# Piecewise additivity for non-expected utility

Research Article

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## Abstract

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.

## Keywords

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

D81## References

- Abdellaoui, M., Wakker, P.P.: The likelihood method for decision under uncertainty. Theory Decis.
**58**, 3–76 (2005)CrossRefGoogle Scholar - 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 - 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 - 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 - Arrow, K.: Essays in the Theory of Risk-Bearing. Markham Publishing Company, Chicago (1970)Google Scholar
- Chateauneuf, A., Maccheroni, F., Marinacci, M., Tallon, J.-M.: Monotone continuous multiple priors. Econ. Theory
**26**, 973–982 (2005)CrossRefGoogle Scholar - 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 - Chew, S.H., Sagi, J.S.: Small worlds: modeling attitudes toward sources of uncertainty. J. Econ. Theory
**139**, 1–24 (2008)CrossRefGoogle Scholar - Chow, C.C., Sarin, R.K.: Comparative ignorance and the Ellsberg paradox. J. Risk Uncertain.
**22**(2), 129–139 (2001)CrossRefGoogle Scholar - Cohen, M.: Security level, potential level, expected utility: a three-criteria decision model under risk. Theory Decis.
**33**, 101–134 (1992)CrossRefGoogle Scholar - Dominiak, A., Lefort, J.-P.: Agreement theorem for neo-additive beliefs. Econ. Theory
**52**, 1–13 (2013)CrossRefGoogle Scholar - Dominiak, A., Eichberger, J., Lefort, J.-P.: Agreeable trade with pessimism and optimism. Math. Soc. Sci.
**46**, 119–126 (2012)CrossRefGoogle Scholar - Eichberger, J., Kelsey, D.: Are the treasures of game theory ambiguous? Econ. Theory
**48**, 313–339 (2011)CrossRefGoogle Scholar - Eichberger, J., Kelsey, D.: Optimism and pessimism in games. Int. Econ. Rev.
**55**, 483–505 (2014)CrossRefGoogle Scholar - Eichberger, J., Grant, S., Lefort, J.-P.: Generalized neo-additive capacities and updating. Int. J. Econ. Theory
**8**(3), 237–257 (2012)CrossRefGoogle Scholar - Ellsberg, D.: Risk, ambiguity and the Savage axioms. QJE
**75**, 643–669 (1961)CrossRefGoogle Scholar - Essid, S.: Choice under risk with certainty and potential effects: a general axiomatic model. Math. Soc. Sci.
**34**, 223–247 (1997)CrossRefGoogle Scholar - 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 - Fox, C.R., Tversky, A.: Ambiguity aversion and comparative ignorance. QJE
**110**, 585–603 (1995)CrossRefGoogle Scholar - Ghirardato, P., Maccheroni, F., Marinacci, M., Siniscalchi, M.: A subjective spin on roulette wheels. Econometrica
**71**(6), 1897–1908 (2003)CrossRefGoogle Scholar - 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
- Gilboa, I.: Expected utility with purely subjective nonadditive probabilities. J. Math. Econ.
**16**, 65–88 (1987)CrossRefGoogle Scholar - Gilboa, I.: A combination of expected utility and maxmin decision criteria. J. Math. Psychol.
**32**, 405–420 (1988)CrossRefGoogle Scholar - Heath, C., Tversky, A.: Preference and belief: ambiguity and competence in choice under uncertainty. J. Risk Uncertain.
**4**, 5–28 (1991)CrossRefGoogle Scholar - Jaffray, J.-Y.: Choice under risk and the security factor: an axiomatic model. Theory Decis.
**24**, 169–200 (1988)CrossRefGoogle Scholar - Kopylov, I.: Subjective probabilities on “small” domains. J. Econ. Theory
**133**, 236–265 (2007)CrossRefGoogle Scholar - 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
- Lopes, L.L.: Between hope and fear: the psychology of risk. Adv. Exp. Psychol.
**20**, 255–295 (1987)CrossRefGoogle Scholar - 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 - Machina, M.J., Schmeidler, D.: A more robust definition of subjective probability. Econometrica
**60**, 745–780 (1992)CrossRefGoogle Scholar - 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
- Romm, A.T.: An interpretation of focal point responses as non-additive beliefs. Judgm. Decis. Mak.
**9**(5), 387–402 (2014)Google Scholar - Savage, L.J.: The Foundation of Statistics. Wiley, New York (1954)Google Scholar
- Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica
**57**, 571–587 (1989)CrossRefGoogle Scholar - Schmidt, U., Zimper, A.: Security and potential preferences with thresholds. J. Math. Psychol.
**51**, 279–289 (2007)CrossRefGoogle Scholar - Teitelbaum, J.C.: A unilateral accident model under ambiguity. J. Leg. Stud.
**36**, 431–477 (2007)CrossRefGoogle Scholar - Villegas, C.: On qualitative probability \(\sigma \)-algebras. Ann. Math. Stat.
**35**, 1787–1796 (1964)CrossRefGoogle Scholar - Wakker, P.P.: Continuous subjective expected utility with nonadditive probabilities. J. Math. Econ.
**18**, 1–27 (1989)CrossRefGoogle Scholar - Wakker, P.P.: Additive representations on rank-ordered sets. I. The algebraic approach. J. Math. Psychol.
**35**, 501–531 (1991)CrossRefGoogle Scholar - Wakker, P.P.: Clarification of some mathematical misunderstandings about Savage’s foundations of statistics, 1954. Math. Soc. Sci.
**25**, 199–202 (1993)CrossRefGoogle Scholar - 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 - Wakker, P.P.: Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle. Econometrica
**69**, 1039–1059 (2001)CrossRefGoogle Scholar - 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 - Wakker, P.P.: Prospect Theory. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
- Webb, C.S., Zank, H.: Accounting for optimism and pessimism in expected utility. J. Math. Econ.
**47**(6), 706–717 (2011)CrossRefGoogle Scholar - 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

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