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Evolutionary game dynamics on networks typically consider the competition among simple strategies such as cooperation and defection in the Prisoner's Dilemma and summarize the effect of population structure as network reciprocity. However, it remains largely unknown regarding the evolutionary dynamics involving multiple powerful strategies typically considered in repeated games, such as the zero-determinant (ZD) strategies that are able to enforce a linear payoff relationship between them and their co-players. Here, we consider the evolutionary dynamics of always cooperate (AllC), extortionate ZD (extortioners), and unbending players in lattice populations based on the commonly used death-birth updating. Out of the class of unbending strategies, we consider a particular candidate, PSO Gambler, a machine-learning-optimized memory-one strategy, which can foster reciprocal cooperation and fairness among extortionate players. We derive analytical results under weak selection and rare mutations, including pairwise fixation probabilities and long-term frequencies of strategies. In the absence of the third unbending type, extortioners can achieve a half-half split in equilibrium with unconditional cooperators for sufficiently large extortion factors. However, the presence of unbending players fundamentally changes the dynamics and tilts the system to favor unbending cooperation. Most surprisingly, extortioners cannot dominate at all regardless of how large their extortion factor is, and the long-term frequency of unbending players is maintained almost as a constant. Our analytical method is applicable to studying the evolutionary dynamics of multiple strategies in structured populations. Our work provides insights into the interplay between network reciprocity and direct reciprocity, revealing the role of unbending strategies in enforcing fairness and suppressing extortion., Comment: 21 pages, 6 figures

Human cooperation depends on indirect reciprocity. In this work, we explore the concept of indirect reciprocity using a donation game in an infinitely large population. In particular, we examine how updating the reputations of recipients influences cooperation. Our work adds a time-scale parameter for updating donor and recipient reputations. We find a trade-off between the level of cooperation and evolutionary stability influenced by social norms. `Forgiving' recipient norms enhance cooperation but increase susceptibility to defectors, whereas `unforgiving' norms reduce cooperation but defend against invasion by defectors. Expanding to include gossip groups allows us to analyze the evolutionary dynamics of the time-scale parameter, identifying `generous' norms that support cooperation, and `strict' norms that discourage such generosity, ultimately showing vulnerability to defector invasions and potential cooperation collapse.

Recent developments of eco-evolutionary models have shown that evolving feedbacks between behavioral strategies and the environment of game interactions, leading to changes in the underlying payoff matrix, can impact the underlying population dynamics in various manners. We propose and analyze an eco-evolutionary game dynamics model on a network with two communities such that players interact with other players in the same community and those in the opposite community at different rates. In our model, we consider two-person matrix games with pairwise interactions occurring on individual edges and assume that the environmental state depends on edges rather than on nodes or being globally shared in the population. We analytically determine the equilibria and their stability under a symmetric population structure assumption, and we also numerically study the replicator dynamics of the general model. The model shows rich dynamical behavior, such as multiple transcritical bifurcations, multistability, and anti-synchronous oscillations. Our work offers insights into understanding how the presence of community structure impacts the eco-evolutionary dynamics within and between niches., Comment: 8 figures, 2 tables

In fighting infectious diseases posing a global health threat, ranging from influenza to Zika, non-pharmaceutical interventions (NPI), such as social distancing and face covering, remain mitigation measures public health can resort to. However, the success of NPI lies in sufficiently high levels of collective compliance, otherwise giving rise to waves of infection incidences that are not only driven by pathogen evolution but also changing vigilance in the population. Here we show that compliance with each NPI measure can be highly dynamic and context-dependent during an ongoing epidemic, where individuals may prefer one to another or even do nothing, leading to intricate temporal switching behavior of NPI adoptions. By characterizing dynamic regimes through the perceived costs of NPI measures and their effectiveness in particular regarding face covering and social distancing, our work offers new insights into overcoming barriers in NPI adoptions.

Evolutionary Game Theory (EGT) and Artificial Intelligence (AI) are two fields that, at first glance, might seem distinct, but they have notable connections and intersections. The former focuses on the evolution of behaviors (or strategies) in a population, where individuals interact with others and update their strategies based on imitation (or social learning). The more successful a strategy is, the more prevalent it becomes over time. The latter, meanwhile, is centered on machine learning algorithms and (deep) neural networks. It is often from a single-agent perspective but increasingly involves multi-agent environments, in which intelligent agents adjust their strategies based on feedback and experience, somewhat akin to the evolutionary process yet distinct in their self-learning capacities. In light of the key components necessary to address real-world problems, including (i) learning and adaptation, (ii) cooperation and competition, (iii) robustness and stability, and altogether (iv) population dynamics of individual agents whose strategies evolve, the cross-fertilization of ideas between both fields will contribute to the advancement of mathematics of multi-agent learning systems, in particular, to the nascent domain of ``collective cooperative intelligence'' bridging evolutionary dynamics and multi-agent reinforcement learning., Comment: 8 pages, 1 figure

Live performances of music are always charming, with the unpredictability of improvisation due to the dynamic between musicians and interactions with the audience. Jazz improvisation is a particularly noteworthy example for further investigation from a theoretical perspective. Here, we introduce a novel mathematical game theory model for jazz improvisation, providing a framework for studying music theory and improvisational methodologies. We use computational modeling, mainly reinforcement learning, to explore diverse stochastic improvisational strategies and their paired performance on improvisation. We find that the most effective strategy pair is a strategy that reacts to the most recent payoff (Stepwise Changes) with a reinforcement learning strategy limited to notes in the given chord (Chord-Following Reinforcement Learning). Conversely, a strategy that reacts to the partner's last note and attempts to harmonize with it (Harmony Prediction) strategy pair yields the lowest non-control payoff and highest standard deviation, indicating that picking notes based on immediate reactions to the partner player can yield inconsistent outcomes. On average, the Chord-Following Reinforcement Learning strategy demonstrates the highest mean payoff, while Harmony Prediction exhibits the lowest. Our work lays the foundation for promising applications beyond jazz: including the use of artificial intelligence (AI) models to extract data from audio clips to refine musical reward systems, and training machine learning (ML) models on existing jazz solos to further refine strategies within the game., Comment: 16 pages, 4 figures

How cooperation evolves and particularly maintains at a large scale remains an open problem for improving humanity across domains ranging from climate change to pandemic response. To shed light on how behavioral norms can resolve the social dilemma of cooperation, here we present a formal mathematical model of individuals' decision making under general social norms, encompassing a variety of concerns and motivations an individual may have beyond simply maximizing their own payoffs. Using the canonical game of the Prisoner's Dilemma, we compare four different norms: compassion, universalizability, reciprocity, and equity, to determine which social forces can facilitate the evolution of cooperation, if any. We analyze our model through a variety of limiting cases, including weak selection, low mutation, and large population sizes. This is complemented by computer simulations of population dynamics via a Fisher process, which confirm our theoretical results. We find that the first two norms lead to the emergence of cooperation in a wide range of games, but the latter two do not on their own. Due to its generality, our framework can be used to investigate many more norms, as well as how norms themselves emerge and evolve. Our work complements recent work on fair-minded learning dynamics and provides a useful bottom-up perspective into understanding the impact of top-down social norms on collective cooperative intelligence.

In the United States the laws regulating the carrying of firearms in public vary state-to-state. In a highly publicized event, the Governor of New Mexico recently issued an emergency order temporarily banning the carrying of firearms in some areas -- and thus rescinding the right-to-carry law in New Mexico -- after a spate of gun violence, citing a statistical threshold of general societal violence under which right-to-carry laws should be superseded. In this paper we frame this policy intervention as a dynamical systems model that measures the incidence of gun violence as a function of gun prevalence. We show that the Governor's emergency order -- fittingly issued as an emergency health order -- is effectively like trying to stop an epidemic after it has become viral in that with this kind of instantaneous stopping condition, under simple assumptions, such a regulation is too little, too late. On the other hand, a graduated response that scales with increasing violence can drive equilibrium gun prevalence to zero. This is a new mathematical model relating gun violence with gun prevalence which we hope continues to spur attention from the modeling community to bring its tools and techniques to bear on this important and challenging social problem. More importantly, our model, despite its simplicity, exhibits complex dynamics and has substantial research and educational value in promoting the application of mathematics in the social sciences, extending beyond gun control issues.

Understanding the impact of human behavior is crucial for successful mitigation of climate change across the globe. To shed light onto this issue, here we couple the forest dieback model with human behaviors. Using evolutionary game theory, we build a time-delay system where forest growth is impacted by both temperature and human mitigation choices, the latter being informed by temperature forecasts. Simulations of the coupled system over 200 years show us the varying outcomes: forest dies out and no one is a mitigator, forest dies out and everyone is a mitigator, or the forest survives and everyone is a mitigator. There exist rare cases where no one is a mitigator and yet the forest survives, but with a low coverage. We also find occasional oscillations where the proportion of mitigators vary between 0 and 1. Our results are based on simple models but have profound insights into determinants of behavior changes desired in social-climate dynamics.

In the wake of epidemics, quarantine measures are typically recommended by health authorities or governments to help control the spread of the disease. Compared with mandatory quarantine, voluntary quarantine offers individuals the liberty to decide whether to isolate themselves in case of infection exposure, driven by their personal assessment of the trade-off between economic loss and health risks as well as their own sense of social responsibility and concern for public health. To better understand self-motivated health behavior choices under these factors, here we incorporate voluntary quarantine into an endemic disease model -- the susceptible-infected-susceptible (SIS) model -- and perform comprehensive agent-based simulations to characterize the resulting behavior-disease interactions in structured populations. We quantify the conditions under which voluntary quarantine will be an effective intervention measure to mitigate disease burden. Furthermore, we demonstrate how individual decision-making factors, including the level of temptation to refrain from quarantine and the degree of social compassion, impact compliance levels of voluntary quarantines and the consequent collective disease mitigation efforts. We find that successful disease control requires either a sufficiently low level of temptation or a sufficiently high degree of social compassion, such that even complete containment of the epidemic is attainable. In addition to well-mixed populations, our simulation results are applicable to other more realistic social networks of contacts, including spatial lattices, small-world networks, and real social networks. Our work offers new insights into the fundamental social dilemma aspect of disease control through non-pharmaceutical interventions, such as voluntary quarantine and isolation, where the collective outcome of individual decision-making is crucial., Comment: 19 pages, 5 figures

The framework of iterated Prisoner's Dilemma (IPD) is commonly used to study direct reciprocity and cooperation, with a focus on the assessment of the generosity and reciprocal fairness of an IPD strategy in one-on-one settings. In order to understand the persistence and resilience of reciprocal cooperation, here we study long-term population dynamics of IPD strategies using the Moran process where stochastic dynamics of strategy competition can lead to the rise and fall of cooperation. Although prior work has included a handful of typical IPD strategies in the consideration, it remains largely unclear which type of IPD strategies is pivotal in steering the population away from defection and providing an escape hatch for establishing cooperation. We use a network-based approach to analyze and characterize networks of evolutionary pathways that bridge transient episodes of evolution dominated by depressing defection and ultimately catalyze the evolution of reciprocal cooperation in the long run. We group IPD strategies into three types according to their stationary cooperativity with an unconditional cooperator: the good (fully cooperative), the bad (fully exploitive), and the ugly (in between the former two types). We consider the mutation-selection equilibrium with rare mutations and quantify the impact of the presence versus absence of any given IPD strategy on the resulting population equilibrium. We identify catalysts (certain IPD strategies) as well as bridges (particular evolutionary pathways) that are most crucial for boosting the abundance of good types and suppressing that of bad types or having the highest betweenness centrality. Our work has practical implications and broad applicability to real-world cooperation problems by leveraging catalysts and bridges that are capable of strengthening persistence and resilience., Comment: 6 pages, 5 figures

Evolutionary game theory provides a mathematical foundation for cross-disciplinary fertilization, especially for integrating ideas from artificial intelligence and game theory. Such integration offers a transparent and rigorous approach to complex decision-making problems in a variety of important contexts, ranging from evolutionary computation to machine behavior. Despite the astronomically huge individual behavioral strategy space for interactions in the iterated Prisoner's Dilemma (IPD) games, the so-called Zero-Determinant (ZD) strategies is a set of rather simple memory-one strategies yet can unilaterally set a linear payoff relationship between themselves and their opponent. Although the witting of ZD strategies gives players an upper hand in the IPD games, we find and characterize unbending strategies that can force ZD players to be fair in their own interest. Moreover, our analysis reveals the ubiquity of unbending properties in common IPD strategies which are previously overlooked. In this work, we demonstrate the important steering role of unbending strategies in fostering fairness and cooperation in pairwise interactions. Our results will help bring a new perspective by means of combining game theory and multi-agent learning systems for optimizing winning strategies that are robust to noises, errors, and deceptions in non-zero-sum games., Comment: 8 pages, 3 figures, 2 tables

Cultures around the world show varying levels of conservatism. While maintaining traditional ideas prevents wrong ones from being embraced, it also slows or prevents adaptation to new times. Without exploration there can be no improvement, but often this effort is wasted as it fails to produce better results, making it better to exploit the best known option. This tension is known as the exploration/exploitation issue, and it occurs at the individual and group levels, whenever decisions are made. As such, it is has been investigated across many disciplines. In this work, we investigate the balance between exploration and exploitation in changing environments by thinking of exploration as mutation in a trait space with a varying fitness function. Specifically, we study how exploration rates evolves by applying adaptive dynamics to the replicator-mutator equation, under two types of fitness functions. For the first, payoffs are accrued from playing a two-player, two-action symmetric game, we consider representatives of all games in this class and find exploration rates often evolve downwards, but can also undergo neutral selection as well. Second, we study time dependent fitness with a function having a single oscillating peak. By increasing the period, we see a jump in the optimal exploration rate, which then decreases towards zero. These results establish several possible evolutionary scenarios for exploration rates, providing insight into many applications, including why we can see such diversity in rates of cultural change.

Following the invasion of Ukraine, the US, UK, and EU governments--among others--sanctioned oligarchs close to Putin. This approach has come under scrutiny, as evidence has emerged of the oligarchs' successful evasion of these punishments. To address this problem, we analyze the role of an overlooked but highly influential group: the secretive professional intermediaries who create and administer the oligarchs' offshore financial empires. Drawing on the Offshore Leaks Database provided by the International Consortium of Investigative Journalists (ICIJ), we examine the ties linking offshore expert advisors (lawyers, accountants, and other wealth management professionals) to ultra-high-net-worth individuals from four countries: Russia, China, the United States, and Hong Kong. We find that resulting nation-level "oligarch networks" share a scale-free structure characterized by heterogeneity of heavy-tailed degree distributions of wealth managers; however, network topologies diverge across clients from democratic versus autocratic regimes. While generally robust, scale-free networks are fragile when targeted by attacks on highly-connected nodes. Our "knock-out" experiments pinpoint this vulnerability to the small group of wealth managers themselves, suggesting that sanctioning these professional intermediaries may be more effective and efficient in disrupting dark finance flows than sanctions on their wealthy clients. This vulnerability is especially pronounced amongst Russian oligarchs, who concentrate their offshore business in a handful of boutique wealth management firms. The distinctive patterns we identify suggest a new approach to sanctions, focused on expert intermediaries to disrupt the finances and alliances of their wealthy clients. More generally, our research contributes to the larger body of work on complexity science and the structures of secrecy.

Understanding the origins of volunteerism and free-riding is crucial in collective action situations where a sufficient number of cooperators is necessary to achieve shared benefits, such as in vaccination campaigns and social change movements. Despite the importance of attaining behavior consistency for successful collective action, the theoretical mechanisms behind this process remain largely elusive. Here, we address this issue by studying the evolutionary dynamics of individual cooperativity levels in multi-round threshold public goods games. By using adaptive dynamics, we explore how individual behavior responses are shaped by incentives, with and without rewards or punishment. We demonstrate that rewarding consistent cooperators can lead to the emergence of two distinct populations: volunteers who consistently cooperate and free-riders who consistently defect. In contrast, punishing consistent defectors does not lead to similar evolutionary branching. Our results help offer insights into designing effective interventions that promote collective action and address collective risk dilemmas ranging from climate mitigation to pandemic control.

The Iterated Prisoner's Dilemma (IPD) is a well studied framework for understanding direct reciprocity and cooperation in pairwise encounters. However, measuring the morality of various IPD strategies is still largely lacking. Here, we partially address this issue by proposing a suit of plausible morality metrics to quantify four aspects of justice. We focus our closed-form calculation on the class of reactive strategies because of their mathematical tractability and expressive power. We define reactive means as a tool for studying how actors in the IPD and Iterated Snowdrift Game (ISG) behave under typical circumstances. We compute reactive means for four functions intended to capture human intuitions about ``goodness'' and ``fair play''. Two of these functions are strongly anticorrelated with success in the IPD and ISG, and the other two are weakly anticorrelated with success. Our results will aid in evaluating and comparing powerful IPD strategies based on machine learning algorithms, using simple and intuitive morality metrics., Comment: 18 pages, 3 figures

Incorporating social factors into disease prevention and control efforts is an important undertaking of behavioral epidemiology. The interplay between disease transmission and human health behaviors, such as vaccine uptake, results in complex dynamics of biological and social contagions. Maximizing intervention adoptions via network-based targeting algorithms by harnessing the power of social contagion for behavior and attitude changes largely remains a challenge. Here we address this issue by considering a multiplex network setting. Individuals are situated on two layers of networks: the disease transmission network layer and the peer influence network layer. The disease spreads through direct close contacts while vaccine views and uptake behaviors spread interpersonally within a potentially virtual network. The results of our comprehensive simulations show that network-based targeting with pro-vaccine supporters as initial seeds significantly influences vaccine adoption rates and reduces the extent of an epidemic outbreak. Network targeting interventions are much more effective by selecting individuals with a central position in the opinion network as compared to those grouped in a community or connected professionally. Our findings provide insight into network-based interventions to increase vaccine confidence and demand during an ongoing epidemic., Comment: 16 pages, 8 figures. Comments are welcome

The secretary problem has been a focus of extensive study with a variety of extensions that offer useful insights into the theory of optimal stopping. The original solution is to set one stopping threshold that gives rise to an immediately rejected sample $r$ out of the candidate pool of size $n$ and to accept the first candidate that is subsequently interviewed and bests the prior $r$ rejected. In reality, it is not uncommon to draw a line between job candidates to distinguish those above the line vs those below the line. Here we consider such satisficing sectary problem that views suboptimal choices (finding any of the top $d$ candidates) also as hiring success. We use a multiple stopping criteria $(r_1, r_2, \cdots, r_d)$ that sequentially lowers the expectation when the prior selection criteria yields no choice. We calculate the probability of securing the top $d$ candidate in closed form solutions which are in excellent agreement with computer simulations. The exhaustive search for optimal stopping has to deal with large parameter space and thus can quickly incur astronomically large computational times. In light of this, we apply the effective computational method of simulated annealing to find optimal $r_i$ values which performs reasonably well as compared with the exhaustive search despite having significantly less computing time. Our work sheds light on maximizing the likelihood of securing satisficing (suboptimal) outcomes using adaptive search strategy in combination with computation methods., Comment: 16 pages, 2 figures. Comments are welcome

Population heterogeneity, especially in individuals' contact networks, plays an important role in transmission dynamics of infectious diseases. For vaccine-preventable diseases, outstanding issues like vaccine hesitancy and availability of vaccines further lead to nonuniform coverage among groups, not to mention the efficacy of vaccines and the mixing pattern varying from one group to another. As the ongoing COVID-19 pandemic transitions to endemicity, it is of interest and significance to understand the impact of aforementioned population heterogeneity on the emergence and persistence of epidemics. Here we analyze epidemic thresholds and characterize bifurcation dynamics by accounting for heterogeneity caused by group-dependent characteristics, including vaccination rate and efficacy as well as disease transmissibility. Our analysis shows that increases in the difference in vaccination coverage among groups can render multiple equilibria of disease burden to exist even if the overall basic reproductive ratio is below one (also known as backward bifurcation). The presence of other heterogeneity factors such as differences in vaccine efficacy, transmission, mixing pattern, and group size can each exhibit subtle impacts on bifurcation. We find that heterogeneity in vaccine efficacy can undermine the condition for backward bifurcations whereas homophily tends to aggravate disease endemicity. Our results have practical implications for improving public health efforts by addressing the role of population heterogeneity in the spread and control of diseases., Comment: 14 pages, 8 figures

We establish a theoretical framework to address evolutionary dynamics of spatial games under strong selection. As the selection intensity tends to infinity, strategy competition unfolds in the deterministic way of winners taking all. We rigorously prove that the evolutionary process soon or later either enters a cycle and from then on repeats the cycle periodically, or stabilizes at some state almost everywhere. This conclusion holds for any population graph and a large class of finite games. This framework suffices to reveal the underlying mathematical rationale for the kaleidoscopic cooperation of Nowak and May's pioneering work on spatial games: highly symmetric starting configuration causes a very long transient phase covering a large number of extremely beautiful spatial patterns. For all starting configurations, spatial patterns transit definitely over generations, so cooperators and defectors persist definitely. This framework can be extended to explore games including the snowdrift game, the public goods games (with or without loner, punishment), and repeated games on graphs. Aspiration dynamics can also be fully addressed when players deterministically switch strategy for unmet aspirations by virtue of our framework. Our results have potential implications for exploring the dynamics of a large variety of spatially extended systems in biology and physics., Comment: 35 pages, 10 figures, and supplementary information

Feedbacks between strategies and the environment are common in social-ecological, evolutionary-ecological, and even psychological-economic systems. Utilizing common resources is always a dilemma for community members, like tragedy of the commons. Here we consider replicator dynamics with feedback-evolving games, where the payoffs switch between two different matrices. Although each payoff matrix on its own represents an environment where cooperators and defectors can't coexist stably, we show that it's possible to design appropriate switching control laws and achieve persistent oscillations of strategy abundance. This result should help guide the widespread problem of population state control in microbial experiments and other social problems with eco-evolutionary feedback loops.

Under constant selection, each trait has a fixed fitness, and small mutation rates allow populations to efficiently exploit the optimal trait. Therefore it is reasonable to expect mutation rates will evolve downwards. However, we find this need not be the case, examining several models of mutation. While upwards evolution of mutation rate has been found with frequency or time dependent fitness, we demonstrate its possibility in a much simpler context. This work uses adaptive dynamics to study the evolution of mutation rate, and the replicator-mutator equation to model trait evolution. Our approach differs from previous studies by considering a wide variety of methods to represent mutation. We use a finite string approach inspired by genetics, as well as a model of local mutation on a discretization of the unit intervals, handling mutation beyond the endpoints in three ways. The main contribution of this work is a demonstration that the evolution of mutation rate can be significantly more complicated than what is usually expected in relatively simple models.

The advent of Zero-Determinant (ZD) strategies has reshaped the study of reciprocity and cooperation in the iterated Prisoner's Dilemma games. The ramification of ZD strategies has been demonstrated through their ability to unilaterally enforce a linear relationship between their own average payoff and that of their co-player. Common practice conveniently represents this relationship by a straight line in the parametric plot of pairwise payoffs. Yet little attention has been paid to studying the actual geometry of the strategy space of all admissible ZD strategies. Here, our work offers intuitive geometric relationships between different classes of ZD strategies as well as nontrivial geometric interpretations of their specific parameterizations. Adaptive dynamics of ZD strategies further reveals the unforeseen connection between general ZD strategies and the so-called equalizers that can set any co-player's payoff to a fixed value. We show that the class of equalizers forming a hyperplane is the critical equilibrium manifold, only part of which is stable. The same hyperplane is also a separatrix of the cooperation-enhancing region where the optimum response is to increase cooperation for each of the four payoff outcomes. Our results shed light on the simple but elegant geometry of ZD strategies that is previously overlooked.

To curb the spread of fake news on social media platforms, recent studies have considered an online crowdsourcing fact-checking approach as one possible intervention method to reduce misinformation. However, it remains unclear under what conditions crowdsourcing fact-checking efforts deter the spread of misinformation. To address this issue, we model such distributed fact-checking as `peer policing' that will reduce the perceived payoff to share or disseminate false information (fake news) and also reward the spread of trustworthy information (real news). By simulating our model on synthetic square lattices and small-world networks, we show that the presence of social network structure enables fake news spreaders to be self-organized into echo chambers, thereby providing a boost to the efficacy of fake news and thus its resistance to fact-checking efforts. Additionally, to study our model in a more realistic setting, we utilize a Twitter network dataset and study the effectiveness of deliberately choosing specific individuals to be fact-checkers. We find that targeted fact-checking efforts can be highly effective, seeing the same level of success with as little as a fifth of the number of fact-checkers, but it depends on the structure of the network in question. In the limit of weak selection, we obtain closed-form analytical conditions for critical threshold of crowdsourced fact-checking in terms of the payoff values in our fact-checker/fake news game. Our work has practical implications for developing model-based mitigation strategies for controlling the spread of misinformation that interferes with the political discourse., Comment: 18 pages, 5 figures

Recent theory shows that extortioners taking advantage of the zero-determinant (ZD) strategy can unilaterally claim an unfair share of the payoffs in the Iterated Prisoner's Dilemma. It is thus suggested that against a fixed extortioner, any adapting co-player should be subdued with full cooperation as their best response. In contrast, recent experiments demonstrate that human players often choose not to accede to extortion out of concern for fairness, actually causing extortioners to suffer more loss than themselves. In light of this, here we reveal fair-minded strategies that are unbending to extortion such that any payoff-maximizing extortioner ultimately will concede in their own interest by offering a fair split in head-to-head matches. We find and characterize multiple general classes of such unbending strategies, including generous zero-determinant strategies and Win-Stay, Lose-Shift as particular examples. When against fixed unbending players, extortioners are forced with consequentially increasing losses whenever intending to demand more unfair share. Our analysis also pivots to the importance of payoff structure in determining the superiority of zero-determinant strategies and in particular their extortion ability. We show that an extortionate ZD player can be even outperformed by, for example, Win-Stay Lose-Shift, if the total payoff of unilateral cooperation is smaller than that of mutual defection. Unbending strategies can be used to outlearn evolutionary extortioners and catalyze the evolution of Tit-for-Tat-like strategies out of ZD players. Our work has implications for promoting fairness and resisting extortion so as to uphold a just and cooperative society., Comment: 11 pages of main text, 5 figures, 70 pages of supplementary information

The COVID-19, the disease caused by the novel coronavirus 2019 (SARS-CoV-2), has caused graving woes across the globe since first reported in the epicenter Wuhan, Hubei, China, December 2019. The spread of COVID-19 in China has been successfully curtailed by massive travel restrictions that put more than 900 million people housebound for more than two months since the lockdown of Wuhan on 23 January 2020 when other provinces in China followed suit. Here, we assess the impact of China's massive lockdowns and travel restrictions reflected by the changes in mobility patterns before and during the lockdown period. We quantify the synchrony of mobility patterns across provinces and within provinces. Using these mobility data, we calibrate movement flow between provinces in combination with an epidemiological compartment model to quantify the effectiveness of lockdowns and reductions in disease transmission. Our analysis demonstrates that the onset and phase of local community transmission in other provinces depends on the cumulative population outflow received from the epicenter Hubei. As such, infections can propagate further into other interconnected places both near and far, thereby necessitating synchronous lockdowns. Moreover, our data-driven modeling analysis shows that lockdowns and consequently reduced mobility lag a certain time to elicit an actual impact on slowing down the spreading and ultimately putting the epidemic under check. In spite of the vastly heterogeneous demographics and epidemiological characteristics across China, mobility data shows that massive travel restrictions have been applied consistently via a top-down approach along with high levels of compliance from the bottom up., Comment: 22 pages, 5 figures