Rock-paper-scissors Research Articles

The Level Sets of Typical Games

W hat is a noncooperative game? You probably know the Rock-PaperScissors (RPS) game. Let’s play: ready, set, go! What did you choose? I chose rock. If you chose scissors, then I win, because rock crushes scissors. If you chose paper, then you win, because paper covers rock. If you also chose rock, then it’s a draw. This is an example of a two-player game in which each player has the same three pure strategies: rock, paper, and scissors. In order for this game to have any significance, we ought to define a payoff function. For example, we could say the winner receives $1, which the loser must pay to the winner, so the loser’s payoff is −$1. If it’s a draw, then neither of us receives anything, so the payoffs are both 0. This is an example of what we’ll call a discrete game. A more general notion is a continuous game, also called a mixed game, which we will simply call a game. Rather than just thinking about playing the game once, we think of the game being repeated an arbitrary or possibly infinite number of times. Instead of deciding upon one of rock, paper, or scissors, we decide upon a probability distribution which is a list of three numbers corresponding to the probabilities of drawing rock, paper, or scissors. The sum of these three numbers is 1, because we assume that we must draw something. One example is (1/3,1/3,1/3), which means the probabilities of drawing rock, paper, or scissors are equal to 1/3. If you only want to draw rock, then your probability distribution would be (1,0,0). We can use these to compute our expected payoffs; these are known as expected values in probability theory. The expected payoff is the sum of the probabilities of each possible outcome multiplied with the payoff

Cumulative frequency-dependent selective episodes allow for rapid morph cycles and rock-paper-scissors dynamics in species with overlapping generations.

Rock-paper-scissors (RPS) dynamics, which maintain genetic polymorphisms over time through negative frequency-dependent (FD) selection, can evolve in short-lived species with no generational overlap, where they produce rapid morph frequency cycles. However, most species have overlapping generations and thus, rapid RPS dynamics are thought to require stronger FD selection, the existence of which yet needs to be proved. Here, we experimentally demonstrate that two cumulative selective episodes, FD sexual selection reinforced by FD selection on offspring survival, generate sufficiently strong selection to generate rapid morph frequency cycles in the European common lizard Zootoca vivipara, a multi-annual species with major generational overlap. These findings show that the conditions required for the evolution of RPS games are fulfilled by almost all species exhibiting genetic polymorphisms and suggest that RPS games may be responsible for the maintenance of genetic diversity in a wide range of species.

Social cycling and conditional responses in the Rock-Paper-Scissors game.

How humans make decisions in non-cooperative strategic interactions is a big question. For the fundamental Rock-Paper-Scissors (RPS) model game system, classic Nash equilibrium (NE) theory predicts that players randomize completely their action choices to avoid being exploited, while evolutionary game theory of bounded rationality in general predicts persistent cyclic motions, especially in finite populations. However as empirical studies have been relatively sparse, it is still a controversial issue as to which theoretical framework is more appropriate to describe decision-making of human subjects. Here we observe population-level persistent cyclic motions in a laboratory experiment of the discrete-time iterated RPS game under the traditional random pairwise-matching protocol. This collective behavior contradicts with the NE theory but is quantitatively explained, without any adjustable parameter, by a microscopic model of win-lose-tie conditional response. Theoretical calculations suggest that if all players adopt the same optimized conditional response strategy, their accumulated payoff will be much higher than the reference value of the NE mixed strategy. Our work demonstrates the feasibility of understanding human competition behaviors from the angle of non-equilibrium statistical physics.

Spatiotemporal dynamics of early cortical gesture processing

Gesture processing has been consistently shown to be associated with activation of the inferior parietal lobe (IPL); however, little is known about the integration of IPL activation into the temporal dynamics of early sensory areas. Using a temporally graded repetition suppression paradigm, we examined the activation and time course of brain areas involved in hand gesture processing. We recorded event-related potentials in response to stimulus pairs of static hand images forming gestures of the popular rock–paper–scissors game and estimated their neuronal generators. We identified two main components associated with adaptive patterns related to stimulus repetition. The N190 component elicited at temporo-parietal sites adapted to repetitions of the same gesture and was associated with right-hemispheric extrastriate body area activation. A later component at parieto-occipital sites demonstrated temporally graded adaptation effects for all gestures with a left-hemispheric dominance. Source localization revealed concurrent activations of the right extrastriate body area, fusiform gyri bilaterally, and the left IPL at about 250ms. The adaptation pattern derived from the graded repetition suppression paradigm demonstrates the functional sensitivity of these sources to gesture processing. Given the literature on IPL contribution to imitation, action recognition, and action execution, IPL activation at about 250ms may represent the access into specific cognitive routes for gesture processing and may thus be involved in integrating sensory information from cortical body areas into subsequent visuo-motor transformation processes.

Algorithmic rationality: Game theory with costly computation

We develop a general game-theoretic framework for reasoning about strategic agents performing possibly costly computation. In this framework, many traditional game-theoretic results (such as the existence of a Nash equilibrium) no longer hold. Nevertheless, we can use the framework to provide psychologically appealing explanations of observed behavior in well-studied games (such as finitely repeated prisoner's dilemma and rock–paper–scissors). Furthermore, we provide natural conditions on games sufficient to guarantee that equilibria exist.

Stochastic patterns in a 1D Rock–Paper–Scissors model with mutation

In the framework of a 1D cyclic competition model, the Rock–Paper–Scissors model, where three kinds of generic agents are allowed to mutate, to interact and to move in space, we study the formation of stochastic patterns, where all the agents do coexist. We modelled the problem using an individual-based setting and we used the system size van Kampen expansion to deal with the Master Equation. We have hence been able to characterise the spatio-temporal patterns using the power spectrum of the fluctuations. We proved that such patterns are robust against the intrinsic noise and they can be found for parameter values beyond the ones fixed by the deterministic approach (mean field approximation). We complement such analytical results with numerical simulations based on the Gillespie algorithm.

Globally synchronized oscillations in complex cyclic games.

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 etal. [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 (χ=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.

The dynamics of generalized reinforcement learning

We consider reinforcement learning in games with both positive and negative payoffs. The Cross rule is the prototypical reinforcement learning rule in games that have only positive payoffs. We extend this rule to incorporate negative payoffs to obtain the generalized reinforcement learning rule. Applying this rule to a population game, we obtain the generalized reinforcement dynamic which describes the evolution of mixed strategies in the population. We apply the dynamic to the class of Rock–Scissor–Paper (RSP) games to establish local convergence to the interior rest point in all such games, including the bad RSP game.

Population interaction structure and the coexistence of bacterial strains playing ‘rock–paper–scissors’

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.

Species diversity in rock—paper—scissors game coupling with Levy flight

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.

Multicoloured greenbeards, bacteriocin diversity and the rock‐paper‐scissors game

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.

Chaotic Time Series Prediction for Rock-Paper-Scissors using Adaptive Social Behaviour Optimization (ASBO)

Time series prediction involves analyzing a set of data from past and current occurrences in order to predict the future set of data. In dynamic systems, chaotic behaviour is intrinsically observable and the resulting chaotic time series have nonlinear characteristics. Nevertheless, such data can be optimized to make sense out of the chaos. Multiple algorithms exist to this end, which have various applications. In this paper, this phenomenon is illustrated by making use of the well-known game rock-paper-scissors(R-P-S) played between two agents, one real and one adaptive. It is possible to identify and predict patterns in the choices made by the real agent during the course of play by analyzing the sequence of chaotic data using the Adaptive Social Behaviour Optimization (ASBO) algorithm. This optimization method makes use of a self-adaptive mutation strategy which takes into consideration dynamic factors such as leadership, confidence and competition, which are all functions of time.

The effect of opponent type on human performance in a three-alternative choice task

Adult participants played computerised games of “Paper Scissors Rock”. Participants in one group were told that they were playing against the computer, and those in the other group were told that they were playing against another participant in the adjacent room. The participant who won the most games would receive a $50 prize. For both groups however, the opponent's responses (paper, scissors, or rock) were generated by the computer, and the distribution of these responses was varied across four blocks of 126 trials. Results were analysed using the generalised matching law for the three possible pairs of alternatives (paper vs. scissors, paper vs. rock, and scissors vs. rock) across all participants in each group. Overall, significantly higher estimates of sensitivity to the distribution of opponent's responses were obtained from participants who were told their opponent was a computer compared to participants who were told their opponent was another participant. While adding to the existing literature showing that the generalised matching law is an adequate descriptor of human three-alternative choice behaviour, these findings show that external factors such as perceived opponent type can affect the efficacy of reinforcer contingencies on human behaviour. This suggests that generalising the results from tasks performed against a computer to real-life human-to-human interactions warrants some caution.

Evolution of repeated prisoner's dilemma play under logit dynamics

In an evolutionary set-up, we append an ecology of iterated prisoner's dilemma (IPD) game strategies, consisting of unconditional cooperators (AllC), unconditional defectors (AllD) and reactive players (TFT) with two repeated strategies that have received less attention in the evolutionary IPD game literature: the error-proof, “generous” tit-for-tat (GTFT) which, with a certain probability, re-establishes cooperation after a (possibly by mistake) defection of the opponent and the penitent, “stimulus–response” (WSLS) strategy that resets cooperation after the opponent punished for defection. An abundance of rock–paper–scissors like patterns is discovered in the 3×3 ecologies comprising Pavlovian and “generous” players. Interestingly, the evolutionary success of Pavlov seems to depend on the absence of unconditional cooperators in the ecologies investigated.

Cycles with undistinguished actions and extended Rock–Paper–Scissors games

This paper examines zero-sum games that are based on a cyclic preference relation defined over undistinguished actions. For each of these games, the set of Nash equilibria is characterized. When the number of actions is odd, a unique Nash equilibrium is always obtained. On the other hand, in the case of an even number of actions, every such game exhibits an infinite number of Nash equilibria. Our results give some insights as to the robustness of Nash equilibria with respect to perturbations of the action set.

Cycle frequency in standard Rock–Paper–Scissors games: Evidence from experimental economics

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 N=6 players. We found strong evidence supporting the existence of persistent cycles. The mean cycling frequency was measured to be 0.029±0.009 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.

Relation between Parametric Change of the Workload and Prefrontal Cortex Activity during a Modified Version of the ‘Rock, Paper, Scissors' Task

Background/Aims: Modified rock, paper, scissors (RPS) tasks have previously been used in neuroscience to investigate activity of the prefrontal cortex (PFC). In this study, we investigated hemodynamic changes in the PFC using near-infrared spectroscopy (NIRS) during a modified RPS task in which each subject's successful performance rate was equalized; the workload was increased parametrically in order to reveal the resulting pattern of PFC activation. Methods: The subjects were 20 healthy adults. During RPS, the player uses hand gestures to represent rock, paper, and scissors. Rock beats scissors, paper beats rock, and scissors beats paper. In the modified RPS task, the player is instructed to lose intentionally against the computer hand; the computer goes first and the player follows. The interstimulus interval (ISI) level was adjusted with 11 steps. If the level rose, the ISI decreased and the workload increased parametrically. The maximal level (maxLv: the shortest ISI and the biggest workload) in which a subject could perform the task correctly was determined for every subject during rehearsal of the task prior to the experiment. Lowering the level from the maxLv made the task easier. Hemodynamic changes were measured by NIRS over 4 task levels (maxLv-3, maxLv-2, maxLv-1 and maxLv). Results: The hemodynamic changes in the left lateral PFC and bilateral Brodmann area 6 rose significantly with the increase in workload and presented a linear trend. Conclusion: These results suggest that PFC activation may linearly increase with increased workload during a modified RPS task in which successful performance rates of subjects are equalized.

Self-organization of five species in a cyclic competition game

Cyclic competition game models, particularly the “rock–paper–scissors” model, play important roles in exploring the problem of multi-species coexistence in spatially ecological systems. We propose an extended “rock–paper–scissors” game to model cyclic interactions among five species, and find that two of the five can coexistent when biodiversity disappears, which is different from the “rock–paper–scissors” game. As the number of fingers is five, we named the new model the “fingers” game, where the thumb, forefinger, middle finger, ring finger, and little finger cyclically dominate their subsequent species and are dominated by their former species. We investigate the “fingers” model in two ways: direct simulations and nonlinear partial differential equations. An important finding is that the number of species in a cyclic competition game has an influence on the emergence of biodiversity. To be specific, the “rock–paper–scissors” model is in favor of maintaining biodiversity in comparison with the “fingers” model when the variables (population size, reproduction rate, selection rate, and migration rate) are the same. It is also shown that the mobility and reproduction rate can promote or jeopardize biodiversity.

Examining belief and confidence in schizophrenia

People with psychoses often report fixed, delusional beliefs that are sustained even in the presence of unequivocal contrary evidence. Such delusional beliefs are the result of integrating new and old evidence inappropriately in forming a cognitive model. We propose and test a cognitive model of belief formation using experimental data from an interactive 'Rock Paper Scissors' (RPS) game. Participants (33 controls and 27 people with schizophrenia) played a competitive, time-pressured interactive two-player game (RPS). Participants' behavior was modeled by a generative computational model using leaky integrator and temporal difference methods. This model describes how new and old evidence is integrated to form a playing strategy to beat the opponent and to provide a mechanism for reporting confidence in one's playing strategy to win against the opponent. People with schizophrenia fail to appropriately model their opponent's play despite consistent (rather than random) patterns that can be exploited in the simulated opponent's play. This is manifest as a failure to weigh existing evidence appropriately against new evidence. Furthermore, participants with schizophrenia show a 'jumping to conclusions' (JTC) bias, reporting successful discovery of a winning strategy with insufficient evidence. The model presented suggests two tentative mechanisms in delusional belief formation: (i) one for modeling patterns in other's behavior, where people with schizophrenia fail to use old evidence appropriately, and (ii) a metacognitive mechanism for 'confidence' in such beliefs, where people with schizophrenia overweight recent reward history in deciding on the value of beliefs about the opponent.

Spatial heterogeneity promotes coexistence of rock–paper–scissors metacommunities

The rock–paper–scissors game–which is characterized by three strategies R,P,S, satisfying the non-transitive relations S excludes P, P excludes R, and R excludes S–serves as a simple prototype for studying more complex non-transitive systems. For well-mixed systems where interactions result in fitness reductions of the losers exceeding fitness gains of the winners, classical theory predicts that two strategies go extinct. The effects of spatial heterogeneity and dispersal rates on this outcome are analyzed using a general framework for evolutionary games in patchy landscapes. The analysis reveals that coexistence is determined by the rates at which dominant strategies invade a landscape occupied by the subordinate strategy (e.g. rock invades a landscape occupied by scissors) and the rates at which subordinate strategies get excluded in a landscape occupied by the dominant strategy (e.g. scissors gets excluded in a landscape occupied by rock). These invasion and exclusion rates correspond to eigenvalues of the linearized dynamics near single strategy equilibria. Coexistence occurs when the product of the invasion rates exceeds the product of the exclusion rates. Provided there is sufficient spatial variation in payoffs, the analysis identifies a critical dispersal rate d∗ required for regional persistence. For dispersal rates below d∗, the product of the invasion rates exceeds the product of the exclusion rates and the rock–paper–scissors metacommunities persist regionally despite being extinction prone locally. For dispersal rates above d∗, the product of the exclusion rates exceeds the product of the invasion rates and the strategies are extinction prone. These results highlight the delicate interplay between spatial heterogeneity and dispersal in mediating long-term outcomes for evolutionary games.