Computational Models for Cumulative Prospect Theory: Application to the Knapsack Problem Under Risk
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.
KeywordsCumulative Prospect Theory Knapsack Problem Risk aversion Mixed-integer linear programming
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