Individual Rationality in Constrained Hedonic Games: Additively Separable and Fractional Preferences

TL;DR

This paper investigates the computational complexity of finding individually rational stable coalitions in constrained hedonic games, revealing both tractable cases and intractable challenges even under simplified preference models.

Contribution

It provides a comprehensive analysis of the complexity landscape for IR coalition existence in constrained hedonic games, including novel tractability results and hardness proofs.

Findings

Tractable cases identified using preference graph structure

Intractability results for certain constrained scenarios

Complete complexity landscape of IR coalition existence

Abstract

Hedonic games are an archetypal problem in coalition formation, where a set of selfish agents want to partition themselves into stable coalitions. In this work, we focus on two natural constraints on the possible outcomes. First, we require that exactly k coalitions are created. Then, loosely following the model of Bil\`o et al. (AAAI 2022), we assume that each of the k coalitions is additionally associated with a lower and upper bound on its size. The notion of stability that we study is that of individual rationality (IR), which requires that no agent strictly prefers to be alone compared to being in his or her coalition. Although IR is trivially satisfiable even in the most general models of hedonic games, the complexity picture of deciding whether an IR allocation exists, considering the above constraints, is unexpectedly rich. We reveal that tractable fragments of this computational problem require surprisingly nontrivial arguments, even if we restrict ourselves to additively separable and fractional hedonic games. Our tractability results, achieved by exploiting the structure of the underlying preference graph, are also complemented by their intractability counterparts, painting a fairly complete picture of the tractability landscape of this problem.