Preference Structures and Their Numerical Representations

Abstract This paper presents a selective survey of numerical representations of preference structures from the perspective of the representational theory of measurement. It reviews historical contributions to ordinal, additive, and expected utility theories, then describes recent contributions in these areas.

[1]  Henry E. Kyburg,et al.  Studies in Subjective Probability , 1965 .

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

[3]  Peter C. Fishburn,et al.  A Nonlinear, Nontransitive and Additive-Probability Model for Decisions Under Uncertainty , 1987 .

[4]  P. Fishburn,et al.  Rank- and sign-dependent linear utility models for finite first-order gambles , 1991 .

[5]  A. Tversky Intransitivity of preferences. , 1969 .

[6]  P. Vincke LINEAR UTILITY FUNCTIONS ON SEMIORDERED MIXTURE SPACES , 1980 .

[7]  P. Fishburn Cancellation Conditions for Multiattribute Preferences on Finite Sets , 1997 .

[8]  P. Fishburn,et al.  State-independent subjective expected lexicographic utility , 1992 .

[9]  M. Machina "Expected Utility" Analysis without the Independence Axiom , 1982 .

[10]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[11]  P. Fishburn Nontransitive preferences in decision theory , 1991 .

[12]  M. A. Croon The Axiomatization of Additive Difference Models for Preference Judgments , 1984 .

[13]  P. Fishburn Subjective expected utility: A review of normative theories , 1981 .

[14]  D. Scott Measurement structures and linear inequalities , 1964 .

[15]  R. Thaler,et al.  Gambling with the house money and trying to break even: the effects of prior outcomes on risky choice , 1990 .

[16]  Peter C. Fishburn,et al.  Joint receipt and Thaler's hedonic editing rule , 1995 .

[17]  G. W. Fischer,et al.  Preferences for separating or combining events. , 1991, Journal of personality and social psychology.

[18]  Peter C. Fishburn,et al.  Finite linear qualitative probability , 1996 .

[19]  Yutaka Nakamura Expected utility with an interval ordered structure , 1988 .

[20]  E. Degreef,et al.  Trends in mathematical psychology , 1984 .

[21]  C. L. Sheng A Theory of Value , 1991 .

[22]  P. Fishburn,et al.  Utility theory , 1980, Cognitive Choice Modeling.

[23]  C. Kraft,et al.  Intuitive Probability on Finite Sets , 1959 .

[24]  Kennan T. Smith,et al.  Linear Topological Spaces , 1966 .

[25]  A. Tversky,et al.  Rational choice and the framing of decisions , 1990 .

[26]  R. Sugden,et al.  Some implications of a more general form of regret theory , 1987 .

[27]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[28]  P. Fishburn Nontransitive additive conjoint measurement , 1991 .

[29]  A. Tversky,et al.  An axiomatization of cumulative prospect theory , 1993 .

[30]  Patrick Suppes,et al.  Foundational aspects of theories of measurement , 1958, Journal of Symbolic Logic.

[31]  D. Schmeidler Subjective Probability and Expected Utility without Additivity , 1989 .

[32]  Begoña Subiza,et al.  Demand correspondence for pseudo-transitive preferences , 1995 .

[33]  Peter C. Fishburn,et al.  Human Decision Making and Ordered Sets , 1989 .

[34]  B. Peleg UTILITY FUNCTIONS FOR PARTIALLY ORDERED TOPOLOGICAL SPACES. , 1970 .

[35]  G. Debreu,et al.  Theory of Value , 1959 .

[36]  H. Raiffa,et al.  The Foundations of Decision under Uncertainty: An Elementary Exposition , 1964 .

[37]  D. Stanley Tarbell,et al.  Colloques Internationaux du Centre National de la Recherche Scientifique. , 1958 .

[38]  Peter C. Fishburn,et al.  Nontransitive measurable utility with constant threshold , 1991 .

[39]  Jutta Mitas Interval orders based on arbitrary ordered sets , 1995, Discret. Math..

[40]  J. Quiggin,et al.  Generalized Expected Utility Theory. The Rank-Dependent Model , 1996 .

[41]  Kenneth P. Bogart,et al.  Bipartite tolerance orders , 1994, Discret. Math..

[42]  E. Malinvaud,et al.  Note on von Neumann-Morgenstern's Strong Independence Axiom , 1952 .

[43]  G. Shafer,et al.  Expected Utility Hypotheses and the Allais Paradox. , 1982 .

[44]  A. B. Freeman,et al.  State-Independent Subjective Expected Lexicographic Utility , 2022 .

[45]  F. J. Anscombe,et al.  A Definition of Subjective Probability , 1963 .

[46]  Sergiu Hart,et al.  Games in extensive and strategic forms , 1992 .

[47]  P. Vincke,et al.  Biorder families, valued relations and preference modelling , 1986 .

[48]  R. M. Adelson,et al.  Utility Theory for Decision Making , 1971 .

[49]  B. D. Finetti La prévision : ses lois logiques, ses sources subjectives , 1937 .

[50]  K. May Intransitivity, Utility, and the Aggregation of Preference Patterns , 1954 .

[51]  Peter C. Fishburn,et al.  Interprofile Conditions And Impossibility , 1987 .

[52]  P. Vincke,et al.  Preference structures and threshold models , 1993 .

[53]  M. Allais Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .

[54]  Peter C. Fishburn,et al.  Failure of cancellation conditions for additive linear orders , 1997 .

[55]  Peter C. Fishbur A general axiomatization of additive measurement with applications , 1992 .

[56]  A. V. Balakrishnan Control theory and the calculus of variations , 1969 .

[57]  Paul Slovic,et al.  Importance of variance preferences in gambling decisions , 1968 .

[58]  Peter C. Fishburn,et al.  The Invention of the Independence Condition for Preferences , 1995 .

[59]  P. Fishburn,et al.  TWO‐PIECE VON NEUMANN‐MORGENSTERN UTILITY FUNCTIONS* , 1979 .

[60]  Peter C. Fishburn,et al.  Uniqueness in Finite Measurement , 1989 .

[61]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[62]  I. Rival Algorithms and Order , 1988 .

[63]  P. Fishburn Transitive measurable utility , 1983 .

[64]  Peter C. Fishburn,et al.  Additive differences and simple preference comparisons , 1992 .

[65]  R. Luce,et al.  Tests of assumptions about the joint receipt of gambles in rank- and sign-dependent utility theory. , 1994 .

[66]  A. Tversky,et al.  Prospect theory: analysis of decision under risk , 1979 .

[67]  D. Bernoulli Exposition of a New Theory on the Measurement of Risk , 1954 .

[68]  V. Klee SEPARATION AND SUPPORT PROPERTIES OF CONVEX SETS: A SURVEY. , 1969 .

[69]  Peter C. Fishburn,et al.  Continuous nontransitive additive conjoint measurement , 1990 .

[70]  P. Fishburn,et al.  A note on deriving rank-dependent utility using additive joint receipts , 1995 .

[71]  B. Finetti Sul significato soggettivo della probabilità , 1931 .

[72]  Irving H. Lavalle,et al.  Structuring and assessing matrix-probability distributions , 1996 .

[73]  Peter C. Fishburn,et al.  Nonlinear preference and utility theory , 1988 .

[74]  P. Fishburn,et al.  On matrix probabilities in nonarchimedean decision theory , 1993 .

[75]  Jaap Van Brakel,et al.  Foundations of measurement , 1983 .

[76]  E. Szpilrajn Sur l'extension de l'ordre partiel , 1930 .

[77]  P. Fishburn,et al.  Structuring and assessing linear lexicographic utility , 1996 .

[78]  M. Kane Measurement theory. , 1980, NLN publications.

[79]  Young-Hee Cho,et al.  Tests of Hypotheses about Certainty Equivalents and Joint Receipt of Gambles , 1995 .

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

[81]  P. Fishburn,et al.  Lexicographic state-dependent subjective expected utility , 1991 .

[82]  Fred S. Roberts,et al.  Applications of combinatorics and graph theory to the biological and social sciences , 1989 .

[83]  Louis Narens,et al.  Classification of concatenation measurement structures according to scale type , 1985 .

[84]  A. Tversky,et al.  Prospect Theory : An Analysis of Decision under Risk Author ( s ) : , 2007 .

[85]  A. Chateauneuf Continuous representation of a preference relation on a connected topological space , 1987 .

[86]  Moncef Abbas Any complete preference structure without circuit admits an interval representation , 1995 .

[87]  K. MacCrimmon,et al.  Utility Theory: Axioms Versus ‘Paradoxes’ , 1979 .

[88]  Peter C. Fishburn,et al.  Utility and subjective probability , 1994 .

[89]  Peter P. Wakker,et al.  Additive Representations of Preferences , 1989 .

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

[91]  I. Gilboa Expected utility with purely subjective non-additive probabilities , 1987 .

[92]  P. Fishburn Preference-based Definitions of Subjective Probability , 1967 .

[93]  G. Bosi,et al.  Representing preferences with nontransitive indifference by a single real-valued function☆ , 1995 .

[94]  P.-C.-F. Daunou,et al.  Mémoire sur les élections au scrutin , 1803 .

[95]  J. Leeuw,et al.  Abstract Measurement Theory. , 1986 .

[96]  Jonathan Barzilai,et al.  On the foundations of measurement , 2001, 2001 IEEE International Conference on Systems, Man and Cybernetics. e-Systems and e-Man for Cybernetics in Cyberspace (Cat.No.01CH37236).

[97]  R. Duncan Luce,et al.  Rank- and sign-dependent linear utility models for binary gambles , 1991 .

[98]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[99]  R. Duncan Luce Joint receipt and certainty equivalents of gambles , 1995 .

[100]  I. Gilboa,et al.  Numerical representations of imperfectly ordered preferences (a unified geometric exposition) , 1992 .

[101]  D. Sondermann Utility representations for partial orders , 1980 .

[102]  Niels Erik Jensen,et al.  An Introduction to Bernoullian Utility Theory: II. Interpretation, Evaluation and Application; A Critical Survey , 1967 .

[103]  Peter C. Fishburn,et al.  On the varieties of matrix probabilities in nonarchimedean decision theory , 1996 .

[104]  S. Chew A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox , 1983 .

[105]  J. Neumann,et al.  Theory of Games and Economic Behavior. , 1945 .

[106]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .