On the duality between prior beliefs and trading demands

In an abstract model with asymmetric information, we show that there is a duality relationship between the prior beliefs and trading demands of bets for any given individual. Then we aggregate all the agents to obtain a second duality relationship between common prior beliefs and trading possibilities. We easily derive from these relationships the no trade theorem and its converse. General efficiency results can be obtained. Moreover, our framework is sufficiently general to cover special cases proved previously (for example, Econometrica 62 (1994) 1327; Discussion Paper 83, Center for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem, 1995; J. Econom. Theory 91 (2000) 127; Games Econom. Behav. 24 (1998) 172. Yet, our arguments are both simple and intuitive.