Sharing Beliefs: Between Agreeing and Disagreeing

When is it Pareto optimal for risk averse agents to take bets? Under what conditions do they choose to introduce uncertainty into an otherwise certain economic environment? One obvious case is where they do not share beliefs. As in the classical (theoretical) example of horse lotteries, people who do not agree on probability assessments do nd it mutually bene cial to engage in uncertainty-generating trade. If the agents involved are Bayesian expected utility maximizers and strictly risk averse, it is not hard to see that disagreement on probabilities is the only way that betting, understood as trade of an uncertain asset, may be Pareto improving when starting from a full insurance allocation. On the other hand, any such disagreement induces betting. Put di erently, Pareto optimality dictates either that there be no betting (in case beliefs are common to all agents) or that there be betting (in case of disagreement). This is somewhat puzzling, because there is no lack of allocation-neutral, \sunspot" sources of uncertainty in the world around us. If every disagreement on probabilities of states of the world suggests a Pareto improving trade, one might have expected to see much more betting taking place.