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Highlights • We considered the effects of time-varying migration path on the metapopulation rock-paper-scissors game. • The dynamics are represented by the reaction-diffusion equations with time-dependent diffusion coefficients in diffusively coupled reactors. • We numerically obtained the solutions of the time-dependent reaction-diffusion-equations. • We found that phase transitions occur by varying a mutation rate. • The phase transitions depend highly on the frequency change of migration path. Abstract Migration paths of animals are rarely the same. The paths may change according to seasonal and circadian rhythms. We study the effect of temporal migration on population dynamics of rock-paper-scissors (RPS) games with mutation by using the metapopulation dynamic model with two patches. Via mutation, an individual R changes to S with rate μ. All agents move by random walk between two patches and the RPS game is performed in each patch. The migration path between two patches is switched on or off periodically. The dynamics are represented by the reaction-diffusion equations with time-dependent diffusion coefficients in diffusively coupled reactors. We obtain the solutions of time-dependent reaction-diffusion equations numerically and analytically. The time-varying migration path induces complex behavior for the RPS dynamics, depending on the frequency of the periodical path. We find that the phase transitions occur by varying mutation rate μ. The phase transition depends highly on the frequency. [ABSTRACT FROM AUTHOR]

Spatial and temporal behaviors of the rock–paper–scissors (RPS) game is key to understanding not only biodiversity but also a variety of cyclic systems. It has been demonstrated that, in the stochastic cellular automaton of RPS game, three species cannot survive on one-dimensional (1-d) lattice; only a single species survives. Previous studies have shown that three species are able to coexist if the migration of species is considered. However, their definitions of migration are the swapping of two species or the random walk of species, which rarely occurs in nature. Here, we investigate the effect of migration by using the 1-d lattice traffic model in which species can move rightward if the site ahead is empty. Computer simulations reveal that three species can survive at the same time within the wide range of parameter values. At low densities, all species can coexist. In contrast, the extinction of two species occurs if the density exceeds the critical limit of the jamming transition. This dynamical phase transition between the coexistence and single (non-coexistence) phase clearly separates due to the self-organized pattern: condensation and rarefaction in the stripe-pattern of three species. [ABSTRACT FROM AUTHOR]

We consider discrete dynamical systems coming from the models of evolution of populations playing rock-paper-scissors game. Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved. [ABSTRACT FROM AUTHOR]

In evolutionary game theory interactions between individuals are often assumed obligatory. However, in many real-life situations, individuals can decide to opt out of an interaction depending on the information they have about the opponent. We consider a simple evolutionary game theoretic model to study such a scenario, where at each encounter between two individuals the type of the opponent (cooperator/defector) is known with some probability, and where each individual either accepts or opts out of the interaction. If the type of the opponent is unknown, a trustful individual accepts the interaction, whereas a suspicious individual opts out of the interaction. If either of the two individuals opt out both individuals remain without an interaction. We show that in the prisoners dilemma optional interactions along with suspicious behaviour facilitates the emergence of trustful cooperation. [ABSTRACT FROM AUTHOR]

Aims To understand processes placing individuals at risk for stimulant (amphetamine and cocaine) use disorder. Design Longitudinal study. Setting University of California, San Diego Department of Psychiatry, CA, USA. Participants Occasional stimulant users (OSU; n = 184) underwent a baseline clinical interview and a functional magnetic resonance imaging (fMRI) session. On the basis of a follow-up clinical interview completed 3 years later, OSU ( n = 147) were then categorized as problem stimulant users (PSU: n = 36; those who developed stimulant use disorders in the interim) or desisted stimulant users (DSU: n = 74; those who stopped using). OSU who did not meet criteria for PSU or DSU ( n = 37) were included in dimensional analyses. Measurements fMRI blood oxygen level-dependent (BOLD) contrast percentage signal change from baseline collected during a Paper-Scissors-Rock task was examined during three decision-making conditions, those resulting in: (1) wins, (2) ties and (3) losses. These data were used as dependent variables in categorical analyses comparing PSU and DSU, as well as dimensional analyses including interim drug use as predictors, controlling for baseline drug use. Findings PSU exhibited lower anterior cingulate, middle insula, superior temporal, inferior parietal, precuneus and cerebellum activation than DSU across all three conditions (significant brain clusters required > 19 neighboring voxels to exceed F(1,108) = 5.58, P < 0.01 two-tailed; all Cohen's d > 0.83). Higher interim marijuana use was linked to lower pre-central and superior temporal activation during choices resulting in wins (> 19 neighboring voxels to exceed t = 2.61, P < 0.01 two-tailed; R2 change > 0.11). Conclusions Individuals who transition from stimulant use to stimulant use disorder appear to show alterations in neural processing of stimulus valuation and outcome monitoring, patterns also evident in chronic stimulant use disorder. Attenuated anterior cingulate and insular processing may constitute a high-risk neural processing profile, which could be used to calculate risk scores for individuals experimenting with stimulants. [ABSTRACT FROM AUTHOR]

Environmental stochasticity and climate affect outcomes in evolutionary games, which can thereby affect biological diversity. Our maximum likelihood (ML) estimates of replicator dynamics for morph frequency data from control (25 years) and three experimentally perturbed populations (14 years) of side-blotched lizards yield a 3 × 3 payoff matrix in the generalized Rock-Paper-Scissors family; it has intransitive best replies, and each strategy is its own worst reply. ML estimates indicate significant interactive effects of density and temperature on morph frequency. Implied dynamics feature a powerful interior attractor and recover (for the first time) observed 4-5 year oscillations. Our evolutionary experiment on morph frequency confirms that oscillations are driven by frequency dependent selection, but climate entrains the cycles across the perturbed and control populations within 10 generations. Applying the model across the species range, we find that climate also accounts for morph fixation and mating system diversity, suggesting climate may similarly impact ecosystem diversity. [ABSTRACT FROM AUTHOR]

Polymorphisms are usually associated with defenses and mating strategies, affecting the individual’s fitness. Coexistence of different morphs is, therefore, not expected, since the fittest morph should outcompete the others. Nevertheless, coexistence is observed in many natural systems. For instance, males of the side-blotched lizards ( Uta stansburiana ) present three morphs with throat colors orange, yellow and blue, which are associated with mating strategies and territorial behavior. The three male morphs compete for females in a system that is well described by the rock-paper-scissors dynamics of game theory. Previous studies have modeled the lizards as hermaphroditic populations whose individual’s behavior were determined only by their phenotypes. Here we consider an extension of this dynamical system where diploidy and sexual reproduction are explicitly taken into account. Similarly to the lizards we represent the genetic system by a single locus with three alleles, o, y , and b in a diploid chromosome with dominance of o over y and of y over b . We show that this genotypic description of the dynamics results in the same equilibrium phenotype frequencies as the phenotypic models, but affects the stability of the system, changing the parameter region where coexistence of the three morphs is possible in a rock-paper-scissors game. [ABSTRACT FROM AUTHOR]

Social dilemmas lead to natural conflict between cooperation and self interests among individuals in large populations. The emergence of cooperation and its maintenance is the key for the understanding of fundamental concepts about the evolution of species. In order to comprehend the mechanisms involved in this framework, here we study the Optional Public Good Games with focus on the effects of diffusive aspects in the emergent patterns of cyclic dominance between the strategies. Differently from other works, we showed that rock–paper–scissors (RPS) patterns occur by introducing a simple kind of random mobility in a lattice sparsely occupied. Such pattern has been revealed to be very important in the conservation of the species in ecological and social environments. The goal of this paper is to show that we do not need more elaborated schemes for construction of the neighbourhood in the game to observe RPS patterns as suggested in the literature. As an interesting additional result, in this contribution we also propose an alternative method to quantify the RPS density in a quantitative context of the game theory which becomes possible to perform a finite size scaling study. Such approach can be very interesting to be applied in other games generically. [ABSTRACT FROM AUTHOR]

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]

Discretizing a differential equation may change the qualitative behaviour drastically, even if the stepsize is small. We illustrate this by looking at the discretization of a piecewise continuous differential equation that models a population of agents playing the Rock-Paper-Scissors game. The globally asymptotically stable equilibrium of the differential equation turns, after discretization, into a repeller surrounded by an annulus shaped attracting region. In this region, more and more periodic orbits emerge as the discretization step approaches zero. [ABSTRACT FROM AUTHOR]

Iterated games provide a framework to describe social interactions among groups of individuals. This body of work has focused primarily on individuals who face a simple binary choice, such as "cooperate" or "defect." Real individuals, however, can exhibit behavioral diversity, varying their input to a social interaction both qualitatively and quantitatively. Here we explore how access to a greater diversity of behavioral choices impacts the evolution of social dynamics in populations. We show that, in public goods games, some simple strategies that choose between only two possible actions can resist invasion by all multichoice invaders, even while engaging in relatively little punishment. More generally, access to a larger repertoire of behavioral choices results in a more "rugged" fitness landscape, with populations able to stabilize cooperation at multiple levels of investment. As a result, increased behavioral choice facilitates cooperation when returns on investments are low, but it hinders cooperation when returns on investments are high. Finally, we analyze iterated rock-paper-scissors games, the nontransitive payoff structure of which means that unilateral control is difficult to achieve. Despite this, we find that a large proportion of multichoice strategies can invade and resist invasion by single-choice strategies--so that even well-mixed populations will tend to evolve and maintain behavioral diversity. [ABSTRACT FROM AUTHOR]

Cyclic dominance models, such as the classic rock-paper-scissors (RPS) game, have found real-world applications in biology, ecology, and sociology. A key quantity of interest in such models is the coexistence time, i.e., the time until at least one population type goes extinct. Much recent research has considered conditions that lengthen coexistence times in an RPS model. A general finding is that coexistence is promoted by localized spatial interactions (low mobility), while extinction is fostered by global interactions (high mobility). That is, there exists a mobility threshold which separates a regime of long coexistence from a regime of rapid collapse of coexistence. The key finding of our paper is that if zealots (i.e., nodes able to defeat others while themselves being immune to defeat) of even a single type exist, then system coexistence time can be significantly prolonged, even in the presence of global interactions. This work thus highlights a crucial determinant of system survival time in cyclic dominance models. [ABSTRACT FROM AUTHOR]

When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hop! bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point. [ABSTRACT FROM AUTHOR]

We show that the extension of Rock-Paper-Scissors to include Lizard and Spock is the unique such five move fair game up to isomorphism. We also categorize all analogous six move fair games [ABSTRACT FROM AUTHOR]

The article looks into the structures of directed graphs through the Borromean rings and the game rock-paper-scissors. Topics discussed include the graph representations for the two-player game rock-paper-scissors (RPS(3)), in which in mathematics is referred to as a balanced tournament, for the five-weapon game (RPS(5)), and for the hexagonal pyramid. The isomorphism and automorphisms of graph are also mentioned.

We analyze the replicator-mutator equations for the rock-paper-scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game. [ABSTRACT FROM AUTHOR]

Intransitive competition occurs when competing strategies cannot be listed in a hierarchy, but rather form loops—as in the game rock–paper–scissors. Due to its cyclic competitive replacement, competitive intransitivity promotes strategy coexistence, both in rock–paper–scissors and in higher-richness communities. Previous work has shown that this intransitivity-mediated coexistence is strongly influenced by spatially explicit interactions, compared to when populations are well mixed. Here, we extend and broaden this line of research and examine the impact on coexistence of intransitive competition taking place on a continuum of small-world networks linking spatial lattices and regular random graphs. We use simulations to show that the positive effect of competitive intransitivity on strategy coexistence holds when competition occurs on networks toward the spatial end of the continuum. However, in networks that are sufficiently disordered, increasingly violent fluctuations in strategy frequencies can lead to extinctions and the prevalence of monocultures. We further show that the degree of disorder that leads to the transition between these two regimes is positively dependent on population size; indeed for very large populations, intransitivity-mediated strategy coexistence may even be possible in regular graphs with completely random connections. Our results emphasize the importance of interaction structure in determining strategy dynamics and diversity. [ABSTRACT FROM AUTHOR]

The article presents the actuarial artificial intelligence in playing the rock-paper-scissors game. Topics discussed include an overview of the rock-paper-scissors game, an outline of its different strategies, the definition of its variables, its basics, the reset of its probabilities, and the flexibility of its probabilities.

The article reports on a study which examined the attitude of chimpanzees (Pan troglodytes) learning to play rock-paper-scissors game. Topics discussed include the relationship among the three different hand signals used in the game, ability of the chimpanzees to learn the circular pattern of the game, and comparing the learning process of chimpanzees to children ages three to six years old.

A review of the article "Rock-Paper-Scissors Meets Borromean Rings," by M. Chamberland and E.A. Herman, that was published in the 2015 issue of the journal "The Mathematical Intelligencer," is presented.

This work deals with a system of three distinct species that changes in time under the presence of mobility, selection, and reproduction, as in the popular rock-paper-scissors game. The novelty of the current study is the modification of the mobility rule to the case of directional mobility, in which the species move following the direction associated to a larger (averaged) number density of selection targets in the surrounding neighborhood. Directional mobility can be used to simulate eyes that see or a nose that smells, and we show how it may contribute to reduce the probability of coexistence. [ABSTRACT FROM AUTHOR]

This work deals with the effects of an apex predator on the cyclic competition among three distinct species that follow the rules of the rock-paper-scissors game. The investigation develops standard stochastic simulations but is motivated by a procedure which is explained in the work. We add the apex predator as the fourth species in a system that contains three species that evolve following the standard rules of migration, reproduction, and predation, and study how the system evolves in this new environment, in comparison with the case in the absence of the apex predator. The results show that the apex predator engenders the tendency to spread uniformly in the lattice, contributing to destroy the spiral patterns, keeping biodiversity but diminishing the average size of the clusters of the species that compete cyclically. [ABSTRACT FROM AUTHOR]

The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. First, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and/or interaction randomness, and this regardless of whether the later is quenched or annealed. While even a tiny fraction of zealots brings significant benefits, at 5% occupancy zealots practically destroy all oscillations regardless of the intensity of mobility, and regardless of the type and strength of randomness in the interaction structure. Interestingly, by annealed randomness the impact of zealots is qualitatively the same as by mobility, which highlights that fast diffusion does not necessarily destroy the coexistence of species, and that zealotry thus helps to recover the stable mean-field solution. Our results strengthen the important role of zealots in models of cyclic dominance, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior. [ABSTRACT FROM AUTHOR]

The classical game of rock-paper-scissors has inspired experiments and spatial model systems that address the robustness of biological diversity. In particular, the game nicely illustrates that cyclic interactions allow multiple strategies to coexist for long-time intervals. When formulated in terms of a one-dimensional cellular automata, the spatial distribution of strategies exhibits coarsening with algebraically growing domain size over time, while the two-dimensional version allows domains to break and thereby opens the possibility for long-time coexistence. We consider a quasi-one-dimensional implementation of the cyclic competition, and study the long-term dynamics as a function of rare invasions between parallel linear ecosystems. We find that increasing the complexity from two to three parallel subsystems allows a transition from complete coarsening to an active steady state where the domain size stays finite. We further find that this transition happens irrespective of whether the update is done in parallel for all sites simultaneously or done randomly in sequential order. In both cases, the active state is characterized by localized bursts of dislocations, followed by longer periods of coarsening. In the case of the parallel dynamics, we find that there is another phase transition between the active steady state and the coarsening state within the three-line system when the invasion rate between the subsystems is varied. We identify the critical parameter for this transition and show that the density of active boundaries has critical exponents that are consistent with the directed percolation universality class. On the other hand, numerical simulations with the random sequential dynamics suggest that the system may exhibit an active steady state as long as the invasion rate is finite. [ABSTRACT FROM AUTHOR]

We study an evolutionary game model based on a transition matrix approach, in which the total change in the proportion of a population playing a given strategy is summed directly over contributions from all other strategies. This general approach combines aspects of the traditional replicator model, such as preserving unpopulated strategies, with mutation-type dynamics, which allow for nonzero switching to unpopulated strategies, in terms of a single transition function. Under certain conditions, this model yields an endemic population playing non-Nash-equilibrium strategies. In addition, a Hopf bifurcation with a limit cycle may occur in the generalized rock-scissors-paper game, unlike the replicator equation. Nonetheless, many of the Folk Theorem results are shown to hold for this model. [ABSTRACT FROM AUTHOR]

A correction to the article "Rock-Paper-Scissors Meets Borromean Rings" that was published in a previous issue is presented.