Noise Robust Core-Stable Coalitions of Hedonic Games

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 and obtain the prediction probability, defined as the probability that the estimated PAC core-stable partition of the noisy game is also PAC core-stable for the 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 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.