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In 1965, T.S. Motzkin and E. G. Straus established an elegant connection between the clique number of a graph and the global maxima of a quadratic program defined on the standard simplex. Over the years, this seminal finding has inspired a number of studies aimed at characterizing the properties of the (local and global) solutions of the Motzkin–Straus program. The result has also been generalized in various ways and has served as the basis for establishing new bounds on the clique number and developing powerful clique-finding heuristics. Despite the extensive work done on the subject, apart from a few exceptions, the existing literature pays little or no attention to the Karush–Kuhn–Tucker (KKT) points of the program. In the conviction that these points might reveal interesting structural properties of the graph underlying the program, this paper tries to fill in the gap. In particular, we study the generalized KKT points of a parameterized version of the Motzkin–Straus program, which are defined via a relaxation of the usual first-order optimality conditions, and we present a number of results that shed light on the symmetries and regularities of certain substructures associated with the underlying graph. These combinatorial structures are further analyzed using barycentric coordinates, thereby providing a link to a related quadratic program that encodes local structural properties of the graph. This turns out to be particularly useful in the study of the generalized KKT points associated with a certain class of graphs that generalize the notion of a star graph. Finally, we discuss the associations between the generalized KKT points of the Motzkin–Straus program and the so-called replicator dynamics, thereby offering an alternative, dynamical-system perspective on the results presented in the paper. [ABSTRACT FROM AUTHOR]

Vaccination is an effective strategy to prevent the spread of diseases. However, hesitancy and rejection of vaccines, particularly in childhood immunizations, pose challenges to vaccination efforts. In that case, according to rational decision-making and classical utility theory, parents weigh the costs of vaccination against the costs of not vaccinating their children. Social norms influence these parental decision-making outcomes, deviating their decisions from rationality. Additionally, variability in values of utilities stemming from stochasticity in parents' perceptions over time can lead to further deviations from rationality. In this paper, we employ independent white noises to represent stochastic fluctuations in parental perceptions of utility functions of the decisions over time, as well as in the disease transmission rates. This approach leads to a system of stochastic differential Eqs of a susceptible-infected-recovered (SIR) model coupled with a stochastic replicator Eq. We explore the dynamics of these Eqs and identify new behaviors emerging from stochastic influences. Interestingly, incorporating stochasticity into the utility functions for vaccination and nonvaccination leads to a decision-making model that reflects the bounded rationality of humans. Noise, like social norms, is a two-sided sword that depends on the degree of bounded rationality of each group. We also perform a stochastic optimal control as a discount to the cost of vaccination to counteract bounded rationality.

Rewarding cooperators and punishing defectors are effective measures for promoting cooperation in evolutionary game theory. Given that previous models treated rewards as constants, this does not reflect real-world dynamics changes. Therefore, this paper focused on the classical payoff matrix and examined the dynamic variable rewards affected by cooperation and defection strategies, as well as the impact of time delays. First, for the system without a time delay, we analyzed the existence and stability of numerous equilibrium points and explored transcritical bifurcations under various conditions. Second, for the time-delay system, we discussed a series of delayed dynamical behaviors including Hopf bifurcation, period, and the stability and direction of bifurcation. Finally, the changes of cooperation strategy were observed by numerical simulation, and some interesting results were obtained: (ⅰ) Under certain circumstances, even if the reward given to the cooperators reaches the maximum, the proportion of cooperators is still zero, which means that increasing rewards does not always promote cooperation. (ⅱ) The initial state can affect the choice of cooperation strategy and defection strategy. (ⅲ) The increase of the time delay makes the stable equilibrium point disappear and forms a stable limit cycle.

Rewarding cooperators and punishing defectors are effective measures for promoting cooperation in evolutionary game theory. Given that previous models treated rewards as constants, this does not reflect real-world dynamics changes. Therefore, this paper focused on the classical payoff matrix and examined the dynamic variable rewards affected by cooperation and defection strategies, as well as the impact of time delays. First, for the system without a time delay, we analyzed the existence and stability of numerous equilibrium points and explored transcritical bifurcations under various conditions. Second, for the time-delay system, we discussed a series of delayed dynamical behaviors including Hopf bifurcation, period, and the stability and direction of bifurcation. Finally, the changes of cooperation strategy were observed by numerical simulation, and some interesting results were obtained: (ⅰ) Under certain circumstances, even if the reward given to the cooperators reaches the maximum, the proportion of cooperators is still zero, which means that increasing rewards does not always promote cooperation. (ⅱ) The initial state can affect the choice of cooperation strategy and defection strategy. (ⅲ) The increase of the time delay makes the stable equilibrium point disappear and forms a stable limit cycle. [ABSTRACT FROM AUTHOR]

Featured Application: This article mainly utilizes the advantages and characteristics of evolutionary game theory and, based on the ideas and methods of evolutionary game theory, describes the relationship between the renewable energy generation enterprise group and the power grid enterprise group as a "learning" progressive evolution system, focusing on the evolution process of the relationship between various stakeholders and the influencing factors of evolutionary stability. It provides a reasonable explanation for the spontaneous formation of interest equilibrium between power generation enterprises and power grid companies and provides theoretical reference and policy recommendations for government regulation of the electricity bidding market. The insights derived from the simulations offer a framework that can inform practical applications, particularly in improving grid stability and promoting renewable energy adoption through effective peak shaving mechanisms and electricity pricing strategies. This paper presents a novel approach to optimizing the coordination between renewable energy generation enterprises and power grid companies using evolutionary game theory. The research focuses on resolving conflicts and distributing benefits between these key stakeholders in the context of large-scale renewable energy integration. A theoretical model based on replicator dynamics is developed to simulate and analyze the evolutionary stable strategies of power generation enterprises and grid companies with particular emphasis on peak shaving services and electricity bidding. These simulations are based on theoretical models and do not incorporate real-world data directly, but they aim to replicate scenarios that reflect realistic behaviors within the electricity market. The model is validated through dynamic simulation under various scenarios, demonstrating that the final strategic choices of both thermal power and renewable energy enterprises tend to evolve towards either high-price or low-price bidding strategies, significantly influenced by initial system parameters. Additionally, this study explores how the introduction of peak shaving compensation affects the coordination process and stability of renewable energy integration, providing insights into improving grid efficiency and enhancing renewable energy adoption. Although the results are simulation-based, they are designed to offer practical recommendations for grid management and policy development, particularly for the integration of renewable energies such as wind power in competitive electricity markets. The findings suggest that effective government regulation, alongside well-designed compensation mechanisms, can help establish a balanced interest distribution between stakeholders. By offering a clear framework for analyzing the dynamics of renewable energy integration, this work provides valuable policy recommendations to promote cooperation and stability in electricity markets. This study contributes to the understanding of the complex interactions in the electricity market and offers practical solutions for enhancing the integration of renewable energy into the grid. [ABSTRACT FROM AUTHOR]

Abstract Measuring the dynamics of microbial communities results in high-dimensional measurements of taxa abundances over time and space, which is difficult to analyze due to complex changes in taxonomic compositions. This paper presents a new method to investigate and visualize the intrinsic hierarchical community structure implied by the measurements. The basic idea is to identify significant intersection sets, which can be seen as sub-communities making up the measured communities. Using the subset relationship, the intersection sets together with the measurements form a hierarchical structure visualized as a Hasse diagram. Chemical organization theory (COT) is used to relate the hierarchy of the sets of taxa to potential taxa interactions and to their potential dynamical persistence. The approach is demonstrated on a data set of community data obtained from bacterial 16S rRNA gene sequencing for samples collected monthly from four groundwater wells over a nearly 3-year period (n = 114) along a hillslope area. The significance of the hierarchies derived from the data is evaluated by showing that they significantly deviate from a random model. Furthermore, it is demonstrated how the hierarchy is related to temporal and spatial factors; and how the idea of a core microbiome can be extended to a set of interrelated core microbiomes. Together the results suggest that the approach can support developing models of taxa interactions in the future.

Measuring the dynamics of microbial communities results in high-dimensional measurements of taxa abundances over time and space, which is difficult to analyze due to complex changes in taxonomic compositions. This paper presents a new method to investigate and visualize the intrinsic hierarchical community structure implied by the measurements. The basic idea is to identify significant intersection sets, which can be seen as sub-communities making up the measured communities. Using the subset relationship, the intersection sets together with the measurements form a hierarchical structure visualized as a Hasse diagram. Chemical organization theory (COT) is used to relate the hierarchy of the sets of taxa to potential taxa interactions and to their potential dynamical persistence. The approach is demonstrated on a data set of community data obtained from bacterial 16S rRNA gene sequencing for samples collected monthly from four groundwater wells over a nearly 3-year period (n = 114) along a hillslope area. The significance of the hierarchies derived from the data is evaluated by showing that they significantly deviate from a random model. Furthermore, it is demonstrated how the hierarchy is related to temporal and spatial factors; and how the idea of a core microbiome can be extended to a set of interrelated core microbiomes. Together the results suggest that the approach can support developing models of taxa interactions in the future. [ABSTRACT FROM AUTHOR]

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This article, titled "Historic Behavior in Nonautonomous Rock-Paper-Scissors Dynamics," explores the concept of historic behavior in dynamical systems. The author challenges the widely held belief that replicator equations in evolutionary game theory satisfy the "Folk Theorem of Evolutionary Game Theory." The paper proposes a class of replicator equations that exhibit historic behavior, where the time averages of the orbit do not converge. The article also presents some auxiliary results and provides a proof for the main theorem. The research was supported by ongoing institutional funding, and the author declares no conflicts of interest. [Extracted from the article]

Evolutionary game theory (EGT) is one of several major developments of game theory. EGT covers ecology and population genetics, among other fields in biology. Most studies in EGT were on a two-player game but non-linearities in biology often occur that need to be considered. Huntington's disease (HD), named after the person who wrote the first detailed description of the disease in 1872, is a neurodegenerative disease that is inherited. This is a case in population genetics, which follows the inheritance pattern called the dominant lethal. In this study, we presented this disease as a multiplayer game among the alleles of the HD gene. We utilized Gokhale and Traulsen's model, wherein a payoff matrix for a four-player game was reduced into a payoff matrix for a two-player game. Depending on the fitness values of each genotype, we have determined that populations consisting of both Huntington and normal alleles may converge to either a purely Huntington, a purely normal, or a mixed composition where both types of genes coexist. If the normal genotype produces more surviving offspring than the other genotypes, then even if a small frequency of normal alleles is injected into a purely Huntington population, the population will be replaced by the normal genotype over time. Such a result was obtained using replicator dynamics and analysis of the stability of equilibrium points. Similar analyses on other genotypes were provided in relation to the inheritance pattern of HD. [ABSTRACT FROM AUTHOR]

The emergence and growth of drug-resistant cancer cell subpopulations during anti-cancer treatment is a major challenge for cancer therapies. Combination therapies are usually applied for overcoming drug resistance. In the present paper, we explored the evolution outcome of tumor cell populations under different combination schedules of chemotherapy and p53 vaccine, by construction of replicator dynamical model for sensitive cells, chemotherapy-resistant cells and p53 vaccine-resistant cells. The local asymptotic stability analysis of the evolutionary stable points revealed that cancer population could evolve to the population with single subpopulation, or coexistence of sensitive cells and p53 vaccine-resistant cells, or coexistence of chemotherapy-resistant cells and p53 vaccine-resistant cells under different monotherapy or combination schedules. The design of adaptive therapy schedules that maintain the subpopulations under control is also demonstrated by sequential and periodic application of combination treatment strategies based on the evolutionary velocity and evolutionary absorbing regions. Applying a new replicator dynamical model, we further explored the supportive effects of sensitive cancer cells on targeted therapy-resistant cells revealed in mice experiments. It was shown that the supportive effects of sensitive cells could drive the evolution of cell population from sensitive cells to coexistence of sensitive cells and one type of targeted therapy-resistant cells.

Common resources are often overexploited and appear subject to critical transitions from one stable state to another antagonistic state. Many times resulting in tragedy of the commons (TOC)—exploitation of shared resources for personal gain/payoffs, leading to worse outcomes or extinction. An adequate response would be strategic interaction, such as inspection and punishment by institutions to avoid TOC. This strategic interaction is often coupled with dynamically changing common resources. However, effect of strategic interaction in complex, coupled socio-ecological systems is less studied. Here, we develop replicator equations using evolving games in which strategy and common resources co-evolve. We consider the shared commons as fish dynamics governed by the intrinsic growth rate, predation and harvesting. The joint dynamics exhibit an oscillatory TOC, revealing that institutions need to pay special attention to intrinsic growth rate and nonlinear interaction. Our research shows that the co-evolving system exhibits a broader range of dynamics when predation is present compared to the disengaged fishery system. We conclude that the usefulness, chances and challenges of modelling co-evolutionary games to create sustainable systems merit further research.

In the digital era, smart-home users face growing threats from cyberattacks that threaten their privacy and security. Hence, it is essential for smart-home users to prioritize cybersecurity education and training to secure their homes. Despite this, the high cost of such training often presents a barrier to widespread adoption and accessibility. This study aims to analyze the costs and benefits associated with various cybersecurity investment strategies for smart-home users in the context of cyberattacks. The study utilizes evolutionary game theory to model a game comprised of three populations: smart-home users, stakeholders, and attackers. We derive and analyze the replicator dynamics of this game to determine the evolutionarily stable strategy (ESS). Furthermore, we investigate the impacts of the costs and benefits of cybersecurity investment and cyberattack costs on the ESS. The findings indicate that incurring costs for cybersecurity training is beneficial for smart-home users to protect their homes and families. However, the training costs must be low and affordable for smart-home users in order to ensure their participation and engagement. Additionally, providing rewards for commitment to cybersecurity is crucial in sustaining interest and investment over the long term. To promote cybersecurity awareness and training for smart-home users, governments can incorporate it as a priority in national cybersecurity plans, provide subsidies for training costs, and incentivize good cybersecurity practices. [ABSTRACT FROM AUTHOR]

The paper analyzes a dynamic evolutionary game between banks and firms whose interaction has always been characterized by conflictual relationships. Banks would like that the funding is spent to achieve the objectives of the projects submitted, whereas firms would allocate these loans to obtain private benefits. Following replicator dynamics, we show that banks and firms have predator-prey interactions of the Lotka–Volterra type. Misbehaving firms who seek private benefits are "predators" and banks are their "prey". We analyze the dynamics emerging from the model and we prove that the stability of equilibria depending on the fundamental parameters which describe the banks–firms interaction. In addition, we compare equilibria in terms of Pareto efficiency computing welfare through the average profits with some numerical applications. Finally, we propose a stochastic replicator dynamics approach in order to assume a perturbation in the population growth rate and we suppose as endogenous the monitoring intensity. [ABSTRACT FROM AUTHOR]

It is estimated that half of all the water extracted, both in developed and developing countries, is unauthorized. This phenomenon makes the management of a groundwater even more difficult to avoid over-exploitation. To study the interaction between farmers, that could be compliant and non-compliant, and a water agency, we built a leader-follower differential game. However, we assumed that the water agency does not know neither ex-ante nor ex-post the number of compliant farmers. After illustrating the results of the dynamic game through numerical simulation using the Western La Mancha (Spain) data, we endogenize the types’ choice in an evolutionary context. Finally, we perform comparative dynamics in the steady state to understand the role of the sanction to counter illegal behaviors. [ABSTRACT FROM AUTHOR]

Male territorial-sneaker polymorphisms are common in nature. To understand how these polymorphisms evolve, we developed a game theoretical model analogous to the classical Hawk-Dove model, but with two important differences. First, we allowed non-uniform interaction rates of strategies to account for the possibility that some interactions between male strategies are disproportionately more frequent than others. Second, we allowed females to exhibit a preference for one type of male and thereby choose mates adaptively. Selection dynamics were modeled using coupled replicator equations. The model confirms that there is a broad range of conditions under which a male polymorphism will arise. We applied the model to understand the genetic polymorphism in adult male Mnais damselflies (Zygoptera). Here, orange-winged adult males defend oviposition sites and mate with females when they arrive, while clear-winged 'sneaker' males are typically non-territorial and opportunistically mate with females. Intriguingly, in allopatry, the males of Mnais costalis and M. pruinosa both exhibit the same orange-clear winged polymorphism but where the species co-occur, males of M. costalis evolve orange wings while males of M. pruinosa tend to evolve clear wings. To understand this phenomenon and evaluate the importance of female choice in mediating it, we extended our game-theoretical model to two interacting species. While both competitive and reproductive interference can explain the male monomorphisms in sympatry, reproductive interference explains the phenomenon under a wider set of conditions. When females of the rarer species change their male preferences to facilitate species discrimination, it can generate runaway selection on male phenotypes., Competing Interests: The authors are not aware of any competing or financial interests., (© The Author(s) 2025. Published by Oxford University Press on behalf of the International Society for Behavioral Ecology.)

This paper proposes a decentralized control scheme that mitigates floods in urban drainage systems (UDSs). First, we develop a partitioning algorithm of the UDS relying on a graph model of the system. Once this is done, we design a local controller for each partition based on the replicator dynamics model (a set of differential equations that describes the evolution of a population of players involved in a strategic game). The decentralized nature of the proposed strategy makes it suitable for applying it in large-scale systems. Stability of the closed-loop system is proved by using Lyapunov theory. Furthermore, we simulate the performance of the decentralized control scheme in two case studies. One of them models part of the Bogotá (Colombia) stormwater UDS. Finally, we compare the proposed technique with two widely used methods for-real time control of UDSs, i.e., constrained linear quadratic regulator (LQR) and model predictive control (MPC).

The article analyzes how conventionalists, pioneers and criminals choose between a national currency (e.g. a central bank digital currency) and a global currency (e.g. a cryptocurrency such as Bitcoin) that both have specific characteristics in an economy. Conventionalists favor what is traditional and historically common. They tend to prefer the national currency. Pioneers (early adopters) tend to break away from tradition, and criminals prefer not to get caught. They both tend to prefer the global currency. Each player has a Cobb-Douglas utility with one output elasticity for each of the two currencies, comprised of backing, convenience, confidentiality, transaction efficiency, financial stability, and security. The replicator equation is used to illustrate the evolution of the fractions of the three kinds of players through time, and how they choose among the two currencies. Each player’s expected utility is inverse U-shaped in the volume fraction of transactions in each currency, skewed towards the national currency for conventionalists, and towards the global currency for pioneers and criminals. Conventionalists on the one hand typically compete against pioneers and criminals on the other hand. Fifteen parameter values are altered to illustrate sensitivity. For parameter values where conventionalists go extinct, pioneers and criminals compete directly with each other. Players choose volume fractions of each currency and which kind of player to be. Conventionalists go extinct when criminals gain more from criminal behavior, and when the parameter values in the conventionalists’ expected utility are unfavorable, causing competition between pioneers and criminals.

This paper proposes a possible mechanism for obtaining sizeable behavioral structures by simulating a network–agent dynamic on an evolutionary public good game with available social.learning. The model considers a population with a fixed number of players. In each round, the chosen players may contribute part of their value to a common pool. Then, each player may imitate the strategy of another player based on relative payoffs (whoever has the lower payoff adopts the strategy of the other player) and change his or her strategy using different exploratory variables. Relative payoffs are subject to incentives, including participation costs, but may also be subject to mutation, whose rate is sensitized by the network characteristics (social ties). The process discussed in this report is interesting and relevant across a broad range of disciplines that use game theory, including cultural evolutionary dynamics. [ABSTRACT FROM AUTHOR]

In our information age, information alone has become a driver of social growth. Information is the fuel of “big data” companies, and the decision-making compass of policy makers. Can we quantify how much information leads to how much social growth potential? Information theory is used to show that information (in bits) is effectively a quantifiable ingredient of growth. The article presents a single equation that allows both to describe hands-off natural selection of evolving populations and to optimize population fitness in uncertain environments through intervention. The setup analyzes the communication channel between the growing population and its uncertain environment. The role of information in population growth can be thought of as the optimization of information flow over this (more or less) noisy channel. Optimized growth implies that the population absorbs all communicated environmental structure during evolutionary updating (measured by their mutual information). This is achieved by endogenously adjusting the population structure to the exogenous environmental pattern (through bet-hedging/portfolio management). The setup can be applied to decompose the growth of any discrete population in stationary, stochastic environments (economic, cultural, or biological). Two empirical examples from the information economy reveal inherent trade-offs among the involved information quantities during growth optimization.

This paper proposes models of learning process in teams of individuals whocollectively execute a sequence of tasks and whose actions are determined byindividual skill levels and networks of interpersonal appraisals and influence.The closely-related proposed models have increasing complexity, starting with acentralized manager-based assignment and learning model, and finishing with asocial model of interpersonal appraisal, assignments, learning, and influences.We show how rational optimal behavior arises along the task sequence for eachmodel, and discuss conditions of suboptimality. Our models are grounded inreplicator dynamics from evolutionary games, influence networks frommathematical sociology, and transactive memory systems from organizationscience.

We consider an evolutionary game on a graph under the condition that players perception of uncertainty and decision making can follow either an objective expected utility theory (EUT) model or a subjective prospect theory (PT) model. Each player chooses one of two strategies with probabilities x and 1-x, where the subjective players bias their choices of the probabilities to be w(x) and w(1-x) to reflect the probability weighting effect of PT. We derive the conditions to achieve a locally asymptotically stable Nash equilibrium (NE) and the necessary conditions to achieve an evolutionary stable strategy (ESS). We illustrate these results through a simple random access channel game. We find that players' behavior is affected by the total number of players as well as the number of each type of players (objective or subjective). We show that increasing the number of interacting players induces an aggressive transmission strategy and the behavior becomes similar for both types of players. Moreover, we let the objective and subjective players adjust their transmission under throughput rewards, delay penalties, and energy costs. The EUT players' throughput is higher than the PT players' throughput for a small number of interacting players and equal to if the number of interacting players increases.

To determine the optimal strategies of multi-participants in a regional energy system, a tripartite game reciprocity model is developed for the government, the energy service company and the energy user by employing the evolutionary game theory. The strategy selection probability of the three-party participants under the conditions of balanced strategy and unbalanced strategy is analyzed. According to the simulation results, the government is more inclined to increase the subsidies for regional distributed energy stations, the energy service companies choose to implement preferential energy prices to increase revenues, and the energy users will choose the energy supply provided by the energy service companies.

In the digital era, smart-home users face growing threats from cyberattacks that threaten their privacy and security. Hence, it is essential for smart-home users to prioritize cybersecurity education and training to secure their homes. Despite this, the high cost of such training often presents a barrier to widespread adoption and accessibility. This study aims to analyze the costs and benefits associated with various cybersecurity investment strategies for smart-home users in the context of cyberattacks. The study utilizes evolutionary game theory to model a game comprised of three populations: smart-home users, stakeholders, and attackers. We derive and analyze the replicator dynamics of this game to determine the evolutionarily stable strategy (ESS). Furthermore, we investigate the impacts of the costs and benefits of cybersecurity investment and cyberattack costs on the ESS. The findings indicate that incurring costs for cybersecurity training is beneficial for smart-home users to protect their homes and families. However, the training costs must be low and affordable for smart-home users in order to ensure their participation and engagement. Additionally, providing rewards for commitment to cybersecurity is crucial in sustaining interest and investment over the long term. To promote cybersecurity awareness and training for smart-home users, governments can incorporate it as a priority in national cybersecurity plans, provide subsidies for training costs, and incentivize good cybersecurity practices.

We elucidate the mechanism of the self-organization of square agglomeration patterns that are described by spatial economic models on a square lattice with periodic boundary conditions. Focusing on the symmetry of the square lattice, we conduct a group-theoretic analysis and obtain bifurcating patterns from the uniform distribution. Furthermore, for the replicator dynamics, which are widely used in economics, we pay attention to the existence of invariant patterns that are solutions to the governing equation for any value of the bifurcation parameter (the trade freeness for spatial economic models). We advance invariant patterns on the square lattice as candidates of stable equilibria. Using a prototype spatial economic model proposed by Forslid and Ottaviano [2003], we numerically show a tendency that bifurcating solutions arrive at invariant patterns after bifurcation. This tendency is advanced as the underlying mechanism of the progress of economic agglomerations that is to be considered in the study of spatial economic agglomerations. [ABSTRACT FROM AUTHOR]

Information transfer is a basic feature of life that includes signaling within and between organisms. Owing to its interactive nature, signaling can be investigated by using game theory. Game theoretic models of signaling have a long tradition in biology, economics, and philosophy. For a long time the analyses of these games has mostly relied on using static equilibrium concepts such as Pareto optimal Nash equilibria or evolutionarily stable strategies. More recently signaling games of various types have been investigated with the help of game dynamics, which includes dynamical models of evolution and individual learning. A dynamical analysis leads to more nuanced conclusions as to the outcomes of signaling interactions. Here we explore different kinds of signaling games that range from interactions without conflicts of interest between the players to interactions where their interests are seriously misaligned. We consider these games within the context of evolutionary dynamics (both infinite and finite population models) and learning dynamics (reinforcement learning). Some results are specific features of a particular dynamical model, whereas others turn out to be quite robust across different models. This suggests that there are certain qualitative aspects that are common to many real-world signaling interactions.

The aim of the paper is to analyse the cooperation and competition relationship in the oil and gas industry in Romania using the game theory. The players are the companies, the payoffs are the profits obtained by the entities and there are two strategies as cooperation and competition. Two cases are considered: duopolistic and triopolistic market. The mathematical models have as variables the probabilities of choosing cooperation and competition by each player. They are described by two and three nonlinear differential equations. The time delay is introduced in order to highlight the time between choosing a strategy and its effect. The case studies use real data for two and three companies, respectively, with two scenarios related to the obtained payoffs if they cooperate or not. The time evolutions of the variables are carried out using Wolfram Mathematica 9. Finally, some conclusions and future research are drawn.

For one variable–supply currency in isolation, one player’s Cobb–Douglas utility depends on the current supply divided by the initial supply, multiplied by the inverse of the accumulative inflation/deflation. With equal weight assigned to both factors, money printing outweighs inflation, and money withdrawal outweighs deflation. The study design is to analyze how competition between one variable–supply and one fixed–supply currency impacts the player’s choice of currency. Applying the 1959–2021 US M2 money supply data and the 1635–2021 US inflation data, the player’s utility increases over time when assigning high weight to money printing/withdrawal and increases less or decreases overall when assigning high weight to inflation/deflation. With different player support for the two currencies, depending on each currency’s backing, convenience, confidentiality, transaction efficiency, financial stability, and security, replicator dynamics is used to determine the player’s volume fraction of transactions in each currency. Low, high, increasing, and decreasing support of a currency are analyzed. Each fraction may increase, decrease, be inverse U–shaped, U–shaped, and approach low or high levels over time. For example, high weight assigned to money printing may cause the player to eventually prefer the variable–supply currency unless the player supports the fixed–supply currency highly and increasingly.

The concept of a face of population states arises naturally in evolutionary games. This paper studies faces of profiles in asymmetric evolutionary games with infinite strategy space. The concepts of strong immovable and immutable faces of profiles are introduced and stability results for these faces are discussed. [ABSTRACT FROM AUTHOR]

In the current blockchain network, many participants rationally migrate the pool to receive a better compensation according to their contribution in situations where the pools they engage encounter undesirable attacks. The Nash equilibria of attacked pool has been widely analyzed, but the analysis of practical methodology for obtaining it is still inadequate. In this paper, we propose an evolutionary game theoretic analysis of Proof-of-Work (PoW) based blockchain network in order to investigate the mining pool dynamics affected by malicious infiltrators and the feasibility of autonomous migration among individual miners. We formulate a revenue model for mining pools which are implicitly allowed to launch a block withholding attack. Under our mining game, we analyze the evolutionary stability of Nash equilibrium with replicator dynamics, which can explain the population change with time between participated pools. Further, we explore the statistical approximation of successful mining events to show the necessity of artificial manipulation for migrating. Finally, we construct a better response learning based on the required block size which can lead to our evolutionarily stable strategy (ESS) with numerical results that support our theoretical discoveries.

A hierarchical approach for the energy management of geographically close microgrids connected through a dedicated AC power network is proposed in this paper. The proposed approach consists of a two-layer energy management system (EMS) for networked microgrids. In the lower layer, each microgrid solves its own economic dispatch problem through a distributed model predictive control approach that respects capacity limits and ramp-rate constraints of distributed generation. In the upper layer, the energy trading in the network of microgrids decides how to optimally trade the energy based on the marginal cost information from the lower layer in order to improve global optimization objectives, e.g., social welfare. In order to solve the trading problem, a consensus-based algorithm and a replicator dynamics algorithm are proposed assuming that the marginal cost function of the microgrid is known and linear. It is shown that both algorithms converge to the same solution, which is equivalent to the minimization of operation costs. The consensus-based algorithm is extended in order to tackle more general marginal cost functions and trading network constraints. Moreover, the effect of ramp constraints and network limits is studied. Simulations show the effectiveness of the proposed algorithms for three interconnected microgrids with different characteristics.

This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group S N [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics. [ABSTRACT FROM AUTHOR]

We revisit some concepts of superiority and weak stability of population states in evolutionary games with continuous strategy space. We prove a general stability result for replicator trajectories by introducing the concept of superiority with respect to a given closed set. Some important results in the literature regarding weak stability turn out to be special cases of our main result. [ABSTRACT FROM AUTHOR]

In this article, we study in detail asymptotic stability of strongly uninvadable faces generated by finite Borel sets in a continuous strategy space of an evolutionary game. It is proved that such a face is an asymptotically stable set for the associated replicator dynamics. This result is illustrated using examples. [ABSTRACT FROM AUTHOR]

This paper provides several theoretical results for empirical game theory. Specifically, we introduce bounds for empirical game theoretical analysis of complex multi-agent interactions. In doing so we provide insights in the empirical meta game showing that a Nash equilibrium of the estimated meta-game is an approximate Nash equilibrium of the true underlying meta-game. We investigate and show how many data samples are required to obtain a close enough approximation of the underlying game. Additionally, we extend the evolutionary dynamics analysis of meta-games using heuristic payoff tables (HPTs) to asymmetric games. The state-of-the-art has only considered evolutionary dynamics of symmetric HPTs in which agents have access to the same strategy sets and the payoff structure is symmetric, implying that agents are interchangeable. Finally, we carry out an empirical illustration of the generalised method in several domains, illustrating the theory and evolutionary dynamics of several versions of the AlphaGo algorithm (symmetric), the dynamics of the Colonel Blotto game played by human players on Facebook (symmetric), the dynamics of several teams of players in the capture the flag game (symmetric), and an example of a meta-game in Leduc Poker (asymmetric), generated by the policy-space response oracle multi-agent learning algorithm. [ABSTRACT FROM AUTHOR]

There is a plethora of instances of interactions between players, in both biological and socio-economical context, that can be modeled as the paradigmatic PGG. However, in such interactions, arguably the PGG is often nonlinear in nature. This is because the increment in benefit generated, owing to additional cost contributed by the players, is realistically seldom linear. Furthermore, sometimes a social good is created due to interspecific interactions, e.g. in cooperative hunting by animals of two different species. In this paper, we study the evolutionary dynamics of a heterogenous population of cooperators and defectors playing nonlinear PGG; here we define heterogenous population as the one composed of distinct subpopulations with interactions among them. We employ the replicator equations for this investigation, and present the non-trivial effects of nonlinearity and size of the groups involved in the game. We report the possibility of discoordination, and coexistence of coordination and anti-coordination in such nonlinear PGG.

As one of the key technologies for the fifth generation (5G) mobile networks, device-todevice (D2D) communications offer promising benefits such as high spectrum efficiency, traffic offloading, and enhanced coverage. Depending on the resource sharing between D2D users and regular cellular users, a D2D user equipment (UE) can dynamically switch its communication mode to improve the quality of service (QoS) and user experience. Mode selection is an essential issue to ensure the QoS of D2D UEs while maximizing the system capacity. In this paper, we investigate the D2D mode selection problem from a novel perspective. In addition to the classic cellular mode and direct reuse mode, we further consider a relay mode for D2D UEs. Moreover, in order to address a potentially large population of D2D UEs, we propose an evolutionary game-based approach for D2D mode selection. The evolutionary game is formulated with a utility function that takes into account both the achievable throughput of D2D UEs and the radio resource consumption. Based on the evolutionary game formulation, we implement selection dynamics, i.e., replication by imitation, in a device-controlled mode selection algorithm. To evaluate the performance of the proposed mode selection algorithm, we conduct simulations to compare it with three baseline schemes, including an approach based on maximum signal-to-interference-plus-noise, a distance-based approach, and a random approach. As shown in the simulation results, the proposed approach achieves higher utilities than the baseline schemes.

In this paper, we present a price-based approach for offloading macro users to small-cells in a two-tier heterogeneous network (HetNet). We have constructively exploited the small-cell density in a HetNet to harness offloading opportunities for macro users (MUs), the goal of which is to solve two problems simultaneously. It aims to shun the MUs, who have a lower received data rate due to the interference perceived from the small-cell tier. The scheme also fends off congestion in a macrocell, by offloading macro users to the small-cell tier. We have proposed a novel threshold pricing scheme, which a macrocell adopts, with a view to influencing low data rate MUs to join a small-cell network. Small-cell networks also charge a price, which includes an access price and an interference compensation price, proportional to the number of MUs who choose the small-cells instead of the macrocellular network. We assume that the small-cells adopt existing cell range expansion techniques to accommodate MUs. We formulate an evolutionary game to model and analyze the behavioral dynamics of the large number of MUs under the proposed pricing strategies of both networks. Replicator dynamics is used to find the evolutionary equilibrium of the evolutionary game. Sequentially, we provide the proof of the existence, uniqueness and stability of the evolutionary equilibrium through extensive analysis. Numerical results are provided to demonstrate that the proposed pricing strategies are able to shape the network dynamics by fine-tuning the rate-threshold and price. The ability to control the macrocell population share by itself with an application of the proposed pricing scheme remains the prime contribution of this paper.

In this paper, it is shown that any migratory flow, imposed by the imitation of the like, can be modeled by a dynamic system, once the rules governing imitative behavior are established. The parameters that characterize this system are determined by the social and economic conditions existing in the countries between which it is established and can be modified from economic policy considerations. These modifications, in turn, will cause changes in the solutions of the dynamic system, which reflect the evolution of the migratory flow.

DeFinetti is a Mathematica program that enables the user to visualize the replicator dynamics of three-strategy games with collective interactions, to study cooperation at different levels of biological complexity. The program computes the replicator dynamics of a game for multiple initial frequencies and shows trajectories, fitness and end points on a ternary plot (a DeFinetti diagram). The user only needs to specify the payoffs, and can change the parameters, visualize their effect, customize the style of the graphs and export the results.

As marine plastic pollution threatens the marine ecosystem seriously, the government needs to find an effective way to clean marine plastics. Due to the advantages of easy operation and high efficiency, autonomous underwater vehicles (AUVs) have been applied to clean marine plastics. As for the large-scale marine environment, the marine plastic cleaning task needs to be accomplished through the collaborative work of multiple AUVs. Assigning the cleaning task to each AUV reasonably and effectively has an essential impact on improving cleaning efficiency. The coordination of AUVs is subjected to harsh communication conditions. Therefore, to release the dependence on the underwater communications among AUVs, proposing a reliable multi-robot task allocation (MRTA) model is necessary. Inspired by the evolutionary game theory, this paper proposes a novel multi-robot task allocation (MRTA) model based on replicator dynamics for marine plastic cleaning. This novel model not only satisfies the minimization of the cost function, but also reaches a relatively stable state of the task allocation. A novel optimization algorithm, equilibrium optimizer (EO), is adopted as the optimizer. The simulation results validate the correctness of the results achieved by EO and the applicability of the proposed model. At last, several valuable conclusions are obtained from the simulations on the three different assumed AUVs.