Prajit K. Dutta

The Tragedy of the Commons

SummaryWe provide a complete characterization of the set of Markov-Perfect Equilibria (MPE) of dynamic common-property resource games a la Levhari and Mirman (1980). We find that all MPE of such games exhibit remarkably regular dynamic behavior. Surprisingly, however, and despite their memoryless nature, MPE need not result in a “tragedy of the commons”, i.e., overexploitation of the resource relative to the first-best solutions. We show through an example that MPE could, in fact, lead to the reverse phenomenon of underexploitation of the resource. Nonetheless, we demonstrate that, in pay off space, MPE are always suboptimal.

Markovian equilibrium in a class of stochastic games: existence theorems for discounted and undiscounted models

SummaryWe study a strategic version of the neoclassical growth model under possible production uncertainty. For a general specification of the problem, we establish (i) the existence of stationary Markov equilibria in pure strategies for the discounted game, and (ii) the convergence, under a boundedness condition, of discounted equilibrium strategies to a pure strategy stationary Markovian equilibrium of the undiscounted game as the discount factor tends to unity. The same techniques can be used to prove that such convergence also obtains in all finitestate, finite-action stochastic games satisfying a certain “full communicability” condition. These results are of special interest since there are well known examples in the literature in which the limit of discounted equilibria fails to be an equilibrium of the undiscounted game.

What do discounted optima converge to?: A theory of discount rate asymptotics in economic models

Abstract This paper considers a sequence of stochastic dynamic problems in which the discount rate goes to zero. Sufficient conditions are given under which the limit of discounted optimal policies (and normalized values) are a (i) long-run average optimal and (ii) catching-up optimal policy (and value). These conditions are shown to be satisfied in many economic models. A new decision criterion for the undiscounted problem, the strong long-run average, is introduced. This criterion refines the long-run average, is often equivalent to the catching-up, and is easier to use than the latter since it has a recursive representation. A generalization of Fatous lemma, which has wider applicability, is also proved.

A Theory of stopping Time Games with Applications to Product Innovations and Asset Sales

SummaryIn this paper, the pure strategy subgame perfect equilibria of a general class of stopping time games are studied. It is shown that there always exists a natural class of Markov Perfect Equilibria, called stopping equilibria. Such equilibria can be computed as a solution of a single agent stopping time problem, rather than of a fixed point problem. A complete characterization of stopping equilibria is presented. Conditions are given under which the outcomes of such equilibria span the set of all possible outcomes from perfect equilibria. Two economic applications of the theory, product innovations and the timing of asset sales, are discussed.

A game-theoretic approach to global warming

In the absence of a world government, stopping the advance of global warming requires implementation of self-enforcing treaties among the countries of the world. In the language of game theory, such treaties are Nash equilibria of an underlying dynamic “climate change game.” In this paper, we report on the progress of a project to formulate and analyze models of such a game. The players are the sovereign countries of the world (say the roughly 200 members of the United Nations). The rules of this game are determined by the laws of physics and chemistry, and by the economic resources of the various countries. An important property of our models is the large multiplicity of equilibria. Indeed, this property enables us to find “Pareto-improving” equilibria, i.e., that improve the outcome for every country relative to the “business-as-usual equilibrium” we seem to be in at the present time. In each model we describe the set of equilibria, the business-as-usual equilibrium, and equilibria that are Pareto-improving relative to business-as-usual. Since much of the global wanning is caused by the accumulation of greenhouse gases (GHGs) in the earth’s atmosphere, and the GHGs dissipate very slowly, an appropriate model must be in the form of a dynamic game, with state variables that change over time as a consequence of the actions of the individual countries. Thus, the state variables include the global stock of GHG and the state of the relevant technology in each country.

Optimal management of an R&D budget

Abstract This paper characterizes the optimal allocation of a budget between several stages of an R&D project. When intermediate stages are profitable, the optimal allocation involves greater expenditures in early stages. Increasing the budget does not necessarily increase expenditures in every stage but does increase the budget that remains unspent at each stage. A simple necessary and sufficient condition is given for the optimality of ‘bold play’ — the spending of all remaining budget on the current stage. If profits are realized only after all stages are successfully completed, decreasing returns in the innovation production function implies that the optimal strategy is to spread the budget evenly between the stages of the R&D project.

Parametric continuity in dynamic programming problems

Abstract We examine conditions under which the solutions to parametric families of dynamic programming problems are continuous in the parameters. Our main results are that parametric continuity obtains whenever either (a) the family of dynamic programming problems satisfies strong (joint-) continuity properties in the parameter and state or (b) if it satisfies weaker (separate-) continuity requirements, provided these are supplemented by either stronger assumptions on the transition probabilities or by monotonicity restrictions such as are common in economic modelling. The usefulness of our results is illustrated by applying them to some commonly used dynamic economic models.

Bankruptcy and expected utility maximization

Abstract This paper introduces a utility formulation to the well-known gamblers ruin problem. An agent who maximizes lifetime expected utility has to trade off short-term utility against longer-term survival prospects. The optimal tradeoff is established by way of characterizing the agents value and optimal policy functions. Further, the scope of expected utility maximization is examined by contrasting the bankruptcy probabilities of an agent employing such a criterion with those of an agent who is more directly interested in survival. Economic applications of the theory are also discussed.

Coordination need not be a problem

In a game of common interest there is one action vector that all players prefer to every other. Yet there may be multiple Pareto-ranked Nash equilibria in the game and the “coordination problem” refers to the fact that rational equilibrium play cannot rule out Pareto-dominated equilibria. In this paper, I prove that two elements — asynchronicity and a finite horizon — are sufficient to uniquely select the Pareto-dominant action vector (in subgame perfect equilibrium play). Asynchronicity may be exogenously specified by the rules of the game. Alternatively, in a game where players choose when to move, asynchronicity may emerge as an equilibrium move outcome.

On Continuity of the Utility Function in Intertemporal Allocation Models: An Example

A standard model of intertemporal allocation (described by a technology set and a welfare function defined on consumption) can be reduced to one described by a technology set and a utility function defined on this set. The authors present an example to show that even when the welfare function is concave, monotone, and continuous, the utility function can be discontinuous. They also provide sufficient conditions on the technology set and the welfare function under which the utility function is continuous. Their results indicate that the common practice of assuming continuity of the utility function is more restrictive than might be apparent. Copyright 1989 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.