Rock-paper-scissors Research Articles

The elimination of hierarchy in a completely cyclic competition system

Interactions among competing units are crucial to maintaining biodiversity, and non-hierarchical interactions can promote biodiversity in cyclic competing systems. In the present study, we explore the role of hierarchical interactions, existing ubiquitously in reality, in the co-evolution of a cyclic competing system. In systems composed of cyclic competing species with hierarchy interactions in which one predator species has more than one prey, we find that hierarchy disappears in a rather short evolving time. In the process of co-evolution, a hierarchical competing system tends to transit to a cyclic non-hierarchical competing system described by the rock–paper–scissors game. In other words, the cyclic competing interactions appear to eradicate hierarchy. This conclusion is analyzed by a mean-field approach and is tested by stochastic simulations.

Replicator dynamics in public goods games with reward funds

Which punishment or rewards are most effective at maintaining cooperation in public goods interactions and deterring defectors who are willing to freeload on others' contribution? The sanction system is itself a public good and can cause problematic “second-order free riders” who do not contribute to the provisions of the sanctions and thus may subvert the cooperation supported by sanctioning. Recent studies have shown that public goods games with punishment can lead to a coercion-based regime if participation in the game is optional. Here, we reveal that even with compulsory participation, rewards can maintain cooperation within an infinitely large population. We consider three strategies for players in a standard public goods game: to be a cooperator or a defector in a standard public goods game, or to be a rewarder who contributes to the public good and to a fund that rewards players who contribute during the game. Cooperators do not contribute to the reward fund and are therefore classified as second-order free riders. The replicator dynamics for the three strategies exhibit a rock–scissors–paper cycle, and can be analyzed fully, despite the fact that the expected payoffs are nonlinear. The model does not require repeated interaction, spatial structure, group selection, or reputation. We also discuss a simple method for second-order sanctions, which can lead to a globally stable state where 100% of the population are rewarders.

Automatic imitation in a strategic context: players of rock–paper–scissors imitate opponents' gestures

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.

Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities

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.

Is there switching for replicator dynamics and bimatrix games?

We consider heteroclinic networks for replicator dynamics and bimatrix games, that is, in a simplex or product of simplices, with equilibria at the vertices and connections at the edges–edge networks. Switching dynamics near a heteroclinic network occurs whenever every (infinite) sequence of connections in the network is shadowed by at least one trajectory in its neighborhood. Aguiar and Castro [M.A.D. Aguiar, S.B.S.D. Castro Chaotic switching in a two-person game, Physica D 239 (16), 1598–1609] prove switching near an edge network for the dynamics of the rock–scissors–paper game. Here we give conditions for switching dynamics in general bimatrix games and show that switching near an edge network can never occur for replicator dynamics.

From genes to games: Cooperation and cyclic dominance in meiotic drive

Evolutionary change can be described on a genotypic level or a phenotypic level. Evolutionary game theory is typically thought of as a phenotypic approach, although it is frequently argued that it can also be used to describe population genetic evolution. Interpreting the interaction between alleles in a diploid genome as a two player game leads to interesting alternative perspectives on genetic evolution. Here we focus on the case of meiotic drive and illustrate how meiotic drive can be directly and precisely interpreted as a social dilemma, such as the prisoners dilemma or the snowdrift game, in which the drive allele takes more than its fair share. Resistance to meiotic drive can lead to the well understood cyclic dominance found in the rock–paper–scissors game. This perspective is well established for the replicator dynamics, but there is still considerable ground for mutual inspiration between the two fields. For example, evolutionary game theorists can benefit from considering the stochastic evolutionary dynamics arising from finite population size. Population geneticists can benefit from game theoretic tools and perspectives on genetic evolution.

Rock Paper and Scissors Image Identification Scheme by using an Angular Criteria Technique

The In this paper, several image processing techniques are used to achieve identifying hand signals of Rock, Paper, and Scissors. We extract the hand images from the video sequences. The skin color, and a generalized threshold value is used to map the hand image into a binary format. However, several kinds of Scissors states are difficult to judge correctly. Therefore, we use the distance between the two finger tips to improve or even get rid of these obvious error states. The segmentation, histogram, and angular criteria methods are then used to distinguish the exactly Scissors hand signal. It is the basic rule that the extended line of the hand (segment CD) shown in Fig. 8 must fall within the included angle (segment AD and BD). Finally, the scheme calculates the conditions of AD CD BD m m m   and BD CD AD m m m   , which are used to judge if the Scissors is correct or not. Simulated results show that our scheme is an effective method to identify the hand signals of Rock, Paper, and Scissors.

Cyclic dominance in adaptive networks

The Rock-Paper-Scissors (RPS) game is a paradigmatic model for cyclic dominance in biological systems. Here we consider this game in the social context of competition between opinions in a networked society. In our model, every agent has an opinion which is drawn from the three choices: rock, paper or scissors. In every timestep a link is selected randomly and the game is played between the nodes connected by the link. The loser either adopts the opinion of the winner or rewires the link. These rules define an adaptive network on which the agents’ opinions coevolve with the network topology of social contacts. We show analytically and numerically that nonequilibrium phase transitions occur as a function of the rewiring strength. The transitions separate four distinct phases which differ in the observed dynamics of opinions and topology. In particular, there is one phase where the population settles to an arbitrary consensus opinion. We present a detailed analysis of the corresponding transitions revealing an apparently paradoxical behavior. The system approaches consensus states where they are unstable, whereas other dynamics prevail when the consensus states are stable.

Laboratory RPS Game Experimental Evaluations of the Evolutionary Dynamics Theories

The evolution equation determines how frequencies of strategies in the population change (Traulsen & Hauert (2009)), where the change indicates velocities (Bowles (2004),Weibull (1997),Sigmund (2010)). In this paper, based on the laboratory Rock-Paper-Scissors (RPS) game experiments and from the velocity vector fields perspective, we empirically study the five most popular evolutionary dynamics theories (Sandholm (2009)), i.e., replicator(Taylor & Jonker (1978)), logit (Fudenberg & Levine (1998)), Brown-von Neumann-Nash (Brown & Von Neumann (1950)) (BNN), best responde (Gilboa & Matsui (1991)) and Smith (Smith (1984)), respectively, which describe different theoretical patterns of the velocity vector fields in strategy space. We conduct a standard Rock-Paper-Scissors (RPS) game in an experimental economics laboratory. The experiments include 12 independent sessions and each session involves 6 human subjects. Each session consists of 300 rounds of the game repeatedly with a random matching for each round. Since we have 6 human subjects in the RPS game experiments, we have 28 possible states. With the data collected from the experiments, we measure the velocity vector at all of the 28 measurable states x(i; j;k) in the strategy space of the game. Three metrics are employed to measure the difference between the experimental result and each specific mode. Main result and conclusion: This is the first time in the literature of experimental economics to reveal the global dynamic patterns in the strategy space using laboratory RPS game study. This is also the first time in the literature to apply the velocity vector fields to study the comparison between the experimental data and the classic evolutionary dynamics theories, which for the first time to show that none of the classic theories completely confirms the experimental data.

Evolutionary dynamics of a system with periodic coefficients

We investigate a problem in evolutionary game theory based on replicator equations with periodic coefficients. This approach to evolution combines classical game theory with differential equations. The RPS (Rock-Paper-Scissors) system studied has application to the population biology of lizards and to bacterial dynamics. The presence of periodic coefficients models variations in the environment due to seasonal effects and results in parametric excitation which is studied through the use of perturbation series and numerical integration.

2P2-L13 Wired to Win : Behavior effect of fast Robot on human during Rock Paper Scissor Game(Neurorobotics & Cognitive Robotics)

2P2-L13 Wired to Win : Behavior effect of fast Robot on human during Rock Paper Scissor Game(Neurorobotics & Cognitive Robotics)

P26-23 An fMRI study of the cortex related to the movements of toes in SCI patients during performance of loss resulting pursuant paper rock scissors

P26-23 An fMRI study of the cortex related to the movements of toes in SCI patients during performance of loss resulting pursuant paper rock scissors

Testing the TASP: An experimental investigation of learning in games with unstable equilibria

Abstract We report experiments studying mixed strategy Nash equilibria that are theoretically stable or unstable under learning. The Time Average Shapley Polygon (TASP) predicts behavior in the unstable case. We study two versions of Rock–Paper–Scissors that include a fourth strategy, Dumb. The unique Nash equilibrium is identical in the two games, but the predicted frequency of Dumb is much higher in the game where the NE is stable. Consistent with TASP, the observed frequency of Dumb is lower and play is further from Nash in the high payoff unstable treatment. However, Dumb is played too frequently in all treatments.

20. Task difficulty dependency of prefrontal activation in four conditions of the Rock–Paper–Scissors task revealed using near-infrared spectroscopy

20. Task difficulty dependency of prefrontal activation in four conditions of the Rock–Paper–Scissors task revealed using near-infrared spectroscopy

Noisy non-transitive quantum games

We study the effect of quantum noise in 3 × 3 entangled quantum games. By taking into account different noisy quantum channels, we analyze how a two-player, three-strategy Rock–Scissor–Paper game is influenced by the quantum noise. We consider the winning non-transitive strategies R, S and P such that R beats S, S beats P and P beats R. The game behaves as a noiseless game for the maximum value of the quantum noise. It is seen that Alice's payoff is heavily influenced by the depolarizing noise as compared to the amplitude damping noise. A depolarizing channel causes a monotonic decrease in players' payoffs as we increase the amount of quantum noise. In the case of the amplitude damping channel, Alice's payoff function reaches its minimum for α = 0.5 and is symmetrical. This means that larger values of quantum noise influence the game weakly. On the other hand, the phase damping channel does not influence the game. Furthermore, the Nash equilibrium and non-transitive character of the game are not affected under the influence of quantum noise.

The role of the dorsolateral prefrontal cortex in the inhibition of stereotyped responses

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.

Chaotic switching in a two-person game

We study robust long-term complex behaviour in the Rock–Scissors–Paper game with two players, played using reinforcement learning. The complex behaviour is connected to the existence of a heteroclinic network for the dynamics. This network is made of three heteroclinic cycles consisting of nine equilibria and the trajectories connecting them. We provide analytical proof both for the existence of chaotic switching near the heteroclinic network and for the relative asymptotic stability of at least one cycle in the network, leading to some trajectories paying alternate visits to only two nodes while others follow a complicated sequence of visits to all nodes in the network. Our results are obtained by making use of the symmetry of the original problem, a new approach in the context of learning, and they provide an explanation of numerical results previously obtained by other authors.

Effect of swarming on biodiversity in non-symmetric rock–paper–scissor game

Cyclic dominance of species is a potential mechanism for maintaining biodiversity. The author investigates the generalised scenario when the cyclic dominance of three or more interacting species is described by a non-symmetric matrix game that has multiple Nash equilibria. Modified Lotka-Volterra equations are proposed to incorporate the effects of swarming, and the condition for biodiversity is derived. The species are modelled using replicator equations, where each member of the species is assigned a fitness value. The authors show, for the first time, that the 'swarming effect' has an important role to play in the maintenance of biodiversity. The authors have also discovered the existence of a critical value of the swarming parameter for a given mobility, above which there is a high probability of existence of biodiversity.

Outcome and view of the player modulate motor area activity during observation of a competitive game

In a previous study, we demonstrated that the mirror neuron system is involved in watching competitive games performed by multiple players [Shimada, S., & Abe, R. (2009). Modulation of the motor area activity during observation of a competitive game. NeuroReport, 20, 979–983]. In the present study, we investigated whether mirror neuron system activity was modulated by the outcome of a competitive game as well as by the spatial relationship between the player and the observer. We measured the activity of motor areas using near-infrared spectroscopy when a subject watched a two-player competitive game, rock–paper–scissors (RPS). Some parts of the motor area that was activated while performing the RPS action were also activated while observing the RPS game. Interestingly, the magnitude of this activity (during observation) was modulated by the outcome and the view of the particular player whom the subject was supporting. In the dorsal anterior region of the motor area measured, the activity was significantly greater when the supported player won rather than when he lost, regardless of the view. It was also greater when the player was presented with an incongruent view than with a congruent view, regardless of the outcome. In contrast, the dorsal posterior region of the motor area was activated only when the supported player won and was observed with a congruent view. We suggest that mirror neuron system activity during observation of a competitive game reflects the subject's internal assimilation of the actions of the observed player.

Sociobiological Control of Plasmid Copy Number in Bacteria

All genes critical for plasmid replication regulation are located on the plasmid rather than on the host chromosome. It is possible therefore that there can be copy-up “cheater” mutants. In spite of this possibility, low copy number plasmids appear to exist stably in host populations. We examined this paradox using a multilevel selection model. Simulations showed that, a slightly higher copy number mutant could out-compete the wild type. Consequently, another mutant with still higher copy number could invade the first invader. However, the realized benefit of increasing intra-host fitness was saturating whereas that of inter-host fitness was exponential. As a result, above a threshold, intra-host selection was overcompensated by inter-host selection and the low copy number wild type plasmid could back invade a very high copy number plasmid. This led to a rock-paper-scissor (RPS) like situation that allowed the coexistence of plasmids with varied copy numbers. Furthermore, another type of cheater that had lost the genes required for conjugation but could hitchhike on a conjugal plasmid, could further reduce the advantage of copy-up mutants. These sociobiological interactions may compliment molecular mechanisms of replication regulation in stabilizing the copy numbers.