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# Probably approximately correct stability of allocations in uncertain coalitional games with private sampling

*Proceedings of the 6th Annual Learning for Dynamics & Control Conference*, PMLR 242:1702-1714, 2024.

#### Abstract

We study coalitional games with exogenous uncertainty in the coalition value, in which each agent is allowed to have private samples of the uncertainty. As a consequence, the agents may have a different perception of stability of the grand coalition. In this context, we propose a novel methodology to study the out-of-sample coalitional rationality of allocations in the set of stable allocations (i.e., the core). Our analysis builds on the framework of probably approximately correct learning. Initially, we state a priori and a posteriori guarantees for the entire core. Furthermore, we provide a distributed algorithm to compute a compression set that determines the generalization properties of the a posteriori statements. We then refine our probabilistic robustness bounds by specialising the analysis to a single payoff allocation, taking, also in this case, both a priori and a posteriori approaches. Finally, we consider a relaxed zeta-core to include nearby allocations and also address the case of empty core. For this case, probabilistic statements are given on the eventual stability of allocations in the zeta-core.