Abstract
Risk measures are commonly used to capture the risk preferences of decision-makers (DMs). The decisions of DMs can be nudged or manipulated when their risk preferences are influenced by factors such as the availability of information about the uncertainties. This work proposes a Stackelberg risk preference design (STRIPE) problem to capture a designer’s incentive to influence DMs’ risk preferences. STRIPE consists of two levels. In the lower level, individual DMs in a population, known as the followers, respond to uncertainties according to their risk preference types. In the upper level, the leader influences the distribution of the types to induce targeted decisions and steers the follower’s preferences to it. Our analysis centers around the solution concept of approximate Stackelberg equilibrium that yields suboptimal behaviors of the players. We show the existence of the approximate Stackelberg equilibrium. The primitive risk perception gap, defined as the Wasserstein distance between the original and the target type distributions, is important in estimating the optimal design cost. We connect the leader’s optimality compromise on the cost with her ambiguity tolerance on the follower’s approximate solutions leveraging Lipschitzian properties of the lower level solution mapping. To obtain the Stackelberg equilibrium, we reformulate STRIPE into a single-level optimization problem using the spectral representations of law-invariant coherent risk measures. We create a data-driven approach for computation and study its performance guarantees. We apply STRIPE to contract design problems under approximate incentive compatibility. Moreover, we connect STRIPE with meta-learning problems and derive adaptation performance estimates of the meta-parameters.
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References
Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance 26(7), 1505–1518 (2002)
Acerbi, C., Simonetti, P.: Portfolio optimization with spectral measures of risk. arXiv:cond-mat/0203607 (2002)
Acerbi, C., Szekely, B.: Back-testing expected shortfall. Risk 27(11), 76–81 (2014)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer, Berlin (2008)
Anscombe, F.J., Aumann, R.J., et al.: A definition of subjective probability. Ann. Math. Stat. 34(1), 199–205 (1963)
Artacho, F.J.A., Mordukhovich, B.S.: Metric regularity and Lipschitzian stability of parametric variational systems. Nonlinear Anal. Theory Methods Appl. 72(3–4), 1149–1170 (2010)
Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Springer, Berlin (2009)
Balseiro, S.R., Besbes, O., Castro, F.: Mechanism design under approximate incentive compatibility. Oper. Res. 72, 355–372 (2022)
Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Birkhäuser, Basel (1982)
Barseghyan, L., Prince, J., Teitelbaum, J.C.: Are risk preferences stable across contexts? Evidence from insurance data. Am. Econ. Rev. 101(2), 591–631 (2011)
Başar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1998)
Bennett, K.P., Hu, J., Ji, X., Kunapuli, G., Pang, J.S.: Model selection via bilevel optimization. In: The 2006 IEEE International Joint Conference on Neural Network Proceedings, pp. 1922–1929. IEEE (2006)
Bensalem, S., Santibáñez, N.H., Kazi-Tani, N.: Prevention efforts, insurance demand and price incentives under coherent risk measures. Insur. Math. Econ. 93, 369–386 (2020)
Bertsimas, D., Brown, D.B.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57(6), 1483–1495 (2009)
Bonnans, J.F., Ioffe, A.D.: Quadratic growth and stability in convex programming problems with multiple solutions. J. Convex Anal. 2(1–2), 41–57 (1995)
Burtscheidt, J., Claus, M., Dempe, S.: Risk-averse models in bilevel stochastic linear programming. SIAM J. Optim. 30(1), 377–406 (2020)
Chade, H., De Serio, V.N.V.: Risk aversion, moral hazard, and the principal’s loss. Econ. Theor. 20(3), 637–644 (2002)
Chen, T., Sun, Y., Xiao, Q., Yin, W.: A single-timescale method for stochastic bilevel optimization. In: International Conference on Artificial Intelligence and Statistics, pp. 2466–2488. PMLR (2022)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)
Delage, E., Li, J.Y.M.: Minimizing risk exposure when the choice of a risk measure is ambiguous. Manag. Sci. 64(1), 327–344 (2018)
Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)
Dempe, S.: Foundations of Bilevel Programming. Springer, Berlin (2002)
Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. 138(1), 447–473 (2013)
Denneberg, D.: Premium calculation: why standard deviation should be replaced by absolute deviation1. ASTIN Bull. J. IAA 20(2), 181–190 (1990)
Dentcheva, D., Ruszczynski, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14(2), 548–566 (2003)
Dohmen, T., Lehmann, H., Pignatti, N.: Time-varying individual risk attitudes over the great recession: a comparison of Germany and Ukraine. J. Comp. Econ. 44(1), 182–200 (2016)
Finn, C., Abbeel, P., Levine, S.: Model-agnostic meta-learning for fast adaptation of deep networks. In: International Conference on Machine Learning, pp. 1126–1135. PMLR (2017)
Föllmer, H., Schied, A.: Stochastic Finance. de Gruyter, Berlin (2016)
Föllmer, H., Weber, S.: The axiomatic approach to risk measures for capital determination. Annu. Rev. Financ. Econ. 7, 301–337 (2015)
Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. In: Gilboa, I. (ed.) Uncertainty in Economic Theory, pp. 141–151. Routledge, London (2004)
Gneiting, T.: Making and evaluating point forecasts. J. Am. Stat. Assoc. 106(494), 746–762 (2011)
Guo, S., Xu, H.: Robust spectral risk optimization when the subjective risk aversion is ambiguous: a moment-type approach. Math. Program. 194, 305–340 (2021)
Haezendonck, J., Goovaerts, M.: A new premium calculation principle based on Orlicz norms. Insur. Math. Econ. 1(1), 41–53 (1982)
Hanaoka, C., Shigeoka, H., Watanabe, Y.: Do risk preferences change? Evidence from the great east japan earthquake. Am. Econ. J. Appl. Econ. 10(2), 298–330 (2018)
Hanin, L.G.: Kantorovich–Rubinstein norm and its application in the theory of Lipschitz spaces. Proc. Am. Math. Soc. 115(2), 345–352 (1992)
Johnson, J.: Lipschitz spaces. Pac. J. Math. 51(1), 177–186 (1974)
Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. In: Handbook of the Fundamentals of Financial Decision Making: Part I, pp. 99–127. World Scientific (2013)
Kantorovich, L.V., Rubinshtein, G.S.: On a functional space and certain extremum problems. In: Doklady Akademii Nauk, vol. 115, pp. 1058–1061. Russian Academy of Sciences (1957)
Karni, E.: A definition of subjective probabilities with state-dependent preferences. Econom. J. Econom. Soc. 61, 187–198 (1993)
Kien, B.: On the lower semicontinuity of optimal solution sets. Optimization 54(2), 123–130 (2005)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications, vol. 60. Springer, Berlin (2006)
Kuhn, D., Esfahani, P.M., Nguyen, V.A., Shafieezadeh-Abadeh, S.: Wasserstein distributionally robust optimization: theory and applications in machine learning. In: Operations Research & Management Science in the Age of Analytics, pp. 130–166. Informs (2019)
Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis. Springer, Berlin (2006)
Kusuoka, S.: On law invariant coherent risk measures. In: Maruyama, T. (ed.) Advances in Mathematical Economics, pp. 83–95. Springer, Berlin (2001)
Levin, I.P., Hart, S.S., Weller, J.A., Harshman, L.A.: Stability of choices in a risky decision-making task: a 3-year longitudinal study with children and adults. J. Behav. Decis. Mak. 20(3), 241–252 (2007)
Levy, H.: Stochastic dominance and expected utility: survey and analysis. Manag. Sci. 38(4), 555–593 (1992)
Li, J.Y.M.: Inverse optimization of convex risk functions. Manag. Sci. 67(11), 7113–7141 (2021)
Li, M., Tong, X., Xu, H.: Randomization of spectral risk measure and distributional robustness. arXiv preprint arXiv:2212.08871 (2022)
Lin, G.H., Xu, M., Jane, J.Y.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144(1), 277–305 (2014)
Liu, S., Zhu, Q.: Robust and stochastic optimization with a hybrid coherent risk measure with an application to supervised learning. IEEE Control Syst. Lett. 5(3), 965–970 (2020)
Liu, S., Zhu, Q.: Mitigating moral hazard in cyber insurance using risk preference design. arXiv preprint arXiv:2203.12001 (2022)
Liu, S., Zhu, Q.: On the role of risk perceptions in cyber insurance contracts. In: 2022 IEEE Conference on Communications and Network Security (CNS), pp. 377–382. IEEE (2022)
Liu, Y., Xu, H.: Stability analysis of stochastic programs with second order dominance constraints. Math. Program. 142(1), 435–460 (2013)
Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74(6), 1447–1498 (2006)
Meyer, D.J., Meyer, J.: Relative risk aversion: what do we know? J. Risk Uncertain. 31(3), 243–262 (2005)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory, vol. 330. Springer, Berlin (2006)
Mordukhovich, B.S.: Failure of metric regularity for major classes of variational systems. Nonlinear Anal. Theory Methods Appl. 69(3), 918–924 (2008)
Mordukhovich, B.S., Wang, B.: Restrictive metric regularity and generalized differential calculus in Banach spaces. Int. J. Math. Math. Sci. 2004(50), 2653–2680 (2004)
Pichler, A.: A quantitative comparison of risk measures. Ann. Oper. Res. 254(1), 251–275 (2017)
Pichler, A., Shapiro, A.: Minimal representation of insurance prices. Insur. Math. Econ. 62, 184–193 (2015)
Pichler, A., Xu, H.: Quantitative stability analysis for minimax distributionally robust risk optimization. Math. Program. (2018). https://doi.org/10.1007/s10107-018-1347-4
Rockafellar, R.T., Uryasev, S., et al.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer, Berlin (2009)
Römisch, W.: Stability of stochastic programming problems. Handb. Oper. Res. Manag. Sci. 10, 483–554 (2003)
Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31(3), 433–452 (2006)
Saghai, Y.: Salvaging the concept of nudge. J. Med. Ethics 39(8), 487–493 (2013)
Schied, A., Föllmer, H., Weber, S.: Robust preferences and robust portfolio choice. Handb. Numer. Anal. 15, 29–87 (2009)
Schildberg-Hörisch, H.: Are risk preferences stable? J. Econ. Perspect. 32(2), 135–54 (2018)
Shapiro, A.: Quantitative stability in stochastic programming. Math. Program. 67(1), 99–108 (1994)
Shapiro, A.: On Kusuoka representation of law invariant risk measures. Math. Oper. Res. 38(1), 142–152 (2013)
Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2021)
Slovic, P.: The construction of preference. Am. Psychol. 50(5), 364 (1995)
Slovic, P., Peters, E.: Risk perception and affect. Curr. Dir. Psychol. Sci. 15(6), 322–325 (2006)
Stole, L.: Lectures on the theory of contracts and organizations. Unpublished monograph (2001)
Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5(4), 297–323 (1992)
Vanschoren, J.: Meta-learning: a survey. arXiv preprint arXiv:1810.03548 (2018)
Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2009)
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (2007)
Wang, S.: Premium calculation by transforming the layer premium density. ASTIN Bull. J. IAA 26(1), 71–92 (1996)
Wang, S.S., Young, V.R.: Ordering risks: Expected utility theory versus Yaari’s dual theory of risk. Insur. Math. Econ. 22(2), 145–161 (1998)
Wang, W., Xu, H.: Robust spectral risk optimization when information on risk spectrum is incomplete. SIAM J. Optim. 30(4), 3198–3229 (2020)
Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)
Yaari, M.E.: The dual theory of choice under risk. Econom. J. Econom. Soc. 55, 95–115 (1987)
Yassin, A., AlOmari, M., Al-Azzam, S., Karasneh, R., Abu-Ismail, L., Soudah, O.: Impact of social media on public fear, adoption of precautionary behaviors, and compliance with health regulations during Covid-19 pandemic. Int. J. Environ. Health Res. 32(9), 2027–2039 (2022)
Ye, J., Zhu, D., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7(2), 481–507 (1997)
Ye, J.J., Yuan, X., Zeng, S., Zhang, J.: Difference of convex algorithms for bilevel programs with applications in hyperparameter selection. arXiv preprint arXiv:2102.09006 (2021)
Yuen, K.F., Wang, X., Ma, F., Li, K.X.: The psychological causes of panic buying following a health crisis. Int. J. Environ. Res. Public Health 17(10), 3513 (2020)
Zhao, J.: The lower semicontinuity of optimal solution sets. J. Math. Anal. Appl. 207(1), 240–254 (1997)
Zheng, X., Li, W., Wong, S.W., Lin, H.C.: Social media and e-cigarette use among us youth: longitudinal evidence on the role of online advertisement exposure and risk perception. Addict. Behav. 119, 106916 (2021)
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Liu, S., Zhu, Q. Stackelberg risk preference design. Math. Program. (2024). https://doi.org/10.1007/s10107-024-02083-2
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DOI: https://doi.org/10.1007/s10107-024-02083-2