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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 spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility. [ABSTRACT FROM AUTHOR]

The current study examined the influence of redundant stimulus features on our ability to build and update representations of our environment. We hypothesized that our ability to process redundant spatial features would speed our ability to adapt to changingnonspatialregularities. Using a computerized version of the children's game “rock–paper–scissors”, undergraduates were instructed to win as often as possible against a computer opponent. The computer's plays were repeating sequences of five choices that were presented eitherwithspatial regularity (i.e., “rock” would always appear on the left, “paper” in the middle, and “scissors” on the right) orwithoutspatial regularity (i.e., the items were equally likely to appear in any of the three locations). Once participants learned a sequence, the computer switched to a different sequence without participants being informed that a switch had occurred. Redundant spatial regularity improved a participant's ability both to learn sequences of plays and to update their plays to reflect new computer sequences. Our results suggest that our perceptual system is sensitive to redundant spatial stimulus features and that this information can improve learning and updating. [ABSTRACT FROM AUTHOR]

We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance, too, the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies [ABSTRACT FROM AUTHOR]

In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships in the context of cyclic predator-prey models with an even number of species N ≥ 8. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behavior associated with the symmetric or asymmetric evolution of the interface profiles depending on whether N/2 is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive to the internal structure of the interfaces. Although this work focuses on cyclic predator-prey models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models. [ABSTRACT FROM AUTHOR]

The simplest example of non-transitive competition is the game rock-paper-scissors (RPS), which exhibits characteristic cyclic strategy replacement: paper beats rock, which in turn beats scissors, which in turn beats paper. In addition to its familiar use in understanding human decision-making, rock-paper-scissors is also played in many biological systems. Among other reasons, this is important because it potentially provides a mechanism whereby species- or strain coexistence can occur in the face of intense competition. Kerr et al. (2002, Nature 418: 171-174) use complementary experiments and simulations to show that RPS-playing toxic, resistant, and susceptible E. coli bacteria can coexist when interactions between the strains are spatially explicit. This raises the question of whether limited interactions associated with space are sufficient to allow strain coexistence, or whether space per se is crucial. I approach this question by extending the Kerr et al. model to include different (aspatial) population network structures with the same degree distributions as corresponding spatial lattice models. I show that the coexistence that occurs for some parameter combinations when simulated bacterial strains compete on lattices is absent when they compete on random regular graphs. Further, considering small-world networks of intermediate 'quenched randomness' between lattices and random regular graphs, I show that only small deviations from pure spatial interactions are sufficient to prevent strain coexistence. These results emphasize the explicit role of space, rather than merely limited interactions, as being decisive in allowing the coexistence of toxic, resistant, and susceptible strains in this model system. [ABSTRACT FROM AUTHOR]

The rock-paper-scissors game and its generalizations with S > 3 species are well-studied models for cyclically interacting populations. Four is, however, the minimum number of species that, by allowing other interactions beyond the single, cyclic loop, breaks both the full intransitivity of the food graph and the one-predator, one-prey symmetry. Lütz et al. [J. Theor. Biol. 317, 286 (2013)] have shown the existence, on a square lattice, of two distinct phases, with either four or three coexisting species. In both phases, each agent is eventually replaced by one of its predators, but these strategy oscillations remain localized as long as the interactions are short ranged. Distant regions may be either out of phase or cycling through different food-web subloops (if any). Here we show that upon replacing a minimum fraction Q of the short-range interactions by long-range ones, there is a Hopf bifurcation, and global oscillations become stable. Surprisingly, to build such long-distance, global synchronization, the four-species coexistence phase requires fewer long-range interactions than the three-species phase, while one would naively expect the opposite to be true. Moreover, deviations from highly homogeneous conditions (x = 0 or 1) increase Qc, and the more heterogeneous is the food web, the harder the synchronization is. By further increasing Q, while the three-species phase remains stable, the four-species one has a transition to an absorbing, single-species state. The existence of a phase with global oscillations for S > 3, when the interaction graph has multiple subloops and several possible local cycles, leads to the conjecture that global oscillations are a general characteristic, even for large, realistic food webs. [ABSTRACT FROM AUTHOR]

We report laboratory experiments that use new, visually oriented software to explore the dynamics of 3×3 games with intransitive best responses. Each moment, each player is matched against the entire population, here 8 human subjects. A “heat map” offers instantaneous feedback on current profit opportunities. In the continuous slow adjustment treatment, we see distinct cycles in the population mix. The cycle amplitude, frequency and direction are consistent with the standard learning models. Cycles are more erratic and higher frequency in the instantaneous adjustment treatment. Control treatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak or no cycles. Average play is approximated fairly well by Nash equilibrium, and an alternative point prediction, “TASP” (Time Average of the Shapley Polygon), captures some regularities that Nash equilibrium misses. [ABSTRACT FROM PUBLISHER]

The rock—paper—scissors (RPS) game is a nice model to study the biodiversity in an ecosystem. However, in the previous studies only the nearest-neighbor interaction among the species was considered. In this paper, taking the long-range migration into account, the effects of the interplay between nearest-neighbor-interaction and long-range-interaction given by Levy flight with distance distribution lh (−3 ≤ h < −1) in the spatial RPS game are investigated. Taking the probability, exchange rate, and power-law exponent of Levy flight as parameters, the coexistence conditions of three species are given. The critical curves for stable coexistence of three species in the parameter space are presented. It is also found that Levy flight has interesting effects on the final spatiotemporal pattern of the system. The results reveal that the long-range-interaction given by Levy flight exhibits pronounced effects on biodiversity of the ecosystem. [ABSTRACT FROM AUTHOR]

Abstract: The Rock–Paper–Scissors (RPS) game is a widely used model system in game theory. Evolutionary game theory predicts the existence of persistent cycles in the evolutionary trajectories of the RPS game, but experimental evidence has remained to be rather weak. In this work, we performed laboratory experiments on the RPS game and analyzed the social-state evolutionary trajectories of twelve populations of players. We found strong evidence supporting the existence of persistent cycles. The mean cycling frequency was measured to be period per experimental round. Our experimental observations can be quantitatively explained by a simple non-equilibrium model, namely the discrete-time logit dynamical process with a noise parameter. Our work therefore favors the evolutionary game theory over the classical game theory for describing the dynamical behavior of the RPS game. [Copyright &y& Elsevier]

Greenbeard genes identify copies of themselves in other individuals and cause their bearer to behave nepotistically towards those individuals. Bacterial toxins (bacteriocins) exemplify the greenbeard effect because producer strains carry closely linked genes for immunity, such that toxicity is limited to nonproducer strains. Bacteriocin producers can be maintained in a dynamic polymorphism, known as rock-paper-scissors ( RPS) dynamics, with immune and susceptible strains. However, it is unclear whether and how such dynamics will be maintained in the presence of multiple toxin types (multiple beard 'colours'). Here, we analyse strain dynamics using models of recurrent patch colonization and population growth. We find that (i) polymorphism is promoted by a small number of founding lineages per patch, strong local resource competition and the occurrence of mutations; (ii) polymorphism can be static or dynamic, depending on the intensity of local interactions and the costs of toxins and immunity; (iii) the occurrence of multiple toxins can promote RPS dynamics; and (iv) strain diversity can be maintained even when toxins differ in toxicity or lineages can exhibit multitoxicity/multi-immunity. Overall, the factors that maintain simple RPS dynamics can also promote the coexistence of multiple toxin types (multiple beard colours), thus helping to explain the remarkable levels of bacteriocin diversity in nature. More generally, we contrast these results with the maintenance of marker diversity in genetic kin recognition. [ABSTRACT FROM AUTHOR]

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the "replicator" equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual "Rock-Paper-Scissors" equilibrium for parameters located in a resonant tongue in pa-rameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately. [ABSTRACT FROM AUTHOR]

We study adaptive dynamics in games where players abandon the population at a given rate and are replaced by naive players characterized by a prior distribution over the admitted strategies. We demonstrate how such a process leads macroscopically to a variant of the replicator equation, with an additional term accounting for player turnover. We study how Nash equilibria and the dynamics of the system are modified by this additional term for prototypical examples such as the rock-paper-scissors game and different classes of two-action games played between two distinct populations. We conclude by showing how player turnover can account for nontrivial departures from Nash equilibria observed in data from lowest unique bid auctions. [ABSTRACT FROM AUTHOR]

There is a large body of evidence of apparently spontaneous mimicry in humans. This phenomenon has been described as "automatic imitation" and attributed to a mirror neuron system, but there is little direct evidence that it is involuntary rather than intentional. Cook et al. supplied the first such evidence in a unique strategic game design that gave all subjects a pecuniary incentive to avoid imitation [Cook R, Bird G, Lünser G, Huck S, Heyes C (2012) Proc Biol Sci 279(1729):780—786]. Subjects played Rock-Paper-Scissors repeatedly in matches between fixed pairs, sometimes with one and sometimes with both subjects blindfolded. The frequency of draws in the blind-blind condition was at chance, but in the blind-sighted condition it was significantly higher, suggesting automatic imitation had occurred. Automatic imitation would raise novel issues concerning how strategic interactions are modeled in game theory and social science; however, inferring automatic imitation requires significant incentives to avoid it, and subjects' incentives were less than 3 US cents per 60-game match. We replaced Cook et al.'s Rock-Paper-Scissors with a Matching Pennies game, which allows far stronger incentives to avoid imitation for some subjects, with equally strong incentives to imitate for others. Our results are important in providing evidence of automatic imitation against significant incentives. That some of our subjects had incentives to imitate also enables us clearly to distinguish intentional responding from automatic imitation, and we find evidence that both occur. Thus, our results strongly confirm the occurrence of automatic imitation, and illuminate the way that automatic and intentional processes interact in a strategic context. [ABSTRACT FROM AUTHOR]

Abstract: Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species allowing other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the parameter setting the rate at which one species invades another changes, from a coexistence to a state in which one species gets extinct. Such a dependence on the invasion rates shows that the interaction graph structure alone is not enough to predict the outcome of such models. In addition, different invasion rates permit to tune the level of transitiveness, indicating that for the coexistence of all species to persist, there must be a minimum amount of intransitivity. [Copyright &y& Elsevier]

The purpose of this study is to examine prospective mathematics specialists' engagement in an instructional sequence designed to elicit and develop their understandings of random processes. The study was conducted with two different sections of a probability and statistics course for K-8 teachers. Thirty-two teachers participated. Video analyses within a collaborative course design were used to support a teaching experiment about teachers' conceptions of random processes. In particular, teachers were asked whether the outcomes ofRock-Paper-Scissors(RPS) are generated randomly or not, were presented with a definition for randomselection,and were asked to come to a conclusion about RPS. Teachers struggled to reconcile the equality of winning outcomes for each player with the potential for human interference in the process of generating outcomes. Ultimately, teachers concluded the outcomes were not generated randomly, but encountered a variety of unexpected obstacles along the way. [ABSTRACT FROM PUBLISHER]

We investigate the population dynamics in generalized rock-paper-scissors models with an arbitrary number of species Ν. We show that spiral patterns with Ν arms may develop both for odd and even N, in particular in models where a bidirectional prédation interaction of equal strength between all species is modified to include one -cyclic predator-prey rule. While the former case gives rise to an interface network with Y-type junctions obeying the scaling law L ∝ t1/2, where L is the characteristic length of the network and t is the time, the latter can lead to a population network with N-armed spiral patterns, having a roughly constant characteristic length scale. We explicitly demonstrate the connection between interface junctions and spiral patterns in these models and compute the corresponding scaling laws. This work significantly extends the results of previous studies of population dynamics and could have profound implications for the understanding of biological complexity in systems with a large number of species. [ABSTRACT FROM AUTHOR]

We introduce a family of rock-paper-scissors-type models with ZN symmetry (N is the number of species), and we show that it has a very rich structure with many completely different phases. We study realizations that lead to the formation of domains, where individuals of one or more species coexist, separated by interfaces whose (average) dynamics is curvature driven. This type of behavior, which might be relevant for the development of biological complexity, leads to an interface network evolution and pattern formation similar to the ones of several other nonlinear systems in condensed matter and cosmology. [ABSTRACT FROM AUTHOR]

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies. [ABSTRACT FROM AUTHOR]

The rock-paper-scissors game is a model example of the ongoing cyclic turnover typical of many ecosystems, ranging from the terrestrial and aquatic to the microbial. Here we explore the evolution of a rock-paper-scissors system where three species compete for space. The species are allowed to mutate and change the speed by which they invade one another. In the case when all species have similar mutation rates, we observe a perpetual arms race where no single species prevails. When only two species mutate, their aggressions increase indefinitely until the ecosystem collapses and only the nonmutating species survives. Finally we show that when only one species mutates, group selection removes individual predators with the fastest growth rates, causing the growth rate of the species to stabilize. We explain this group selection quantitatively. [ABSTRACT FROM AUTHOR]

A compelling body of evidence indicates that observing a task-irrelevant action makes the execution of that action more likely. However, it remains unclear whether this ‘automatic imitation’ effect is indeed automatic or whether the imitative action is voluntary. The present study tested the automaticity of automatic imitation by asking whether it occurs in a strategic context where it reduces payoffs. Participants were required to play rock–paper–scissors, with the aim of achieving as many wins as possible, while either one or both players were blindfolded. While the frequency of draws in the blind–blind condition was precisely that expected at chance, the frequency of draws in the blind–sighted condition was significantly elevated. Specifically, the execution of either a rock or scissors gesture by the blind player was predictive of an imitative response by the sighted player. That automatic imitation emerges in a context where imitation reduces payoffs accords with its ‘automatic’ description, and implies that these effects are more akin to involuntary than to voluntary actions. These data represent the first evidence of automatic imitation in a strategic context, and challenge the abstraction from physical aspects of social interaction typical in economic and game theory. [ABSTRACT FROM AUTHOR]

Abstract: We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model. [Copyright &y& Elsevier]

We investigate the impact of cyclic competition on pattern formation in the rock-paper-scissors game. By separately considering random and prepared initial conditions, we observe a critical influence of the competition rate p on the stability of spiral waves and on the emergence of biodiversity. In particular, while increasing values of p promote biodiversity, they may act detrimentally on spatial pattern formation. For random initial conditions, we observe a phase transition from biodiversity to an absorbing phase, whereby the critical value of mobility grows linearly with increasing values of p on a log-log scale but then saturates as p becomes large. For prepared initial conditions, we observe the formation of single-armed spirals, but only for values of p that are below a critical value. Once above that value, the spirals break up and form disordered spatial structures, mainly because of the percolation of vacant sites. Thus there exists a critical value of the competition rates pc for stable single-armed spirals in finite populations. Importantly though, pc increases with increasing system size because noise reinforces the disintegration of ordered patterns. In addition, we also find that pc increases with the mobility. These phenomena are reproduced by a deterministic model that is based on nonlinear partial differential equations. Our findings indicate that competition is vital for the sustenance of biodiversity and the emergence of pattern formation in ecosystems governed by cyclical interactions. [ABSTRACT FROM AUTHOR]

Abstract: Understanding the emergence of cooperation is a central issue in evolutionary game theory. The hardest setup for the attainment of cooperation in a population of individuals is the Public Goods game in which cooperative agents generate a common good at their own expenses, while defectors “free-ride” this good. Eventually this causes the exhaustion of the good, a situation which is bad for everybody. Previous results have shown that introducing reputation, allowing for volunteer participation, punishing defectors, rewarding cooperators or structuring agents, can enhance cooperation. Here we present a model which shows how the introduction of rare, malicious agents – that we term jokers – performing just destructive actions on the other agents induce bursts of cooperation. The appearance of jokers promotes a rock-paper-scissors dynamics, where jokers outbeat defectors and cooperators outperform jokers, which are subsequently invaded by defectors. Thus, paradoxically, the existence of destructive agents acting indiscriminately promotes cooperation. [Copyright &y& Elsevier]

The Rock-Scissors-Paper is a game in which two players choose one out of three possible options: rock, scissors or paper. Rock breaks scissors, scissors cuts paper, and paper covers a rock. We use this game model to motive our students, with heterogeneous backgrounds, to obtain a basic knowledge of Complex Networks and Ecology. In particular, we analyze the behavior of the rock-scissors-paper model, a basic cyclic antagonist ecological model, played at different network topologies: random, scale-free and entangled. One of our main goals is that students scrutinize the topological properties determining optimal persistence of Biodiversity. In particular, our students have studied the extinction time of the 3 species when the size of the complex networks change, keeping the same properties in the nodes of the network. They have performed the simulations on networks with the same average degree = 4, and with an initial random distribution of the species (rock, scissor, paper). Substantially different behaviors can be found playing in the different topologies. We want the students to understand the influence of the network topology in the outcome of the game; and more general the relevance of the underlying network topology in Game Theory. [ABSTRACT FROM AUTHOR]

The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool punishers as the three competing strategies. Fines and costs of pool punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects a payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single- and two-strategy stationary states, a rock-paper-scissors type of cyclic dominance can emerge in strikingly different ways. [ABSTRACT FROM AUTHOR]

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system. [ABSTRACT FROM AUTHOR]

Stereotyped behaviors should be inhibited under some circumstances in order to encourage appropriate behavior. Psychiatrists have used the modified rock-paper-scissors (RPS) task to examine the inhibition of stereotyped behavior. When subjects are required to lose in response to a gesture, it is difficult for them to lose, and they have a tendency to win involuntarily. It is thought that the win response is the stereotyped response in the RPS task, and the difficulty in making positive attempts to lose is due to the requirement for inhibition of the stereotyped response. In this study, we investigated the brain regions related to inhibition of the stereotyped response using functional magnetic resonance imaging (fMRI). Subjects were assigned to one of two groups: the “win group” or the “lose group.” The lose group showed higher activation of the left dorsolateral prefrontal cortex (DLFPC) when compared to the win group. We also delivered transcranial magnetic stimulation (TMS) while the subjects performed the modified RPS task to investigate whether the left DLPFC (middle frontal gyrus, Brodmann area, BA 9) was directly involved in the inhibition of the stereotyped response. When TMS was delivered before onset of the visual stimulus, the subjects displayed increased response errors. In particular, the subjects had a tendency to win erroneously in a lose condition even though they were required to lose. These results indicate involvement of the left DLPFC in inhibition of the stereotyped responses, which suggests that this region is associated with inhibition of the preparatory setting for stereotyped responses rather than inhibition of ongoing processing to produce a stereotyped response. [ABSTRACT FROM AUTHOR]

Abstract: We study the oscillatory dynamics in the generic three-species rock–paper–scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude. [Copyright &y& Elsevier]

Processes occurring within small areas (patch-scale) that influence species richness and spatial heterogeneity of larger areas (landscape-scale) have long been an interest of ecologists. This research focused on the role of patch-scale deterministic chaos arising in phytoplankton assemblages characteristic of “Rock-Paper-Scissors” population dynamics (i.e., competitively non-hierarchical). We employed a simple 2-patch model configuration with lateral mixing and through-flow, and tested the robustness of species richness at the scale of the landscape and spatial heterogeneity. Three different assemblages were used that in a dimensionless box model configuration exhibited chaotic behavior. Our results showed that when a spatial dimension was added to the model configuration, and when all species were shared between patches (i.e., no invading populations), chaos-induced species richness and spatial heterogeneity were quickly reduced with the onset of mixing. While assemblages in each patch were comprised of exactly the same species, they differed in their proportional population densities due to differing stages of succession and the incidence of alternative assemblage structures. Even at very low mixing rates (0.001d−1), which produced low passive migration rates (0.1% of the total biomass per day), the incidence of high richness and heterogeneity decreased by ∼80%. Interestingly, this sensitivity was not the same for the three assemblages tested. Declines in species richness and spatial heterogeneity associated with mixing were greater in assemblages comprised of competitively dissimilar species (based on the area occupied in the resource-tradeoff space defined by the R* model). The underlying mechanisms may involve the degree to which nutrient dynamics are altered with the arrival of immigrants. Our findings suggest that in partially to well-mixed aquatic systems, the roles of patch-scale non-hierarchical competition and chaos as factors maintaining species richness and spatial heterogeneity may be limited. However, in aquatic systems that experience periods of very low mixing, or even disconnection, non-hierarchical competition and chaos might indeed contribute significantly to biodiversity. [Copyright &y& Elsevier]

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.