature, namely, Friends Appreciation (FA) and Enemies Aversion (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 EA, we show that our objective is hard to approximate within $O(n^{1-\epsilon})$, for any fixed $\epsilon>0$, and provide a polynomial-time $(n-1)$-approximation. For FA, we obtain an NP-hardness result and a polynomial-time approximation algorithm. Our algorithm achieves a ratio better than $2-\Theta(\frac{1}{n})$ 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 $2-\Theta(\frac{1}{n})$ approximation ratio for two important variants: when randomization is allowed and when the friendship relationship is symmetric. Additionally, for both EA and FA we identify special cases where the optimal egalitarian partition can be computed in polynomial time.

Area: Game Theory and Economic Paradigms (GTEP)

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Submission Number: 1679