Generative models for human decision making are studied extensively in behavioral economics and psychology. The classical formalisms of human decision making are the Expected Utility models of VonNeumann and Morgenstern. Despite the successes of these models, numerous experimental findings, most notably those of Kahneman and Tverksy, have demonstrated violations of the proposed decision making axioms. There have since been subsequent efforts to develop axiomatic systems which encompass wider ranges of human behavior, such as the Prospect Theory. However, given the complexity of human psychology and behavior it is no surprise that current models still have points of failure. The theory of ‘Quantum Decision Making’ has emerged as a new paradigm which is capable of generalizing current models and accounting for certain violations of axiomatic assumptions. For example, it has been empirically shown that humans routinely violate Savage’s ‘Sure Thing Principle’, which is equivalent to violation of the law of total probability, and that human decision making is affected by the order of presentation of information (“order effects”). These violations are natural motivators for treating the decision making agent’s mental state as a quantum state in Hilbert Space; The mathematics of quantum probability was developed as an explanation of observed selfinterfering and noncommutative behaviors of physical systems, directly analogous to the findings which Quantum Decision Theory (QDT) aims to treat.
Remark: QDT models in psychology do not claim that the brain is acting as a quantum device in any physical sense. Instead QDT serves as a parsimonious generative blackbox model for human decision making that is backed up by extensive experimental studies.
Within Quantum Decision Theory, several recent advances have utilized quantum dynamical systems to model timeevolving decision preferences. The classical model for this type of timeevolving mental state is a Markovian model, but in [Busemeyer et. al. 2009] an alternative formulation based on Schrodinger’s Equation is developed. This model is shown to both reconcile observed violations of the law of total probability via quantum interference effects and model choiceinduced preference changes via quantum projection. This is further advanced in [Asano et. al. 2012], and [Martinez et. al. 2016] where the mental state is modeled as an openquantum system. This openquantum system representation allows for a generalization of the widely used Markovian model of preference evolution, while maintaining these advantages of the quantum framework. [Busemeyer et. al. 2021] provide empirical analysis which supports the use of openquantum models and conclude “An open system model that incorporates elements of both classical and quantum dynamics provides the
best available single system account of these three characteristics: evolution, oscillation, and choiceinduced preference change”.
Motivated by these considerations, we have implemented the openquantum decision making model of [Martinez et. al. 2016] in several humanmachine interaction frameworks, including a stochastic control interface in which a machine adaptively signals a human decision maker so as to converge to an optimal decision, and a statistical sequential change detection framework in which an analyst attempts to detect an underlying system parameter change by observing human decisions which are influenced by this parameter. Below we outline our work in these areas.
Lyapunov based Stochastic Stability of a Quantum Decision System for HumanMachine Interaction Shashwat Jain, Luke Snow, Vikram Krishnamurthy, ArXiv Preprint, 2022
Consider the humanmachine collaborative decisionmaking framework in the figure above. An underlying state of nature exists which parameterizes the system reward obtained given the human decisions. This state is observed in noise by both the human (‘Psychological State’ box) and machine (‘SensorController’ box). At each time step, the human takes an action according to the quantum decision process of [Martinez et. al. 2016], which is parameterized by the noisy state observation and a recommendation signal given by the machine. For the sake of generality, we consider when these humanmachine interactions occur over T timestep intervals, for random variable T, which represents variability in the particular decision making timescale. The decisionmaking process of [Martinez et. al. 2016] is particularly useful in modeling suboptimal or irrational human decisions, such as those made in stressful or complex environments. Thus, we consider the case where the machine attempts to steer the sequential human decisions from suboptimal ones to a particular decision which it considers optimal, perhaps in the sense of maximizing an expected utility. Given this framework, our results include:

 We prove that, for any target action, there exists a machine policy for which the human action sequence is guided to the target action with probability one.
 To prove (1), we generalize an existing finitestep stochastic Lyapunov stability argument to fit our framework. This generalization is also useful on its own.