Humans have developed considerable machinery used at scale to create policies and to distribute incentives, yet we are forever seeking ways in which to improve upon these, our institutions. Especially when funding is limited, it is imperative to optimise spending without sacrificing positive outcomes, a challenge which has often been approached within several areas of social, life and engineering sciences. These studies often neglect the availability of information, cost restraints or the underlying complex network structures, which define real-world populations. Here, we have extended these models, including the aforementioned concerns, but also tested the robustness of their findings to stochastic social learning paradigms. Akin to real-world decisions on how best to distribute endowments, we study several incentive schemes, which consider information about the overall population, local neighbourhoods or the level of influence which a cooperative node has in the network, selectively rewarding cooperative behaviour if certain criteria are met. Following a transition towards a more realistic network setting and stochastic behavioural update rule, we found that carelessly promoting cooperators can often lead to their downfall in socially diverse settings. These emergent cyclic patterns not only damage cooperation, but also decimate the budgets of external investors. Our findings highlight the complexity of designing effective and cogent investment policies in socially diverse populations.

We analyze the implications of strategic interactions between two heterogeneous groups (i.e., young and old, men and women) in a macroeconomic-epidemiological framework. The interactions between groups determine the overall prevalence of a communicable disease, which in turn affects the level of economic activity. Individuals may lower their disease exposure by reducing their mobility, but since changing mobility patterns is costly, each group has an incentive to free ride negatively affecting the chances of disease containment at the aggregate level. By focusing on an early epidemic setting, we explicitly characterize the cooperative and noncooperative equilibria, determining how the inefficiency induced by noncooperation (i.e., failure to internalize epidemic externalities) depends both on economic and epidemiological parameters. We show that long-run eradication may be possible even in the absence of coordination, but coordination leads to a faster reduction in the number of infectives in finite time. Moreover, the inefficiency induced by noncooperation increases (decreases) with the factors increasing (decreasing) the pace of the disease spread.

We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented with a graphon, which can be viewed as the limit of a dense random graph. A player's transition rates between the states depend on their control and the strength of interaction with the other players. We develop a rigorous mathematical framework for the game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equations characterizing Nash equilibria. Moreover, we propose a numerical approach based on machine learning methods and we present experimental results on different applications to compartmental models in epidemiology.

We study a multi-type SIR epidemic process within a heterogeneous population that interacts through a network. We base social contact on a random graph with given vertex degrees, and we give limit theorems on the fraction of infected individuals. For given social distancing individual strategies, we establish the epidemic reproduction number , which can be used to identify network vulnerability and inform vaccination policies. In the second part of the paper, we study the equilibrium of the social distancing game. Individuals choose their social distancing level according to an anticipated global infection rate, which must equal the actual infection rate following their choices. We give conditions for the existence and uniqueness of an equilibrium. In the case of random regular graphs, we show that voluntary social distancing will always be socially sub-optimal.

The most serious threat to ecosystems is the global climate change fueled by the uncontrolled increase in carbon emissions. In this project, we use mean field control and mean field game models to analyze and inform the decisions of electricity producers on how much renewable sources of production ought to be used in the presence of a carbon tax. The trade-off between higher revenues from production and the negative externality of carbon emissions is quantified for each producer who needs to balance in real time reliance on reliable but polluting (fossil fuel) thermal power stations versus investing in and depending upon clean production from uncertain wind and solar technologies. We compare the impacts of these decisions in two different scenarios: (1) the producers are competitive and hopefully reach a ; (2) they cooperate and reach a . In the model, the producers have both time dependent and independent controls. We first propose nonstandard forward-backward stochastic differential equation systems that characterize the Nash equilibrium and the social optimum. Then, we prove that both problems have a unique solution using these equations. We then illustrate with numerical experiments the producers' behavior in each scenario. We further introduce and analyze the impact of a regulator in control of the carbon tax policy, and we study the resulting Stackelberg equilibrium with the field of producers.

COVID-19 comes out as a sudden pandemic disease within human population. The pandemic dynamics of COVID-19 needs to be studied in detail. A pandemic model with hierarchical quarantine and time delay is proposed in this paper. In the COVID-19 case, the virus incubation period and the antibody failure will cause the time delay and reinfection, respectively, and the hierarchical quarantine strategy includes home isolation and quarantine in hospital. These factors that affect the spread of COVID-19 are well considered and analyzed in the model. The stability of the equilibrium and the nonlinear dynamics is studied as well. The threshold value of the bifurcation is deduced and quantitatively analyzed. Numerical simulations are performed to establish the analytical results with suitable examples. The research reveals that the COVID-19 outbreak may recur over a period of time, which can be helpful to increase the number of tested people with or without symptoms in order to be able to early identify the clusters of infection. And before the effective vaccine is successfully developed, the hierarchical quarantine strategy is currently the best way to prevent the spread of this pandemic.

This preface introduces the special issue on Dynamic Games for Modeling and Control of Epidemics. It showcases 12 papers with timely contributions to dynamic games and their applications to the modeling, analysis, and control of epidemics. The papers in this collection connect dynamic games and epidemic models to address the recent challenges related to screening, containment, and mitigation strategies for epidemics. This collection covers broad application areas in networks, human behaviors, and epidemiology as well as a diverse range of dynamic game methods, including evolutionary games, differential games, and mean-field games.

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of -player, -strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an -partite graph to describe games.

Masks save lives. Therefore, while the culture of wearing masks is promoted, it is critical to understand the various aspects of how that culture is adopted. The main contribution of this paper is in the modeling of the mask game. Wearing a mask provides partial protection against epidemics at some cost of comfort. Players can be differentiated according to both their risk state as well as their health state (susceptible, infected and removed). We formulate the problem as a Bayesian game in which players know their own risk state and ignore their own health state and the health and risk states of their counterparts. Using ideas from evolutionary games, we reduce the problem to a one-shot equivalent game and describe the structure of the symmetric equilibria. We prove that the policies adopted by the players at such equilibria admit a threshold structure. More specifically, players wear masks only if their risk state is equal to or bigger than a given threshold.

This review presents and reviews various solved and open problems in developing, analyzing, and mitigating epidemic spreading processes under human decision-making. We provide a review of a range of epidemic models and explain the pros and cons of different epidemic models. We exhibit the art of coupling between epidemic models and decision models in the existing literature. More specifically, we provide answers to fundamental questions in human decision-making amid epidemics, including what interventions to take to combat the disease, who are decision-makers, and when and how to take interventions, and how to make interventions. Among many decision models, game-theoretic models have become increasingly crucial in modeling human responses or behavior amid epidemics in the last decade. In this review, we motivate the game-theoretic approach to human decision-making amid epidemics. This review provides an overview of the existing literature by developing a multi-dimensional taxonomy, which categorizes existing literature based on multiple dimensions, including (1) types of games, such as differential games, stochastic games, evolutionary games, and static games; (2) types of interventions, such as social distancing, vaccination, quarantine, and taking antidotes; (3) the types of decision-makers, such as individuals, adversaries, and central authorities at different hierarchical levels. A fine-grained dynamic game framework is proposed to capture the essence of game-theoretic decision-making amid epidemics. We showcase three representative frameworks with unique ways of integrating game-theoretic decision-making into the epidemic models from a vast body of literature. Each of the three frameworks has their unique way of modeling and analyzing and develops results from different angles. In the end, we identify several main open problems and research gaps left to be addressed and filled.

The recent COVID-19 pandemic has led to an increasing interest in the modeling and analysis of infectious diseases. The pandemic has made a significant impact on the way we behave and interact in our daily life. The past year has witnessed a strong interplay between human behaviors and epidemic spreading. In this paper, we propose an evolutionary game-theoretic framework to study the coupled evolution of herd behaviors and epidemics. Our framework extends the classical degree-based mean-field epidemic model over complex networks by coupling it with the evolutionary game dynamics. The statistically equivalent individuals in a population choose their social activity intensities based on the fitness or the payoffs that depend on the state of the epidemics. Meanwhile, the spreading of the infectious disease over the complex network is reciprocally influenced by the players' social activities. We analyze the coupled dynamics by studying the stationary properties of the epidemic for a given herd behavior and the structural properties of the game for a given epidemic process. The decisions of the herd turn out to be strategic substitutes. We formulate an equivalent finite-player game and an equivalent network to represent the interactions among the finite populations. We develop a structure-preserving approximation technique to study time-dependent properties of the joint evolution of the behavioral and epidemic dynamics. The resemblance between the simulated coupled dynamics and the real COVID-19 statistics in the numerical experiments indicates the predictive power of our framework.

Evolutionary game theory mathematically conceptualizes and analyzes biological interactions where one's fitness not only depends on one's own traits, but also on the traits of others. Typically, the individuals are not overtly rational and do not select, but rather inherit their traits. Cancer can be framed as such an evolutionary game, as it is composed of cells of heterogeneous types undergoing frequency-dependent selection. In this article, we first summarize existing works where evolutionary game theory has been employed in modeling cancer and improving its treatment. Some of these game-theoretic models suggest how one could anticipate and steer cancer's eco-evolutionary dynamics into states more desirable for the patient via evolutionary therapies. Such therapies offer great promise for increasing patient survival and decreasing drug toxicity, as demonstrated by some recent studies and clinical trials. We discuss clinical relevance of the existing game-theoretic models of cancer and its treatment, and opportunities for future applications. Moreover, we discuss the developments in cancer biology that are needed to better utilize the full potential of game-theoretic models. Ultimately, we demonstrate that viewing tumors with evolutionary game theory has medically useful implications that can inform and create a lockstep between empirical findings and mathematical modeling. We suggest that cancer progression is an evolutionary competition between different cell types and therefore needs to be viewed as an evolutionary game.

Trade changes incentives to protect an open-access natural resource independently of its effect on the resource price. General equilibrium linkages cause resource policy to affect the price of privately owned assets regardless of whether they are used in the resource sector. In the closed economy, the asset market in our overlapping generations setting creates incentives for currently living agents to protect the natural resource. The interplay of the asset market and general equilibrium effects causes trade to reverse these incentives. Trade liberalization and the establishment of formal property rights are policy complements: the former makes the latter more important.

We model a society with two types of citizens: healthy and vulnerable individuals. While both types can be exposed to the virus and contribute to its spread, the vulnerable people tend to be more cautious as being exposed to the virus can be fatal for them due to their conditions, e.g., advanced age or prior medical conditions. We assume that both types would like to participate in in-person social activities as freely as possible and they make this decision based on the total number of infected people in the society. In this model, we assume that a local governmental authority imposes and administers social distancing regulations based on the infection status of the society and revises it accordingly in each time period. We model and solve for the steady state in four scenarios: (i) non-cooperative (Nash), (ii) cooperative, (iii) egoistic, and (iv) altruistic. The results show that the Altruistic scenario is the best among the four, i.e., the healthy citizens put the vulnerable citizens' needs first and self-isolate more strictly which results in more flexibility for the vulnerable citizens. We use a numerical example to illustrate that the Altruistic scenario will assist with pandemic control for both healthy and vulnerable citizens in the long run. The objective of this research is not to find a way to resolve the pandemic but to optimally live in a society which has been impacted by pandemic restrictions, similar to what was experienced in 2020 with the spread of COVID-19.

During epidemics, the population is asked to socially distance, with pairs of individuals keeping two meters apart. We model this as a new optimization problem by considering a team of agents placed on the nodes of a network. Their common aim is to achieve pairwise graph distances of at least ,  a state we call . (If they want to be at distinct nodes; if they want to be non-adjacent.) We allow only a simple type of motion called a lazy random walk: with probability (called the parameter), they remain at their current node next period; with complementary probability , they move to a random adjacent node. The team seeks the common value of which achieves social distance in the least expected time, which is the absorption time of a Markov chain. We observe that the same Markov chain, with different goals (absorbing states), models the gathering, or multi-rendezvous problem (all agents at the same node). Allowing distinct laziness for two types of agents (searchers and hider) extends the existing literature on predator-prey search games to multiple searchers. We consider only special networks: line, cycle and grid.

Since Downs proposed that the act of voting is irrational in 1957, myriad models have been proposed to explain voting and account for observed turnout patterns. We propose a model in which partisans consider both the instrumental and expressive benefits of their vote when deciding whether or not to abstain in an election, introducing an asymmetry that most other models do not consider. Allowing learning processes within our electorate, we analyze what evolutionarily stable strategies are rationalizable under various conditions. Upon varying electorate size, the partisan split of the electorate, and the degree to which an electorate takes underdog considerations into account in its payoff structure, we find that different equilibria arise. Our model predicts comparative statics that are consistent with voter behavior, specifically affirming a "size effect," in which turnout decreases as electorate size increases. Furthermore, relaxing some of our preliminary assumptions eliminates some of the discrepancies between the predictions of our model and empirical voter behavior. In particular, our work demonstrates that misperceptions about the partisan split of an electorate may account for high turnout behavior .

Zika fever is an emerging mosquito-borne disease. While it often causes no or only mild symptoms that are similar to dengue fever, Zika virus can spread from a pregnant woman to her baby and cause severe birth defects. There is no specific treatment or vaccine, but the disease can be mitigated by using several control strategies, generally focusing on the reduction in mosquitoes or mosquito bites. In this paper, we model Zika virus transmission and incorporate a game-theoretical approach to study a repeated population game of DEET usage to prevent insect bites. We show that the optimal use effectively leads to disease elimination. This result is robust and not significantly dependent on the cost of the insect repellents.

Individual behaviors play an essential role in the dynamics of transmission of infectious diseases, including COVID-19. This paper studies a dynamic game model that describes the social distancing behaviors during an epidemic, assuming a continuum of players and individual infection dynamics. The evolution of the players' infection states follows a variant of the well-known SIR dynamics. We assume that the players are not sure about their infection state, and thus, they choose their actions based on their individually perceived probabilities of being susceptible, infected, or removed. The cost of each player depends both on her infection state and on the contact with others. We prove the existence of a Nash equilibrium and characterize Nash equilibria using nonlinear complementarity problems. We then exploit some monotonicity properties of the optimal policies to obtain a reduced-order characterization for Nash equilibrium and reduce its computation to the solution of a low-dimensional optimization problem. It turns out that, even in the symmetric case, where all the players have the same parameters, players may have very different behaviors. We finally present some numerical studies that illustrate this interesting phenomenon and investigate the effects of several parameters, including the players' vulnerability, the time horizon, and the maximum allowed actions, on the optimal policies and the players' costs.

Social stigma can effectively prevent people from going out and possibly spreading COVID-19. Using the framework of replicator dynamics, we analyze the interaction between self-restraint behavior, infection with viruses such as COVID-19, and stigma against going out. Our model is analytically solvable with respect to an interior steady state in contrast to the previous model of COVID-19 with stigma. We show that a non-legally binding policy reduces the number of people going out in a steady state.

We study how payoffs and network structure affect reaching the payoff-dominant equilibrium in a [Formula: see text] coordination game that actors play with their neighbors in a network. Using an extensive simulation analysis of over 100,000 networks with 2-25 actors, we show that the importance of network characteristics is restricted to a limited part of the payoff space. In this part, we conclude that the payoff-dominant equilibrium is chosen more often if network density is larger, the network is more centralized, and segmentation of the network is smaller. Moreover, it is more likely that heterogeneity in behavior persists if the network is more segmented and less centralized. Persistence of heterogeneous behavior is not related to network density.

We present a dynamic population game model to capture the behavior of a large population of individuals in presence of an infectious disease or epidemic. Individuals can be in one of five possible infection states at any given time: susceptible, asymptomatic, symptomatic, recovered and unknowingly recovered, and choose whether to opt for vaccination, testing or social activity with a certain degree. We define the evolution of the proportion of agents in each epidemic state, and the notion of best response for agents that maximize long-run discounted expected reward as a function of the current state and policy. We further show the existence of a stationary Nash equilibrium and explore the transient evolution of the disease states and individual behavior under a class of evolutionary learning dynamics. Our results provide compelling insights into how individuals evaluate the trade-off among vaccination, testing and social activity under different parameter regimes, and the impact of different intervention strategies (such as restrictions on social activity) on vaccination and infection prevalence.