Bayesian Group Decisions: Algorithms and Complexity

We address the computations that Bayesian agents undertake to realize their optimal actions, as they repeatedly observe each other's actions, following an initial private observation. We use iterated eliminations of infeasible signals (IEIS) to model the thinking process as well as the calculations of a Bayesian agent in a group decision scenario. We show that IEIS runs in exponential time; however, when the group structure is a partially ordered set, the Bayesian calculations simplify and polynomial-time computation of the Bayesian recommendations is possible. We next shift attention to the case where agents reveal their beliefs (instead of actions) at every decision epoch. We analyze the computational complexity of the Bayesian belief formation in groups and show that it is NP-hard. We also investigate the factors underlying this computational complexity and show how belief calculations simplify in special network structures or cases with strong inherent symmetries. We finally give insights about the statistical efficiency (optimality) of the beliefs and its relations to computational efficiency.