Expected utility analysis of infinite compound lotteries

ABSTRACT Lotteries can be used to model alternatives with uncertain outcomes. Decision theory uses compound ordinary lotteries to represent a structure of lotteries within lotteries, but can only rank the finite compound lottery structure. We expand upon this approach to introduce solutions for infinite compound ordinary lotteries (ICOL). We describe a novel procedure to simplify any ICOL as much as possible to a maximum reduced ICOL, which is not a unique representation. We limit our discussion to ICOLs of first order, which are defined as maximum reduced ICOLs with a single maximum reduced ICOL in their direct outcome. Two special cases of ICOLs of first order are discussed. These are recursive and semi-recursive ICOLs. We provide an analytical approach to find the expected utility of recursive ICOLs, and a numerical algorithm for semi-recursive ICOLs. We demonstrate our solution methods by evaluating example decision problems involving: a randomizing device with unsuccessful trials, the St. Petersburg paradox, and training with virtual reality.

[1]  Ralph L. Keeney,et al.  Decisions with multiple objectives: preferences and value tradeoffs , 1976 .

[2]  Daniel John Zizzo.  Choices Between Simple and Compound Lotteries: Experimental Evidence and Neural Network Modelling , 2001 .

[3]  Anatol Rapoport,et al.  Decision Theory and Decision Behaviour: Normative and Descriptive Approaches , 2010 .

[4]  J. Quiggin.  Generalized expected utility theory : the rank-dependent model , 1994 .

[5]  R. Clemen,et al.  Soft Computing , 2002 .

[6]  K. Tenekedjiev,et al.  Local Risk Proneness in Analytically Approximated Utility Functions under Monotonically Decreasing Preferences , 2018 .

[7]  R. L. Keeney,et al.  Decisions with Multiple Objectives: Preferences and Value Trade-Offs , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[8]  Mark R. McCord,et al.  Lottery Equivalents: Reduction of the Certainty Effect Problem in Utility Assessment , 1986 .

[9]  Babak Daneshvar Rouyendegh,et al.  Multi‐criteria decision making approach for evaluation of the performance of computer programming languages in higher education , 2018, Comput. Appl. Eng. Educ..

[10]  Colin Camerer,et al.  Recent developments in modeling preferences: Uncertainty and ambiguity , 1992 .

[11]  Thomas L. Saaty,et al.  Models, Methods, Concepts & Applications of the Analytic Hierarchy Process , 2012 .

[12]  M. Brunelli.  Introduction to the Analytic Hierarchy Process , 2014 .

[13]  S. R. Jammalamadaka,et al.  Against the Gods: The Remarkable Story of Risk , 1999 .

[14]  Glenn W. Harrison,et al.  Reduction of Compound Lotteries with Objective Probabilities: Theory and Evidence , 2015 .

[15]  O. Peters,et al.  The time resolution of the St Petersburg paradox , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  S. French.  Decision Theory: An Introduction to the Mathematics of Rationality , 1986 .

[17]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[18]  S. R. Jammalamadaka,et al.  Against the Gods: The Remarkable Story of Risk , 1999 .

[19]  A. Hama.  Predictably Irrational: The Hidden Forces That Shape Our Decisions , 2010 .

[20]  S. Stearns.  Daniel Bernoulli (1738): evolution and economics under risk , 2000, Journal of Biosciences.

[21]  A. Tversky,et al.  Advances in prospect theory: Cumulative representation of uncertainty , 1992 .

[22]  P. Wakker,et al.  Eliciting von Neumann-Morgenstern Utilities When Probabilities Are Distorted or Unknown , 1996 .

[23]  H. Raiffa,et al.  Introduction to Statistical Decision Theory , 1996 .

[24]  Evangelos Triantaphyllou,et al.  Multi-criteria Decision Making Methods: A Comparative Study , 2000 .

[25]  Kiril Tenekedjiev,et al.  Fuzzy rationality and parameter elicitation in decision analysis , 2010, Int. J. Gen. Syst..

[26]  Kannan Govindan,et al.  ELECTRE: A comprehensive literature review on methodologies and applications , 2016, Eur. J. Oper. Res..

[27]  Songfa Zhong,et al.  An Experimental Study of Attitude towards Compound Lottery , 2012 .

[28]  Kiril Tenekedjiev,et al.  Ranking discrete outcome alternatives with partially quantified uncertainty , 2008, Int. J. Gen. Syst..