The Stochastic Shapley Value for coalitional games with externalities

A long debated but still open question in the game theory literature is that of how to extend the Shapley Value to coalitional games with externalities. While previous work predominantly focused on developing alternative axiomatizations, in this article we propose a novel approach which centers around the coalition formation process and the underlying probability distribution from which a suitable axiomatization naturally follows. Specifically, we view coalition formation in games with externalities as a discrete-time stochastic process. We focus, in particular, on the Chinese Restaurant Process – a well-known stochastic process from probability theory. We show that reformulating Shapley's coalition formation process as the Chinese Restaurant Process yields in games with externalities a unique value with various desirable properties. We then generalize this result by proving that all values that satisfy the direct translation of Shapley's axioms to games with externalities can be obtained using our approach based on stochastic processes.