Self-fulfilling mechanisms and rational expectations
In a Bayesian game G, the players first receive private information on the state of nature and then simultaneously choose an action. We assume that the vector of actions a generates a signal g(a). A mechanism for G is a mapping [ mu ] from the set of states of nature S to the product sert of playersâ€™ actions A. [ mu ] is self-fulfilling if, given the information revealed by [ mu ] (namely, g([ mu ] )(s)) if the state of nature is s), no player can gain in unilaterally deviating from the action prescribed by the mechanism. Let SF(G) denote the set of payoffs achievable through an incentive compatible self-fulfilling mechanism. Examples show that SF(G) may not intersect the set N(G) of Nash equilibrium payoffs of G. Obviously, SF(G) and N(G) coincide if G is a game of complete information. Let E be an exchange economy with differential information. We associate a ( Bayesian) market game GE with E. In GE, the signal generated by the playersâ€™ actions is a vector of prices. We prove that the allocations achieved through a self-fulfilling mechanism in GE coincide with the rational expectations equilibrium allocations in E. In order to understand how self-fulfillingness can be achieved in a dynamic framework, we analyze the relationship between SF(G) and the Nash equilibria of the infinitely repeated game G [ infinity] generated by G. We show in particular that SF(G) can be interpreted as a set of inert solutions of G [ infinity].
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