Graphical Models for Groups: Belief Aggregation and Risk Sharing

We investigate the use of graphical models in two fundamental problems of group coordination: (1) reaching a consensus on beliefs, and (2) allocating risk. On the negative side, we prove that under mild assumptions, even if every member of a group agrees on a graphical topology, no method of combining their beliefs can maintain that structure. Even weaker conditions rule out local aggregation within the conditional probability tables of the graphical models. We show that the linear opinion pool (LinOP) and the logarithmic opinion pool (LogOP) are both NP-hard to compute, even for queries easy to compute for every individual. In terms of risk sharing, we show that securities markets structured like graphical models are generally no more tractable than complete securities markets, the unattainable gold standard for optimal risk allocation. On the positive side, we show that if