By Boualem Djehiche and Hamidou Tembine
Figure 1. The main steps of the mean-field-type game (MFTG) method. Figure courtesy of [7].
One can conceive the term “mean field” as a physics concept that attempts to describe the effect of an infinite number of particles on the motion of a single particle. This concept has seen widespread use in population dynamics and evolutionary game theory [6], and researchers began to apply it to the social sciences in the early 1960s to study the way in which an infinite number of factors affect individual decisions. However, the key ingredient in a game-theoretic context is the influence of the distribution of states and/or control actions on decision-makers’ performance criteria; there is no need for a large population of decision-makers.
Mean-field-type game (MFTG) theory is an extended form of game theory that studies interactions between two or more entities, which may include people, animals, devices, machines, companies, nations, networks, genes, populations, and so on. Decisionmakers in MFTG theory can be atomic, non-atomic, or a mixture of both. MFTG interactions can be fully cooperative, partially altruistic, partially cooperative, selfish, selfless, co-opetitive, spiteful, or a mixture of these types [8]. The key extension feature of MFTG theory is the integration of higher-order performance criteria like variance, quantile, and other risk-measures; such integration is not necessarily linear in the measure of the state or state action [3]. MFTG has found applications in a variety of scenarios, including the evacuation of high-level buildings, smart energy systems, next-generation wireless networks, meta learning in communication networks, smart transportation systems [5], epidemiology, predictive maintenance, marriage [2], and blockchains [1].
Data-driven MFTG theory aims to incorporate certain data sets and real measurements into MFTG settings in a closed-loop fashion. Researchers use the measured data to learn and extract additional useful information from the field that the model then takes into consideration. One can thus utilize the model to study emerging dynamics and features and conduct new measurements, intervention measures, and actions. Scientists have developed mean-field-type filters, mean-field-type forecasting, and risk-aware filtering and forecasting based on MFTG theory; they apply these techniques to intelligent transportation systems.
In the context of the ongoing COVID-19 pandemic, data-driven MFTG theory can model interactions wherein the decision-makers are individuals, authorities, institutions, and firms (see Figure 1). Each individual has a multi-layer interaction. The epidemiological layer consists of the individual’s internal state, which is comprised of the following factors: age, gender, preexisting health conditions, location, mask and sanitation adherence, and health status. The latter includes many contributing elements: susceptible; exposed; tested (positive or negative) or untested (active or recovered); re-tested; number of contacts; contact-tracing; hospitalized; recovered; vaccinated with the first, second, or booster dose; up-to-date vaccine; and tested or untested deceased.
Figure 2. COVID-19 immunity map of cases per 100,000 U.S. residents. Figure courtesy of [7].
The individual’s economic layer consists of consumption; family revenue (both ex-ante and current); employment; and work status (i.e., working from home, working part-time, working partially from home, and workload), including individual economic supports and benefits from the authority incentive policies. The mobility layer involves the mobility pattern (i.e., going to work, marketplaces, grocery stores, hotspots, drug stores, etc.); travel status (i.e., a recent trip); transportation usage (i.e., utilizing public transport, a personal vehicle, carsharing, cabs, and Uber-like services); and in-city, inter-city, and international movement.
Each authority has multiple layers of interactions that include mitigation policies; testing, hospital, and treatment capacities; and vaccination capabilities. The firms also have various layers that interact with the other categories. Figure 2 illustrates the estimated temporary protection index (temporary immunity map) against COVID-19 in the U.S., and Figure 3 provides an MFTG filtering illustration of multiple waves and sub-variants of SARS-CoV-2. One can evaluate the MFTG filter by its global error: measurement/reporting error, estimation error, modeling error, optimization error, computational error, and decision-making error.
Figure 3. Mean-field-type game (MFTG) filtering illustration of multiple waves and sub-variants of SARS-CoV-2 in India. Figure courtesy of [7].
The development of data-driven MFTG theory is timely due to the rise of high-dimensional systems and data science, both of which encompass a broad range of techniques that range from strategic deep learning and filtering to compressed sensing and useful information extraction from data. The methodology is flexible enough to capture multi-class interactions in epidemic propagation wherein multiple authorities and firms are risk-aware atomic decision-makers and individuals are risk-aware, non-atomic decision-makers. This work has also boosted other scientific endeavors. We continue to advance human-centered artificial intelligence-driven methods to accelerate knowledge extraction for both COVID-19 and a diverse set of diseases that includes Ebola, malaria, and cancer. These approaches also have applications in other related areas like food security, agriculture, and demand-supply matching in underserved locations.
This piece is based on several recent papers [4, 7]. Hamidou Tembine also presented the work during a minisymposium on “Mean Field Game Models in Finance” at the 2021 SIAM Annual Meeting, which took place virtually this July.
References
[1] Barreiro-Gomez, J., & Tembine, H. (2019). Blockchain token economics: A mean-field-type game perspective. IEEE Access, 7, 64603-64613.
[2] Bauso, D., Dia, B.M., Djehiche, B., Tembine, H., & Tempone, R. (2014). Mean-field games for marriage. PLOS ONE, 9(5), e94933.
[3] Djehiche, B., Tcheukam, A., & Tembine, H. (2017). Mean-field-type games in engineering. AIMS Electron. Elect., Eng., 1(1), 18-73.
[4] Frihi, Z.E.O., Barreiro-Gomez, J., Choutri, S.E., & Tembine, H. (2021). Toolbox to simulate and mitigate COVID-19 propagation. SoftwareX, 14, 100673.
[5] Gao, J., & Tembine, H. (2019). Distributed mean-field-type filters for traffic networks. IEEE Trans. Intell. Transp. Syst., 20(2), 507-521.
[6] Hofbauer, J., & Sigmund, K. (1998). Evolutionary games and population dynamics. Cambridge, U.K.: Cambridge University Press.
[7] Tembine, H. (2020). COVID-19: Data-driven mean-field-type game perspective. Games J., 11(4), 51.
[8] Tembine, H. (2021). Master adjoint systems in mean-field-type games. Commun. Inf. Syst., 21(4), 623-650.
Boualem Djehiche is a professor of mathematical statistics in the Department of Mathematics’ Division of Mathematical Statistics at the KTH Royal Institute of Technology. His current research interests are in the area of stochastic analysis and include stochastic control and differential games, insurance mathematics, and mathematical finance. Hamidou Tembine was the director of the Learning & Game Theory Laboratory (L&G Lab) at New York University. His main research interests are learning, evolution, and games. In 2014, Tembine received the IEEE ComSoc Outstanding Young Researcher Award for his promising research activities for the benefit of society. He was previously a Simons Participant and a Senior Fellow of the Institute for Pure and Applied Mathematics. He is also a Next Einstein Fellow (class of 2017).