Technical Report . A Market Framework for Pooling

Consider a group of Bayesians, each with a subjective probability distribution over a set of uncertain events. An opinion pool derives a single consensus distribution over the events, representative of the group as a whole. Several pooling functions have been proposed, each sensible under particular assumptions or measures. Many researchers over many years have failed to form a consensus on which method is best. We propose a market-based pooling procedure, and analyze its properties. Participants bet on securities, each paying oo contingent on an uncertain event, so as to maximize their own expected utilities. The consensus probability of each event is deened as the corresponding security's equilibrium price. The market framework provides explicit monetary incentives for participation and honesty, and allows agents to maintain individual rationality and limited privacy. \No arbitrage" arguments ensure that the equilibrium prices form legal probabilities. We show that, when events are disjoint and all participants have exponential utility for money, the market derives the same result as the logarithmic opinion pool; similarly, logarithmic utility for money yields the linear opinion pool. In both cases, we prove that the group's behavior is, to an outside observer, indistinguishable from that of a rational individual, whose beliefs equal the equilibrium prices.