Evolutionary Games and Population Dynamics
Journal of the American Statistical Association
Abstract
Every form of behaviour is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realised how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centred not on the concept of rational players but on the population dynamics of behavioural programmes. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behaviour, and of the closely related interactions between species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions which can alter the basis of their success, i.e. to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions which punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms.
... . For an introduction to the vast literature surrounding the replicator dynamics, see Hofbauer and Sigmund (1998), Weibull (1995), Sandholm (2010) and references therein. ...
... In this case, the payoff to agents playing i ∈ A at state x is u i (x) = ∑ j∈A M ij x j , so the game's payoff field can be written in concise form as v(x) = Mx . Following standard conventions in the field, we will refer to this scenario as symmetric random matching (Hofbauer and Sigmund 1998;Weibull 1995;Sandholm 2010;Hadikhanloo et al. 2022). ...
... In addition, for small , a simple first-order Taylor expansion yields so, in the continuous-time limit → 0 , we get In the above, the asymptotic equality sign " ∼ " is to be interpreted loosely and is only meant to suggest that Model I represents an Euler discretization of (RD) up to a higher-order O ( 2 ) correction term. Because this term is negligible in the continuous-time limit → 0 , (RD) is commonly regarded in the literature as the mean dynamics of Model I (Hofbauer and Sigmund 1998;Sandholm 2010). ...
We consider three distinct discrete-time models of learning and evolution in games: a biological model based on intra-species selective pressure, the dynamics induced by pairwise proportional imitation, and the exponential/multiplicative weights algorithm for online learning. Even though these models share the same continuous-time limit—the replicator dynamics—we show that second-order effects play a crucial role and may lead to drastically different behaviors in each model, even in very simple, symmetric games. Specifically, we study the resulting discrete-time dynamics in a class of parametrized congestion games, and we show that (i) in the biological model of intra-species competition, the dynamics remain convergent for any parameter value; (ii) the dynamics of pairwise proportional imitation exhibit an entire range of behaviors for larger time steps and different equilibrium configurations (stability, instability, and even Li–Yorke chaos); while (iii) in the exponential/multiplicative weights algorithm, increasing the time step (almost) inevitably leads to chaos (again, in the formal, Li–Yorke sense). This divergence of behaviors comes in stark contrast to the globally convergent behavior of the replicator dynamics, and serves to delineate the extent to which the replicator dynamics provide a useful predictor for the long-run behavior of their discrete-time origins.
... The same principle extends also to human social systems. For example, in a collective action problem, when given a choice between maximizing personal gains or improving societal benefits at a personal cost, a purely rational approach would dictate that individuals defect against one another [7,8]. ...
... Combining game theory with the Darwinian principle of the survival of the fittest, evolutionary game theory extends this paradigm to consider repeated interactions in a population. The fraction of individuals adopting each strategy changes over time based on how fit the strategy is in the population, where the fitness is determined by the average payoff earned by players adopting that particular strategy [8,9,15,21]. Over the years extensive numerical and theoretical work for various games and social dilemma situations have identi-fied several different mechanisms contributing to the success of cooperative behaviour. ...
... Repeated interactions with the same set of individuals (or neighbours) is at the heart of network reciprocity which has gathered substantial attention in recent years as a computational and mathematical framework able to explain cooperation in socio-structural interaction patterns [8,19,25,[30][31][32][33]. In particular, the effect of different interaction structures has been investigated, from lattices [9,19,34], to scale-free networks [31,35], and multi-layer networks [36][37][38] enhancing our understanding of how pro-sociality evolves in large-scale structured populations with realistic features. ...
Understanding cooperation in social dilemmas requires models that capture the complexity of real-world interactions. While network frameworks have provided valuable insights to model the evolution of cooperation, they are unable to encode group interactions properly. Here, we introduce a general higher-order network framework for multi-player games on structured populations. Our model considers multi-dimensional strategies, based on the observation that social behaviours are affected by the size of the group interaction. We investigate dynamical and structural coupling between different orders of interactions, revealing the crucial role of nested multilevel interactions, and showing how such features can enhance cooperation beyond the limit of traditional models with uni-dimensional strategies. Our work identifies the key drivers promoting cooperative behaviour commonly observed in real-world group social dilemmas.
... In this section, we consider an ecological network of n species whose dynamics are governed by generalized Lotka-Volterra (GLV) equations [16] of the forṁ ...
... 38 to z i = y i (y 1 + · · · + y n ) −1 , i.e. to relative abundances. Having into account Hypothesis (iii) and after a change in velocity, we obtain that the z i satisfy the replicator equations [16] ...
... , z n ) T and (Bz) i denotes the i-th component of vector Bz. This is Exercise 7.5.2 of [16]. Eqs. ...
With improvements in data resolution and quality, researchers can now construct detailed representations of complex systems as signed, weighted, and directed networks. In this article, we introduce a framework for measuring net and indirect effects without simplifying these information-rich networks. Building on a generalization of Katz centrality, this framework captures both direct and indirect interactions, the effect of the whole network on a node and its reverse, the effect of a node on the whole network, while accommodating the complexity of signed, weighted, and directed edges. To contextualize our contribution, we propose a taxonomy that unifies existing approaches and measures from the literature. We then apply our measure to ecological networks, where net and indirect effects remain critical yet difficult to quantify factors influencing coexistence. Specifically, we observe a strong correlation between negative net effects and species extinction in generalized Lotka-Volterra dynamics. Additionally, we test our framework on a real-world social network, where it effectively identifies informative importance rankings, providing insights into influence propagation and power dynamics.
... The approach is centered around a classic result from graph theory due to Motzkin and Straus (1965), and variations thereof, which allow us to formulate the MCP as a standard quadratic program-namely, a continuous quadratic optimization problem with simplex (or probability) constraints, to solve which replicator equations have been remarkably effective despite their simplicity. These are well-known continuous-and discrete-time dynamical systems developed and studied in evolutionary game theory, a discipline pioneered by J. Maynard Smith (1982) that aims to model the evolution of animal behavior using the principles and tools of noncooperative game theory (Hofbauer & Sigmund, 1998). Evolutionary game-theoretic models are also gaining increasing popularity in economics since they elegantly get rid of the much-debated assumptions of traditional game theory concerning the full rationality and complete knowledge of players (Weibull, 1995;Samuelson, 1997;Fudenberg & Levine, 1998). ...
... They act instead according to a preprogrammed behavior pattern, or pure strategy, and it is supposed that some evolutionary selection process operates over time on the distribution of behaviors. (We refer the reader to Hofbauer &Sigmund, 1998, andWeibull, 1995, for excellent introductions to this rapidly expanding field.) ...
... where κ is a positive constant. As κ tends to 0, the orbits of this dynamics approach those of the standard, first-order replicator model, equation 2.6, slowed down by the factor κ; moreover, for large values of κ, the model approximates the so-called best-reply dynamics (Hofbauer, 1995;Hofbauer & Sigmund, 1998). ...
Evolutionary game-theoretic models and, in particular, the so-called replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems. The approach is centered around a classic result from graph theory that formulates the maximum clique problem as a standard (continuous) quadratic program and exploits the dynamical properties of these models, which, under a certain symmetry assumption, possess a Lyapunov function. In this letter, we generalize previous work along these lines in several respects. We introduce a wide family of game-dynamic equations known as payoff-monotonic dynamics, of which replicator dynamics are a special instance, and show that they enjoy precisely the same dynamical properties as standard replicator equations. These properties make any member of this family a potential heuristic for solving standard quadratic programs and, in particular, the maximum clique problem. Extensive simulations, performed on random as well as DIMACS benchmark graphs, show that this class contains dynamics that are considerably faster than and at least as accurate as replicator equations. One problem associated with these models, however, relates to their inability to escape from poor local solutions. To overcome this drawback, we focus on a particular subclass of payoff-monotonic dynamics used to model the evolution of behavior via imitation processes and study the stability of their equilibria when a regularization parameter is allowed to take on negative values. A detailed analysis of these properties suggests a whole class of annealed imitation heuristics for the maximum clique problem, which are based on the idea of varying the parameter during the imitation optimization process in a principled way, so as to avoid unwanted inefficient solutions. Experiments show that the proposed annealing procedure does help to avoid poor local optima by initially driving the dynamics toward promising regions in state space. Furthermore, the models outperform state-of-the-art neural network algorithms for maximum clique, such as mean field annealing, and compare well with powerful continuous-based heuristics.
... However, it raises the fundamental question: 3 why would an individual sacrifice personal gain to support potential rivals in a 4 competitive struggle? To understand the complex balance between cooperation and 5 defection, evolutionary game theory offers valuable insights. It serves as a profound 6 framework to describe the dynamic interactions in societal and economic behaviors [5][6][7]. ...
... To understand the complex balance between cooperation and 5 defection, evolutionary game theory offers valuable insights. It serves as a profound 6 framework to describe the dynamic interactions in societal and economic behaviors [5][6][7]. 7 While natural selection generally favors defectors in well-mixed populations [8,9], the 8 structure of a population -defining the scope of individual interactions -can 9 significantly influence evolutionary outcomes [10,11]. Networks, where nodes 10 (representing individuals) are connected by edges (indicating interactions), are a key 11 tool for exploring complex structured systems [12][13][14][15][16][17][18][19]; And network reciprocity has 12 been proposed as a fundamental mechanism to address the cooperative dilemma, 13 suggesting that certain population structures can foster local clusters of cooperators 14 capable of resisting exploitation by defectors [4,6,10,20,21]. ...
Cooperation is fundamental to human societies, and the interaction structure among individuals profoundly shapes its emergence and evolution. In real-world scenarios, cooperation prevails in multi-group (higher-order) populations, beyond just dyadic behaviors. Despite recent studies on group dilemmas in higher-order networks, the exploration of cooperation driven by higher-order strategy updates remains limited due to the intricacy and indivisibility of group-wise interactions. Here we investigate four categories of higher-order mechanisms for strategy updates in public goods games and establish their mathematical conditions for the emergence of cooperation. Such conditions uncover the impact of both higher-order strategy updates and network properties on evolutionary outcomes, notably highlighting the enhancement of cooperation by overlaps between groups. Interestingly, we discover that the group-mutual comparison update – selecting a high-fitness group and then imitating a random individual within this group – can prominently promote cooperation. Our analyses further unveil that, compared to pairwise interactions, higher-order strategy updates generally improve cooperation in most higher-order networks. These findings underscore the pivotal role of higher-order strategy updates in fostering collective cooperation in complex social systems.
Author summary
Human societies often organize and cooperate within social groups, where relatives, friends, neighbors, and colleagues influence behavior at both group and individual levels. Individuals may exhibit biased or neutral attitudes when selecting a neighboring group and a peer within it for imitation or comparison, a process termed as higher-order strategy update. These selection preferences originate from four personality types: aggressive, open-minded, myopic, and passive. This work demonstrates that the open-minded type – indiscriminately imitating a peer within a well-functioning group – significantly promotes cooperation. The mathematical framework proposed in this study deepens the understanding of how decision-making within higher-order structures affects the emergence and spread of cooperative behaviors.
... It has found its way into several applications, notably in economics and biology. See [6,25,32] for general works that covers several distinct features of it. A game, in normal form, is defined by a set of players, a set of elementary strategies available for each player, and a set of (real) payoff functions, one for each player. ...
... For the sake of completeness, we recall that the pair of strategies (x,ŷ) is called a Nash Equilibrium ifx T Aŷ ≥ x T Aŷ andŷ T Bx ≥ y T Bx, for all strategies x, y, cf. [25,32]. ...
Real populations are seldom found at the Nash equilibrium strategy. The present work focuses on how population size can be a relevant evolutionary force diverting the population from its expected Nash equilibrium. We introduce the concept of insuperable strategy, a strategy that guarantees that no other player can have a larger payoff than the player that adopts it. We show that this concept is different from the rationality assumption frequently used in game theory and that for small populations the insuperable strategy is the most probable evolutionary outcome for any dynamics that equal game payoff and reproductive fitness. We support our ideas with several examples and numerical simulations. We finally discuss how to extend the concept to multiplayer games, introducing, in a limited way, the concept of game reduction.
... The above-mentioned phase transition found in common ecological models, such as Lotka-Volterra or resource-competition models [14], is one example in a broader family: highdimensional dynamical systems, of interest in neuroscience [15], game theory [16] and economics [17], also exhibit a phase transition between two such phases. In many cases, this transition has been linked to a loss of fixed point stability and the emergence of unstable fixed points whose number is exponentially large in the dimension of the system [18][19][20]. ...
... where W λ (z) = exp(| ln λ|z). Here δξ(s) ≡ δξ(t) is a zero-mean Gaussian noise with welldefined correlations δĈ + (s) when λ → 0 + , as follows from Eq. (14). Note that when z > 0, W λ (z) → +∞ when λ → 0 + , while W λ (z) → 0 for z < 0. Therefore, in the limit λ → 0 + , the process z(s) follows a well-defined stochastic differential equation [11], ...
Many-variable differential equations with random coefficients provide powerful models for the dynamics of many interacting species in ecology. These models are known to exhibit a dynamical phase transition from a phase where population sizes reach a fixed point, to a phase where they fluctuate indefinitely. Here we provide a theory for the critical behavior close to the phase transition. We show that timescales diverge at the transition and that temporal fluctuations grow continuously upon crossing it. We further show the existence of three different universality classes, with different sets of critical exponents, highlighting the importance of the migration rate coupling the system to its surroundings.
... Appendix A shows that (t * 1 , t * 2 ) is an evolutionarily stable strategy [58], i.e., no mutant can invade the resident population that uses these residence times. The corresponding population distribution between patches is ...
... , t 1 > 0, t 2 > 0. This is the condition for evolutionary stability [58] of equilibrium (13). ...
This article studies patch retention time dynamics for which dispersal dynamics converge on the Ideal Free Distribution. Two types of dispersal dynamics are considered: one that assumes immigration rates are random and the second where immigration rates depend on patch distances. Emigration rates are inversely proportional to patch retention times. Both these dispersal dynamics converge to a unique and stable population distribution equilibrium. Assuming that animal distribution tracks current retention times instantaneously, retention time dynamics are modeled by the canonical equation of adaptive dynamics. This general framework is applied to two negative density-dependent patch payoff functions, hyperbolic and linear. Patch retention time dynamics are unbounded for low population densities, meaning dispersal tends to stop, and the resulting population distribution is identical to the classic Ideal Free Distribution. For higher densities, retention time dynamics converge on an equilibrium where animal dispersal is balanced in that the net dispersal stops.
... It has found its way into several applications, notably in economics and biology. See Gintis (2009) ;Hofbauer and Sigmund (1998) ;Broom and Rychtář (2013) for general works that covers several distinct features of it. ...
... For the sake of completeness, we recall that the pair of strategies (x,ŷ) is called a Nash Equilibrium ifx T Aŷ ≥ x T Aŷ andŷ T Bx ≥ y T Bx, for all strategies x, y, cf. Gintis (2009);Hofbauer and Sigmund (1998). ...
Real populations are seldom found at the Nash equilibrium strategy. The present work focuses on how population size can be a relevant evolutionary force diverting the population from its expected Nash equilibrium. We introduce the concept of insuperable strategy, a strategy that guarantees that no other player can have a larger payoff than the player that adopts it. We show that this concept is different from the rationality assumption frequently used in game theory and that for small populations the insuperable strategy is the most probable evolutionary outcome for any dynamics that equal game payoff and reproductive fitness. We support our ideas with several examples and numerical simulations. We finally discuss how to extend the concept to multiplayer games, introducing, in a limited way, the concept of game reduction.
... , 6), considered analytically in Seno et al. (2020), and numerically in Kȓivan (2014). Hence it should be remarked that Theorem 1 does not necessarily show what is called persistence mathematically defined for the solution of system (1) (as for the mathematically defined persistence or related permanence, for example, see Hofbauer and Sigmund 1998;Thieme 2003). In the following arguments, we will focus on the feature of the system (1) with respect to which prey species goes extinct or persists with the persistent shared single predator. ...
... , n) has been shown with a Lyapunov function in Kȓivan (2014); Seno et al. (2020). Theorem 2 indicates that the system (1) is what is mathematically called persistent if the equilibrium E * [n] exists as an interior equilibrium in D (as for the definition, for example, see Hofbauer and Sigmund 1998;Thieme 2003). ...
We analyze the Lotka–Volterra n prey-1 predator system with no direct interspecific interaction between prey species, in which every prey species undergoes the effect of apparent competition via a single shared predator with all other prey species. We prove that the considered system necessarily has a globally asymptotically stable equilibrium, and we find the necessary and sufficient condition to determine which of feasible equilibria becomes asymptotically stable. Such an asymptotically stable equilibrium shows which prey species goes extinct or persists, and we investigate the composition of persistent prey species at the equilibrium apparent competition system. Making use of the results, we discuss the transition of apparent competition system with a persistent single shared predator through the extermination and invasion of prey species. Our results imply that the long-lasting apparent competition system with a persistent single shared predator would tend toward an implicit functional homogenization in coexisting prey species, or would transfer to a 1 prey-1 predator system in which the predator must be observed as a specialist (monophagy).
... To model an early ecological system where to shed light on the importance of neutrality in early ecologies, we use a quintessential model for replicating entities, which has been applied to the modeling of polynucleotides in a container, the dynamics of gene frequencies, selection (Price equation), relative densities of interacting populations (Lotka-Volterra model), the frequency of different strategies in a population (game theoretical models), and adaptive dynamics (Taylor and Jonker 1978;Schuster and Sigmund 1983;Hofbauer and Sigmund 1998;Page and Nowak 2002). One of the most basic and general equations for all these dynamics is the deterministic equation introduced by Taylor and Jonker (1978) and dubbed replicator equation by Schuster and Sigmund (1983), Taylor and Jonker (1978). ...
... , d, the entry A i j is interpreted as the payoff from using strategy i against an individual using strategy j. The term − x, Ax 1 modulates the fitness variation in such a way that the fitness of a player of type i increases if the payoff is greater than the average payoff of the community (see Hofbauer and Sigmund 1998;Hofbauer 1981;Hofbauer et al. 1981 for a thorough study of these systems in the context of evolutionary game theory). Many generalizations of the original replicator dynamics can be considered. ...
We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class of associated McKean–Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology.
... The discrete urn process described in [10] can be viewed as a discrete, stochastic approximation to the continuous, deterministic replicator dynamics [24,40,44] found in evolutionary game theory. The replicator equation models changes in the frequencies of different species over time based on their fitness. ...
AI alignment, the challenge of ensuring AI systems act in accordance with human values, has emerged as a critical problem in the development of systems such as foundation models and recommender systems. Still, the current dominant approach, reinforcement learning with human feedback (RLHF) faces known theoretical limitations in aggregating diverse human preferences. Social choice theory provides a framework to aggregate preferences, but was not developed for the multidimensional applications typical of AI. Leveraging insights from a recently published urn process, this work introduces a preference aggregation strategy that adapts to the user's context and that inherits the good properties of the maximal lottery, a Condorcet-consistent solution concept.
... For (3.1) restricted to R n + , one can make the coordinate transformation y j = x 2 j , and a rescaling of time, to get the equations where χ = n j=1 y j . These equations are the Lotka-Volterra equations [15] and are used in population dynamics contexts, where negative values of the coordinates have non-physical meanings. These equations are not equivariant, but still contain the same flow invariant coordinate axes and (hyper-) planes. ...
The concept of stability has a long history in the field of dynamical systems: stable invariant objects are the ones that would be expected to be observed in experiments and numerical simulations. Heteroclinic networks are invariant objects in dynamical systems associated with intermittent cycling and switching behaviour, found in a range of applications. In this article, we note that the usual notions of stability, even those developed specifically for heteroclinic networks, do not provide all the information needed to determine the long-term behaviour of trajectories near heteroclinic networks. To complement the notion of stability, we introduce the concept of visibility, which pinpoints precisely the invariant objects that will be observed once transients have decayed. We illustrate our definitions with examples of heteroclinic networks from the literature.
... The conceptual deterministic Rickertype model of two species with competition was introduced by R. May in [21] and has been studied in detail in many successive papers (see, e.g., [11,22]). Other mathematical models have also been used to analyze the effects of competition, among which the most famous is the Lotka-Volterra competition model [23][24][25]. In populations coupled by competition, investigating synchronization regimes [26] is an attractive subject from both empirical and theoretical perspectives [27][28][29][30][31]. ...
The problem of mathematical modeling and analysis of stochastic phenomena in population systems with competition is considered. This problem is investigated based on a discrete system of two populations modeled by the Ricker map. We study the dependence of the joint dynamic behavior on the parameters of the growth rate and competition intensity. It is shown that, due to multistability, random perturbations can transfer the population system from one attractor to another, generating stochastic P-bifurcations and transformations of synchronization modes. The effectiveness of a mathematical approach, based on the stochastic sensitivity technique and the confidence domain method, in the parametric analysis of these stochastic effects is demonstrated. For monostability zones, the phenomenon of stochastic generation of the phantom attractor is found, in which the system enters the trigger mode with alternating transitions between states of almost complete extinction of one or the other population. It is shown that the noise-induced effects are accompanied by stochastic D-bifurcations with transitions from order to chaos.
... In other words, two species competing for the same limited resource cannot coexist: the fittest one will dominate in the long term, thus leading either to extinction or an evolutionary shift towards another niche. Among this class of models, Lotka-Volterra equations deserve a special place Lotka (1925); Wangersky (1978); Hofbauer and Sigmund (1998). They were introduced in the early 20th century -in terms of a pair of first-order nonlinear differential equations -postulating that the demographic rates of a species depend not only on its abundance but also on the abundance of other species in the community. ...
The study of ecological systems is gaining momentum in modern scientific research, driven by an abundance of empirical data and advancements in bioengineering techniques. However, a full understanding of their dynamical and thermodynamical properties, also in light of the ongoing biodiversity crisis, remains a formidable endeavor. From a theoretical standpoint, modeling the interactions within these complex systems -- such as bacteria in microbial communities, plant-pollinator networks in forests, or starling murmurations -- presents a significant challenge. Given the characteristic high dimensionality of the datasets, alternative elegant approaches employ random matrix formalism and techniques from disordered systems. In these lectures, we will explore two cornerstone models in theoretical ecology: the MacArthur/Resource-Consumer model, and the Generalized Lotka-Volterra model, with a special focus on systems composed of a large number of interacting species. In the second part, we will highlight timely directions, particularly to bridge the gap with empirical observations and detect macroecological patterns.
... EGT, or Evolutionary Game Theory, models how individuals adapt their strategies based on interactions with others, which can be applied to social dynamics where cultural or learned behaviors are predominant. For instance, individuals may imitate successful strategies or explore new ones, akin to cultural evolution rather than genetic mutation [12]. These models can illustrate how the environment and past interactions influence future social dynamics; they are predictive and, therefore, very well suited for policy analysis. ...
This paper explores the dynamics of political competition, resource availability, and conflict through a simulation-based approach. Utilizing agent-based models (ABMs) within an evolutionary game theoretical framework, we investigate how individual behaviors and motivations influence collective outcomes in civil conflicts. Our study builds on the theoretical model developed by Basuchoudhary et al. (2023), which integrates factors such as resource availability, state capacity, and political entrepreneurship to explain the evolution of civil conflict. By simulating boundedly rational agents, we demonstrate how changes in resource availability can alter the nature of civil conflict, leading to different equilibrium outcomes. The findings highlight the importance of understanding individual motivations and adaptive behaviors in predicting the stability and resolution of conflicts. This research contributes to the growing body of literature on the use of agent-based models in evolutionary game theory and provides valuable insights into the complex interactions that drive civil violence.
... Biological evolution has been a particularly rich source of ideas about dynamically adapting behavior. The field of evolutionary game theory (EGT) applies such ideas to strategic interaction, building dynamic accounts of adaptive game play [Börgers and Sarin, 1997, Tuyls et al., 2003, Tuyls and Parsons, 2007, based on biological operators such as natural selection and mutation [Maynard Smith and Price, 1973, Zeeman, 1980, 1981, Weibull, 1997, Hofbauer and Sigmund, 1998]. The simplicity and concreteness of these operators provides a constructive basis for determination of joint behavior in complex strategic environments. ...
In the empirical approach to game-theoretic analysis (EGTA), the model of the game comes not from declarative representation, but is derived by interrogation of a procedural description of the game environment. The motivation for developing this approach was to enable game-theoretic reasoning about strategic situations too complex for analytic specification and solution. Since its introduction over twenty years ago, EGTA has been applied to a wide range of multiagent domains, from auctions and markets to recreational games to cyber-security. We survey the extensive methodology developed for EGTA over the years, organized by the elemental subproblems comprising the EGTA process. We describe key EGTA concepts and techniques, and the questions at the frontier of EGTA research. Recent advances in machine learning are accelerating progress in EGTA, and promise to significantly expand our capacities for reasoning about complex game situations.
... Historically, the emergence of cooperation was examined with well-mixed population models that assume all-to-all interactions [46], with evolutionary dynamics most commonly described by the replicator equation, wherein strategies providing higher payoffs increase in frequency [108] (as in Fig. 1b). Nevertheless, these models fail to capture the complexity Fig. 1 a In public good games, individuals are presented with a choice to either contribute an amount c to a common pot (Cooperate) or not (Defect). ...
Cooperation arises in nature at every scale, from within cells to entire ecosystems. Public goods games (PGGs) are used to represent scenarios characterised by the conflict/dilemma between choosing cooperation as a socially optimal strategy and defection as an individually optimal strategy. Evolutionary game theory is often used to analyse the dynamics of behaviour emergence in this context. Here, we focus on PGGs arising in the disease modelling of cancer evolution and the spread of infectious diseases. We use these two systems as case studies for the development of the theory and applications of PGGs, which we succinctly review. We also posit that applications of evolutionary game theory to decision-making in cancer, such as interactions between a clinician and a tumour, can learn from the PGGs studied in epidemiology, where cooperative behaviours such as quarantine and vaccination compliance have been more thoroughly investigated. Furthermore, instances of cellular-level cooperation observed in cancers point to a corresponding area of potential interest for modellers of other diseases, be they viral, bacterial or otherwise. We aim to demonstrate the breadth of applicability of PGGs in disease modelling while providing a starting point for those interested in quantifying cooperation arising in healthcare.
... Eco-evolutionary dynamics [1][2][3] explains the interplay between such ecological and evolutionary processes. Evolutionary game theory [4][5][6][7][8] explores how strategic interactions between individuals shape the evolution of behaviours in populations, applying population dynamics to game theory [9,10]. The principles of game theory help model strategic interactions among species, providing insight into how these interactions influence community structures. ...
Eco-evolutionary game dynamics explores the intricate interplay between ecological processes and evolutionary strategies. Here we unveil various transition mechanisms from transitive dominance to cyclic dominance within ecological communities, using evolutionary game theory to model strategic interactions among species. Specifically, we choose a multigame framework that incorporates games that exhibit transitive and cyclic dominance. By integrating appropriate models for transitive dominance and cyclic dominance, and through rigorous numerical simulations and analytical approaches, we explore evolutionary game dynamics under varying conditions. We show that, under specific conditions, communities can change from a linear hierarchy, or transitive dominance, to a cyclical structure, or cyclic dominance, which significantly affects ecosystem community structures. We also show that not only game-independent factors, but also intrinsic game parameters can drive a transition from transitive dominance to cyclic dominance. More precisely, we show how changes in the probability of game selection can lead to this transition. Moreover, we demonstrate that even benefits arising from free space can facilitate the shift from transitive to cyclic dominance. Our study thus explores the mechanisms that drive transitions between these dominance patterns and, in so doing, we underscore the importance of multiple game dynamics for accurately describing the intricacies of natural ecosystems.
... In evolutionary game theory, replicator dynamics is the most commonly utilized approach to describe the evolutionary process of cooperation in an infinite well-mixed population [14,15,16,17,18,19]. Nevertheless, populations frequently exhibit a structural characteristic where individuals only interact with their immediate neighbors, reflecting a common pattern observed in various social and biological systems [20,21,22]. ...
The interdependence between an individual strategy decision and the resulting change of environmental state is often a subtle process. Feedback-evolving games have been a prevalent framework for studying such feedback in well-mixed populations, yielding important insights into the coevolutionary dynamics. However, since real populations are usually structured, it is essential to explore how population structure affects such coevolutionary dynamics. Our work proposes a coevolution model of strategies and environmental state in a structured population depicted by a regular graph. We investigate the system dynamics, and theoretically demonstrate that there exist different evolutionary outcomes including oscillation, bistability, the coexistence of oscillation and dominance, as well as the coexistence of cooperation and defection. Our theoretical predictions are validated through numerical calculations. By using Monte Carlo simulations we examine how the number of neighbors influences the coevolutionary dynamics, particularly the size of the attractive domain of the replete environmental state in the cases of bistability or cooperation-defection coexistence. Specifically, in the case of bistability, a larger neighborhood size may be beneficial to save the environment when the environmental enhancement rate by cooperation / degradation rate by defection is high. Conversely, if this ratio is low, a smaller neighborhood size is more beneficial. In the case of cooperator-defector coexistence, environmental maintenance is basically influenced by individual payoffs. When the ratio of temptation minus reward versus punishment minus sucker's payoff is high, a larger neighborhood size is more favorable. In contrast, when the mentioned ratio is low, a smaller neighborhood size is more advantageous.
... where ϵ provides a relative time-scale of individual updating their strategies relative to the strength of environmental feedback, r describes a background growth rate of the resource and a describes a conversion efficiency of harvester effort to the extraction of the resource. Equation (2.6a) is a replicator equation, commonly used to describe the changing strategic composition of the population that can be derived from many individual-based rules for either social learning [73] or natural selection [74,75]. Equation (2.6b) characterizes how the environmental quality index changes in response to the harvesting efforts of the population, and was derived by Tilman and coauthors [33] based on a model of harvesting a renewable resource undergoing background logistic growth that is often used to study the management of common-pool resources [76][77][78]. ...
The sustainable management of common resources often leads to a social dilemma known as the tragedy of the commons: individuals benefit from rapid extraction of resources, while communities as a whole benefit from more sustainable extraction strategies. Such a social dilemma can be further complicated by the role played by space for both resources and harvesters, where spatial diffusion of resources and directed motion of harvesters can potentially feature the emergence of clusters of environmental resource and sustainable harvesting strategies. In this paper, we explore a PDE model of evolutionary game theory with environmental feedback, describing how the spatial distribution of resource extraction strategies and environmental resources can change due to both local eco-evolutionary dynamics and environmental-driven directed motion of harvesters. Through linear stability analysis, we show that this biased motion towards higher-quality environments can lead to spatial patterns in the distribution of extraction strategies, creating local regions with improved environmental quality and increase payoff for resource extractors. However, by measuring the average payoff and environmental quality across the spatial domain, we see that this pattern-forming mechanism can actually decrease the overall success of the population relative to the equilibrium outcome in the absence of spatial motion. This suggests that environmental-driven motion can produce a spatial social dilemma, in which biased motion towards more beneficial regions can produce emergent patterns featuring a worse overall environment for the population.
... The noise in perceived payoff results in the introduction of a new mutation term κ 2 x 1 x 2 (ν 2 2 x 2 − ν 2 1 x 1 ) to replicator dynamics; see, e.g., [36,38]. A mutation term induces irrational choices of a strategy that might not be optimal. ...
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.
... However, in contrast to standard frameworks of evolution, such as those used to study game dynamics (e.g. [42][43][44]), and consistently with a new eco-evolutionary approach [37], no fitness function is explicitly defined. Instead, evolution takes place as a direct consequence of ecological interactions and is inseparable from the latter. ...
Ecological processes and evolutionary change are increasingly recognized as intimately linked. Here, we introduce an eco-evolutionary model of trophic interactions between predators and prey and show that the flow of resources in the ecosystem results in the scale-invariant spatial and temporal structure of ecosystems. In contrast to conventional approaches that rely on fitness-based selection, evolution in our eco-evolutionary framework is a direct consequence of ecological interactions. To illustrate this, we combine trophic interactions with evolutionary games by allowing individuals to play a game within the population where they can adopt aggressive or non-aggressive strategies. We show that individuals develop consistent personalities and their life-history trade-offs become intertwined with the scale-invariant ecological dynamics. Aggressive individuals tend to live faster, more reproduction-focused lives, whereas nonaggressive individuals favor slower, longer-lived strategies. These patterns emerge naturally, rather than being imposed as model assumptions. Furthermore, we demonstrate that the nonequilibrium dynamics of resource flow play a decisive role in driving the evolution of consistent personalities within and across populations. We identify a new class of aggression scaling laws arising from the interplay of ecological and evolutionary processes. The model relates predator–prey scaling laws with food web control and shows that small offspring size, high relative prey mobility, low predator conversion efficiency, predator competition, and prey competition all favor prey control over the food web. Our findings illuminate how large-scale ecological patterns—including power laws in predator–prey biomass and avalanche-like resource pulses—can relate to evolutionary outcomes such as consistent personalities, life-history trade-offs, and density-dependent growth. This perspective strengthens the emerging view that ecology and evolution are two faces of the same coin, each shaping the other in a self-organized, energy-driven system.
... After setting the initial conditions, all participants in each round engage in strategy interactions with their opponents to obtain corresponding payoffs, then update their strategies based on learning rules. Repeating this process, all individuals in the system ultimately reach a dynamic evolutionary stable equilibrium, which corresponds to the Nash equilibrium in classical game theory [2]. Common models widely applied in evolutionary game theory include the Prisoner's Dilemma [3], Snowdrift Game [4], and Public Goods Game [5]. ...
In recent years, coupled double-layer networks have played an increasingly critical role in evolutionary game theory. Research indicates that these networks more accurately reflect real-world relationships between individuals. However, current studies mainly focus on unidirectional influence within double-layer networks. Based on this, we propose a strongly coupled double-layer network cooperation evolution model. Strength individuals are located in the upper network layer, influencing the strategy choices of ordinary individuals in the lower layer, and vice versa. Monte Carlo simulations show that strength individuals can effectively enhance overall group cooperation. Under low temptation to defect, the group maintains a high cooperation rate; under high temptation, the presence of strength individuals prevents the group from falling into total defection, helping ordinary individuals escape the defection dilemma and improve cooperation levels.
... The densities of populations directly influence their rates of encounter and therefore how much they interact. The dynamical evolution of populations must account for the trajectory of densities and should stem from local births and deaths altering densities, instead of being measured by relative fitness on proportions (i.e., frequencies) (Hofbauer and Sigmund (1998)). These shortcomings of standard evolutionary game theory make it incomplete as a framework for population dynamics. ...
Despite being a powerful tool to model ecological interactions, traditional evolutionary game theory can still be largely improved in the context of population dynamics. One of the current challenges is to devise a cohesive theoretical framework for ecological games with density-dependent (or concentration-dependent) evolution, especially one defined by individual-level events. In this work, I use the notation of reaction networks as a foundation to propose a framework and show that classic two-strategy games are a particular case of the theory. The framework exhibits a strong versatility and provides a standardized language for model design, and I demonstrate its use through a simple example of mating dynamics and parental care. In addition, reaction networks provide a natural connection between stochastic and deterministic dynamics and therefore are suitable to model noise effects on small populations, also allowing the use of stochastic simulation algorithms such as Gillespie’s with game models. The methods I present can help to bring evolutionary game theory to new reaches in ecology, facilitate the process of model design, and put different models on a common ground.
... Note 4. On evolutionary dynamics, see Hofbauer and Sigmund (1998), Vega-Redondo (1996) and Weibull (1995). ...
There is considerable evidence that heterogeneity in tax compliance behavior is persistent and pervasive. This paper develops an evolutionary analytical framework in which taxpayers periodically choose between to comply or not to comply with their tax obligations. Aggregate demand formation arising from private and public expenditures depends on the frequency distribution of tax compliance behavior across taxpayers, so that the macrodynamic of the rates of capacity utilization and output growth is coevolutionarily coupled to the microdynamic of tax compliance across individuals. The analytical framework set forth here replicates several pieces of empirical evidence on tax evasion. First, the proportion of non-complying taxpayers (and hence the volume of tax evasion) depends on the tax rate and the expected cost of tax evasion. Second, heterogeneity in tax compliance behavior across taxpayers is evolutionarily persistent instead of temporary. Third, the immediate impact of a change in the proportion of tax evading individuals on the rates of capacity utilization and output growth is non-linear. Fourth, the proportion of non-complying taxpayers and the rates of capacity utilization and output growth vary positively with the tax rate in the evolutionary equilibrium.
... And these differences permit (or preclude) coexistence along the corresponding niche axes. Via ecological and evolutionary dynamics these games of community organization model species diversity, coevolution and speciation (Hofbauer & Sigmund, 1998;Maynard Smith, 1982;Nowak, 2006). Such games are usually built upon models from population ecology (e.g. ...
In mathematical models of eco-evolutionary dynamics with a quantitative trait, two species with different strategies can coexist only if they are separated by a valley or peak of the adaptive landscape. A community is ecologically and evolutionarily stable if each species’ trait sits on global, equal fitness peaks, forming a saturated ESS community. However, the adaptive landscape may allow communities with fewer ( undersaturated ) or more ( hypersaturated ) species than the ESS. Non-ESS communities at ecological equilibrium exhibit invasion windows of strategies that can successfully invade. Hypersaturated communities can arise through mutual invasibility where each non-ESS species’ strategy lies in another’s invasion window. Hypersaturation in ESS communities with >1 species remains poorly understood. We use the G -function approach to model niche coevolution and Darwinian dynamics in a Lotka-Volterra competition model. We confirm that up to 2 (or 3) species can coexist in a hypersaturated community with a single-species ESS if the strategy is scalar-valued (or bivariate). We conjecture that at most n*(s+1) species can form a hypersaturated community, where n is the number of ESS species at the strategy’s dimension. For a scalar-valued 2-species ESS, four species coexist by “straddling” the would-be ESS traits. In a 5-species ESS, 7 or 8, but not 10, species can coexist in the hypersaturated community. In a bivariate model with a single-species ESS, an infinite number of 3-species hypersaturated communities can exist. We offer conjectures and discuss their relevance to ecosystems that may be non-ESS due to invasive species, climate change, and human-altered landscapes.
... To do so, suppose that (FTRL+) performs r steps from n k so y n k +r = y n k + η r j=1v n k +1/2 (D.23) where, to ease notation, we have made the simplifying assumption that η i = η for all i ∈ N . 12 Then, by invoking Lemma B.4 with y ← y n k and y + ← y n k +r , we obtain η r j=1v n k +1/2 , p − x n k +r ≤ ⟨∇h(x n k +r ) − y n k , p − x n k +r ⟩ = ⟨∇h(x n k +r ) − z n k , p − x n k +r ⟩ (D. 24) where, in the second line, we have used the fact that ⟨y, x ′ − x⟩ = ⟨Π(y), x ′ − x⟩ for all x, x ′ ∈ X and all y ∈ Y. Thus, letting k → ∞, we get from Step 3 and the continuity of ∇h that ηr⟨v( 25) for all r = 1, 2, . . . and all x ∈ X . ...
The long-run behavior of multi-agent learning - and, in particular, no-regret learning - is relatively well-understood in potential games, where players have aligned interests. By contrast, in harmonic games - the strategic counterpart of potential games, where players have conflicting interests - very little is known outside the narrow subclass of 2-player zero-sum games with a fully-mixed equilibrium. Our paper seeks to partially fill this gap by focusing on the full class of (generalized) harmonic games and examining the convergence properties of follow-the-regularized-leader (FTRL), the most widely studied class of no-regret learning schemes. As a first result, we show that the continuous-time dynamics of FTRL are Poincar\'e recurrent, that is, they return arbitrarily close to their starting point infinitely often, and hence fail to converge. In discrete time, the standard, "vanilla" implementation of FTRL may lead to even worse outcomes, eventually trapping the players in a perpetual cycle of best-responses. However, if FTRL is augmented with a suitable extrapolation step - which includes as special cases the optimistic and mirror-prox variants of FTRL - we show that learning converges to a Nash equilibrium from any initial condition, and all players are guaranteed at most O(1) regret. These results provide an in-depth understanding of no-regret learning in harmonic games, nesting prior work on 2-player zero-sum games, and showing at a high level that harmonic games are the canonical complement of potential games, not only from a strategic, but also from a dynamic viewpoint.
We develop a dynamic evolutionary model of innovation culture and its organizational effects at three levels. At the frontline level, informal bottom-up innovation ideas are driven by unplanned interactions governed by cultural norms, as frontline employees sort themselves into idea generator or implementor behaviors. Management cannot directly control innovation interactions and thus culture, but it can imperfectly nudge it by influencing a sharing norm at the second level. At the third level, the culture influences the innovation performance and productivity of the firm and is thus relevant for the organization’s competitiveness. This modeling approach disentangles how top-down strategy interacts with implicit frontline dynamics of cultural evolution, in the context of innovation. Unguided culture does not lead to maximum firm performance. Management guidance may or may not improve performance, and industry competition may select higher-performing cultures, which then spread through the industry under some circumstances. We characterize under which conditions management has a strong influence on culture and performance and under which conditions performance is driven more by forces outside of management’s control.
This paper was accepted by Alfonso Gambardella, business strategy.
Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2024.05087 .
Microbial communities are complex ecological systems of organisms that evolve in time, with new variants created, while others disappear. Understanding how species interact within communities can help us shed light into the mechanisms that drive ecosystem processes. We studied systems with serial propagation, where the community is kept alive by taking a subsample at regular intervals and replating it in fresh medium. The data that are usually collected consist of the % of the population for each of the species, at several time points. In order to utilize this type of data, we formulated a system of equations (based on the generalized Lotka–Volterra model) and derived conditions of species noninteraction. This was possible to achieve by reformulating the problem as a problem of finding feasibility domains, which can be solved by a number of efficient algorithms. This methodology provides a cost‐effective way to investigate interactions in microbial communities.
To promote the reasonable allocation of tasks and improve the occurrence of cooperative dilemma, we consider the interference of the third strategy based on the two-strategy division of labor game. When the cost difference of performing tasks exceeds the cooperation benefits, the strategies (players performing task A, players performing task B, and destroyers) coexist. The system also experiences bistability and occurs a transcritical bifurcation at . Information often cannot be transmitted in a timely manner, so the effects of the delays in obtaining fitness information for task performers and destroyers is investigated, respectively. We perform the existence of local Hopf bifurcation occurring at and the stability of in the delayed system. We determine the direction, stability, and period of these periodic solutions bifurcating from steady state by the central manifold theorem and the normal form rule. The results illustrate that a small information delay does not affect the final evolutionary outcomes of the three strategies. The information delay exceeds the critical value, the frequencies of the three strategies oscillate. It is worth noting that a sufficiently large information delay leads to a mutual transformation between the existence of a single strategy and the coexistence of two strategies. Based on the above results, we can establish effective mechanisms to promote the completion of all tasks and maximize the benefits of the group.
In this paper, we classify all global dynamics of the three-dimensional type- K monotone Lotka–Volterra system with the identical intrinsic growth rate inside the compactification of the positive octant of . By means of the replicator equations, it is proved that this system can have exactly 35 topologically different phase portraits. As a consequence, we obtain the necessary and sufficient condition for the system to be bounded in the positive octant and verify that the limit set of any orbit of the compactified vector field associated with the system is an equilibrium.
Host–pathogen interactions consist of an attack by the pathogen, frequently a defense by the host and possibly a counterdefense by the pathogen. Here, we present a game-theoretical approach to describe such interactions. We consider a game where the host and pathogen are players and can choose between the strategies of defense (or counterdefense) and no response. Specifically, they may or may not produce a toxin and an enzyme degrading the toxin, respectively. We consider that the host and pathogen must also incur a cost for toxin or enzyme production. We highlight both the sequential and non-sequential versions of the game and determine the Nash equilibria. Furthermore, we resolve a paradox occurring in that interplay. If the inactivating enzyme is very efficient, producing the toxin becomes useless, leading to the enzyme being no longer required. Then, the production of the defense becomes useful again. In game theory, such situations can be described by a generalized matching pennies game. As a novel result, we find under which conditions the defense cycle leads to a steady state or an oscillation. We obtain, for saturating dose–response kinetics and considering monotonic cost functions, “partial (counter)defense” strategies as pure Nash equilibria. This implies that producing a moderate amount of toxin and enzyme is the stable situation in this game.
In this study, to show the conditions for stochastic evolutionary stability in evolutionary game dynamics with more than two pure strategies, the results from two-phenotype models with two pure strategies are extended to the situation with three pure strategies. Our main results enrich our understanding of the stochastic evolutionary stability in situations with multiple pure strategies. In particular, it should be noted that the mathematical complexity in determining the conditions for stochastic evolutionary stability, especially for the degenerate cases with multiple pure strategies, should be given a great attention in the future study.
The hypercycling replicator system with infinite many members is investigated. The existence and uniqueness theorem are proved. It is shown that dynamis of system present non damping nonlinear waves.
Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation–linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation.
Evolutionary game theory offers an interesting avenue of exploration for populations that are subdivided into smaller groups based on shared traits. Despite being self-contained, interactions between individuals within each group are crucial. These interactions lead to a game with a block-diagonal payoff matrix having blocks of order two or three. A constant negative payoff is assigned to each player, while the background fitness function is inversely proportional to the density of players in the given territory. Through the lens of reaction–diffusion systems, we examine the circumstances necessary for diffusion-driven instability or Turing instability. We derive a set of necessary conditions for Turing instability around the interior equilibrium state. These results reveal that Turing instability occurs when some diagonal elements are positive, or diagonal cofactors of 3-order blocks are negative in the payoff matrix of the game. In summary, this article explores the dynamics of group interactions in population games and identifies key conditions that lead to instability.
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 modelled 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 when 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.
The selected effects theory is supposed to provide a fully naturalistic basis for statements about what biological traits or processes are for without appeal to final causes or intelligent design. On the selected effects theory, biologists are allowed to say, for instance, that hindwing eyespots on butterfly wings serve to deflect predators’ attacks away from vital organs because a similar fitness-enhancing effect explains why eyespots themselves were favoured by natural selection and persisted in the population. This is known as the explanatory dimension of the selected effects theory. According to it, appealing to the fitness-enhancing effect of a certain trait or process is sufficient to explain its current presence in a population, namely, why it persisted and still exists in that population. In this paper, however, I will call such a claim into question, and I will do so by discussing a mathematical Hawk-Dove example and a real case scenario taken from evolutionary biology, that of Perissodus microlepis. These are scenarios in which the selective filter does not allow variants with the highest fitness at a certain moment to prevail over their available alternatives. In similar cases, I will argue, citing fitness-enhancing effects does not represent an adequate explanation of what happens in the population, undermining the explanatory dimension of the selected effects theory.
Green technological innovation in enterprises is a key driving force for achieving sustainable development and industrial transformation. However, how enterprises formulate effective innovation strategies by integrating subjective factors with objective market conditions within complex industrial networks remains an area requiring further exploration. This study aims to construct a complex network evolution model based on prospect theory and examines how subjective factors, such as reference points, risk preferences, and loss aversion, influence the adoption of green technology innovation in different network environments. It further explores how network characteristics, including topology, size, and node degree, affect the diffusion of innovation. Numerical analysis results indicate that in scale‐free networks, lowering reference points for enterprise gains, reducing loss aversion, increasing risk preference, expanding network size, and raising average node degree generally promote higher adoption of green technology innovation. In small‐world networks, the dependence on reference points is relatively lower, risk preference and average node degree demonstrate more complex impacts. Additionally, a moderate rewiring probability can enhance green technology innovation adoption in small‐world networks. These findings provide new insights and practical implications for understanding the driving mechanisms of green technological innovation in enterprises. They further emphasize the importance of government interventions tailored to the specific characteristics of industrial networks to effectively facilitate the diffusion of green technological innovation.
Networked evolutionary game theory is a well-established framework for modeling the evolution of social behavior in structured populations. Most of the existing studies in this field have focused on 2-strategy games on heterogeneous networks or n-strategy games on regular networks. In this paper, we consider n-strategy games on arbitrary networks under the pairwise comparison updating rule. We show that under the limit of weak selection, the short-run behavior of the stochastic evolutionary process can be approximated by replicator equations with a transformed payoff matrix that involves both the average value and the variance of the degree distribution. In particular, strongly heterogeneous networks can facilitate the evolution of the payoff-dominant strategy. We then apply our results to analyze the evolutionarily stable strategies in an n-strategy minimum-effort game and two variants of the prisoner’s dilemma game. We show that the cooperative equilibrium becomes evolutionarily stable when the average degree of the network is low and the variance of the degree distribution is high. Agent-based simulations on quasi-regular, exponential, and scale-free networks confirm that the dynamic behaviors of the stochastic evolutionary process can be well approximated by the trajectories of the replicator equations.
Linear compartmental models are often employed to capture the change in cell type composition of cancer cell populations. Yet, these populations usually grow in a nonlinear fashion. This begs the question of how linear compartmental models can successfully describe the dynamics of cell types. Here, we propose a general modeling framework with a nonlinear part capturing growth dynamics and a linear part capturing cell type transitions. We prove that dynamics in this general model are asymptotically equivalent to those governed only by its linear part under a wide range of assumptions for nonlinear growth.
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