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Graphical Models for Groups: Belief Aggregation and Risk Sharing

Yahoo! Research, 74 N. Pasadena Avenue, 3rd Floor, Pasadena, California 91103
University of Michigan AI Laboratory, 1101 Beal Avenue, Ann Arbor, Michigan 48109-2110
pennockd{at}yahoo-inc.com
wellman{at}umich.edu

We investigate the practical value of using graphical models to aid in two fundamental problems of group coordination: (1) belief aggregation and (2) risk sharing. We identify restrictive conditions under which graphical models can be useful in both settings. We show that the output of the logarithmic opinion pool (LogOP) can be represented as a Markov network (MN) or a decomposable Bayesian network (BN), and give an algorithm for doing so. We show that a securities market structured like a decomposable BN can support optimal risk sharing, if all agents have exponential utility and all of their Markov independencies coincide with the market structure. On the other hand, most of our results are negative, taking the form of impossibility theorems. We show that no belief aggregation function can maintain all independencies representable in a BN. Neither can an aggregation computation be decomposed into local computations on graph subsets. We show that computing query outputs of LogOP or the linear opinion pool (LinOP) is NP-hard. Except in fairly restrictive settings, structuring securities markets according to unanimously agreed upon independencies may be of no help in supporting optimal risk sharing because agents’ behavioral independencies change as they engage in securities trade.

Key Words: belief aggregation; opinion pools; consensus probabilities; risk sharing and allocation; securities markets; Bayesian networks; Markov networks; graphical models; multiagent systems
History: Received on June 15, 2004. Accepted on September 12, 2005.