Allan Muir

On extensive form implementation of contracts in differential information economies

In the context of differential information economies, with and without free disposal, we consider the concepts of Radner equilibrium, rational expectations equilibrium, private core, weak fine core and weak fine value. We look into the possible implementation of these concepts as perfect Bayesian or sequential equilibria of noncooperative dynamic formulations. We construct relevant game trees which indicate the sequence of decisions and the information sets, and explain the rules for calculating ex ante expected payoffs. The possibility of implementing an allocation is related to whether or not it is incentive compatible. Implementation through an exogenous third party or an endogenous intermediary is also considered.

An extensive form interpretation of the private core

The private core of an economy with differential information, (Yannelis (1991)), is the set of all state-wise feasible and private information measurable allocations which cannot be dominated, in terms of ex ante expected utility functions, by state-wise feasible and private information measurable net trades of any coalition. It is coalitionally Bayesian incentive compatible and also takes into account the information superiority of an individual. We provide a noncooperative extensive form interpretation of the private core for three person games. We construct game trees which indicate the sequence of decisions and the information sets, and explain the rules for calculating ex ante expected payoffs. In the spirit of the Nash programme, the private core is thus shown to be supported by the perfect Bayesian equilibrium of a noncooperative game. The discussion contributes not only to the development of ideas but also to the understanding of the dynamics of how coalitionally incentive compatible contracts can be realized.

The compactness of Pr (K)

We prove the compactness of Pr(K), the set of Borel probability measures on a compactum K endowed with the weak* topology, without embedding this set in rca(K), the space of regular, countably-additive, signed measures with their finite total variation as norm. Pr(K) can be extended to a convex, Hausdorff, linear topological space. Then Glicksberg’s fixed point theorem is applied to prove the existence of Nash equilibria.

On the solution of a stochastic optimal growth model

This note discusses a stochastic optimal growth model in which the optimal paths can be obtained by a simple direct argument. The structural characteristics of the model are the infinite horizon, the form of the instantaneous utility function, and uncertainty as a Wiener process in a linear production constraint. The note explains that, for optimality, at each point in time a formally identical problem must be solved. This implies that the optimal saving ratio must be constant.A proof, employing the rules of stochastic calculus, that the ensuing paths are the unique globally optimal paths is also given.

On extensive form implementation of equilibria in differential information economies

We investigate the possibility of a dynamic explanation of the equilibrium ideas in terms of the perfect Bayesian equilibrium (PBE) (or sequential equilibrium). In particular, we take an equilibrium outcome which has been found by means of a static optimizing behavior and ask the following question. Can this outcome be supported, (or implemented), as a PBE of an extensive form game of a reasonable form? We provide a positive answer for solution concepts which are incentive compatible and a negative one for those which are not.

AN APPROACH TO BARGAINING FOR GENERAL PAYOFFS REGIONS

This paper discusses an approach to bargaining which provides a solution to a demand game even when the payoffs region is not convex. The game is played repeatedly but the demands of the two players at each play are only used as information in subsequent calculations. This process goes on until convergence to a feasible vector is established. This will be the negotiated outcome of the game. The properties of such solutions are considered and the relation with the axiomatically justified Nash bargaining solution is discussed.