Uncertainty from the small to the large

Related decisions are often observed in isolation without direct measurement of correlation in beliefs across state spaces or complementarity in tastes across prize spaces. We introduce a novel model with two decision problems with distinct states and prizes, which we call small worlds, without observation of bets that are contingent on the realization of both worlds. We characterize an appropriate version of subjective expected utility, where choices are made as if there is a joint distribution over the product of the state spaces and a joint utility index over pairs of prizes from both prize spaces. Turning to identification, the joint utility index over pairs of prizes and the marginal belief over each small world is identified, but the uniqueness of the joint distribution is more subtle. If the utility index is separable across prize spaces, then the correlation across state spaces is unidentified; but if there is any complementarity across prizes, then the joint distribution is exactly identified.