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Open AccessOpen Access JUSTC Mathematics 22 November 2022

Second-order stochastic dominance with respect to the rank-dependent utility model

  • Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2022-0097
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  • Author Bio:

    Qinyu Wu is a Ph.D. candidate at the School of Management, University of Science and Technology of China. His research interests focus on risk management and mathematical finance

  • Corresponding author: E-mail: wu051555@mail.ustc.edu.cn
  • Received Date: 16 July 2022
  • Accepted Date: 18 August 2022
    • Available Online: 22 November 2022
    Abstract

  • Abstract

    A generalized family of partial orders is studied, which is semiparametrized by a utility function $ f $ and a distortion function $ g $ , namely, $ (f,g) $ -utility and distorted stochastic dominance ( $ (f,g) $ -UDSD). Such a family is especially suitable for representing a decision maker’s preferences in terms of risk aversion. We characterize the monotonicity of the partial order in the rank-dependent utility model, and the isotonic classes of rank-dependent utility with $ (f,g) $ -UDSD are also established. Inspired by the concept of the congruent utility class, we introduce the definition of the congruent distortion class. The characterization of the congruent utility class or distortion class of $ (f,g) $ -utility and distorted stochastic dominance is investigated. Based on the main results in this paper, we unify some related results in the existing literature. As an application, we propose a general approach to develop a continuum between first-order stochastic dominance and second-order stochastic dominance based on the partial order.

    Graphical abstract

    A new stochastic dominance based on a fixed utility function and a fixed distortion function.

    Abstract

    A generalized family of partial orders is studied, which is semiparametrized by a utility function $ f $ and a distortion function $ g $ , namely, $ (f,g) $ -utility and distorted stochastic dominance ( $ (f,g) $ -UDSD). Such a family is especially suitable for representing a decision maker’s preferences in terms of risk aversion. We characterize the monotonicity of the partial order in the rank-dependent utility model, and the isotonic classes of rank-dependent utility with $ (f,g) $ -UDSD are also established. Inspired by the concept of the congruent utility class, we introduce the definition of the congruent distortion class. The characterization of the congruent utility class or distortion class of $ (f,g) $ -utility and distorted stochastic dominance is investigated. Based on the main results in this paper, we unify some related results in the existing literature. As an application, we propose a general approach to develop a continuum between first-order stochastic dominance and second-order stochastic dominance based on the partial order.

    • Public Summary

    • We introduce a new generalized class of partial orders that contains some rules of stochastic dominance as its special cases.
    • We investigate the properties and characterizations of the new partial order through the RDU model.
    • The characterization of utility congruency or distortion congruency of the new partial order is presented.
    • We provide a general approach to interpolate first-order stochastic dominance and second-order stochastic dominance based on the new partial order.

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    • References(21)
  • [1]
    Yaari M E. The dual theory of choice under risk. Econometrica, 1987, 9: 5–115. doi: 10.2307/1911158
    [2]
    Friedman M, Savage L J. The utility analysis of choices involving risk. Journal of Political Economy, 1948, 56 (4): 279–304. doi: 10.1086/256692
    [3]
    Markowitz H. The utility of wealth. Journal of Political Economy, 1952, 60 (2): 151–158. doi: 10.1086/257177
    [4]
    Leshno M, Levy H. Preferred by “all” and preferred by “most” decision makers: Almost stochastic dominance. Management Science, 2002, 48 (8): 1074–1085. doi: 10.1287/mnsc.48.8.1074.169
    [5]
    Meyer J. Second degree stochastic dominance with respect to a function. International Economic Review, 1977, 47: 7–487. doi: 10.2307/2525760
    [6]
    Pratt J W. Risk aversion in the small and in the large. Econometrica, 1964, 32: 122–136. doi: 10.2307/1913738
    [7]
    Lando T, Bertoli-Barsotti L. Distorted stochastic dominance: A generalized family of stochastic orders. Journal of Mathematical Economics, 2020, 90: 132–139. doi: 10.1016/j.jmateco.2020.07.005
    [8]
    Quiggin J. A theory of anticipated utility. Journal of Economic Behavior and Organization, 1982, 3: 323–343. doi: 10.1016/0167-2681(82)90008-7
    [9]
    Fishburn P C. Continua of stochastic dominance relations for bounded probability distributions. Journal of Mathematical Economics, 1976, 3 (3): 295–311. doi: 10.1016/0304-4068(76)90015-x
    [10]
    Tsetlin I, Winkler R L, Huang R J, et al. Generalized almost stochastic dominance. Operations Research, 2015, 63 (2): 363–377. doi: 10.1287/opre.2014.1340
    [11]
    Müller A, Scarsini M, Tsetlin I, et al. Between first- and second-order stochastic dominance. Management Science, 2017, 63 (9): 2933–2947. doi: 10.1287/mnsc.2016.2486
    [12]
    Huang R J, Tzeng L Y, Zhao L. Fractional degree stochastic dominance. Management Science, 2020, 66 (10): 4630–4647. doi: 10.1287/mnsc.2019.3406
    [13]
    Thistle P D. Ranking distributions with generalized Lorenz curves. Southern Economic Journal, 1989, 56 (1): 1–12. doi: 10.2307/1059050
    [14]
    Shaked M, Shanthikumar J G. Stochastic Orders. New York: Springer, 2007.
    [15]
    Liu P, Schied A, Wang R. Distributional transforms, probability distortions, and their applications. Mathematics of Operations Research, 2021, 46 (4): 1490–1512. doi: 10.1287/moor.2020.1090
    [16]
    Chew S H, Karni E, Safra Z. Risk aversion in the theory of expected utility with rank dependent probabilities. Journal of Economic Theory, 1987, 42: 370–381. doi: 10.1016/0022-0531(87)90093-7
    [17]
    Wang S S, Young V R. Ordering risks: Expected utility theory versus Yaari’s dual theory of risk. Insurance:Mathematics and Economics, 1998, 22 (2): 145–161. doi: 10.1016/s0167-6687(97)00036-x
    [18]
    Mao T, Wu Q, Hu T. Further properties of fractional stochastic dominance. Journal of Applied Probability, 2022, 59 (1): 202–223. doi: 10.1017/jpr.2021.44
    [19]
    Mao T, Wang R. A critical comparison of three notions of fractional stochastic dominance. 2020. https://ssrn.com/abstract=3642983. Accessed July 1, 2022.
    [20]
    Levy H, Wiener Z. Stochastic dominance and prospect dominance with subjective weighting functions. Journal of Risk and Uncertainty, 1998, 16 (2): 147–163. doi: 10.1023/a:1007730226688
    [21]
    Müller A, Stoyan D. Comparison Methods for Stochastic Models and Risks. Hoboken, NJ: Wiley, 2022.
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    [1]
    Yaari M E. The dual theory of choice under risk. Econometrica, 1987, 9: 5–115. doi: 10.2307/1911158
    [2]
    Friedman M, Savage L J. The utility analysis of choices involving risk. Journal of Political Economy, 1948, 56 (4): 279–304. doi: 10.1086/256692
    [3]
    Markowitz H. The utility of wealth. Journal of Political Economy, 1952, 60 (2): 151–158. doi: 10.1086/257177
    [4]
    Leshno M, Levy H. Preferred by “all” and preferred by “most” decision makers: Almost stochastic dominance. Management Science, 2002, 48 (8): 1074–1085. doi: 10.1287/mnsc.48.8.1074.169
    [5]
    Meyer J. Second degree stochastic dominance with respect to a function. International Economic Review, 1977, 47: 7–487. doi: 10.2307/2525760
    [6]
    Pratt J W. Risk aversion in the small and in the large. Econometrica, 1964, 32: 122–136. doi: 10.2307/1913738
    [7]
    Lando T, Bertoli-Barsotti L. Distorted stochastic dominance: A generalized family of stochastic orders. Journal of Mathematical Economics, 2020, 90: 132–139. doi: 10.1016/j.jmateco.2020.07.005
    [8]
    Quiggin J. A theory of anticipated utility. Journal of Economic Behavior and Organization, 1982, 3: 323–343. doi: 10.1016/0167-2681(82)90008-7
    [9]
    Fishburn P C. Continua of stochastic dominance relations for bounded probability distributions. Journal of Mathematical Economics, 1976, 3 (3): 295–311. doi: 10.1016/0304-4068(76)90015-x
    [10]
    Tsetlin I, Winkler R L, Huang R J, et al. Generalized almost stochastic dominance. Operations Research, 2015, 63 (2): 363–377. doi: 10.1287/opre.2014.1340
    [11]
    Müller A, Scarsini M, Tsetlin I, et al. Between first- and second-order stochastic dominance. Management Science, 2017, 63 (9): 2933–2947. doi: 10.1287/mnsc.2016.2486
    [12]
    Huang R J, Tzeng L Y, Zhao L. Fractional degree stochastic dominance. Management Science, 2020, 66 (10): 4630–4647. doi: 10.1287/mnsc.2019.3406
    [13]
    Thistle P D. Ranking distributions with generalized Lorenz curves. Southern Economic Journal, 1989, 56 (1): 1–12. doi: 10.2307/1059050
    [14]
    Shaked M, Shanthikumar J G. Stochastic Orders. New York: Springer, 2007.
    [15]
    Liu P, Schied A, Wang R. Distributional transforms, probability distortions, and their applications. Mathematics of Operations Research, 2021, 46 (4): 1490–1512. doi: 10.1287/moor.2020.1090
    [16]
    Chew S H, Karni E, Safra Z. Risk aversion in the theory of expected utility with rank dependent probabilities. Journal of Economic Theory, 1987, 42: 370–381. doi: 10.1016/0022-0531(87)90093-7
    [17]
    Wang S S, Young V R. Ordering risks: Expected utility theory versus Yaari’s dual theory of risk. Insurance:Mathematics and Economics, 1998, 22 (2): 145–161. doi: 10.1016/s0167-6687(97)00036-x
    [18]
    Mao T, Wu Q, Hu T. Further properties of fractional stochastic dominance. Journal of Applied Probability, 2022, 59 (1): 202–223. doi: 10.1017/jpr.2021.44
    [19]
    Mao T, Wang R. A critical comparison of three notions of fractional stochastic dominance. 2020. https://ssrn.com/abstract=3642983. Accessed July 1, 2022.
    [20]
    Levy H, Wiener Z. Stochastic dominance and prospect dominance with subjective weighting functions. Journal of Risk and Uncertainty, 1998, 16 (2): 147–163. doi: 10.1023/a:1007730226688
    [21]
    Müller A, Stoyan D. Comparison Methods for Stochastic Models and Risks. Hoboken, NJ: Wiley, 2022.
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