Noise Robust Core-stable Coalitions of Hedonic Games

Prashant Trivedi, Nandyala Hemachandra

Abstract: In this work, we consider the coalition formation games with an additional component, ‘noisy preferences’. Moreover, such noisy preferences are available only for a sample of coalitions. We propose a multiplicative noise model (equivalent to an additive noise model) and obtain the prediction probability, defined as the probability that the estimated PAC core-stable partition of the \emph{noisy} game is also PAC core-stable for the \emph{unknown noise-free} game. This prediction probability depends on the probability of a combinatorial construct called an ‘agreement event’. We explicitly obtain the agreement probability for $n$ agent noisy game with $l\geq 2$ support noise distribution. For a user-given satisfaction value on this probability, we identify the noise regimes for which an estimated partition is noise robust; that is, it is PAC core-stable in both the noisy and noise-free games. We obtain similar robustness results when the estimated partition is not PAC core-stable. These noise regimes correspond to the level sets of the agreement probability function and are non-convex sets. Moreover, an important fact is that the prediction probability can be high even if high noise values occur with a high probability. Further, for a class of top-responsive hedonic games, we obtain the bounds on the extra noisy samples required to get noise robustness with a user-given satisfaction value. We completely solve the noise robustness problem of a $2$ agent hedonic game. In particular, we obtain the prediction probability function for $l=2$ and $l=3$ noise support cases. For $l=2$, the prediction probability is convex in noise probability, but the noise robust regime is non-convex. Its minimum value, called the safety value, is 0.62; so, below 0.62, the noise robust regime is the entire probability simplex. However, for $l \geq 3$, the prediction probability is non-convex; so, the safety value is the global minima of a non-convex function and is computationally hard.