Two-strategy Games Research Articles

Evolutionary dynamics in bilingual games

In two-strategy coordination games with distinct payoff- and risk-dominant equilibria, existing results show that the inefficient risk-dominant equilibrium is uniquely selected under many evolutionary dynamics. In the above class of coordination games, we study the effect of introducing a bilingual strategy that is compatible with both of the existing strategies. An agent playing the bilingual strategy incurs an additional adoption cost but never miscoordinates with any other agent. We show that if the adoption cost of the bilingual strategy is low, then the efficient payoff-dominant equilibrium can be uniquely selected under many evolutionary dynamics.

Transitions between equilibria in Bilingual Games under Probit Choice

We study the effect of introducing a bilingual strategy on the long run equilibrium outcome in a class of two-strategy coordination games with distinct payoff- and risk-dominant equilibria under the probit choice rule. Existing results show that in the class of two-strategy games under consideration, the inefficient risk dominant equilibrium is selected in the long run under noisy best response models. We show that if the cost of the bilingual option is sufficiently low then the efficient payoff dominant equilibrium will be selected in the long run under the probit choice rule.

Behavioral vaccination policies and game-environment feedback in epidemic dynamics

Many policymakers have adopted voluntary vaccination policies to alleviate the consequences of contagious diseases. Such policies have several well-established feathers, i.e. they are seasonal, depending on an individual’s decision, adaptive, and control epidemic activity. Here, we study ideas from behavioral epidemiology embedded with a vaccination game and pairwise two-player two-strategy game to represent the environmental feedback in an SVIR model by using a composite information index including disease incidence, vaccine factors and cooperative behavior on a global time scale (repeated season). In its turn, the information index’s game dynamics to participate in the vaccine program (cooperation) is supposed to reflect the feedback-evolving dynamics of competitive cognitions and the environment. The assuming model is described by two different evolutionary game systems connected by an unknown external public opinion environment feedback. The embedded model is described by an inherited system showing a behavioral aspect, i.e. pairwise game indicates an individual’s cooperative behavior, and a vaccine game refers to vaccine-cost influence. This is a novel attempt to stabilize the two different decision processes to pool them into a single index. Extensive simulations suggest a rich spectrum of achievable results, including epidemic control, human behavior, social dilemma, and policy suggestions.

On a finite population variation of the Fisher–KPP equation

In this paper, we formulate a finite population variation of the Fisher–KPP equation using the fact that the reaction term can be generated from the replicator dynamic using a two-player two-strategy skew-symmetric game. We use prior results from Ablowitz and Zeppetella to show that the resulting system of partial differential equations admits a travelling wave solution, and that there are closed form solutions for this travelling wave. Interestingly, the closed form solution is constructed from a sign-reversal of the known closed form solution of the classic Fisher equation. We also construct a closed form solution approximation for the corresponding equilibrium problem on a finite interval with Dirichlet and Neumann boundary conditions. Two conjectures on these corresponding equilibrium problems are presented and analysed numerically.

Prediction of groundwater pollution diffusion path based on multi-source data fusion

In order to improve the prediction accuracy of groundwater pollution diffusion path, this paper combines multivariate data fusion technology to predict and analyze the groundwater pollution diffusion path. Under the special two-strategy swarm game model of water pollution particle swarm, this paper introduces replication dynamics with bounded continuous time-delay. Moreover, considering the dynamic behavior in both cases of constant kernel function and exponential kernel function, the hawk-dove game model is a special case of the model we are discussing. In addition, this paper proposes a method combining leaching surface and flux concentration, and applies numerical simulation method to simulate and analyze transient leakage monitoring of similar point and line source pollution in planar two-dimensional heterogeneous aquifers. The experimental study verifies that the multivariate data fusion proposed in this paper can play an important role in the prediction of groundwater pollution diffusion path.

Judgment on the evolution state of asymmetric games

In this article, we probe into two-role two-player two-strategy asymmetric evolutionary games by utilizing replicator dynamic equations and present all equilibrium points and their stability conditions. Further, the resulting evolution state of the system can be depicted directly according to the specific payoff matrix and initial values. This study facilitates and perfects the research of asymmetric evolutionary games.

Evolutionary dynamics of multi-player snowdrift games based on the Wright-Fisher process

Although cooperative behavior is ubiquitous in biological and social systems, the causes and mechanisms of cooperation are a basic problem in evolutionary theory. The snowdrift game is considered as an effective evolutionary game model to describe cooperative behavior in a competitive situation. Thus, this paper studies the evolutionary dynamics of cooperative behavior in multi-player snowdrift games. This work establishes a stochastic two-strategy multi-player snowdrift game based on the Wright-Fisher (W-F) update process. Next, a specific analytical expression for fixation probabilities of cooperation and defection is considered, and the conditions under which cooperative strategies take root in a population and become an evolutionarily stable strategy are given. Finally, the relationships between the fixation probability of cooperation and each parameter involved in the game are obtained via simulation analysis. A simulation analysis reveals that the fixation probability of cooperation decreases with selection intensity, the number of players playing in multi-player snowdrift games, and population size but increases with the benefit-cost ratio. The present work promotes an understanding of the evolutionary dynamics of cooperative behavior and the theory of multi-player snowdrift games with the W-F update process.

Optimal strategies and cost-benefit analysis of the {varvec{n}}-player weightlifting game

The study of cooperation has been extensively studied in game theory. Especially, two-player two-strategy games have been categorized according to their equilibrium strategies and fully analysed. Recently, a grand unified game covering all types of two-player two-strategy games, i.e., the weightlifting game, was proposed. In the present study, we extend this two-player weightlifting game into an n-player game. We investigate the conditions for pure strategy Nash equilibria and for Pareto optimal strategies, expressed in terms of the success probability and benefit-to-cost ratio of the weightlifting game. We also present a general characterization of n-player games in terms of the proposed game. In terms of a concrete example, we present diagrams showing how the game category varies depending on the benefit-to-cost ratio. As a general rule, cooperation becomes difficult to achieve as group size increases because the success probability of weightlifting saturates towards unity. The present study provides insights into achieving behavioural cooperation in a large group by means of a cost–benefit analysis.

On two-player games with pure strategies on intervals $ [a, \; b] $ and comparisons with the two-player, two-strategy matrix case

<p style='text-indent:20px;'>We consider games of two-players with utility functions which are not necessarily linear on the product of convex and compact intervals of <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}^2 $\end{document}</tex-math></inline-formula>. An issue is how far an analogy can be drawn with two-player, two-strategy matrix games with linear utility functions, where [0, 1] registers probabilities and equilibria are at the intersection of reaction functions. Now, the idea of <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> functions is exploited to construct mixed strategies to look for Nash equilibria (NE). "Reaction" functions are constructed and results are obtained graphically. They are related to topological theorems on NE. The games chosen make specific points in relation to existence conditions and properties of solutions. It is a distinguishing feature that an interval [a, b] now registers both pure and mixed strategies. For NE a choice has to be justified. Also "reaction" functions are more complicated and their intersection does not guarantee an equilibrium.</p>

Tuning Cooperative Behavior in Games With Nonlinear Opinion Dynamics

We examine the tuning of cooperative behavior in repeated multi-agent games using an analytically tractable, continuous-time, nonlinear model of opinion dynamics. Each modeled agent updates its real-valued opinion about each available strategy in response to payoffs and other agents’ opinions, as observed over a network. We show how the model provides a principled and systematic means to investigate behavior of agents that select strategies using rationality and reciprocity, key features of human decision-making in social dilemmas. For two-strategy games, we use bifurcation analysis to prove conditions for the bistability of two equilibria and conditions for the first (second) equilibrium to reflect all agents favoring the first (second) strategy. We prove how model parameters, e.g., level of attention to opinions of others, network structure, and payoffs, influence dynamics and, notably, the size of the region of attraction to each stable equilibrium. We provide insights by examining the tuning of the bistability of mutual cooperation and mutual defection and their regions of attraction for the repeated prisoner’s dilemma and the repeated multi-agent public goods game. Our results generalize to games with more strategies, heterogeneity, and additional feedback dynamics, such as those designed to elicit cooperation or coordination.

Behavioural response to heterogeneous severity of COVID-19 explains temporal variation of cases among different age groups.

Together with seasonal effects inducing outdoor or indoor activities, the gradual easing of prophylaxis caused second and third waves of SARS-CoV-2 to emerge in various countries. Interestingly, data indicate that the proportion of infections belonging to the elderly is particularly small during periods of low prevalence and continuously increases as case numbers increase. This effect leads to additional stress on the health care system during periods of high prevalence. Furthermore, infections peak with a slight delay of about a week among the elderly compared to the younger age groups. Here, we provide a mechanistic explanation for this phenomenology attributable to a heterogeneous prophylaxis induced by the age-specific severity of the disease. We model the dynamical adoption of prophylaxis through a two-strategy game and couple it with an SIR spreading model. Our results also indicate that the mixing of contacts among the age groups strongly determines the delay between their peaks in prevalence and the temporal variation in the distribution of cases.This article is part of the theme issue ‘Data science approaches to infectious disease surveillance’.

Proposal of an apposite strategy-updating rule for the vaccination game where hubs refer to hubs and lower-degree agents refer to lower-degree agents.

In the vaccination game, the spread of disease and the human decision-making to obtain pre-emptive vaccinations are coordinately united in successive seasons. This is backed both by epidemiological models, such as SIR, and by evolutionary game theory, assuming a given strategy-updating rule. Several rules have been proposed by the community and rely on either the imitation concept or the switching-action concept. The latter directly stipulates whether or not an agent commits to a course of action based on a rule, such as the aspiration concept. In contrast, the former borrowed its fundamental idea from the spatial version of a two-player, two-strategy (2 × 2) game, such as the spatial prisoner's dilemma (SPD). The pairwise Fermi (PW-Fermi) strategy has been heavily employed as the most representative idea. The present study modifies PW-Fermi, which consists of two processes: one for selecting a pairwise opponent to imitate and the other giving the probability of copying from the opponent. Instead of a random selection, our proposed model applies a stochastically skewed selection in which a neighbor who has a similar degree to the focal player is preferentially selected. This specific rule allows us to establish a quite efficient society, in which hub agents spontaneously obtain vaccination, but lower-degree agents do not. To this end, a small number of higher-degree agents, who are exposed to higher infection risk, are urged to be vaccinated, whereas many other agents enjoy free-riding. This produces a relatively small vaccination cost as a social sum and also effectively suppresses the spread of disease, resulting in a small disease cost for society as a whole.

Almost sure exponential stability of two-strategy evolutionary games with multiplicative noise

In this paper, stochastic two-strategy evolutionary games in which the players are interfered by multiplicative noise are studied. We propose a stochastic replicator dynamic model to investigate the almost sure exponential stability (ASES) of general two-strategy games. Compared with the deterministic games and stochastic games with additive noise, our most interesting results concern the impact of multiplicative noise on the cooperation behavior in large well-mixed populations. We establish sufficient conditions on stochastic noise which can ensure the stable equilibrium points of general two-strategy games are indeed ASES. The stochastic Lyapunov framework is used to prove the stochastic stability, where finding the suitable Lyapunov function is the key and challenging in our case. Three different types of two-strategy game models (snowdrift games, hunt stag games and prisoner’s dilemmas) are simulated to verify our theoretical results.

Cyclical behavior of evolutionary dynamics in coordination games with changing payoffs

The paper presents a model of two-speed evolution in which the payoffs in the population game (or, alternatively, the individual preferences) slowly adjust to changes in the aggregate behavior of the population. The model investigates how, for a population of myopic agents with homogeneous preferences, changes in the environment caused by current aggregate behavior may affect future payoffs and hence alter future behavior. The interaction between the agents is based on a symmetric two-strategy game with positive externalities and negative feedback from aggregate behavior to payoffs, so that at every point in time the population has an incentive to coordinate, whereas over time the more popular strategy becomes less appealing. Under the best response dynamics and the logit dynamics with small noise levels the joint trajectories of preferences and behavior converge to closed orbits around the unique steady state, whereas for large noise levels the steady state of the logit dynamics becomes a sink. Under the replicator dynamics the unique steady state of the system is repelling and the trajectories are unbounded unstable spirals.

Behavioral decision-making in power demand-side response management: A multi-population evolutionary game dynamics perspective

Traditional single-agent decision-based optimization theory system is gradually hard to address long-term dynamic interaction problems in power demand-side response management (DRM). To this end, this paper thoroughly investigates the behavioral decision-making issues in power DRM from a perspective of multi-population evolutionary game dynamics. First, the evolutionary dynamics of general two-strategy three-population evolutionary games (2s3pEGs) is discussed, and relative net payoffs (RNPs) are defined for them in engineering. Discussion reveals that long-term evolutionary stable equilibrium (ESE) achieved in 2s3pEGs is only determined by RNPs. Further, the modeling idea of general two-strategy n-population (n ≥ 2) evolutionary games (npEGs) is elaborated. Second, a two-strategy npEG-based power DRM model is developed, as well as an npEG algorithm. Based on these, this paper investigates the long-term ESE of user engagement in power DRM using six subcases, which verifies the practicality and effectiveness of the obtained findings. Moreover, the case study reveals that incentive pricing from utility companies plays a major role in increasing user engagement in power DRM, thereby promoting different user populations to participate in smart power consumption, dispatching and distribution. Finally, the future work is prospected. This paper attempts to apply npEG dynamics to power DRM. The findings could provide guidelines for the investigations on bounded rationality and limited information-based behavioral decision-making issues, especially in the power DRM field.

Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games

This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection-as a result of competition fashioned by game-theoretic consideration-into the Wright-Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker-Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.

General Three-Population Multi-Strategy Evolutionary Games for Long-Term On-Grid Bidding of Generation-Side Electricity Market

Founded on bounded rationality and limited information, evolutionary game theory has been preliminarily applied in many fields, such as electricity market (EM). To address the complex behavioral decision-making issues in the more-common three-population multi-strategy evolutionary game (3PmSEG) scenarios in EM. This paper explores the long-term evolutionarily stable equilibrium (ESE) characteristics of general 3PmSEG systems with the aim of systematically investigating the evolution process of long-term on-grid bidding of a generation-side EM based on these features. First, the long-term ESE characteristics of general three-population two-strategy and three-strategy evolutionary games are thoroughly investigated. Complete relative net payoff (RNP) parameters are defined for these games. Then, the modeling idea of general 3PmSEGs is elaborated. Research shows that the game can be guided to evolve toward an expected long-term ESE point by properly adjusting its RNP parameters. To verify this, finally, the long-term on-grid bidding of power generation is investigated for a tripartite generation-side EM. The case study reveals that effective government supervision can effectively promote new energy accommodation of the market. Overall, the models developed in this paper are relatively universal and practical, which can provide some theoretical and methodological references for complex evolutionary game issues in related fields.

Optimal Management of Public Perceptions During A Flu Outbreak: A Game-Theoretic Perspective.

Public perceptions and sentiments play a crucial role in the success of vaccine uptake in the community. While vaccines have proven to be the best preventive method to combat the flu, the attitude and knowledge about vaccines are a major hindrance to higher uptake in most of the countries. The yearly coverage, especially in the vulnerable groups in the population, often remains below the herd immunity level despite the Flu Awareness Campaign organized by WHO every year worldwide. This brings immense challenges to the nation’s public health protection agency for strategic decision-making in controlling the flu outbreak every year. To understand the impact of public perceptions and vaccination decisions while designing optimal immunization policy, we model the individual decision-making as a two-strategy pairwise contest game, where pay-off is considered as a function of public health effort for the campaign. We use Pontryagin’s maximum principle to identify the best possible strategy for public health to implement vaccination and reduce infection at a minimum cost. Our optimal analysis shows that the cost of public health initiatives is qualitatively and quantitatively different under different public perceptions and attitudes towards vaccinations. When individual risk perception evolves with vaccine uptake or disease induced death, our model demonstrates a feed-forward mechanism in the dynamics of vaccination and exhibits an increase in vaccine uptake. Using numerical simulation, we also observe that the optimal cost can be minimized by putting the effort in the beginning and later part of the outbreak rather than during the peak. It confers that public health efforts towards disseminating disease severity or actual vaccination risk might accelerate the vaccination coverage and mitigate the infection faster.

Equilibrium analysis of general N-population multi-strategy games for generation-side long-term bidding: An evolutionary game perspective

Founded on bounded rationality and limited information, evolutionary game theory can well describe the evolution rule of population behavior and predict individuals’ decision-making behavior. Therefore, this paper focuses on the general N-population multi-strategy evolutionary games, and uses them to investigate the generation-side long-term bidding issues in electricity market. First, the long-term equilibrium characteristics of typical two-population and three-population two-strategy evolutionary game scenarios are thoroughly investigated through theoretical analysis and dynamic simulation, where novel relative net payoff parameters are completely defined for these games in engineering. Research shows that the long-term evolutionary stable equilibria are only determined by the relative net payoff parameters, so that an expected evolutionary stable equilibrium can be obtained by adjusting these parameters. Second, the modeling idea of general N-population multi-strategy evolutionary games is elaborated based on replicator dynamics. In the case study, the evolutionary stable equilibrium of generation-side long-term bidding is investigated for a supply-side market involving different generator populations. This case effectively verifies the evolutionary dynamics of the general N-population multi-strategy evolutionary games built in this paper. Lastly, future investigations of evolutionary game theory are prospected. Overall, this paper explores the long-term equilibrium characteristics of the general N-population multi-strategy evolutionary games, which can provide some inspirations and theoretical reference for researches on complex long-term dynamic interactive decision-making problems of group participants with bounded rationality in some relevant fields, especially in the economics, management, and engineering fields.

Family of chaotic maps from game theory

ABSTRACT From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, for every . This solves a known open question whether there exists a ‘natural’ nontrivial smooth map other than with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.