Mahmoud A. El-Gamal

Are People Bayesian? Uncovering Behavioral Strategies

Abstract Economists and psychologists have recently been developing new theories of decision making under uncertainty that can accommodate the observed violations of standard statistical decision theoretic axioms by experimental subjects. We propose a procedure that finds a collection of decision rules that best explain the behavior of experimental subjects. The procedure is a combination of maximum likelihood estimation of the rules together with an implicit classification of subjects to the various rules and a penalty for having too many rules. We apply our procedure to data on probabilistic updating by subjects in four different universities. We get remarkably robust results showing that the most important rules used by the subjects (in order of importance) are Bayess rule, a representativeness rule (ignoring the prior), and, to a lesser extent, conservatism (overweighting the prior).

A Bayesian sequential experimental study of learning in games

Abstract We apply a sequential Bayesian sampling procedure to study two models of learning in repeated games. In the first model individuals learn only about an opponent when they play her or him repeatedly but do not update from their experience with that opponent when they move on to play the same game with other opponents. We label this the nonsequential model. In the second model individuals use Bayesian updating to learn about population parameters from each of their opponents, as well as learning about the idiosyncrasies of that particular opponent. We call this the sequential model. We sequentially sample observations on the behavior of experimental subjects in the so-called “centipede game.” This game allows for a trade-off between competition and cooperation, which is of interest in many economic situations. At each point in time, the “state” of our dynamic problem consists of our beliefs about the two models and beliefs about the nuisance parameters of the two models. Our “choice” set is to samp...

Bayesian economists... Bayesian agents: An alternative approach to optimal learning

We generalize the results on Bayesian learning based on the martingale convergence theorem to the sequential framework. We show that the variability in the sequential framework is sufficient under mild conditions to circumvent the incomplete learning results that characterize the optimal learning literature. We then give an alternative approach whereby the economist is Bayesian with a prior on the space of agent priors. We illustrate the usefulness of our approach by applying it to two popular economic examples: a monopolist who does not know the demand curve he faces, and the stochastic single-sector growth model with an unknown production function.

Economical experiments: Bayesian efficient experimental design

We propose and implement a Bayesian optimal design procedure. Our procedure takes as its primitives a class of parametric models of strategic behavior, a class of games (experimental designs), and priors on the behavioral parameters. We select the experimental design that maximizes the information from the experiment. We sequentially sample with the given design and models until only one of the models has viable posterior odds. A model which has low posterior odds in a small collection of models will have an even lower posterior odds when compared to a larger class, and hence we can dismiss it. The procedure can be used sequentially by introducing new models and comparing them to the models that survived earlier rounds of experiments. The emphasis is not on running as many experiments as possible, but rather on choosing experimental designs to distinguish between models in the shortest possible time period. We illustrate this procedure with a simple experimental game with one-sided incomplete information.

Investment and Growth in the Middle East and North Africa

The paper considers investment and growth in the Middle East and North Africa (MENA) region. Notwithstanding cross-country differences, investment as a whole has been too low, too heavily tilted toward the public sector, too highly dependent on external influences, and less productive than in many other regions. Improving the region’s investment performance is critical if policymakers are to succeed in increasing the region’s economic growth rate. After discussing the relationship between investment and growth, the paper analyzes the investment responsiveness of various countries in the region and notes the policy priorities for strengthening the basis for rapid and sustained economic growth.

A Consistent Test of Stationary Ergodicity

A formal statistical test of stationary-ergodicity is developed for known Markovian processes on null null This makes it applicable to testing models and algorithms, as well as estimated time series processes ignoring the estimation error. The analysis is conducted by examining the asymptotic properties of the Markov operator on density space generated by the transition in the state space. The test is developed under the null of stationary-ergodicity, and it is shown to be consistent against the alternative of nonstationary-ergodicity. The test can be easily performed using any of a number of standard statistical and mathematical computer packages.

Learning in experimental games

SummaryExperimental games typically involve subjects playing the same game a number of times. In the absence of perfect rationality by all players, the subjects may use the behavior of their opponents in early rounds to learn about the extent of irrationality in the population they face. This makes the problem of finding the Bayes-Nash equilibrium of the experimental game much more complicated than finding the game-theoretic solution to the ideal game without irrationality. We propose and implement a computationally intensive algorithm for finding the equilibria of complicated games with irrationality via the minimization of an appropriate multi-variate function. We propose two hypotheses about how agents learn when playing experimental games. The first posits that they tend to learn about each opponent as they play it repeatedly, but do not learn about the population parameters through their observations of random opponents (myopic learning). The second posits that both types of learning take place (sequential learning). We introduce a computationally intensive sequential procedure to decide on the informational value of conducting additional experiments. With the help of that procedure, we decided after 12 experiments that our original model of irrationality was unsatisfactory for the purpose of discriminating between our two hypotheses. We changed our models, allowing for two different types of irrationality, reanalyzed the old data, and conducted 7 more experiments. The new model successfully discriminated between our two hypotheses about learning. After only 7 more experiments, our approximately optimal stopping rule led us to stop sampling and accept the model where both types of learning occur.

The Role of Priors in Active Bayesian Learning in the Sequential Statistical Decision Framework

This chapter introduces the notions of rational expectations and optimal learning extensively used in economic theory.* It has become well known from recent literature that in active learning situations (where the actions of the statistician, or the person learning about some parameters, influences the draws from the distribution about which he/she is learning), full learning may not take place. This challenges the use of the rational expectations hypothesis which is justified on the basis that agents operating in an economy eventually all learn the true structure of the economy and optimize accordingly. In El-Gamal and Sundaram (1989, 1990) we presented a framework where a Bayesian economist imposes priors on agent-priors and we then study the evolution of those economist beliefs. We showed that generically, the economist limit beliefs generically do not have point mass at any particular agent-belief, let alone the true rational expectations belief. We show, however, that in most cases where there is sufficient variability in the law of motion that the agents are trying to learn, in sequential models that are extensively used in the economic literature, the rational expectations hypothesis may indeed be justified on the basis of optimizing and optimally updating agents.

The Extraction of Information From Multiple Point Estimates

The use of a number of point estimation experiments to construct a least informative prior subject to the information in the estimation experiments is studied. The form of the resulting prior is established. The prior depends on parameters that result from solving a calculus of variations problem. It is shown that simple Gibbs sampler algorithms converge to the desired solution, and simulated annealing algorithms yield the mode of the prior. The Gibbs sampler algorithm is ergodic, and hence Bayes risks can be directly computed using time averages of a single series of draws from the sampler.

Non-parametric estimation of deterministically chaotic systems*

SummaryThis paper studies theoretical and econometric issues that arise in systems characterized by deterministic chaos. Such systems can arise from standard dynamic economic models and are extensively used in Monte Carlo and other simulation-based statistical procedures which use pseudo-random number generators. The virtues of studying chaotic laws of motion in the space of densities over the state space are shown. A complete characterization of deterministic stationary ergodic processes in that space of densities is suggested and proved when the invariant measure is unknown. The asymptotic properties of the kernel estimators of the stationary density and the law of motion in the density space are studied, and shown to hold for chaotic systems. Small sample behavior for the estimators is subjectively shown to be good even when optimal choices of the kernel and smoothing parameters are not exploited.