Advertisement

Synthese

, Volume 194, Issue 9, pp 3393–3419 | Cite as

Do bets reveal beliefs?

A unified perspective on state-dependent utility issues
  • Jean BaccelliEmail author
Article

Abstract

This paper examines the preference-based approach to the identification of beliefs. It focuses on the main problem to which this approach is exposed, namely that of state-dependent utility. First, the problem is illustrated in full detail. Four types of state-dependent utility issues are distinguished. Second, a comprehensive strategy for identifying beliefs under state-dependent utility is presented and discussed. For the problem to be solved following this strategy, however, preferences need to extend beyond choices. We claim that this a necessary feature of any complete solution to the problem of state-dependent utility. We also argue that this is the main conceptual lesson to draw from it. We show that this lesson is of interest to both economists and philosophers.

Keywords

Beliefs Subjective probability State-dependent utility Hypothetical preferences Revealed preference 

Notes

Acknowledgments

The author thanks the Editor-in-Chief Wiebe van der Hoek, two referees, Mikaël Cozic, Éric Danan, Vincent Éli, Itzhak Gilboa, Raphaël Giraud, Brian Hill, Marcus Pivato, Fanyin Zheng and especially Philippe Mongin for most helpful comments and discussions. The author is sole responsible for all remaining errors or omissions. Research for this paper was funded by the Université de Cergy-Pontoise, and before that by the École Normale Supérieure - Ulm (grants ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL*).

References

  1. Allais, M. (1953). Le comportement de l’homme rationnel devant le risque : critique des postulats et axiomes de l’école américaine. Econometrica, 21(4), 503–546.CrossRefGoogle Scholar
  2. Anscombe, F., & Aumann, R. (1963). A definition of subjective probability. The Annals of Mathematical Statistics, 34(1), 199–205.CrossRefGoogle Scholar
  3. Arrow, K. (1953). Le rôle des valeurs boursières pour la répartition la meilleure des risques. Colloques Internationaux du Centre National de la Recherche Scientifique, Econometrie, 40, 41–58.Google Scholar
  4. Arrow, K. (1965). Aspects of the theory of risk-bearing. Helsinki: Yrjö Jahnssonin Säätiö.Google Scholar
  5. Arrow, K. (1973). Optimal insurance and generalized deductibles. Office of Economic Opportunity Report, R-1108-OEO.Google Scholar
  6. Aumann, R. (1976). Agreeing to disagree. The Annals of Statistics, 4(6), 1236–1239.CrossRefGoogle Scholar
  7. Bonanno, G., & Nehring, K. (1997). Agreeing to disagree: A survey. U.C.-Davis Department of Economics Working Paper No. 97-18.Google Scholar
  8. Bradley, R. (2004). Ramsey’s representation theorem. Dialectica, 58(4), 483–497.CrossRefGoogle Scholar
  9. Clemen, R., & Winkler, R. (2007). Aggregating probability distributions. In W. Edwards, R. Miles, & D. von Winterfeldt (Eds.), Advances in decision analysis (pp. 154–176). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  10. de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré, 7(1), 1–68.Google Scholar
  11. Drèze, J. (1961). Les fondements logiques de la probabilité subjective et de l’utilité (pp. 73–87). Paris: La décision, Colloques Internationaux du Centre National de la Recherche Scientifique.Google Scholar
  12. Drèze, J. (1987). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  13. Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 75(4), 643–669.CrossRefGoogle Scholar
  14. Finkelstein, A., Luttmer, E., & Notowidigdo, M. (2013). What good is wealth without health? the effect of health on the marginal utility of consumption. Journal of the European Economic Association, 11(1), 221–258.CrossRefGoogle Scholar
  15. Fishburn, P. (1970). Utility theory for decision making. New York: Wiley.CrossRefGoogle Scholar
  16. Fishburn, P. (1973). A mixture-set axiomatization of conditional subjective expected utility. Econometrica, 41(1), 1–25.CrossRefGoogle Scholar
  17. Genest, C., & Zidek, J. (1986). Combining probability distributions: A critique and an annotated bibliography. Statistical Science, 1(1), 114–135.CrossRefGoogle Scholar
  18. Gilboa, I. (1987). Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics, 16(1), 65–88.CrossRefGoogle Scholar
  19. Gilboa, I. (2009). Theory of decision under uncertainty. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  20. Gilboa, I., Samet, D., & Schmeidler, D. (2004). Utilitarian aggregation of beliefs and tastes. Journal of Political Economy, 112(4), 932–938.CrossRefGoogle Scholar
  21. Gollier, C. (1997). About the insurability of catastrophic risks. Geneva Papers on Risk and Insurance, 22(83), 177–186.CrossRefGoogle Scholar
  22. Good, I. (1950). Probability and the weighing of evidence. London: C. Griffin.Google Scholar
  23. Grant, S., & Karni, E. (2004). A theory of quantifiable beliefs. Journal of Mathematical Economics, 40(5), 515–546.CrossRefGoogle Scholar
  24. Hájek, A. (2008). Dutch book arguments. In P. Anand, P. Pattanaik, & C. Puppe (Eds.), The handbook of rational and social choice (pp. 173–196). Oxford: Oxford University Press.Google Scholar
  25. Hausman, D. (2000). Revealed preference, belief, and game theory. Economics and Philosophy, 16(1), 99–115.CrossRefGoogle Scholar
  26. Hill, B. (2009). When is there state independence? Journal of Economic Theory, 144(3), 1119–1134.CrossRefGoogle Scholar
  27. Hill, B. (2010). An additively separable representation in the Savage frameork. Journal of Economic Theory, 145(5), 2044–2054.CrossRefGoogle Scholar
  28. Horn, A., & Tarski, A. (1948). Measures in Boolean algebras. Transactions of the American Mathematical Society, 64(3), 467–497.CrossRefGoogle Scholar
  29. Joyce, J. (1998). A nonpragmatic vindication of probabilism. Philosophy of Science, 65(4), 575–603.CrossRefGoogle Scholar
  30. Joyce, J. (1999). The foundations of causal decision theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  31. Karni, E. (1985). Decision making under uncertainty: The case of state-dependent preferences. Cambridge: Harvard University Press.CrossRefGoogle Scholar
  32. Karni, E. (1996). Probabilities and beliefs. Journal of Risk and Uncertainty, 13(3), 249–262.CrossRefGoogle Scholar
  33. Karni, E. (1999). Elicitation of subjective probabilities when preferences are state-dependent. International Economic Review, 40(2), 479–486.CrossRefGoogle Scholar
  34. Karni, E. (2003). On the representation of beliefs by probabilities. Journal of Risk and Uncertainty, 26(1), 17–38.CrossRefGoogle Scholar
  35. Karni, E. (2007). Foundations of Bayesian theory. Journal of Economic Theory, 132(1), 167–188.CrossRefGoogle Scholar
  36. Karni, E. (2008). On optimal insurance in the presence of moral hazard. The Geneva Risk and Insurance Review, 33(1), 1–18.CrossRefGoogle Scholar
  37. Karni, E. (2009). A mechanism for eliciting probabilities. Econometrica, 77(2), 603–606.Google Scholar
  38. Karni, E. (2011a). Subjective probabilities on a state space. American Economic Journal: Microeconomics, 3(4), 172–185.Google Scholar
  39. Karni, E. (2011b). A theory of Bayesian decision making with action-dependent subjective probabilities. Economic Theory, 48(1), 125–146.CrossRefGoogle Scholar
  40. Karni, E., & Mongin, P. (2000). On the determination of subjective probability by choices. Management Science, 46(2), 233–248.CrossRefGoogle Scholar
  41. Karni, E., & Schmeidler, D. (1993). On the uniqueness of subjective probabilities. Economic Theory, 3(2), 267–277.CrossRefGoogle Scholar
  42. Karni, E., Schmeidler, D., & Vind, K. (1983). On state dependent preferences and subjective probabilities. Econometrica, 51(4), 1021–1031.CrossRefGoogle Scholar
  43. Karni, E., & Vierø, M.-L. (2013). Reverse Bayesianismø: A choice-based theory of growing awareness. The American Economic Review, 103(7), 2790–2810.CrossRefGoogle Scholar
  44. Koopman, B. (1940). The axioms and algebra of intuitive probability. The Annals of Mathematics, 41(2), 269–292.CrossRefGoogle Scholar
  45. Kraft, C., Pratt, J., & Seidenberg, A. (1959). Intuitive probability on finite sets. The Annals of Mathematical Statistics, 30(2), 408–419.CrossRefGoogle Scholar
  46. Luce, R., & Krantz, D. (1971). Conditional expected utility. Econometrica, 39(2), 253–271.CrossRefGoogle Scholar
  47. Machina, M., & Schmeidler, D. (1992). A more robust definition of subjective probability. Econometrica, 60(4), 745–780.CrossRefGoogle Scholar
  48. Mongin, P. (2001). The impartial observer theorem of social ethics. Economics and Philosophy, 17(2), 147–179.CrossRefGoogle Scholar
  49. Nau, R. (1995). Coherent decision analysis with inseparable probabilities and utilities. Journal of Risk and Uncertainty, 10(1), 71–91.CrossRefGoogle Scholar
  50. Nau, R. (2001). de Finetti was right: Probability does not exist. Theory and Decision, 51(2–4), 89–124.CrossRefGoogle Scholar
  51. Ramsey, F. (1931). Truth and probability. The Foundations of Mathematics and Other Logical Essays, pp. 156–198.Google Scholar
  52. Samuelson, P. (1950). The problem of integrability in utility theory. Economica, 17(68), 355–385.CrossRefGoogle Scholar
  53. Savage, L. (1954/1972). The foundations of statistics. New York: Dover.Google Scholar
  54. Savage, L., & Aumann, R. (1987). Letters between Leonard Savage and Robert Aumann (1971). In J. Drèze (Ed.), Essays on economic decisions under uncertainty (pp. 76–81). Cambridge: Cambridge University Press.Google Scholar
  55. Schervish, M., Seidenfeld, T., & Kadane, J. (1990). State-dependent utilities. Journal of the American Statistical Association, 85(411), 840–847.CrossRefGoogle Scholar
  56. Schlag, K., Tremewan, J., & van der Weele, J. (2014). A penny for your thoughts: A survey of methods for eliciting beliefs. University of Vienna Department of Economics Working Paper No. 1401.Google Scholar
  57. Sen, A. (1971). Choice functions and revealed preference. The Review of Economic Studies, 38(3), 307–317.CrossRefGoogle Scholar
  58. Sen, A. (1973). Behaviour and the concept of preference. Economica, 40(159), 241–259.CrossRefGoogle Scholar
  59. Wakker, P. (1987). Subjective probabilities for state dependent continuous utility. Mathematical Social Sciences, 14(3), 289–298.CrossRefGoogle Scholar
  60. Wakker, P. (1993). Clarification of some mathematical misunderstandings about Savage’s foundations of statistics, 1954. Mathematical Social Sciences, 25(2), 199–202.CrossRefGoogle Scholar
  61. Wakker, P. (1994). Separating marginal utility and probabilistic risk aversion. Theory and Decision, 36(1), 1–44.CrossRefGoogle Scholar
  62. Wakker, P., & Zank, H. (1999). State dependent expected utility for Savage’s state space. Mathematics of Operations Research, 24(1), 8–34.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.THEMAUniversité de Cergy-PontoiseCergy-PontoiseFrance

Personalised recommendations