L. Luthi, E. Pestelacci, M. Tomassini
We simulate the prisoner's dilemma and hawk-dove games on a real social acquaintance network. Using a discrete analogue of replicator dynamics, we show that surprisingly high levels of cooperation can be achieved, contrary to what happens in unstructured mixing populations. Moreover, we empirically show that cooperation in this network is stable with respect to invasion by defectors.
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We show that the success of cooperation in an evolutionary prisoner's dilemma on a complex network can be predicted by a simple, quantitative network analysis using mean field parameters. The criterion is shown to be accurate on a wide variety of networks with degree distributions ranging from regular to Poisson to scale-free. The network analysis uses a parameter that allows for comparisons of networks with both similar, and distinct, topologies. Furthermore, we establish the criterion here as a natural network analogue of Hamilton's classical rule for kin selection, despite arising in an ent…
Maintenance of cooperation was studied for a two-strategy evolutionary Prisoner's Dilemma game where the players are located on a one-dimensional chain and their payoff comes from games with the nearest and next-nearest neighbor interactions. The applied host geometry makes possible to study the impacts of two conflicting topological features. The evolutionary rule involves some noise affecting the strategy adoptions between the interacting players. Using Monte Carlo simulations and the extended versions of dynamical mean-field theory we determined the phase diagram as a function of noise leve…
We investigate an evolutionary prisoner's dilemma game among self-driven agents, where collective motion of biological flocks is imitated through averaging directions of neighbors. Depending on the temptation to defect and the velocity at which agents move, we find that cooperation can not only be maintained in such a system but there exists an optimal size of interaction neighborhood, which can induce the maximum cooperation level. When compared with the case that all agents do not move, cooperation can even be enhanced by the mobility of individuals, provided that the velocity and the size o…
We study an evolutionary prisoner's dilemma game with two layered graphs, where the lower layer is the physical infrastructure on which the interactions are taking place and the upper layer represents the connections for the strategy adoption (learning) mechanism. This system is investigated by means of Monte Carlo simulations and an extended pair-approximation method. We consider the average density of cooperators in the stationary state for a fixed interaction graph, while varying the number of edges in the learning graph. According to the Monte Carlo simulations, the cooperation is modified…
How Committed Individuals Shape Social Dynamics: A Survey on Coordination Games and Social Dilemma Games
该综述梳理了坚定个体对协调博弈、社会困境博弈衍生的社会动态的多方面影响，并提出了实证验证、大语言模型代理、负面影响应对三类未来研究方向。
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 indi…
We study the evolutionary Prisoner's Dilemma game on scale-free networks for different initial distributions. We consider three types of initial distributions for cooperators and defectors: initially random distribution with different frequencies of defectors; intentional organization with defectors initially occupying the most connected nodes with different fractions of defectors; intentional assignment for cooperators occupying the most connected nodes with different proportions of defectors at the beginning. It is shown that initial configurations for cooperators and defectors can influence…
Punishment, especially selfish punishment, has recently been identified as a potent promoter in sustaining or even enhancing the cooperation among unrelated individuals. However, without other key mechanisms, the first-order social dilemma and second-order social dilemma are still two enduring conundrums in biology and the social sciences even with the presence of punishment. In the present study, we investigate a spatial evolutionary four-strategy prisoner's dilemma game model with avoiding mechanism, where the four strategies are cooperation, defection, altruistic and selfish punishment. By …
We study a modified prisoner's dilemma game taking place on two-dimensional disordered square lattices. The players are pure strategists and can either cooperate or defect with their immediate neighbors. In the generations each player update its strategy by following one of the neighboring strategies with a probability dependent on the payoff difference. The neighbor selection obeys a dynamic preferential rule, i.e., the more frequently a neighbor's strategy was adopted by the focal player in the previous rounds, the larger probability it will be chosen to refer to in the subsequent rounds. It…
Prediction of social dilemmas in networked populations via graph neural networks
本文提出TMIFE特征提取方法结合图神经网络，用于预测结构化群体社会困境中的高维集体行为，并通过数值模拟、迁移学习及人类囚徒困境实验验证了其鲁棒性和预测准确性。
Adaptive Punishment in Social Dilemmas
本文提出了一种自适应惩罚机制，通过共演化反馈在社会困境中促进合作，揭示了丰富的动态行为和应用前景。
We investigate two paradigms for studying the evolution of cooperation--Prisoner's Dilemma and Snowdrift game in an online friendship network obtained from a social networking site. We demonstrate that such social network has small-world property and degree distribution has a power-law tail. Besides, it has hierarchical organizations and exhibits disassortative mixing pattern. We study the evolutionary version of the two types of games on it. It is found that enhancement and sustainment of cooperative behaviors are attributable to the underlying network topological organization. It is also sho…
Evolutionary Dynamics of Reputation-Based Voluntary Prisoner's Dilemma Games
本文提出基于声誉的自愿囚徒困境模型，通过分析退出激励机制在不同种群结构中的动态影响，揭示了合作维持的多种共存路径及稳定机制。
Monitoring and reporting incorrect acts are pervasive for maintaining human cooperation, but in theory it is unclear how they influence each other. To explore their possible interactions we consider spatially structured population where individuals face the collective-risk social dilemma. In our minimal model cooperator players report defection according to the loss of their interests. In parallel we assume a monitoring institution that monitors all group member and identifies wrong behavior with a certain probability. In response to these feedbacks a sanctioning institution develops punishmen…
The lowest temptation level (T = 1) is considered a trivial case for the Prisoner's Dilemma. Here, we show that this statement is true only for a very particular case, where the players interact with only one player. Otherwise, if the players interact with more than one player, the system presents the same possible behaviors observed for $T > 1$. In the steady state, the system can reach the cooperative, chaotic or defective phases, when adopting the Darwinian Evolutionary Strategy and the cooperative or quasi-regular phases, adopting the Pavlovian Evolutionary Strategy.
In this paper we extend the investigation of cooperation in some classical evolutionary games on populations were the network of interactions among individuals is of the scale-free type. We show that the update rule, the payoff computation and, to some extent the timing of the operations, have a marked influence on the transient dynamics and on the amount of cooperation that can be established at equilibrium. We also study the dynamical behavior of the populations and their evolutionary stability.
We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy and imitation rule from a randomly chosen neighbor with a probability dependent on the payoff difference when the player's income is collected from games with the neighbors. The players, located on the sites of a two-dimensional lattice, follow unconditional cooperation or defection and use individual strategy adoption rule described by a parameter. If the system is started from a random initial state then the present co-evolutionary rule drives the system towards a state where only one evolutiona…
Dynamics of evolutionary games strongly depend on underlying networks. We study the coevolutionary prisoner's dilemma in which players change their local networks as well as strategies (i.e., cooperate or defect). This topic has been increasingly explored by many researchers. On the basis of active linking dynamics [J. M. Pacheco et al., J. Theor. Biol. 243, 437 (2006), J. M. Pacheco et al., Phys. Rev. Lett. 97, 258103 (2006)], we show that cooperation is enhanced fairly robustly. In particular, cooperation evolves when the payoff of the player is normalized by the number of neighbors; this is…
Prisoner's Dilemma is a game theory model used to describe altruistic behavior seen in various populations. This theoretical game is important in understanding why a seemingly selfish strategy does persist and spread throughout a population that is mixing homogeneously at random. For a population with structure determined by social interactions, Prisoner's Dilemma brings to light certain requirements for the altruistic strategy to become established. Monte Carlo simulations of Prisoner's Dilemma are carried out using both simulated social networks and a dataset of a real social network. In bot…
Seasonal social dilemmas
该研究将博弈论行为模型与带季节性驱动的流行病学模型耦合，发现季节性社会困境会每年出现且行为反应可升降感染峰值，揭示行为与外部动态的复杂依存关系，指出干预需精准择时。
Recent studies on the evolutionary dynamics of the Prisoner's Dilemma game in scale-free networks have demonstrated that the heterogeneity of the network interconnections enhances the evolutionary success of cooperation. In this paper we address the issue of how the characterization of the asymptotic states of the evolutionary dynamics depends on the initial concentration of cooperators. We find that the measure and the connectedness properties of the set of nodes where cooperation reaches fixation is largely independent of initial conditions, in contrast with the behavior of both the set of n…
We focus on the heterogeneity of social networks and its role to the emergence of prevailing cooperation and sustaining cooperators. The social networks are representative of the interaction relationships between players and their encounters in each round of games. We study an evolutionary Prisoner's Dilemma game on a variant of Watts-Strogatz small-world network, whose heterogeneity can be tuned by a parameter. It is found that optimal cooperation level exists at some intermediate topological heterogeneity for different temptations to defect. Moreover, neither the most heterogeneous case nor …
The interplay of social and strategic motivations in human interactions is a largely unexplored question in collective social phenomena. Whether individuals' decisions are taken in a pure strategic basis or due to social pressure without a rational background crucially influences the model outcome. Here we study a networked Prisoner's Dilemma in which decisions are made either based on the replication of the most successful neighbor's strategy (unconditional imitation) or by pure social imitation following an update rule inspired by the voter model. The main effects of the voter dynamics are a…
Social distancing as one of the main non-pharmaceutical interventions can help slow down the spread of diseases, like in the COVID-19 pandemic. Effective social distancing, unless enforced as drastic lockdowns and mandatory cordon sanitaire, requires consistent strict collective adherence. However, it remains unknown what the determinants for the resultant compliance of social distancing and their impact on disease mitigation are. Here, we incorporate into the epidemiological process with an evolutionary game theory model that governs the evolution of social distancing behavior. In our model, …
Although cooperation is central to the organisation of many social systems, relatively little is known about cooperation in situations of collective emergency. When groups of people flee from a danger such as a burning building or a terrorist attack, the collective benefit of cooperation is important, but the cost of helping is high and the temptation to defect is strong. To explore the degree of cooperation in emergencies, we develop a new social game, the help-or-escape social dilemma. Under time and monetary pressure, players decide how much risk they are willing to take in order to help ot…
Restoring spatial cooperation with myopic agents in a three-strategy social dilemma
本文研究结构化种群中三策略社会困境，发现固定伙伴关系可能反常地降低合作水平，但通过引入近视智能体替代模仿策略协议，可在巨大参数空间内有效恢复空间合作。
The world in which we are living is a huge network of networks and should be described by interdependent networks. The interdependence between networks significantly affects the evolutionary dynamics of cooperation on them. Meanwhile, due to the diversity and complexity of social and biological systems, players on different networks may not interact with each other by the same way, which should be described by multiple models in evolutionary game theory, such as the Prisoner's Dilemma and Snowdrift Game. We therefore study the evolutionary dynamics of cooperation on two interdependent networks…
Social Dilemma of Non-Pharmaceutical Interventions
本研究聚焦非药物干预（NPI）的集体依从难题，揭示疫情期间公众对不同NPI措施的依从行为具有动态情境依赖性与复杂切换特征，为提升NPI采纳率提供了新的理论视角。
We introduce an evolutionary game on hypergraphs in which decisions between a risky alternative and a safe one are taken in social groups of different sizes. The model naturally reproduces choice shifts, namely the differences between the preference of individual decision makers and the consensual choice of a group, that have been empirically observed in choice dilemmas. In particular, a deviation from the Nash equilibrium towards the risky strategy occurs when the dynamics takes place on heterogeneous hypergraphs. These results can explain the emergence of irrational herding and radical behav…
Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix, the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is uniqu…
Spatial games are crucial for understanding patterns of cooperation in nature (and to some extent society). They are known to be more sensitive to local symmetries than e.g. spin models. This paper concerns the evolution of the prisoner's dilemma game on regular lattices with three different types of neighborhoods -- the von Neumann-, Moore-, and kagome types. We investigate two kinds of dynamics for the players to update their strategies (that can be unconditional cooperator or defector). Depending on the payoff difference, an individual can adopt the strategy of a random neighbor (a voter-mo…
Stability of pairwise social dilemma games: destructive agents, constructive agents, and their joint effects
本研究在无限充分混合种群中用复制动力学分析成对社会困境博弈，发现破坏性智能体无法促进合作反而破坏均衡，建设性智能体的效应随收益变化，二者组合会产生多稳态复杂演化动态，深化了对自愿参与机制影响合作的理解。
Impact of Committed Minorities: Unveiling Critical Mass of Cooperation in the Iterated Prisoner's Dilemma Game
本研究针对重复囚徒困境博弈，揭示了不同类型坚定少数派（无条件/条件合作者）在不同模仿强度、种群结构下的合作临界点及普遍规律，同时发现高密度条件合作者会阻碍结构化种群的合作，推进了对社会困境合作动力学的认知。
Advancing our understanding of human behavior hinges on the ability of theories to unveil the mechanisms underlying such behaviors. Measuring the ability of theories and models to predict unobserved behaviors provides a principled method to evaluate their merit and, thus, to help establish which mechanisms are most plausible. Here, we propose models and develop rigorous inference approaches to predict strategic decisions in dyadic social dilemmas. In particular, we use bipartite stochastic block models that incorporate information about the dilemmas faced by individuals. We show, combining the…
Whether or not to change strategy depends not only on the personal success of each individual, but also on the success of others. Using this as motivation, we study the evolution of cooperation in games that describe social dilemmas, where the propensity to adopt a different strategy depends both on individual fitness as well as on the strategies of neighbors. Regardless of whether the evolutionary process is governed by pairwise or group interactions, we show that plugging into the "wisdom of groups" strongly promotes cooperative behavior. The more the wider knowledge is taken into account th…
The pursuit of highest payoffs in evolutionary social dilemmas is risky and sometimes inferior to conformity. Choosing the most common strategy within the interaction range is safer because it ensures that the payoff of an individual will not be much lower than average. Herding instincts and crowd behavior in humans and social animals also compel to conformity on their own right. Motivated by these facts, we here study the impact of conformity on the evolution of cooperation in social dilemmas. We show that an appropriate fraction of conformists within the population introduces an effective su…
The effects of networking on the extent of cooperation emerging in a competitive setting are studied. The evolutionary snowdrift game, which represents a realistic alternative to the well-known Prisoner's Dilemma, is studied in the Watts-Strogatz network that spans the regular, small-world, and random networks through random re-wiring. Over a wide range of payoffs, a re-wired network is found to suppress cooperation when compared with a well-mixed or fully connected system. Two extinction payoffs, that characterize the emergence of a homogeneous steady state, are identified. It is found that, …
The spatial Prisoner's Dilemma is a prototype model to show the emergence of cooperation in very competitive environments. It considers players, at site of lattices, that can either cooperate or defect when playing the Prisoner's Dilemma with other z players. This model presents a rich phase diagram. Here we consider players in cells of one-dimensional cellular automata. Each player interacts with other z players. This geometry allows us to vary, in a simple manner, the number of neighbors ranging from one up to the lattice size, including self-interaction. This approach has multiple advantage…
We show that strategy independent adaptations of random interaction networks can induce powerful mechanisms, ranging from the Red Queen to group selection, that promote cooperation in evolutionary social dilemmas. These two mechanisms emerge spontaneously as dynamical processes due to deletions and additions of links, which are performed whenever players adopt new strategies and after a certain number of game iterations, respectively. The potency of cooperation promotion, as well as the mechanism responsible for it, can thereby be tuned via a single parameter determining the frequency of link …
Punishment and partner switching are two well-studied mechanisms that support the evolution of cooperation. Observation of human behaviour suggests that the extent to which punishment is adopted depends on the usage of alternative mechanisms, including partner switching. In this study, we investigate the combined effect of punishment and partner switching in evolutionary prisoner's dilemma games conducted on a network. In the model, agents are located on the network and participate in the prisoner's dilemma games with punishment. In addition, they can opportunistically switch interaction partn…
Evolution of conditional cooperation in collective-risk social dilemma with repeated group interactions
本研究构建重复互动演化博弈模型，探讨集体风险社会困境中条件合作行为对合作演化的影响，发现条件合作策略可促进合作涌现，高风险推动合作，风险超阈值时策略转变利于维持高水平合作。
Situations of conflict giving rise to social dilemmas are widespread in society and game theory is one major way in which they can be investigated. Starting from the observation that individuals in society interact through networks of acquaintances, we model the co-evolution of the agents' strategies and of the social network itself using two prototypical games, the Prisoner's Dilemma and the Stag Hunt. Allowing agents to dismiss ties and establish new ones, we find that cooperation and coordination can be achieved through the self-organization of the social network, a result that is non-trivi…
The emergence of cooperation in the thermodynamic limit of social dilemmas is an emerging field of research. While numerical approaches (using replicator dynamics) are dime a dozen, analytical approaches are rare. A particularly useful analytical approach is to utilize a mapping between the spin-1/2 Ising model in 1-D and the social dilemma game and calculate the magnetization, which is the net difference between the fraction of cooperators and defectors in a social dilemma. In this paper, we look at the susceptibility, which probes the net change in the fraction of players adopting a certain …
When evolutionary games are contested in structured populations, the degree of each player in the network plays an important role. If they exist, hubs often determine the fate of the population in remarkable ways. Recent research based on optimal percolation in random networks has shown, however, that the degree is neither the sole nor the best predictor of influence in complex networks. Low-degree nodes may also be optimal influencers if they are hierarchically linked to hubs. Taking this into account leads to the formalism of collective influence in complex networks, which as we show here, h…
Varying environmental conditions affect relations between interacting individuals in social dilemmas, thus affecting also the evolution of cooperation. Oftentimes these environmental variations are seasonal and can therefore be mathematically described as periodic changes. Accordingly, we here study how periodic shifts between different manifestations of social dilemmas affect cooperation. We observe a non-trivial interplay between the inherent spatiotemporal dynamics that characterizes the spreading of cooperation in a particular social dilemma type and the frequency of payoff changes. In par…
Spatial reciprocity is a well known tour de force of cooperation promotion. A thorough understanding of the effects of different population densities is therefore crucial. Here we study the evolution of cooperation in social dilemmas on different interaction graphs with a certain fraction of vacant nodes. We find that sparsity may favor the resolution of social dilemmas, especially if the population density is close to the percolation threshold of the underlying graph. Regardless of the type of the governing social dilemma as well as particularities of the interaction graph, we show that under…
Vaccination is crucial for the control of epidemics. Yet it is a social dilemma since non-vaccinators can benefit from the herd immunity created by the vaccinators. Thus the optimum vaccination level is not reached via voluntary vaccination at times. Intensive studies incorporate social networks to study vaccination behavior, and it is shown that vaccination can be promoted on some networks. The underlying network, however, is often assumed to be static, neglecting the dynamical nature of social networks. We investigate the vaccination behavior on dynamical social networks using both simulatio…
The functioning of animal as well as human societies fundamentally relies on cooperation. Yet, defection is often favorable for the selfish individual, and social dilemmas arise. Selection by individuals' fitness, usually the basic driving force of evolution, quickly eliminates cooperators. However, evolution is also governed by fluctuations that can be of greater importance than fitness differences, and can render evolution effectively neutral. Here, we investigate the effects of selection versus fluctuations in social dilemmas. By studying the mean extinction times of cooperators and defecto…
Note: Evolutionary Game Theory Focus Informational Health: The Cocktail Party Effect Through Werewolfgame under Incomplete Information and ESS Search Method Using Expected Gains of Repeated Dilemmas
本文在非完美信息博弈框架下，利用复制者方程和收益矩阵建模分析鸡尾酒会效应与假新闻污染风险对狼人游戏中ESS形成的影响，为信息真实性问题提供理论见解。
We present an extensive, systematic study of the Prisoner's Dilemma and Snowdrift games on a square lattice under a synchronous, noiseless imitation dynamics. We show that for both the occupancy of the network and the (random) mobility of the agents there are intermediate values that may increase the amount of cooperators in the system and new phases appear. We analytically determine the transition lines between these phases and compare with the mean field prediction and the observed behavior on a square lattice. We point out which are the more relevant microscopic processes that entitle coope…
Knowing the strategy of an opponent in a competitive environment conveys obvious evolutionary advantages. But this information is costly, and the benefit of being informed may not necessarily offset the additional cost. Here we introduce social dilemmas with informed strategies, and we show that this gives rise to two cyclically dominant triplets that form defensive alliances. The stability of these two alliances is determined by the rotation velocity of the strategies within each triplet. A weaker strategy in a faster rotating triplet can thus overcome an individually stronger competitor. Fas…
In this work we analyze the evolution of voluntary vaccination in networked populations by entangling the spreading dynamics of an influenza-like disease with an evolutionary framework taking place at the end of each influenza season so that individuals take or not the vaccine upon their previous experience. Our framework thus put in competition two well-known dynamical properties of scale-free networks: the fast propagation of diseases and the promotion of cooperative behaviors. Our results show that when vaccine is perfect scale-free networks enhance the vaccination behavior with respect to …
Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graphs defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation $b$ to choose defection, and the cost $c$ of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of …
Drivers of cooperation in social dilemmas on higher-order networks
提出一种用于结构化种群多人博弈的高阶网络框架，通过引入多维策略并研究不同阶交互间的动态与结构耦合，揭示了嵌套多级交互如何超越传统一维策略模型限制以增强合作，识别了促进现实群体社会困境中合作的关键驱动因素。
The network reciprocity is an important dynamic rule fostering the emergence of cooperation among selfish individuals. This was reported firstly in the seminal work of Nowak and May, where individuals were arranged on the regular lattice network, and played the prisoner's dilemma game (PDG). In the standard PDG, one often assumes that the players have perfect rationality. However, in reality, we human are far from rational agents, as we often make mistakes, and behave irrationally. Accordingly, in this work, we introduce the element of noise into the measurement of fitness, which is determined…
There is a tragedy of the traffic analogous to the tragedy of the commons that can be caused by overtaking. We analyze the effect of overtaking in a minimal model of vehicular traffic, the model proposed by Nagel and Schreckenberg, with two types of drivers: drivers that overtake and drivers that do not. We show that, under certain circumstances, overtaking is good because it increases the road capacity and minimizes the mean time spent by the driver on the road. However, when these conditions are not met, overtaking is harmful to all. More specifically, we found that a social dilemma emerges …
Individuals might abstain from participating in an instance of an evolutionary game for various reasons, ranging from lack of interest to risk aversion. In order to understand the consequences of such diverse activity patterns on the evolution of cooperation, we study a weak prisoner's dilemma where each player's participation is probabilistic rather than certain. Players that do not participate get a null payoff and are unable to replicate. We show that inactivity introduces cascading failures of cooperation, which are particularly severe on scale-free networks with frequently inactive hubs. …
A recently introduced concept of "cooperative equilibrium", based on the assumption that players have a natural attitude to cooperation, has been proven a powerful tool in predicting human behaviour in social dilemmas. In this paper, we extend this idea to more general game models, termed "Pareto solvable" games, which in particular include the Nash Bargaining Problem and the Ultimatum Game. We show that games in this class possess a unique pure cooperative equilibrium. Furthermore, for the Ultimatum Game, this notion appears to be strongly correlated with a suitably defined variant of the Dic…
Closeness is a global measure of centrality in networks, and a proxy for how influential actors are in social networks. In most network models, and many empirical networks, closeness is strongly correlated with degree. However, in social networks there is a cost of maintaining social ties. This leads to a situation (that can occur in the professional social networks of executives, lobbyists, diplomats and so on) where agents have the conflicting objectives of aiming for centrality while simultaneously keeping the degree low. We investigate this situation in an adaptive network-evolution model …
We have studied an evolutionary game with spatially arranged players who can choose one of the two strategies (named cooperation and defection for social dilemmas) when playing with their neighbors. In addition to the application of the usual strategies in the present model the players are also characterized by one of the two extreme personal features representing the egoist or fraternal behavior. During the evolution each player can modify both her own strategy and/or personal feature via a myopic update process in order to improve her utility. The results of numerical simulations and stabili…
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