An Ambiguity Measure under EUUP and Its Application to a Portfolio Problem
- 1 Faculty of Business Administration, Kyoto Sangyo University, Kyoto, Japan
Abstract
This paper derives a measure that quantifies the degree of ambiguity under expected utility with uncertain probability (EUUP) by [1] . Here, ambiguity means a situation in which the first-order probabilities, i.e., the probabilities of the states of nature, are not given uniquely, but as random variables. Because EUUP can completely distinguish attitudes toward both risk and ambiguity from beliefs and risk from ambiguity, the derived ambiguity measure is independent of risk and attitudes toward both risk and ambiguity. We show that the degree of ambiguity can be measured by the variance of the first-order probabilities. Although [2] also derives an ambiguity measure based on the variance of the first-order probabilities, our measure is more flexible, and it discriminates between ambiguity in favorable outcomes and in unfavorable ones. Based on the measure, we also discuss effects of ambiguity on a problem of portfolio selection through comparative statics.
- Izhakian, Y. (2017) Expected Utility with Uncertain Probability Theory. Journal of Mathematical Economics, 69, 91-103. https://doi.org/10.1016/j.jmateco.2016.12.004
- Izhakian, Y. (2020) A Theoretical Foundation of Ambiguity Measurement. Journal of Economic Theory, 187, Article ID: 105001. https://doi.org/10.1016/j.jet.2020.105001
- Wakker, P. (2010) Prospect Theory: For Risk and Ambiguity. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511779329
- Knight, F.M. (1921) Risk, Uncertainty and Profit. Houghton Mifflin, Boston.
- Schmeidler, D. (1989) Subjective Probability and Expected Utility without Additivity. Econometrica, 57, 571-587. https://doi.org/10.2307/1911053
- Gilboa, I. and Schmeidler, D. (1989) Maxmin Expected Utility with Non-Unique Prior. Journal of Mathematical Economics, 18, 141-153. https://doi.org/10.1016/0304-4068(89)90018-9
- Tversky, A. and Kahneman, D. (1992) Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty, 5, 297-323. https://doi.org/10.1007/BF00122574
- Ghirardato, P., Klibanoff, P. and Marinacci, M. (1998) Additivity with Multiple Priors. Journal of Mathematical Economics, 30, 405-420. https://doi.org/10.1016/S0304-4068(97)00047-5
- Klibanoff, P., Marinacci, M. and Mukerji, S. (2005) A Smooth Model of Decision Making under Ambiguity. Econometrica, 73, 1849-1892. https://doi.org/10.1111/j.1468-0262.2005.00640.x
- Kocher, M., Lahno, A.M. and Trautmann, S. (2018) Ambiguity Aversion Is Not Universal. European Economic Review, 101, 268-283. https://doi.org/10.1016/j.euroecorev.2017.09.016
- Trautmann, S. and Wakker, P. (2018) Making the Anscombe-Aumann Approach to Ambiguity Suitable for Descriptive Applications. Journal of Risk and Uncertainty, 56, 83-116. https://doi.org/10.1007/s11166-018-9273-7
- Bossaerts, P., Paolo Ghirardato, P., Guarnaschelli, S. and Zame, W.R. (2010) Ambiguity in Asset Markets: Theory and Experiment. The Review of Financial Studies, 23, 1325-1359. https://doi.org/10.1093/rfs/hhp106
- Dow, J. and Werlang, S.R.d.C. (1992) Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio. Econometrica, 60, 197-204. https://doi.org/10.2307/2951685
- Baillon, A., Huang, Z., Selim, A. and Wakker, P.P. (2018) Measuring Ambiguity Attitudes for All (Natural) Events. Econometrica, 86, 1839-1858. https://doi.org/10.3982/ECTA14370