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International Journal of Quantum Information

Quantum-like model on multiple lotteries selection

Changchun Feng, Lin Chen, Junhuan Zhang, Jiaqi Wen, Chenze Ji

Published Oct 25, 2024• 0 citations
Publisher

Abstract

This study proposes an agent-based quantum-like model to investigate the individual selection among three or more lotteries while incorporating the decision-making risk and uncertainty. We extend the classical expected utility functions with quantum probabilities and construct the compound belief state to compare one specific lottery belief state against others. The involved decision-making process is represented formally by the comparison operator, which can be decomposed into a few subprocesses. We give an example of individual lottery selection from three lotteries to illustrate the model. Finally, we propose ways to select from more than three lotteries.

DOI
10.1142/s0219749924500497

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This study proposes an agent-based quantum-like model to investigate the individual selection among three or more lotteries while incorporating the decision-making risk and uncertainty. We extend the classical expected utility functions with quantum probabilities and construct the compound belief state to compare one specific lottery belief state against others. The involved decision-making process is represented formally by the comparison operator, which can be decomposed into a few subprocesses. We give an example of individual lottery selection from three lotteries to illustrate the model. Finally, we propose ways to select from more than three lotteries.
This chapter can be considered as an illustration of applications of the mathematical formalism of QM to modeling the process of decision making by humans (see section 1.3 for an introduction and general discussion). In this chapter we present a very general quantum-like model of lottery selection based on the representation of beliefs of an agent by pure quantum states. Subjective probabilities are mathematically realized in the framework of quantum probability (QP). Utility functions are borrowed from the classical decision theory. An agent processes this information by using the rules of quantum information and QP. This process is very complex; it combines counterfactual reasoning for comparison between preferences for different outcomes of lotteries which are in general complementary. These comparisons induce interference-type effects (constructive or destructive). The decision process is mathematically represented by the comparison operator and the outcome of this process is determined by the sign of the value of a corresponding quadratic form on the belief state. This operational process can be decomposed into a few subprocesses. Each of them can be formally treated as a comparison of subjective expected 202 utilities and interference factors (the latter express, in particular, risks related to lottery selection).
Quantum decision theory, advanced earlier by the authors, and illustrated for lotteries with gains, is generalized to the games containing lotteries with gains as well as losses. The mathematical structure of the approach is based on the theory of quantum measurements, which makes this approach relevant both for the description of decision making of humans and the creation of artificial quantum intelligence. General rules are formulated allowing for the explicit calculation of quantum probabilities representing the fraction of decision makers preferring the considered prospects. This provides a method to quantitatively predict decision-maker choices, including the cases of games with high uncertainty for which the classical expected utility theory fails. The approach is applied to experimental results obtained on a set of lottery gambles with gains and losses. Our predictions, involving no fitting parameters, are in very good agreement with experimental data. The use of quantum decision making in game theory is described. A principal scheme of creating quantum artificial intelligence is suggested.
In this paper, we introduce a new model of selection behavior under risk that describes an essential cognitive process for comparing values of objects and making a selection decision. This model is constructed by the quantum-like approach that employs the state representation specific to quantum theory, which has the mathematical framework beyond the classical probability theory. We show that our quantum approach can clearly explain the famous examples of anomalies for the expected utility theory, the Ellsberg paradox, the Machina paradox and the disparity between WTA and WTP. Further, we point out that our model mathematically specifies the characteristics of the probability weighting function and the value function, which are basic concepts in the prospect theory.
Quantum cognition in decision-making is a recent and rapidely growing field. In this paper we develop an expected utility theory in a context of non-classical (quantum) uncertainty. We replace the classical state space with a Hilbert space which allows introducing the concept of quantum lottery. Within that framework we formulate axioms on preferences over quantum lotteries to establish a representation theorem. We show that demanding the consistency of choice behavior conditional on new information is equivalent to the von Neuman-Lüders postulate applied to beliefs. A dynamically consistent quantum-like agent may violate dynamic recursive consistency, however. This feature suggests interesting applications in behavioral economics as we illustrate in an example of persuasion.
The core assumptions of modern economic utility theories such as rationality have been placed under scrutiny by recent studies in decision theory and behavioural economics and fi?nance. The main questions that were explored in these studies could be expressed as follows: do people always make rational choices? Do they make use of the normative rules of traditional logic and probability when dealing with risk and uncertainty? Are there any alternative methods to capture choices and reasoning when individuals are faced with uncertainty and risk? In this thesis I will elaborate on an interdisciplinary application of so-called QL (quantum-like) models to decision-making problems in ?finance and other socio-economic contexts. The core inquiry of this work is related to experimental tests of decision making with the help of investment related questions. Calculus and logic borrowed from quantum physics will be used to develop probabilistic models for the formation and evolution of decisions in ?finance context with the potential to generalize to other social decision- making settings. The main fi?ndings of this thesis can be summarized as follows: individuals exhibit preference reversals and context-dependent behaviour in different informational contexts. In particular, in condition of risk individuals exhibit uncertainty aversion and thus aversion to participating in other investment task, violating the postulates of classical probability theory. The core principle of rational (consequential) behaviour conceived by Savage (1954) [207], that is the "Savage Sure Thing" principle, was shown to be violated. In mathematical language, this violation is equivalent to the violation of the "Law of total probability". Implying contextuality of human judgement, different events cannot be jointly evaluated in a single probability space. Moreover, additional experimental fi?ndings showed that participants exhibited reference dependence in their risk attitudes depending on the preceding gain respective loss. These fi?ndings were statistically signi?cant and a quantum model for accommodating the above effects of contextual behaviour is introduced. A geometric depiction of the experimental fi?ndings was proposed. A so-called `state reconstruction' procedure from statistical data was performed via the inverse Born's rule (1926) [47], allowing the construction of a complete representation of agents' resolution from indeterminacy.
In this work, we study different parametrizations of Quantum Decision Theory (QDT) using experimental data made of choices between pairs of lotteries. We then compare QDT with two of most famous classical decision theories, Rank-Dependent Utility(RDU) and Cumulative Prospect Theory(CPT). At the aggregate level, assuming homogeneous preferences across subjects, we find that CPT-based QDT wins by far. At the individual level, we classify decision makers as RDU, CPT or QDT. Our major findings are: quantum factor plays a key role in describing subjects’ behaviour; there is a considerable heterogeneity across subjects so that the classic representative agent approach would be completely wrong for this sample.