Annals of Operations Research

, Volume 19, Issue 1, pp 229–246 | Cite as

An overview of lexicographic choice under uncertainty

  • Lawrence Blume
  • Adam Brandenburger
  • Eddie Dekel
Part IV New Results In Nonlinear Preference Theory


This overview focuses on lexicographic choice under conditions of uncertainty. First, lexicographic versions of traditional (von Neumann-Morgenstern) expected utility theory are described where the usual Archimedean axiom is weakened. The role of these lexicographic variants in explaining some well-known “paradoxes” of choice theory is reviewed. Next, the significance of lexicographic choice for game theory is discussed. Finally, some lexicographic extensions of the classical maximin decision rule are described.


Game Theory Decision Rule Choice Theory Utility Theory Expect Utility Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    M. Allais, Le comportement de l'homme rationnel devant le risque: Critique des postulats et axioms de l'ecole Americaine, Econometrica 21(1953)503.Google Scholar
  2. [2]
    F. Anscombe and R. Aumann, A definition of subjective probability, Ann. Math. Stat. 34(1963)199.Google Scholar
  3. [3]
    K. Arrow and L. Hurwicz, An optimality criterion for decision-making under ignorance, in:Uncertainty and Expectations in Economics, ed. C. Carter and J. Ford (Augustus M. Kelley, New Jersey, 1972).Google Scholar
  4. [3]a
    R. Aumann, Utility theory without the completeness axiom, Econometrica 30(1962)445.Google Scholar
  5. [4]
    S. Barbera and M. Jackson, Maximin, leximin, and the protective criterion: Characterizations and comparisons, J. Economic Theory, forthcoming.Google Scholar
  6. [5]
    R. Barrett and P. Pattanaik, Decision-making under complete uncertainty, Research Memorandum 1/86, Department of Economics, University of Birmingham (1986).Google Scholar
  7. [6]
    T. Bewley, Knightian decision theory: parts I and II, Cowles Foundation Discussion Paper Nos. 807 and 835, Cowles Foundation, Yale University (1986 – 87).Google Scholar
  8. [7]
    L. Blume, A. Brandenburger and E. Dekel, Lexicographic probabilities and equilibrium refinements, I and II (1988), in preparation.Google Scholar
  9. [8]
    S. Chew, A generalization of the quasi-linear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox, Econometrica 57 (1983)1065.Google Scholar
  10. [9]
    J. Chipman, The foundations of utility, Econometrica 28(1960)193.Google Scholar
  11. [10]
    J. Chipman, On the lexicographic representation of preference orderings, in:Preference, Utility, and Demand, ed. J. Chipman, L. Hurwicz, M. Richter and H. Sonnenschein (Harcourt Brace Jovanovich, New York, 1971).Google Scholar
  12. [11]
    J. Chipman, Non-Archimedean behavior under risk: An elementary analysis — with application to the theory of assets, in:Preference, Utility, and Demand, ed. J. Chipman, L. Hurwicz, M. Richter and H. Sonnenschein (Harcourt Brace Jovanovich, New York, 1971).Google Scholar
  13. [12]
    M. Cohen and J.-Y. Jaffray, Rational behavior under ignorance, Economctric 48(1980)1281.Google Scholar
  14. [13]
    M. Cohen and J.-Y. Jaffray, Approximations of rational criteria under complete ignorance and the independence axiom, Theory and Decision 15(1983)121.Google Scholar
  15. [14]
    N. Dalkey, Equivalence of information patterns and essentially determinate games, in:Contributions to the Theory of Games, II, ed. H. Kuhn and A. Tucker, Annals of Mathematical Studies 28 (Princeton University Press, Princeton, 1953).Google Scholar
  16. [15]
    G. Debreu, Representation of a preference ordering by a numerical function, in:Decision Processes, ed. R. Thrall, C. Coombs and R. Davis (Wiley, New York, 1954).Google Scholar
  17. [16]
    E. Dekel, An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom, J. Economic Theory 40(1986)304.Google Scholar
  18. [17]
    S. Elmes and P. Reny, The equivalence of games with perfect recall, Department of Economics, Princeton University (1987), unpublished.Google Scholar
  19. [18]
    J. Encarnacion, Constraints and the firm's utility function, Review of Economic Studies 31(1964)113.Google Scholar
  20. [19]
    J. Encarnacion, On decisions under uncertainty, Economic Journal 75(1965)442.Google Scholar
  21. [20]
    C. Ferguson, An essay on cardinal utility, Southern Economic Journal 25(1958)11.Google Scholar
  22. [21]
    C. Ferguson, The theory of multidimensional utility analysis in relation to multiple-goal business behavior: A synthesis, Southern Economic Journal 32(1965)169.Google Scholar
  23. [22]
    P. Fishburn, A study of lexicographic expected utility, Manag. Sci. 17(1971)672.Google Scholar
  24. [23]
    P. Fishburn, Lexicographic orders, utilities, and decision rules: A survey, Manag. Sci. 20(1974)1442.Google Scholar
  25. [24]
    P. Fishburn,The Foundations of Expected Utility (Reidel, Dordrecht, 1982).Google Scholar
  26. [25]
    P. Fishburn, Transitive measurable utility, J. Economic Theory 31(1983)293.Google Scholar
  27. [26]
    M. Friedman and L. Savage, The utility analysis of choices involving risk, J. Political Economy 56(1948)279.Google Scholar
  28. [27]
    I. Gilboa, A combination of expected utility and maximin decision criteria, Working Paper No. 12–86, Foerder Institute for Economic Research, Tel-Aviv University (1986).Google Scholar
  29. [28]
    P. Hammond, Extended probabilities for decision theory and games, Department of Economics, Stanford University (1987), unpublished.Google Scholar
  30. [29]
    M. Hausner, Multidimensional utilities, in:Decision Processes, ed. R. Thrall, C. Coombs and R. Davis (Wiley, New York, 1954).Google Scholar
  31. [30]
    M. Hausner and J. Wendel, Ordered vector spaces, Proc. Amer. Math. Soc. 3(1952)977.Google Scholar
  32. [31]
    D. Kelsey, Decision making under partial uncertainty, Churchill College, Cambridge (1988), unpublished.Google Scholar
  33. [32]
    E. Kohlberg and J.-F. Mertens, On the strategic stability of equilibria, Econometrica 54(1986)1003.Google Scholar
  34. [33]
    D. Kreps and G. Ramey, Structural consistency, consistency, and sequential rationality, Econometrica 55(1987)1331.Google Scholar
  35. [34]
    D. Kreps and R. Wilson, Sequential equilibria, Econometrica 50(1982)863.Google Scholar
  36. [35]
    R. Luce and H. Raiffa,Games and Decisions (Wiley, New York, 1957).Google Scholar
  37. [36]
    M. Machina, Expected utility analysis without the independence axiom, Econometrica 50(1982)277.Google Scholar
  38. [37]
    E. Maskin, Decision-making under ignorance with implications for social choice, Theory and Decision 11(1979)319.Google Scholar
  39. [38]
    A. McLennan, Consistent conditional systems in noncooperative game theory, Discussion Paper 5217-86, Mathematical Sciences Research Institute, University of California at Berkeley (1986).Google Scholar
  40. [39]
    J.-F. Mertens, Ordinality in non cooperative games, CORE Discussion Paper 8728, Université Catholique de Louvain (1987).Google Scholar
  41. [40]
    J. Milnor, Games against nature, in:Decision Processes, ed. R. Thrall, C. Coombs and R. Davis (Wiley, New York, 1954).Google Scholar
  42. [41]
    R. Myerson, Refinements of the Nash equilibrium concept, Int. J. Game Theory 7(1978)73.Google Scholar
  43. [42]
    R. Myerson, Multistage games with communication, Econometrica 54(1986)323.Google Scholar
  44. [43]
    R. Myerson, Axiomatic foundations of Bayesian decison theory, Discussion Paper 671, J.L. Kellogg Graduate School of Management, Northwestern University (1986).Google Scholar
  45. [44]
    R. Myerson, Acceptable and predominant correlated equilibria, Int. J. Game Theory 15 (1986).Google Scholar
  46. [45]
    L. Narens,Abstract Measurement Theory (MIT Press, Cambridge, MA, 1985).Google Scholar
  47. [46]
    M. Richter, Rationa choice, in:Preference, Utility, and Demand, ed. J. Chipman, L. Hurwicz, M. Richter and H. Sonnenschein (Harcourt Brace Jovanovich, New York, 1971).Google Scholar
  48. [47]
    A. Rubinstein, Equilibrium in supergames with the overtaking criterion, J. Economic Theory 21(1979)21.Google Scholar
  49. [48]
    L. Savage,The Foundations of Statistics (Wiley, New York, 1954).Google Scholar
  50. [49]
    R. Selten, Spieltheoretische behandlung eines oligopolmodels mit nachfragetragheit, Zeitschrift für die Gesamte Staatswissenschaft 121(1965)401; 667.Google Scholar
  51. [50]
    R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. Game Theory4(1975)25.Google Scholar
  52. [51]
    H. Skala,Non-Archimedean Utility Theory (Reidel, Dordrecht, 1975).Google Scholar
  53. [52]
    F. Thompson, Equivalence of games in extensive form, Research Memorandum RM-759, The RAND Corporation, Santa Monica (1952).Google Scholar
  54. [53]
    R. Thrall, Applications of multidimensional utility theory, in:Decision Processes, ed. R. Thrall, C. Coombs and R. Davis (Wiley, New York, 1954).Google Scholar
  55. [54]
    J. von Neumann and O. Morgenstern,Theory of Games and Economic Behavior, 2nd ed. (Princeton University Press, Princeton, 1947).Google Scholar
  56. [55]
    M. Yaari, The dual theory of choice under risk, Econometrica 55(1987)95.Google Scholar

Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1989

Authors and Affiliations

  • Lawrence Blume
    • 1
  • Adam Brandenburger
    • 2
  • Eddie Dekel
    • 3
  1. 1.Department of EconomicsUniversity of MichiganAnn ArborUSA
  2. 2.Harvard Business SchoolBostonUSA
  3. 3.Department of EconomicsUniversity of CaliforniaBerkeleyUSA

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