Knowledge that Transforms

economic models ; economic theory

This paper studies the class of denumerable-armed (i.e. finite- or countably infinitearmed) bandit problems with independent arms and geometric discounting over an infinite horizon, in which each arm generates rewards according to one of a finite number of distributions, or "types." The number of types in the support of an arm, as also the types themselves, are allowed to vary across the arms. We derive certain continuity and curvature properties of the dynamic allocation (or Gittins) index of Gittins and Jones (1974), and provide necessary and sufficient conditions under which the Gittins-Jones result identifying all optimal strategies for finite-armed bandits may be extended to infinite-armed bandits. We then establish our central result: at each point in time, the arm selected by an optimal strategy will, with strictly positive probability, remain an optimal selection forever. More specifically, for every such arm, there exists (at least) one type of that arm such that, when conditioned on that type being the arm's "true" type, the arm will survive forever and continuously with nonzero probability. When the reward distributions of an arm satisfy the monotone likelihood ratio property (MLRP), the survival prospects of an arm improve when conditioned on types generating higher expected rewards; however, we show how this need not be the case in the absence of MLRP. Implications of these results are derived for the theories of job search and matching, as well as other applications of the bandit paradigm.

Much macroeconometric discussion has recently emphasized the economic significance of the size of the permanent component in GNP.Consequently, a large literature has developed that tries to estimate this magnitude-measured, essentially, as the spectral density of increments in GNP at frequency zero.This paper shows that unless the permanent component is a random walk this attention has been misplaced: in general, that quantity does not identify the magnitude of the permanent component.Further, by developing bounds on reasonable measures of this magnitude, the paper shows that a random walk specification is biased towards establishing the permanent component as important.

This paper describes a semiparametric estimator for binary response models in which there may be arbitrary heteroskedasticity of unknown form. The estimator is obtained by maximizing a smoothed version of the objective function of C. Manski's maximum score estimator. The smoothing procedure is similar to that used in kernel nonparametric density estimation. The resulting estimator's rate of convergence in probability is the fastest possible under the assumptions that are made. The centered, normalized estimator is asymptotically normally distributed. Methods are given for consistently estimating the parameters of the limiting distribution and for selecting the bandwidth required by the smoothing procedure. Copyright 1992 by The Econometric Society.

THERE HAS BEEN MUCH INTEREST over the years in estimating discrete choice models. However, with the exception of multinomial logit models, agents have been restricted to few choices so that multivariate integrals could be feasibly integrated numerically.2 Pakes and Pollard (1989) and McFadden (1989) have independently developed the method of simulated moments (MSM) to deal with estimating a wide class of models of which high order discrete choice models are a subset. One of the problems a researcher must handle when using MSM for a discrete choice model is how to smooth the discrete simulated random variables. This is necessary to keep small the number of draws required to simulate derivatives and to simulate variation in the data. Geweke (1989) and McFadden (1989) have suggested importance sampling methods (and other methods) to smooth the simulated variables for general error structures. In this paper, I present a factor analytic smoothing method that can be applied to probit problems. Although it is not as general as the methods Geweke and McFadden suggest, it is easy to use and has clear intuition. Furthermore, simulated probabilities will be unbiased, will be bounded between zero and unity, and will have smaller variances than unsmoothed probabilities always and smaller variances than importance sampling estimates for a large class of probabilities. The second section of this paper develops notation, defines the problem, and presents the smoothing method and an algorithm to employ it. The last section presents Monte Carlo comparisons of the smoothing method to the importance sampling method.

The authors consider the problem of cost sharing in the case of a fixed group of agents sharing a one input, one output technology with decreasing returns. They introduce and analyze the serial cost sharing method. Among agents endowed with convex and monotonic preferences, serial cost sharing is dominance solvable and its unique Nash equilibrium is also robust to coalitional deviations. The authors show that no other smooth cost sharing mechanism yields a unique Nash equilibrium at all preference profiles. Copyright 1992 by The Econometric Society.