Morris H. DeGroot

Reaching a Consensus

Abstract Consider a group of individuals who must act together as a team or committee, and suppose that each individual in the group has his own subjective probability distribution for the unknown value of some parameter. A model is presented which describes how the group might reach agreement on a common subjective probability distribution for the parameter by pooling their individual opinions. The process leading to the consensus is explicitly described and the common distribution that is reached is explicitly determined. The model can also be applied to problems of reaching a consensus when the opinion of each member of the group is represented simply as a point estimate of the parameter rather than as a probability distribution.

Rational Expectations and Bayesian Analysis

The approach in this paper is the development of models that describe the process by which rational expectations may be developed within a market. The concept of Bayesian learning is introduced. Consistent and rational expectations are introduced in models where the firms cannot immediately move to equilibrium. Three different models are developed which demonstrate the interaction of Bayesian learning and expectations in achieving a market equilibrium. These models are dynamic and describe the transition process toward equilibrium. Two of the models involve unknown parameters about which the firms learn. The third is a control theory model explicitly involving adjustment costs.

Multiperiod Decision Models with Alternating Choice as a Solution to the Duopoly Problem

In the previous chapter we illustrated some of the ways that Bayesian analysis can be used in duopoly theory. Duopoly and oligopoly theory is characterized by the fact that an infinite number of models can be generated by assuming different values for the conjectural variations term (Cohen and Cyert 1975; Kamien and Schwartz 1983). No general solution exists and there is no basis, either empirical or theoretical, for preferring one of the models over the other.

Doing What Comes Naturally: Interpreting a Tail Area as a Posterior Probability or as a Likelihood Ratio

Abstract Consider a problem in which a certain statistic X has a specified distribution function F(x) if a given hypothesis H is true, and suppose that the hypothesis H is evaluated by calculating the tail area 1 – F(x) corresponding to the observed value x of the statistic X. Examples are given in which this tail area is equal to the posterior probability that H is true and in which it is equal to the likelihood ratio comparing H to a certain class K of alternatives. The purpose of these examples is to render the traditional statistical practice of calculating tail areas consonant with the principles of Bayesian statistics.

Personal probabilities of probabilities

By definition, the subjective probability distribution of a random event is revealed by the (‘rational’) subjects choice between bets — a view expressed by F. Ramsey, B. De Finetti, L. J. Savage and traceable to E. Borel and, it can be argued, to T. Bayes. Since hypotheses are not observable events, no bet can be made, and paid off, on a hypothesis. The subjective probability distribution of hypotheses (or of a parameter, as in the current ‘Bayesian’ statistical literature) is therefore a figure of speech, an ‘as if’, justifiable in the limit. Given a long sequence of previous observations, the subjective posterior probabilities of events still to be observed are derived by using a mathematical expression that would approximate the subjective probability distribution of hypotheses, if these could be bet on. This position was taken by most, but not all, respondents to a ‘Round Robin’ initiated by J. Marschak after M. H. De-Groots talk on Stopping Rules presented at the UCLA Interdisciplinary Colloquium on Mathematics in Behavioral Sciences. Other participants: K. Borch, H. Chernoif, R. Dorfman, W. Edwards, T. S. Ferguson, G. Graves, K. Miyasawa, P. Randolph, L. J. Savage, R. Schlaifer, R. L. Winkler. Attention is also drawn to K. Borchs article in this issue.

Bayesian Analysis and Duopoly Theory

Duopoly theory has a long history in economics and a distinguished list of names associated with that history (Cournot 1897; Frisch 1951; Stackelberg 1952). Nevertheless, the problem has proved to be a frustrating one for economists. The obvious reason for the difficulty is the uncertainty that characterizes the problem. The specific uncertainty revolves around the interrelationship of the two firms and the fact that the decisions of one of them affect the other. Solutions have consisted of finding plausible (or implausible) behavioral assumptions that effectively eliminate the uncertainty. Economists have developed duopoly and oligopoly theory through the years by making different assumptions which produce models explaining some regularity believed to exist in duopoly or oligopoly markets (Bishop 1960).

Information about Hyperparameters in Hierarchical Models

Abstract We consider situations in which the prior distribution of a parameter vector θ1 in the distribution of an observable random vector X contains a hyperparameter vector θ2. The experimenter specifies another distribution for θ2 that contains hyperparameters θ3, and so forth. One wants to learn about the hyperparameters at each level of this hierarchical model. We show that for many measures of information, the gain in information decreases as one moves to higher levels of hyperparameters. These results are illustrated for univariate normal models and a general linear hierarchical model. Examples of measures of information are given for which this property does not hold.

Modeling lake-chemistry distributions: approximate Bayesian methods for estimating a finite-mixture model

A modification of the Laplace method is presented and applied to estimation of posterior functions in a Bayesian analysis of finite-mixture distributions. The technique is nonsequential yet relatively fast and provides estimates of mixture-model parameters and classification probabilities. The method is applied to a regional distribution of lake-chemistry data for north central Wisconsin. A mixture density of two lognormal populations is estimated for the acid-neutralizing capacity of lakes in the region, using several other lake characteristics as explanatory variables for classification into lake subpopulations. The fitted mixture model provides a good representation of the observed distribution. Separation into subpopulations based solely on the other lake characteristics matches the mixture-model classification relatively well.

Interfirm learning and the kinked demand curve

The last three chapters have been concerned primarily with duopoly models. In this chapter we apply Bayesian analysis to an oligopoly model known as the kinked demand curve. The kinked demand curve represents a theoretical dilemma that is not uncommon in the social sciences (Stigler 1978). The dilemma concerns the inclusion or the exclusion of the model from the mixed bag represented by the classification “oligopoly theory.” The model has been in the literature for a number of years and is still in an ambiguous position. Sweezy (1939) proposed the model as an explanation of rigid oligopoly prices, which were taken as an empirical observation (see also Hall and Hitch 1939). The basic assumption underlying the kinked demand curve is that rivals will not follow an attempted increase in price by one of the firms but will follow a decrease. The result is that for each firm the portion of the demand curve above the current price is elastic and the portion below the curve is inelastic. Hence, in the firm’s view, the demand curve appears kinked at the current price and the firm has no incentive to modify its price. Because of the paucity of good alternatives, the model was quickly accepted as the theory of oligopoly by many textbook writers. The theory did not explain how oligopoly prices reached a particular level, but it did offer an explanation of their stability.

Bayesian Analysis of Selection Models

On considere des problemes dans lesquels les observations peuvent etre obtenues seulement a partir de certaines portions selectionnees de la population