J. S. Jordan

Bayesian learning in normal form games

Abstract This paper studies myopic Bayesian learning processes for finite-player, finite-strategy normal form games. Initially, each player is presumed to know his own payoff function but not the payoff functions of the other players. Assuming that the common prior distribution of payoff functions satisfies independence across players, it is proved that the conditional distributions on strategies converge to a set of Nash equilibria with probability one. Under a further assumption that the prior distributions are sufficiently uniform, convergence to a set of Nash equilibria is proved for every profile of payoff functions, that is, every normal form game.

Rational expectations in microeconomic models: An overview

Abstract This paper is an expository introduction to several topics of current research in the general equilibrium theory of rational expectations. More specifically, we discuss the existence of exact and approximate rational expectations equilibria, the implementation of equilibria, the behavior of learning and smoothing processes by which traders construct expectations from repeated observations of the market, and the lagged use of the information revealed by prices in an intertemporal sequence of markets. The purpose of this discussion is to introduce papers on these topics appearing in the Journal of Economic Theory Symposium on Rational Expectations in Microeconomic Models .

The competitive allocation process is informationally efficient uniquely

Abstract This paper establishes that the competitive allocation process is the only informationally decentralized mechanism for exchange environments which (i) achieves Pareto optimal allocations; (ii) gives each consumer an allocation which is, according to his preferences, at least as good as his endowment; (iii) satisfies certain regularity conditions; and (iv) has a message space of the smallest dimension necessary to satisfy (i–iii).

The generic existence of rational expectations equilibrium in the higher dimensional case

Abstract The study of the generic existence of rational expectations equilibrium has concentrated on three cases, according to whether the space of “states of private information” has dimension less than, equal to, or greater than the price space. Previous results have shown that in the first case fully revealing equilibria exist generically and in the second case there are open sets of environments with no equilibrium. This paper shows that in the third case there is a residual set of environments which permits equilibria which are arbitrarily close to fully revealing.

The exponential convergence of Bayesian learning in normal form games

Abstract This paper continues the study of Bayesian learning processes for general finite-player, finite-strategy normal form games. Bayesian learning was introduced in an earlier paper by the present author as an iterative mechanism by which players can learn Nash equilibria. The main result of the present paper is that if prior beliefs are sufficiently uniform and expectations converge to a “regular” Nash equilibrium, then the rate of convergence is exponential.

Instability in the implementation of Walrasian allocations

Abstract The existence of game forms which implement Walrasian allocations as Cournot (Nash) equilibrium outcomes is well known. However, if the equilibria are also required to be locally dynamically stable, at least for environments with unique Walrasian allocations, this paper shows that the requisite game forms do not exist. Our definition of a game form entails certain regularity conditions, and requires the Cournot equilibrium to be unique when the Walrasian equilibrium is unique. The main result is that for such a game form, there does not exist a continuous-time strategy adjustment process which ensures the local stability of Cournot equilibria throughout a certain class of environments having unique Walrasian equilibria. Each trader adjusts his strategy in response to his own characteristics and the observed current strategies of others; but the direction and magnitude of adjustments are not constrained by any behavioral assumptions. The definitions permit the inclusion of an artificial player, such as an auctioneer, so the well-known tatonnement instability emerges as a special case.

Learning Rational Expectations: The Finite State Case

This paper is devoted to the question of whether traders can learn rational expectations from repeated observations of market data in a stationary environment with finitely many exogenous states of the world. The learning problem is placed in the context of an iterative adjustment process which achieves equilibrium if traders have rational expectations. The main result is that even if traders begin with no knowledge of their environment, there exists an estimation procedure which converges to rational expectations when the environment satisfies a certain regularity condition. The regularity condition is shown to be generic.

Expectations equilibrium and informational efficiency for stochastic environments

Abstract This paper studies the existence of expectations equilibria for message processes on stochastic exchange environments. A message process which permits the general existence of expectations equilibria is termed admissible. The main results are: (1) Every admissible process satisfies a weakened version of the Hurwicz “privacy” condition. (2) The competitive process is admissible, and has a message space of locally minimal size among admissible nonwasteful processes. (3) The only admissible condensations of the competitive process are constant functions.

The Asymptotic Optimality of Residual Income Maximization

Residual income subtracts from operating income an interest charge for invested capital. Residual income can be calculated each period from current accounting information, unlike discounted cash flow (DCF), which requires the knowledge of future cash flows. This paper provides a normative justification for residual-income maximization by showing that if investment decisions are made myopically each period to maximize residual income, the resulting path asymptotically maximizes discounted cash flow. Thus, under the assumptions of the model, residual-income maximization is a heuristic that leads to the long-run DCF-optimum.

Locally stable price mechanisms

Abstract This paper develops necessary conditions for a price adjustment mechanism to achieve local stability at regular competitive equilibria. Two principal questions are: how closely must a locally stable mechanism be tailored to particular excess demand functions, and can any such mechanism be interpreted as a market adjustment process. In response to the first question, a variant of the (local) Newton method, termed the ‘orthogonal Newton method’ is shown to require, in a dimensional sense, the minimal information about excess demand functions. The second question is answered in the negative by proving the non-existence of any locally stable mechanism with the property that the price of any given commodity is not changed when its own market is in equilibrium. These and other results are obtained by using convergent price paths to generate a homotopy between the adjustment dictated by the mechanism and the actual direction of the equilibrium.