Rock-paper-scissors Research Articles

Measuring the Non-Transitivity in Chess

In this paper, we quantify the non-transitivity in chess using human game data. Specifically, we perform non-transitivity quantification in two ways—Nash clustering and counting the number of rock–paper–scissor cycles—on over one billion matches from the Lichess and FICS databases. Our findings indicate that the strategy space of real-world chess strategies has a spinning top geometry and that there exists a strong connection between the degree of non-transitivity and the progression of a chess player’s rating. Particularly, high degrees of non-transitivity tend to prevent human players from making progress in their Elo ratings. We also investigate the implications of non-transitivity for population-based training methods. By considering fixed-memory fictitious play as a proxy, we conclude that maintaining large and diverse populations of strategies is imperative to training effective AI agents for solving chess.

Generalized Hamiltonian dynamics and chaos in evolutionary games on networks

We study the network replicator equation and characterize its fixed points on arbitrary graph structures for 2 × 2 symmetric games. We show a relationship between the asymptotic behavior of the network replicator and the existence of an independent vertex set in the graph and also show that complex behavior cannot emerge in 2 × 2 games. This links a property of the dynamical system with a combinatorial graph property. We contrast this by showing that ordinary rock–paper–scissors (RPS) exhibits chaos on the 3-cycle and that on general graphs with ≥3 vertices the network replicator with RPS is a generalized Hamiltonian system. This stands in stark contrast to the established fact that RPS does not exhibit chaos in the standard replicator dynamics or the bimatrix replicator dynamics, which is equivalent to the network replicator on a graph with one edge and two vertices (K2).

Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link.

The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.

Stability of cycling behaviour near a heteroclinic network model of Rock–Paper–Scissors–Lizard–Spock

The well-known game of Rock–Paper–Scissors can be used as a simple model of competition between three species. When modelled in continuous time using differential equations, the resulting system contains a heteroclinic cycle between the three equilibrium solutions representing the existence of only a single species. The game can be extended in a symmetric fashion by the addition of two further strategies (‘Lizard’ and ‘Spock’): now each strategy is dominant over two of the remaining four strategies, and is dominated by the remaining two. The differential equation model contains a set of coupled heteroclinic cycles forming a heteroclinic network. In this paper we carefully consider the dynamics near this heteroclinic network. We develop a technique to use a previously defined definition of stability (known as fragmentary asymptotic stability) in numerical continuation software. We are able to identify regions of parameter space in which arbitrarily long periodic sequences of visits are made to the neighbourhoods of the equilibria, which form a complicated pattern in parameter space.

Disabled throwing shoulder/elbow players have high rates of impaired foot function

ABSTRACT The prevalence of impaired foot function and floating toes of 91 baseball players with disabled throwing shoulder or elbow were retrospectively investigated. Foot function was evaluated by foot “rock paper scissors” and floating toes were confirmed if none of the toes made contact with the mat in standing posture. The prevalence of impaired foot function and floating toes and the relationship between between them were determined. Abnormal foot function was observed in 83 players (91%); of those, 73 players (88%) also had floating toes, and the prevalence was statistically significant compared to those without it (12%) (P < 0.001). Floating toes were observed in 74 players (81%); of those, 73 players (99%) had impaired foot function, and the prevalence was also statistically significant compared to those without them (59%) (P < 0.001). Baseball players with disabled throwing shoulder or elbow have high rates of impaired foot function and floating toes.

The effect of territorial awareness in a three-species cyclic predator\u2013prey model

Recognizing territories is essential to decide behavior of population either human or animals, and interaction between groups or individuals according to the territorial awareness is universal. Understanding various mechanisms which affect on such species behaviors can be possible by evolutionary games, and in particular, the rock–paper–scissors (RPS) game has been played a key role as a paradigmatic model to explore biodiversity from microbiota to societies. Among paramount mechanisms in systems of RPS, the role of intraspecific interaction has been recently noted in terms of promoting biodiversity. Since intraspecific interaction is defined by an invasive reaction between individuals in the same group, the interaction may be also sensitive to the territorial awareness. To explore how territorial awareness-based intraspecific interaction can affect species biodiversity, we endow species with the mechanism in the classic RPS game. By means of the Monte-Carlo method, we find the phenomenon that the presence of species’ territorial awareness has an impact on intraspecific interaction which ultimately affects species biodiversity. At the same time, we also find that territorial awareness can play a significant role to the average waiting time for extinction which is numerically elucidated by exploiting the quantity: interface width statistic. Unlike prior research that concentrated solely on the relationship between interaction frequency and species diversity, our results shed lights on the important role of territorial awareness in models of RPS, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such awareness behavior.

Effect of mobility in the rock-paper-scissor dynamics with high mortality.

In the evolutionary dynamics of a rock-paper-scissor model, the effect of natural death plays a major role in determining the fate of the system. Coexistence, being an unstable fixed point of the model, becomes very sensitive toward this parameter. In order to study the effect of mobility in such a system which has explicit dependence on mortality, we perform Monte Carlo simulation on a two-dimensional lattice having three cyclically competing species. The spatiotemporal dynamics has been studied along with the two-site correlation function. Spatial distribution exhibits emergence of spiral patterns in the presence of mobility. It reveals that the joint effect of death rate and mobility (diffusion) leads to new coexistence and extinction scenarios.

Species coexistence in spatial cyclic game of five species

The evolutionary game of cyclic competition has been extensively studied to gain insight into one of the most fundamental phenomena in nature: biodiversity. The three-species Rock-Paper-Scissors (RPS) game and its extension Rock-Paper-Scissors-Lizard-Spock (RPSLS) game are typical models in this field. However, in real ecosystems, species richness can be easily influenced by the interaction structure among species. In this paper, we investigate species coexistence in modified RPSLS games by removing certain predator-prey interactions. We find that the interaction structure impacts the evolutionary dynamics and different interaction structures allow for different states of multi-species coexistence. We also find that the competition between different three-species-cyclic interactions is crucial for the realization of different asymptotic behaviors at low mobility. Our findings may be useful to understand the subtle effects of competitive structure on species coexistence and evolutionary game outcomes.

Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs

Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs

Transposable elements promote the evolution of genome streamlining.

Eukaryotes and prokaryotes have distinct genome architectures, with marked differences in genome size, the ratio of coding/non-coding DNA, and the abundance of transposable elements (TEs). As TEs replicate independently of their hosts, the proliferation of TEs is thought to have driven genome expansion in eukaryotes. However, prokaryotes also have TEs in intergenic spaces, so why do prokaryotes have small, streamlined genomes? Using an in silico model describing the genomes of single-celled asexual organisms that coevolve with TEs, we show that TEs acquired from the environment by horizontal gene transfer can promote the evolution of genome streamlining. The process depends on local interactions and is underpinned by rock–paper–scissors dynamics in which populations of cells with streamlined genomes beat TEs, which beat non-streamlined genomes, which beat streamlined genomes, in continuous and repeating cycles. Streamlining is maladaptive to individual cells, but improves lineage viability by hindering the proliferation of TEs. Streamlining does not evolve in sexually reproducing populations because recombination partially frees TEs from the deleterious effects they cause.This article is part of the theme issue ‘The secret lives of microbial mobile genetic elements’.

Introduction: the secret lives of microbial mobile genetic elements.

Introduction: the secret lives of microbial mobile genetic elements.

The interplay of rock-paper-scissors competition and environments mediates species coexistence and intriguing dynamics

Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities.

Multi-layered model for rock-paper-scissors game: A swarm intelligence sustains biodiversity

Much literature exists for rock-paper-scissors (RPS) game. It is well known in a single habitat that spatial models are more stable than well-mixed population. However, the mechanism of spatial stabilization has been unclear. In the present article, we study multilayer model of RPS game in a patchy environment; respective patches are set as lattices. The predation probability of one of three species is assumed to be reduced as a parameter. Simulations on two-layer system reveal the opposite result to those on a single-habitat system. Namely, the well-mixed population becomes stable. In contrast, extinction occurs by random migration in spatially-explicit model, when the reduced predation probability is below a certain critical value. The extinction in spatial model is found to be caused by the collapse of a kind of swarm intelligence (SI). In RPS system, large swarms of each species are automatically protected from its predator. Such a self-organized protection is one of the simplest examples of SI. We also discuss some serious problems for random migration.

Influence of the neighborhood on cyclic models of biodiversity

This work deals with the influence of the neighborhood in simple rock–paper–scissors models of biodiversity. We consider the case of three distinct species which evolve under the standard rules of mobility, reproduction and competition. The rule of competition follows the guidance of the rock–paper–scissors game, with the prey being annihilated, leaving an empty site in accordance with the May-Leonard proposal for the predator and prey competition. We use the von Neumann neighborhood, but we consider mobility under the presence of the nearest, next nearest and next to next nearest neighbors in three distinct environments, one with equal probability and the others with probability following power law and exponential profiles. The results are different, but they all show that increasing the neighborhood increases the characteristic length of the system in an important way. We have studied other models, in particular the case where one modifies the manner a specific species competes, breaking the cyclic evolution and unveiling the interesting result in which the strongest individuals may constitute the less abundant population.

Formalizing Opponent Modeling with the Rock, Paper, Scissors Game

In simple dyadic games such as rock, paper, scissors (RPS), people exhibit peculiar sequential dependencies across repeated interactions with a stable opponent. These regularities seem to arise from a mutually adversarial process of trying to outwit their opponent. What underlies this process, and what are its limits? Here, we offer a novel framework for formally describing and quantifying human adversarial reasoning in the rock, paper, scissors game. We first show that this framework enables a precise characterization of the complexity of patterned behaviors that people exhibit themselves, and appear to exploit in others. This combination allows for a quantitative understanding of human opponent modeling abilities. We apply these tools to an experiment in which people played 300 rounds of RPS in stable dyads. We find that although people exhibit very complex move dependencies, they cannot exploit these dependencies in their opponents, indicating a fundamental limitation in people’s capacity for adversarial reasoning. Taken together, the results presented here show how the rock, paper, scissors game allows for precise formalization of human adaptive reasoning abilities.

Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance

AbstractHeteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game.

Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model

We study a rock–paper–scissors model for competing populations that exhibits travelling waves in one spatial dimension and spiral waves in two spatial dimensions. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. The model also has travelling fronts that are heteroclinic connections between two equilibria in a moving frame of reference, but these fronts are unstable. However, we find that large-wavelength travelling waves can be stable in spite of being made up of three of these unstable travelling fronts. In this paper, we focus on determining the essential spectrum (and hence, stability) of large-wavelength travelling waves in a cyclic competition model with one spatial dimension. We compute the curve of transition from stability to instability with the continuation scheme developed by Rademacher et al (2007 Physica D 229 166–83). We build on this scheme and develop a method for computing what we call belts of instability, which are indicators of the growth rate of unstable travelling waves. Our results from the stability analysis are verified by direct simulation for travelling waves as well as associated spiral waves. We also show how the computed growth rates accurately quantify the instabilities of the travelling waves.

Rock-Paper-Scissors Play: Beyond the Win-Stay/Lose-Change Strategy

This research studied the strategies that players use in sequential adversarial games. We took the Rock-Paper-Scissors (RPS) game as an example and ran players in two experiments. The first experiment involved two humans, who played the RPS together for 100 times. Importantly, our payoff design in the RPS allowed us to differentiate between participants who used a random strategy from those who used a Nash strategy. We found that participants did not play in agreement with the Nash strategy, but rather, their behavior was closer to random. Moreover, the analyses of the participants’ sequential actions indicated heterogeneous cycle-based behaviors: some participants’ actions were independent of their past outcomes, some followed a well-known win-stay/lose-change strategy, and others exhibited the win-change/lose-stay behavior. To understand the sequential patterns of outcome-dependent actions, we designed probabilistic computer algorithms involving specific change actions (i.e., to downgrade or upgrade according to the immediate past outcome): the Win-Downgrade/Lose-Stay (WDLS) or Win-Stay/Lose-Upgrade (WSLU) strategies. Experiment 2 used these strategies against a human player. Our findings show that participants followed a win-stay strategy against the WDLS algorithm and a lose-change strategy against the WSLU algorithm, while they had difficulty in using an upgrade/downgrade direction, suggesting humans’ limited ability to detect and counter the actions of the algorithm. Taken together, our two experiments showed a large diversity of sequential strategies, where the win-stay/lose-change strategy did not describe the majority of human players’ dynamic behaviors in this adversarial situation.

FNIRS Signal Classification Based on Deep Learning in Rock-Paper-Scissors Imagery Task

To explore whether the brain contains pattern differences in the rock–paper–scissors (RPS) imagery task, this paper attempts to classify this task using fNIRS and deep learning. In this study, we designed an RPS task with a total duration of 25 min and 40 s, and recruited 22 volunteers for the experiment. We used the fNIRS acquisition device (FOIRE-3000) to record the cerebral neural activities of these participants in the RPS task. The time series classification (TSC) algorithm was introduced into the time-domain fNIRS signal classification. Experiments show that CNN-based TSC methods can achieve 97% accuracy in RPS classification. CNN-based TSC method is suitable for the classification of fNIRS signals in RPS motor imagery tasks, and may find new application directions for the development of brain–computer interfaces (BCI).

Weak species in rock-paper-scissors models

In this letter, we investigate the population dynamics in a May-Leonard formulation of the rock-paper-scissors (RPS) game in which one or two species, which we shall refer to as “weak”, have a reduced predation or reproduction probability. We show that in a non-spatial model the stationary solution where all three species coexist is always unstable, while in a spatial stochastic model coexistence is possible for a wide parameter space. We find that a reduced predation probability results in a significantly higher abundance of “weak” species, in models with either one or two “weak” species, as long as the simulation lattices are sufficiently large for coexistence to prevail. On the other hand, we show that a reduced reproduction probability has a smaller impact on the abundance of “weak” species, generally leading to a slight decrease of its population size —the increase of the population size of one of the “weak” species being more than compensated by the reduction of the other, in the two-species case. We further show that the species abundances in models where both predation and reproduction probabilities are simultaneously reduced may be accurately estimated from the results obtained considering only a reduction of either the predation or the reproduction probability.