Bayesian Group Decisions: Algorithms and Complexity

Many important real-world decision-making problems involve group interactions among individuals with purely informational externalities. Such situations arise for example in jury deliberations, expert commiŠees, medical diagnoses, etc. We model the purely informational interactions of rational agents in a group, where they receive private information and act based on that information while also observing other people’s beliefs or actions. As a Bayesian agent aŠempts to infer the true state of the world from her sequence of observations of actions of others as well as her own private signal, she recursively re€nes her belief on the signals that other players could have observed and actions that they could have taken given the assumption that other players are also rational. Œe existing literature addresses asymptotic and equilibrium properties of Bayesian group decisions and important questions such as convergence to consensus and learning. In this work, we address the computations that the Bayesian agent should undertake to realize the optimal actions at every decision epoch. We use the 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 algorithm 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 shi‰ aŠention 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 €nally give insights about the statistical eciency (optimality) of the beliefs and its relations to computational eciency.