Beyond optimization and dynamic programming: incorporating frequency-dependent effects through mean-field games

Abstract

Optimization approaches have been widely used to study the behavior of organisms. According to this idea, behaviors that lead to a selective advantage in the given environment would proliferate; thus, over time, the optimal behavior would be adopted with the phenotypic gambit assumption. Among optimization approaches, dynamic programming models, in particular, solve for a series of behavioral decisions dependent on the state of the organism, which maximizes its reproductive fitness over a time period. However, organisms exist within populations and interact directly or indirectly with others, sometimes competing or else benefiting from collective behavior. Therefore, a behavioral strategy that is optimal with respect to a fixed environment may no longer provide an advantage if every organism employs the strategy. The resulting frequency-dependent effects call for a game-theoretic approach. In this thesis, I extend the previous class of models based on optimization and dynamic programming to include the frequency-dependent effects through mean-field game models and apply this perspective to a variety of problems. The five chapters in this thesis are mostly self-contained, but Chapter 3-5 follow the theme of extending dynamic programming models, introduced in Chapter 2, to include frequency-dependence. Chapter 2 studies the population and phenological change of migratory birds by means of a dynamic programming model. Chapter 3 then investigates two approaches of including frequency-dependence into the dynamic programming framework by revisiting two previous models. Chapter 4 studies the optimal foraging problem where avoidance of predation is the main driver of choosing a foraging rate. The optimization problem is solved using the calculus of variations; when frequency-dependence is included in the form of competition and collective vigilance, the resulting mean-field game is solved with a numerical method. Chapter 5 applies the mean-field game model to human behavior, in particular, the dilemma of social distancing during an infectious disease epidemic. Finally, the last chapter is unrelated to the overarching theme of the thesis but stems from my internship at the University of Oslo. Here, we formulate a compartmental model of bubonic plague, taking into account recent experimental results in the flea, rat, and human dynamics, as well as the effect of seasonality.