$\varepsilon$-fractional Core Stability in Hedonic Games
Simone Fioravanti, Michele Flammini, Bojana Kodric, Giovanna Varricchio
TL;DR
This work introduces $\varepsilon$-fractional core stability ($\varepsilon$-FC) as a robust relaxation of core stability in Hedonic Games, enabling existence and polynomial-time computation in two fundamental HG classes: Simple Fractional and Anonymous. It provides algorithms that achieve $\varepsilon$-FC under the uniform distribution with $\varepsilon$ exponentially small in the number of agents, and extends to $\lambda$-bounded distributions with PAC-learning-style learning of preferences and high-confidence guarantees. The paper also presents impossibility results showing nonexistence of constant-$\varepsilon$ FC outcomes under arbitrary distributions, establishing a nuanced boundary between hard and tractable settings. Overall, it bridges core-stability concepts with learning and distributional assumptions, offering practical stability notions when full information is unavailable.
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.