Confidence intervals for Shapley value in Markovian dynamic games

We consider a dynamic multiagent system in which several states succeed each other, following a Markov chain process. In each state, a different single stage game among the agents, or players, is played, and cooperation among subsets of players can arise in order to achieve a common goal. We assume that each coalition can ensure a certain value for itself. The Shapley value is a well known method to share the value of the grand coalition, formed by all the players, among the players themselves. It reflects the effective incremental asset brought by each player to the community. Unfortunately, the exact computation of the Shapley value for each player requires an exponential complexity in the number of players. Moreover, we prove that an exponential number of queries is necessary for any deterministic algorithm even to approximate the Shapley value in a Markovian dynamic game with polynomial accuracy. Motivated by these reasons, we propose three different methods to compute a confidence interval for the Shapley value in the Markovian game. Our approaches require a polynomial number of queries to achieve a polynomial accuracy. We compare our confidence intervals in terms of their tightness and we provide a straightforward sampling strategy to optimize the tightness of one of them.