Analysis and Control of Decision Making Dynamics

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Adopting or rejecting new technology, following or ignoring fashion trends, and abiding by or breaking non-pharmaceutical policies are examples of populations of interacting individuals who decide between available actions over time. The evolution of the individuals' decisions results in the so--called "decision- making dynamics." Predicting and ideally controlling the long--term behavior of the dynamics is of great interest to a wide range of scientists and policy makers because of their applications in, for example, social networks, marketing, and health management. Although researchers have successfully performed initial analyses to understand decision--making dynamics, this scientific field remains at its elementary stages. Due to several simplifications in the modeling and analysis, the focus has been mainly on equilibrium states, where all individuals in the population are satisfied with and do not tend to switch their choices. Consequently, little attention has been paid to perpetual, seemingly random, and possibly wild fluctuations in individuals' decisions, which are ubiquitous and frequently reported. This has dramatically limited our prediction and management abilities in decision-making dynamics. I plan to not only continue taking the initial steps towards characterizing the fluctuations and rigorously study their long-term behavior, but also to practically identify them from noisy data, and actually control their lengths and form of fluctuations. As case studies, I use real datasets of twitter and COVID-19 (adherence to non-pharmaceutical policies). I will consider different combinations of decision--makers, find both the equilibria and invariant sets (where fluctuations happen), and study their stability and convergence. I will then consider the case when the agents make "errors" in their decisions and choose an action other than their intended with a certain probability, resulting in stochastic dynamics. I will then study the stochastic stability of the invariant sets. Next, I will address the question of the identifiability of the dynamics and whether and how the parameters of the dynamics can be estimated from data by means of data science and machine-learning techniques. Finally, by providing incentives to the agents to take a certain action, I will design open-loop and closed-loop controllers to lead the population to either the desired equilibrium state or invariant set.  Compared to previous studies, for the first time, the analysis will focus on finding necessary and sufficient conditions for the existence and uniqueness of positively invariant sets rather than equilibrium states. Such sets allow the solution trajectory to fluctuate rather than stay still as in the equilibria-a more realistic situation that is often observed in reality. This also allows controlling the population dynamics by leading the solution trajectory to an invariant set rather than the often unrealistic and costly equilibrium state.