David K. Hildebrand

Prediction logic: A method for empirical evaluation of formal theory†

This paper proposes an approach to data analysis that assists the investigator in discriminating among specific relations corresponding to alternative scientific predictions about qualitative variates.

Lord's MAD Paradox and Jensen's Inequality

Abstract Lord (1983) showed that the mean absolute deviation (MAD) measure of dispersion does not always increase when an additional independent source of random variation is added. Jensens inequality is used to find necessary and sufficient conditions for this paradoxical behavior. The conditions are unlikely to occur in practice.

On the uses of probablistic thinking in game theory

Abstract The purpose of this essay is to interpret the solutions of 2-person, 0-sum, complete-information games from the viewpoint and with the methods of the mathematical theory of probability, in contrast with the more common “linear algebra” approach. The approach presented here is often stated and occasionally applied in Decision Theory [DeGroot, M. H., Optimal Statistical Decisions , McGraw-Hill, New York (1970)]. Here, although the principal concern is with a formalism, an attempt will be made to interpret some of the terms of the “Behavioural” theory of games with the aid of the formal scheme. The following question is raised: If “strategies” are called probabilities, then what is the associated measure-space, as demanded by the mathematical theory of probability? The following answer is given: “outcomes” constitute a fundamental measure-space. It is shown that measures on “acts”-space and on “outcomes”-space are in a natural, mutual correspondence; also, we show that the two players optimal measures on the respective “marginal” “act”-spaces may be taken as stochastically independent . This latter point is usually tacitly assumed in most presentations. To illustrate the usefulness of such an approach, we give a proof that, in a sequence of plays, the overall optimal strategies are independent sequences of individual strategies.