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Quantum-Inspired Approach to Computational Economics

Michail A. Zak

Published Oct 1, 2014• 0 citations
OpenAlex

Abstract

The paper extends the mathematical formalism of quantum physics to include economics. The novelty of the approach is based upon a human-factor-based behavioral model of an intelligent agent. The model is quantum inspired: it is represented by a modified Madelung equation in which the gradient of quantum potential is replaced by a specially chosen information force. It consists of motor dynamics simulating actual behavior of the agent, and mental dynamics representing evolution of the corresponding knowledge base and incorporating this knowledge in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics. All of these departures actually extend and complement the classical methods making them especially successful in analysis of communications of agents represented by new mathematical formalism.

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The paper extends the mathematical formalism of quantum physics to include economics. The novelty of the approach is based upon a human-factor-based behavioral model of an intelligent agent. The model is quantum inspired: it is represented by a modified Madelung equation in which the gradient of quantum potential is replaced by a specially chosen information force. It consists of motor dynamics simulating actual behavior of the agent, and mental dynamics representing evolution of the corresponding knowledge base and incorporating this knowledge in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics. All of these departures actually extend and complement the classical methods making them especially successful in analysis of communications of agents represented by new mathematical formalism.
The paper extends the mathematical formalism of quantum physics to include games of intelligent agents that communicate due to entanglement. The novelty of the approach is based upon a human factor-based behavioral model of an intelligent agent. The model is quantum inspired: it is represented by a modified Madelung equation in which the gradient of quantum potential is replaced by a specially chosen information force. It consists of motor dynamics simulating actual behavior of the agent, and mental dynamics representing evolution of the corresponding knowledge base and incorporating this knowledge in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics; each agent is entangled (in a quantum way) to other agents and makes calculated predictions for future actions; human factor is associated with violation of the second law of thermodynamics: the system can move from disorder to order without external help, and that represents intrinsic intelligence. All of these departures actually extend and complement the classical methods making them especially successful in analysis of communications of agents represented by new mathematical formalism, and in particular, in agent-based economics with a human factor.
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.
This paper discusses quantum-inspired models of Livings from the viewpoint of information processing. The model of Livings consists of motor dynamics simulating actual behavior of the object, and mental dynamics representing evolution of the corresponding knowledge-base and incorporating it in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics The paper concentrates on discovery of a new type of entanglement that correlates the probabilities of actions of Livings rather than the actions themselves.
The paper introduces a new approach to theory of differential games in which entangled players try to predict and influence actions of their adversaries. The entanglement is generated be a joint probability density known by the players. Incase of complex density, its imaginary part represents a measure of uncertainty of the density distribution. The novelty of the approach is in non-Newtonian mathematical formalism thatis based upon a behavioral model of Livings. The model is quantum-inspired: it represented by a modified Madelung equation which the quantum potential is replaced by different, specially chosen "computational" potential. It consists of motor dynamics simulating actual behavior of the object, and mental dynamics representing evolution of the corresponding knowledge-based and incorporating this knowledge in the for m of information flows into the motor dynamics. Due to feedback from mental dynamics, motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics. The model addresses a new type of entanglement that correlates the probabilities of actions of Livings rather than the actions themselves.
The challenge of this work is to connect quantum mechanics with the concept of intelligence. By intelligence we understand a capability to move from disorder to order without external resources, i.e. in violation of the second law of thermodynamics. The objective is to find such a mathematical object described by ODE that possesses such a capability. The proposed approach is based upon modification of the Madelung version of the Schrodinger equation by replacing the force following from quantum potential with non-conservative forces that link to the concept of information. A mathematical formalism suggests that a hypothetical intelligent particle, besides the capability to move against the second law of thermodynamics, acquires such properties like self-image, self-awareness, self- supervision, etc. that are typical for Livings. However since this particle being a quantum-classical hybrid acquires non-Newtonian and non-quantum properties, it does not belong to the physics matter as we know it: the modern physics should be complemented with the concept of an information force that represents a bridge to intelligent particle. It has been suggested that quantum mechanics should be complemented by the intelligent particle as an independent entity, and that will be the necessary step to physics of Life. At this stage, the intelligent particle is introduced as an abstract mathematical concept that is satisfied only mathematical rules and assumptions, and its physical representation is still an open problem.
This paper describes an approach to economics that is inspired by quantum computing, and is motivated by the need to develop a consistent quantum mathematical framework for economics. The traditional neoclassical approach assumes that rational utility-optimisers drive market prices to a stable equilibrium, subject to external perturbations or market failures. While this approach has been highly influential, it has come under increasing criticism following the financial crisis of 2007/8. The quantum approach, in contrast, is inherently probabilistic and dynamic. Decision-makers are described, not by a utility function, but by a propensity function which specifies the probability of transacting. We show how a number of cognitive phenomena such as preference reversal and the disjunction effect can be modelled by using a simple quantum circuit to generate an appropriate propensity function. Conversely, a general propensity function can be quantized,viaan entropic force, to incorporate effects such as interference and entanglement that characterise human decision-making. Applications to some common problems and topics in economics and finance, including the use of quantum artificial intelligence, are discussed.