# Computational Models for Cumulative Prospect Theory: Application to the Knapsack Problem Under Risk

• Hugo Martin
• Patrice Perny
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11940)

## Abstract

Cumulative Prospect Theory (CPT) is a well known model introduced by Kahneman and Tversky in the context of decision making under risk to overcome some descriptive limitations of Expected Utility. In particular CPT makes it possible to account for the framing effect (outcomes are assessed positively or negatively relatively to a reference point) and the fact that people often exhibit different risk attitudes towards gains and losses. We study here computational aspects related to the implementation of CPT for decision making in combinatorial domains. More precisely, we consider the Knapsack Problem under Risk that consists of selecting the “best” subset of alternatives (investments, projects, candidates) subject to a budget constraint. The alternatives’ outcomes may be positive or negative (gains or losses) and are uncertain due to the existence of several possible scenarios of known probability. Preferences over admissible subsets are based on the CPT model and we want to determine the CPT-optimal subset for a risk-averse Decision Maker (DM). The problem requires to optimize a non-linear function over a combinatorial domain. In the paper we introduce two distinct computational models based on mixed-integer linear programming to solve the problem. These models are implemented and tested on randomly generated instances of different sizes to show the practical efficiency of the proposed approach.

## Keywords

Cumulative Prospect Theory Knapsack Problem Risk aversion Mixed-integer linear programming

## References

1. 1.
Chateauneuf, A.: On the use of capacities in modeling uncertainty aversion and risk aversion. J. Math. Econ. 20(4), 343–369 (1991)
2. 2.
Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions, vol. 127. Cambridge University Press, Cambridge (2009)
3. 3.
He, X.D., Zhou, X.Y.: Portfolio choice under cumulative prospect theory: an analytical treatment. Manag. Sci. 57(2), 315–331 (2011)
4. 4.
Hines, G., Larson, K.: Preference elicitation for risky prospects. In: Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1, vol. 1, pp. 889–896. International Foundation for Autonomous Agents and Multiagent Systems (2010)Google Scholar
5. 5.
Jaffray, J., Nielsen, T.: An operational approach to rational decision making based on rank dependent utility. Eur. J. Oper. Res. 169(1), 226–246 (2006)
6. 6.
Jeantet, G., Spanjaard, O.: Computing rank dependent utility in graphical models for sequential decision problems. Artif. Intell. 175(7–8), 1366–1389 (2011)
7. 7.
Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47(2), 263–292 (1979)
8. 8.
Labreuche, C., Grabisch, M.: Generalized choquet-like aggregation functions for handling bipolar scales. Eur. J. Oper. Res. 172(3), 931–955 (2006)
9. 9.
Li, X., Wang, W., Xu, C., Li, Z., Wang, B.: Multi-objective optimization of urban bus network using cumulative prospect theory. J. Syst. Sci. Complex. 28(3), 661–678 (2015)
10. 10.
Mansini, R., Ogryczak, W., Speranza, M.G.: Twenty years of linear programming based portfolio optimization. Eur. J. Oper. Res. 234(2), 518–535 (2014)
11. 11.
Mansini, R., Ogryczak, W., Speranza, M.G.: Linear and Mixed Integer Programming for Portfolio Optimization. EATOR. Springer, Cham (2015).
12. 12.
Martin, H., Perny, P.: Biowa for preference aggregation with bipolar scales: application to fair optimization in combinatorial domains. In: IJCAI (2019)Google Scholar
13. 13.
Ogryczak, W., Śliwiński, T.: On efficient wowa optimization for decision support under risk. Int. J. Approximate Reasoning 50(6), 915–928 (2009)
14. 14.
Perny, P., Spanjaard, O., Storme, L.X.: State space search for risk-averse agents. In: IJCAI, pp. 2353–2358 (2007)Google Scholar
15. 15.
Perny, P., Viappiani, P., Boukhatem, A.: Incremental preference elicitation for decision making under risk with the rank-dependent utility model. In: Proceedings of Uncertainty in Artificial Intelligence (2016)Google Scholar
16. 16.
Prashanth, L., Jie, C., Fu, M., Marcus, S., Szepesvári, C.: Cumulative prospect theory meets reinforcement learning: prediction and control. In: International Conference on Machine Learning, pp. 1406–1415 (2016)Google Scholar
17. 17.
Quiggin, J.: Generalized Expected Utility Theory - The Rank-dependent Model. Kluwer Academic Publisher, Dordrecht (1993)
18. 18.
Rothschild, M., Stiglitz, J.E.: Increasing risk: I. A definition. J. Econ. Theory 2(3), 225–243 (1970)
19. 19.
Savage, L.J.: The Foundations of Statistics. J. Wiley and Sons, New-York (1954)
20. 20.
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)
21. 21.
Schmidt, U., Zank, H.: Risk aversion in cumulative prospect theory. Manag. Sci. 54(1), 208–216 (2008)
22. 22.
Shapley, L.: Cores of convex games. Int. J. Game Theory 1, 11–22 (1971)
23. 23.
Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertainty 5(4), 297–323 (1992)
24. 24.
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, Princeton (1947)
25. 25.
Yaari, M.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)