Economic Theory

, Volume 60, Issue 2, pp 371–392 | Cite as

Piecewise additivity for non-expected utility

  • Craig S. Webb
Research Article


A model of choice under purely subjective uncertainty, Piecewise Additive Choquet Expected utility, is introduced. PACE utility allows for optimism and pessimism simultaneously, but represents a minimal departure from expected utility. It can be seen as a continuous version of NEO-expected utility (Chateauneuf et al. in J Econ Theory 137:538–567, 2007) and, as such, is especially suited for applications with rich state spaces. The main theorem provides a preference foundation for PACE utility in the Savage framework of purely subjective uncertainty with an arbitrary outcome set.


Optimism Pessimism Inverse-S Choquet expected utility NEO-additive capacities Probabilistic sophistication 

JEL Classification



  1. Abdellaoui, M., Wakker, P.P.: The likelihood method for decision under uncertainty. Theory Decis. 58, 3–76 (2005)CrossRefGoogle Scholar
  2. Abdellaoui, M., L’Haridon, O., Zank, H.: Separating curvature and elevation: a parametric probability weighting function. J. Risk Uncertain. 4, 39–65 (2010)CrossRefGoogle Scholar
  3. Abdellaoui, M., Baillon, A., Placido, L., Wakker, P.P.: The rich domain of uncertainty: source functions and their experimental implementation. Am. Econ. Rev. 101, 695–723 (2011)CrossRefGoogle Scholar
  4. Allais, M.: Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine. Econometrica 21, 503–546 (1953)CrossRefGoogle Scholar
  5. Arrow, K.: Essays in the Theory of Risk-Bearing. Markham Publishing Company, Chicago (1970)Google Scholar
  6. Chateauneuf, A., Maccheroni, F., Marinacci, M., Tallon, J.-M.: Monotone continuous multiple priors. Econ. Theory 26, 973–982 (2005)CrossRefGoogle Scholar
  7. Chateauneuf, A., Eichberger, J., Grant, S.: Choice under uncertainty with best and worst in mind: NEO-additive capacities. J. Econ. Theory 137, 538–567 (2007)CrossRefGoogle Scholar
  8. Chew, S.H., Sagi, J.S.: Small worlds: modeling attitudes toward sources of uncertainty. J. Econ. Theory 139, 1–24 (2008)CrossRefGoogle Scholar
  9. Chow, C.C., Sarin, R.K.: Comparative ignorance and the Ellsberg paradox. J. Risk Uncertain. 22(2), 129–139 (2001)CrossRefGoogle Scholar
  10. Cohen, M.: Security level, potential level, expected utility: a three-criteria decision model under risk. Theory Decis. 33, 101–134 (1992)CrossRefGoogle Scholar
  11. Dominiak, A., Lefort, J.-P.: Agreement theorem for neo-additive beliefs. Econ. Theory 52, 1–13 (2013)CrossRefGoogle Scholar
  12. Dominiak, A., Eichberger, J., Lefort, J.-P.: Agreeable trade with pessimism and optimism. Math. Soc. Sci. 46, 119–126 (2012)CrossRefGoogle Scholar
  13. Eichberger, J., Kelsey, D.: Are the treasures of game theory ambiguous? Econ. Theory 48, 313–339 (2011)CrossRefGoogle Scholar
  14. Eichberger, J., Kelsey, D.: Optimism and pessimism in games. Int. Econ. Rev. 55, 483–505 (2014)CrossRefGoogle Scholar
  15. Eichberger, J., Grant, S., Lefort, J.-P.: Generalized neo-additive capacities and updating. Int. J. Econ. Theory 8(3), 237–257 (2012)CrossRefGoogle Scholar
  16. Ellsberg, D.: Risk, ambiguity and the Savage axioms. QJE 75, 643–669 (1961)CrossRefGoogle Scholar
  17. Essid, S.: Choice under risk with certainty and potential effects: a general axiomatic model. Math. Soc. Sci. 34, 223–247 (1997)CrossRefGoogle Scholar
  18. Ford, J., Kelsey, D., Pang, W.: Information and ambiguity: herd and contrarian behaviour in financial markets. Theory Decis. 75(1), 1–15 (2013)CrossRefGoogle Scholar
  19. Fox, C.R., Tversky, A.: Ambiguity aversion and comparative ignorance. QJE 110, 585–603 (1995)CrossRefGoogle Scholar
  20. Ghirardato, P., Maccheroni, F., Marinacci, M., Siniscalchi, M.: A subjective spin on roulette wheels. Econometrica 71(6), 1897–1908 (2003)CrossRefGoogle Scholar
  21. Gilboa, I.: Subjective Distortions of Probabilities and Non-additive Probabilities. Working paper 18-85, Foerder Institute for Economic Research. Tel-Aviv University, Ramat Aviv, Israel (1985)Google Scholar
  22. Gilboa, I.: Expected utility with purely subjective nonadditive probabilities. J. Math. Econ. 16, 65–88 (1987)CrossRefGoogle Scholar
  23. Gilboa, I.: A combination of expected utility and maxmin decision criteria. J. Math. Psychol. 32, 405–420 (1988)CrossRefGoogle Scholar
  24. Heath, C., Tversky, A.: Preference and belief: ambiguity and competence in choice under uncertainty. J. Risk Uncertain. 4, 5–28 (1991)CrossRefGoogle Scholar
  25. Jaffray, J.-Y.: Choice under risk and the security factor: an axiomatic model. Theory Decis. 24, 169–200 (1988)CrossRefGoogle Scholar
  26. Kopylov, I.: Subjective probabilities on “small” domains. J. Econ. Theory 133, 236–265 (2007)CrossRefGoogle Scholar
  27. Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement, vol. 1 (Additive and Polynomial Representations). Academic Press, New York (1971)Google Scholar
  28. Lopes, L.L.: Between hope and fear: the psychology of risk. Adv. Exp. Psychol. 20, 255–295 (1987)CrossRefGoogle Scholar
  29. Ludwig, A., Zimper, A.: Biased Bayesian learning with an application to the risk-free rate puzzle. J. Econ. Dyn. Control 39, 79–97 (2014)CrossRefGoogle Scholar
  30. Machina, M.J., Schmeidler, D.: A more robust definition of subjective probability. Econometrica 60, 745–780 (1992)CrossRefGoogle Scholar
  31. Parry, M.L., Canziani, O.F., Palutikof, J.P., et al.: Technical summary. In: Parry, M.L., Canziani, O.F., Palutikof, J.P., van der Linden, P.J., Hanson, C.E. (eds.) Climate Change 2007: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, pp. 23–78. Cambridge University Press, Cambridge (2007)Google Scholar
  32. Romm, A.T.: An interpretation of focal point responses as non-additive beliefs. Judgm. Decis. Mak. 9(5), 387–402 (2014)Google Scholar
  33. Savage, L.J.: The Foundation of Statistics. Wiley, New York (1954)Google Scholar
  34. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)CrossRefGoogle Scholar
  35. Schmidt, U., Zimper, A.: Security and potential preferences with thresholds. J. Math. Psychol. 51, 279–289 (2007)CrossRefGoogle Scholar
  36. Teitelbaum, J.C.: A unilateral accident model under ambiguity. J. Leg. Stud. 36, 431–477 (2007)CrossRefGoogle Scholar
  37. Villegas, C.: On qualitative probability \(\sigma \)-algebras. Ann. Math. Stat. 35, 1787–1796 (1964)CrossRefGoogle Scholar
  38. Wakker, P.P.: Continuous subjective expected utility with nonadditive probabilities. J. Math. Econ. 18, 1–27 (1989)CrossRefGoogle Scholar
  39. Wakker, P.P.: Additive representations on rank-ordered sets. I. The algebraic approach. J. Math. Psychol. 35, 501–531 (1991)CrossRefGoogle Scholar
  40. Wakker, P.P.: Clarification of some mathematical misunderstandings about Savage’s foundations of statistics, 1954. Math. Soc. Sci. 25, 199–202 (1993)CrossRefGoogle Scholar
  41. Wakker, P.P.: The sure-thing principle and the comonotonic sure-thing principle: an axiomatic analysis. J. Math. Econ. 25, 213–227 (1996)CrossRefGoogle Scholar
  42. Wakker, P.P.: Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle. Econometrica 69, 1039–1059 (2001)CrossRefGoogle Scholar
  43. Wakker, P.P.: Decision-foundations for properties of nonadditive measures; general state spaces or general outcome spaces. Games Econ. Behav. 50, 107–125 (2005)CrossRefGoogle Scholar
  44. Wakker, P.P.: Prospect Theory. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  45. Webb, C.S., Zank, H.: Accounting for optimism and pessimism in expected utility. J. Math. Econ. 47(6), 706–717 (2011)CrossRefGoogle Scholar
  46. Zimper, A.: Asset pricing in a Lucas fruit tree economy with the best and worst in mind. J. Econ. Dyn. Control 36(4), 610–628 (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Economics, School of Social SciencesThe University of ManchesterManchesterUK

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