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arXiv:2212.04286

Machine learning for decision-making under uncertainty

Lizhi Xin, Kevin Z. Xin, Houwen Xin

Published Dec 8, 2022• 0 citations
arXivHTMLPDF

Abstract

We live in a world brimming with uncertainty, where we constantly have to make a lot of decisions under incomplete information. We are firm believers that our subjective belief cannot be computed by rigorous mathematical formula; instead based on Darwin's natural selection (the evolution process is simulated by machine learning with genetic programming), a proposed computational model that incorporates insights from quantum theory to describe and explain decision-making under uncertainty. Unlike other decision-making theories that explain the decision-making process through probability theory, our proposed decision theory discovers "laws" of thought by learning observed historical data. There is no differential equation and no transition probability in our decision theory, our decision model has an emphasis on machine learning, where decision-makers build-up their experience by being rewarded or punished for each decision they make and prepare them for making better decisions in the future. We do not model with the usual utility function, but with quantum decision tree that simulates people's decision process. Each quantum decision tree includes a set of strategies; every time a decision is made, the decision-maker first chooses a strategy from the quantum decision tree's strategy pool, and then chooses an action based on the degree of belief which is obtained by genetic programming based on maximizing expected value.

DOI
10.48550/arxiv.2212.04286

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We live in a world brimming with uncertainty, where we constantly have to make a lot of decisions under incomplete information. We are firm believers that our subjective belief cannot be computed by rigorous mathematical formula; instead based on Darwin's natural selection (the evolution process is simulated by machine learning with genetic programming), a proposed computational model that incorporates insights from quantum theory to describe and explain decision-making under uncertainty. Unlike other decision-making theories that explain the decision-making process through probability theory, our proposed decision theory discovers "laws" of thought by learning observed historical data. There is no differential equation and no transition probability in our decision theory, our decision model has an emphasis on machine learning, where decision-makers build-up their experience by being rewarded or punished for each decision they make and prepare them for making better decisions in the future. We do not model with the usual utility function, but with quantum decision tree that simulates people's decision process. Each quantum decision tree includes a set of strategies; every time a decision is made, the decision-maker first chooses a strategy from the quantum decision tree's strategy pool, and then chooses an action based on the degree of belief which is obtained by genetic programming based on maximizing expected value.
Incorporating insights from quantum theory, we propose a machine learning-based decision-making model, including a logic tree and a value tree; a genetic programming algorithm is applied to optimize both the logic tree and value tree. The logic tree and value tree together depict the entire decision-making process of a decision-maker. We applied this framework to the financial market, and a "machine economist" is developed to study a time series of the Dow Jones index. The "machine economist" will obtain a set of optimized strategies to maximize profits, and discover the efficient market hypothesis (random walk).
Abstract Decision’s biggest challenge is uncertainty. We propose a quantum expected value theory for decision-making under uncertainty. Decision-making under uncertainty is the unification of people’s subjective beliefs and the objective world. Quantum density operator as value operator which we call it quantum circuit is proposed to simulate people’s subjective beliefs. Quantum circuit guides people to choose corresponding actions based on their subjective beliefs through objective world. The quantum circuit is a decision tree constructed from quantum gates and logic operations. The genetic programming can be used to optimize and auto-generate quantum circuits. Our proposed synapse’s quantum circuit hypothesis states that the combination of neurotransmitter and receptor functioned as quantum circuit open or close ion channels probabilistically which determines the excitation or inhibition of synapse.
The application of the mathematical formalism of quantum mechanics to model behavioral patterns in social science and economics is a novel and constantly emerging field. The aim of the so called 'quantum like' models is to model the decision making processes in a macroscopic setting, capturing the particular 'context' in which the decisions are taken. Several subsequent empirical findings proved that when making a decision people tend to violate the axioms of expected utility theory and Savage's Sure Thing principle, thus violating the law of total probability. A quantum probability formula was devised to describe more accurately the decision making processes. A next step in the development of QL-modeling in decision making was the application of Schrödinger equation to describe the evolution of people's mental states. A shortcoming of Schrödinger equation is its inability to capture dynamics of an open system; the brain of the decision maker can be regarded as such, actively interacting with the external environment. Recently the master equation, by which quantum physics describes the process of decoherence as the result of interaction of the mental state with the environmental 'bath', was introduced for modeling the human decision making. The external environment and memory can be referred to as a complex 'context' influencing the final decision outcomes. The master equation can be considered as a pioneering and promising apparatus for modeling the dynamics of decision making in different contexts.
We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans do not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data.
A quantum device simulating the human decision making process is introduced. It consists of quantum recurrent nets generating stochastic processes which represent the motor dynamics, and of classical neural nets describing the evolution of probabilities of these processes which represent the mental dynamics. The autonomy of the decision making process is achieved by a feedback from the mental to motor dynamics which changes the stochastic matrix based upon the probability distribution. This feedback replaces unavailable external information by an internal knowledge- base stored in the mental model in the form of probability distributions. As a result, the coupled motor-mental dynamics is described by a nonlinear version of Markov chains which can decrease entropy without an external source of information. Applications to common sense based decisions as well as to evolutionary games are discussed. An example exhibiting self-organization is computed using quantum computer simulation. Force on force and mutual aircraft engagements using the quantum decision maker dynamics are considered.
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.