$\varepsilon$-fractional Core Stability in Hedonic Games

• URL: http://arxiv.org/abs/2311.11101v2
• Date: Sun, 14 Jan 2024 13:51:06 GMT
• Title: $\varepsilon$-fractional Core Stability in Hedonic Games
• Abstract summary: Hedonic Games (HGs) are a framework for modeling coalitions of strategic agents guided by their individual preferences. We propose the notion of $varepsilon$-fractional core-stability, where at most an $varepsilon$-fraction of all possible coalitions is allowed to formation core-block. Specifically, we design efficient algorithms returning an $varepsilon$-fractional core-stable partition, with $varepsilon$ exponentially decreasing in the number of agents.
• Score: 13.576268908412338
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences. According to these preferences, it is desirable that a coalition structure (i.e. a partition of agents into coalitions) satisfies some form of stability. The most well-known and natural of such notions is arguably core-stability. Informally, a partition is core-stable if no subset of agents would like to deviate by regrouping in a so-called core-blocking coalition. Unfortunately, core-stable partitions seldom exist and even when they do, it is often computationally intractable to find one. To circumvent these problems, we propose the notion of $\varepsilon$-fractional core-stability, where at most an $\varepsilon$-fraction of all possible coalitions is allowed to core-block. It turns out that such a relaxation may guarantee both existence and polynomial-time computation. Specifically, we design efficient algorithms returning an $\varepsilon$-fractional core-stable partition, with $\varepsilon$ exponentially decreasing in the number of agents, for two fundamental classes of HGs: Simple Fractional and Anonymous. From a probabilistic point of view, being the definition of $\varepsilon$-fractional core equivalent to requiring that uniformly sampled coalitions core-block with probability lower than $\varepsilon$, we further extend the definition to handle more complex sampling distributions. Along this line, when valuations have to be learned from samples in a PAC-learning fashion, we give positive and negative results on which distributions allow the efficient computation of outcomes that are $\varepsilon$-fractional core-stable with arbitrarily high confidence.

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