Analysis and Control of Decision Making Dynamics

采用或拒绝新技术，追随或忽视流行趋势，遵守或打破非药物政策，这些都是相互作用的个体群体的例子，这些个体随着时间的推移在可用的行动之间做出决定。个体决策的演化产生了所谓的“决策动力学”。预测和理想地控制动态的长期行为引起了许多科学家和政策制定者的极大兴趣，因为它们在社会网络、市场营销和健康管理等方面都有应用。尽管研究人员已经成功地进行了初步分析，以了解决策动力学，但这一科学领域仍处于初级阶段。由于建模和分析中的一些简化，焦点主要集中在均衡状态上，在这种状态下，种群中的所有个体都对自己的选择感到满意，并且不倾向于改变自己的选择。因此，很少有人注意到个人决策中永恒的、看似随机的、可能是疯狂的波动，这种波动无处不在，而且经常被报道。这极大地限制了我们在决策动力学方面的预测和管理能力。我计划不仅继续采取初步步骤来描述波动并严格研究它们的长期行为，而且还计划从嘈杂的数据中实际识别它们，并实际控制它们的波动长度和形式。作为案例研究，我使用了twitter和COVID-19的真实数据集（遵守非药物政策）。我将考虑不同的决策者组合，找到平衡点和不变集（发生波动的地方），并研究它们的稳定性和收敛性。然后，我将考虑代理在决策中犯“错误”并以一定概率选择与其预期不同的行为，从而导致随机动力学的情况。然后我将研究不变量集的随机稳定性。接下来，我将解决动力学的可识别性问题，以及是否以及如何通过数据科学和机器学习技术从数据中估计动力学参数。最后，通过向agent提供采取某种行动的激励，我将设计开环和闭环控制器，以引导总体达到期望的平衡状态或不变集。与以往的研究相比，本文首次将分析的重点放在寻找正不变集合存在唯一性的充分必要条件上，而不是寻找平衡状态。这样的集合允许解的轨迹波动，而不是像在平衡中那样保持不变——这是一种在现实中经常观察到的更现实的情况。这也允许通过将解轨迹引导到一个不变的集合而不是通常不现实和昂贵的平衡状态来控制种群动态。

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.