Lawrence E. Blume

On the private provision of public goods

Abstract We consider a general model of the non-cooperative provision of a public good. Under very weak assumptions there will always exist a unique Nash equilibrium in our model. A small redistribution of wealth among the contributing consumers will not change the equilibrium amount of the public good. However, larger redistributions of wealth will change the set of contributors and thereby change the equilibrium provision of the public good. We are able to characterize the properties and the comparative statics of the equilibrium in a quite complete way and to analyze the extent to which government provision of a public good ‘crowds out’ private contributions.

Learning to Be Rational

We study the dynamical system of expectations generated by a simple general equilibrium model of an exchange economy in which each agent considers a finite collection of models, each of which specifies a relationship between payoff-relevant information and equilibrium prices. One of the models under consideration is a correct description of the rational expectations equilibrium. We find that under a Bayesian type of learning process the rational expectations equilibrium is locally stable, but that nonrational equilibria may also be locally stable. Journal of Economic Literature Classification Numbers: 021, 026.

Lexicographic Probabilities and Equilibrium Refinements

This paper develops a decision-theoretic approach to normal-form refinements of Nash equilibrium and provides characterizations of (normal-form) perfect equilibrium and proper equilibrium. The approach relies on a theory of single-person decision-making that is a non-Archimedean version of subjective expected utility theory. Copyright 1991 by The Econometric Society.

Equilibrium concepts for social interaction models

This paper describes the relationship between two different binary choice social interaction models. The Brock and Durlauf (2001) model is essentially a static Nash equilibrium model with random utility preferences. In the Blume (2003) model is a population game model similar to Blume (1993), Kandori, Mailath and Rob (1993) and Young (1993). We show that the equilibria of the Brock–Durlauf model are steady states of a differential equation which is a deterministic approximation of the sample-path behavior of Blumes model. Moreover, the limit distribution of this model clusters around a subset of the steady states when the population is large.

Introduction to the stability of rational expectations equilibrium

Abstract This paper surveys the literature which examines the stability of the expectations that agents are assumed to have in a rational expectations equilibrium (REE). This issue is more complex than the usual statistical estimation problem because the relationship between observable variables and payoff relevant variables is endogenous. One approach taken in the literature yields convergence to a REE but requires agents to have extensive knowledge about the structure and dynamics of the model that prevails while they learn. A second approach does not assume that agents have correctly specified likelihood functions and finds that REE may not be stable.

How noise matters

Abstract Recent advances in evolutionary game theory have introduced noise into decisionmaking to select in favor of certain equilibria in coordination games. Noisy decisionmaking is justified on bounded rationality grounds, and consequently the sources of noise are left unmodelled. This methodological approach can only be successful if the results do not depend too much on the nature of the noise process. This paper investigates invariance to noise of these results, both for the random matching paradigm that has characterized much of the recent literature and for a larger class of two-strategy population games where payoffs may vary non-linearly with the distribution of strategies among the population. Several parametrizations of noise reduction are investigated. The results show that a symmetry property of the noise process and, in the case of non-linear payoffs, bounds on the asymmetry of the payoff functions suffice to preserve the known stochastic stability results.

The Algebraic Geometry of Perfect and Sequential Equilibrium

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Seltens (1975) {\it perfect equilibrium\/} and Kreps and Wilsons (1982) more inclusive {\it sequential equilibrium\/}. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. \par We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. \par We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are {\em semi-algebraic sets\/}; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential

Characterization of optimal plans for stochastic dynamic programs

This paper provides general techniques for the characterization of optimal plans resulting from stochastic dynamic programming, We show that under standard assumptions the optimal plans in both finite and infinite horizon problems can be obtained by an application of the Implicit Function Theorem to first order conditions. Further, we show that under certain checkable conditions, optimal plans and value functions are p-times differentiable for any integer p > 0. Finally, we apply our technique to obtain a C” plan and value function in a one sector infinite horizon growth problem under uncertainty. Journal of Economic Literature Classification Numbers: 022, 111. 213. The analysis of many problems in economics requires the consideration of both time and uncertainty. One of the standard tools for solving such problems is stochastic dynamic programming. A frequent criticism of the application of this technique to economic decision problems is that although solutions are shown to exist they are not adequately characterized. The aim of this paper is to rebut this criticism for a broad class of finite and infinite horizon stochastic dynamic programming problems. We show that under

Uniqueness of Nash equilibrium in private provision of public goods : An improved proof

Our 1986 paper ‘Private Provision of Public Goods’ presents a theorem on the uniqueness of Nash equilibrium in the private provision of public goods. Richard Hirth of the University of Pennsylvania and Clive Fraser of Warwick have independently suggested that although our theorem is correct, our proof is incorrect. Each writer proposes an alternative method of proof for the theorem. With characteristic charity to our own deficiencies, we prefer to view our earlier proof as unduly opaque rather than wrong. Whichever view one takes, a more transparent proof is called for. The proof presented here follows the same outline as our 1986 proof, but explains a step of the argument that was far from obvious in the original proof. This proof uses the notation of the 1986 paper, and refers to ‘Facts’ that are proved there.

Rational Expectations and Rational Learning

We provide an overview of the methods of analysis and results obtained, and, most important, an assessment of the success of rational learning dynamics in tying down limit beliefs and limit behavior. We illustrate the features common to rational or Bayesian learning in single agent, game theoretic and equilibrium frameworks. We show that rational learing is possible in each of these environments. The issue is not in whether rational learning can occur, but in what results it produces. If we assume a natural complex parameterization of the choice environment all we know is the rational learner believes that his posteriors will converge somewhere with prior probability one. Alternatively, if we, the modelers, assume the simple parameterization of the choice environment that is necessary to obtain positive results we are closing our models in the ad hoc fashion that rational learning was inroduced to avoid. We believe that a partial resolution of this conundrum is to pay more attention to how learning interacts with other dynamic forces. We show that in a simple economy, the forces of market selection can yield convergence to rational expectations equilibria even without every agent behaving as a rational learner.