Consequentialism, Non-Archimedean Probabilities, and Lexicographic Expected Utility

Earlier work (Hammond, 1988a, b) on dynamically consistent “consequentialist” behaviour in decision trees was unable to treat zero probability events satisfactorily. Here the rational probability functions considered in Hammond (1992), as well as other non-Archimedean probabilities, are incorporated into decision trees. As before, the consequentialist axioms imply the existence of a preference ordering satisfying independence. In the case of rational probability functions, those axioms, together with continuity and a new refinement assumption, imply the maximization of a somewhat novel lexicographic expected utility preference relation. This is equivalent to maximization of expected utility in the ordering of the relevant non-Archimedean field.

[1]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[2]  P. Hammond.  Consequentialist foundations for expected utility , 1988 .

[3]  Eddie Dekel,et al.  Lexicographic Probabilities and Equilibrium Refinements , 1991 .

[4]  Niels-Erik Jensen.  An Introduction to Bernoullian Utility Theory: I. Utility Functions , 1967 .

[5]  P. Hammond.  Elementary Non-Archimedean Representations of Probability for Decision Theory and Games , 1994 .

[6]  R. Selten.  Reexamination of the perfectness concept for equilibrium points in extensive games , 1975, Classics in Game Theory.

[7]  R. Myerson.  MULTISTAGE GAMES WITH COMMUNICATION , 1984 .

[8]  Peter J. Hammond,et al.  Consequentialism and the Independence Axiom , 1988 .

[9]  Seymour Geisser,et al.  Introduction to Probability and Statistics from a Bayesian Viewpoint Part 1. Probability , 1966 .

[10]  P. Whittle,et al.  Introduction to Probability and Statistics from a Bayesian Viewpoint, Part 1: Probability , 1969 .

[11]  Eddie Dekel,et al.  Lexicographic Probabilities and Choice Under Uncertainty , 1991 .

[12]  Abraham Robinson,et al.  Function Theory on Some Nonarchimedean Fields , 1973 .

[13]  A. Rényi.  On a new axiomatic theory of probability , 1955 .

[14]  J. Milnor,et al.  AN AXIOMATIC APPROACH TO MEASURABLE UTILITY , 1953 .

[15]  Ira R. Buchler,et al.  Game Theory in the Behavioral Sciences , 1969 .

[16]  A. McLennan.  Consistent conditional systems in noncooperative game theory , 1989 .

[17]  William Mendenhall,et al.  Introduction to Probability and Statistics , 1961, The Mathematical Gazette.

[18]  Peter J. Hammond,et al.  Changing Tastes and Coherent Dynamic Choice , 1976 .

[19]  Paul A. Samuelson,et al.  Probability, Utility, and the Independence Axiom , 1952 .

[20]  Howard Raiffa,et al.  Decision analysis: introductory lectures on choices under uncertainty. 1968. , 1969, M.D.Computing.

[21]  Andrew McLennan,et al.  The space of conditional systems is a ball , 1989 .

[22]  E. Damme.  Refinements of the Nash Equilibrium Concept , 1983 .