Rock-paper-scissors Research Articles

Dynamics of Reliance on Algorithmic Advice

ABSTRACTThis study examines the dynamics of human reliance on algorithmic advice in a situation with strategic interaction. Participants played the strategic game of Rock–Paper–Scissors (RPS) under various conditions, receiving algorithmic decision support while facing human or algorithmic opponents. Results indicate that participants often underutilize algorithmic recommendations, particularly after early errors, but increasingly rely on the algorithm following successful early predictions. This behavior demonstrates a sensitivity to decision outcomes, with asymmetry: rejecting advice consistently reinforces rejecting advice again while accepting advice leads to varied reactions based on outcomes. We also investigate how personal characteristics, such as algorithm familiarity and domain experience, influence reliance on algorithmic advice. Both factors positively correlate with increased reliance, and algorithm familiarity significantly moderates the relationship between outcome feedback and reliance. Facing an algorithmic opponent increases advice rejection frequencies, and the determinants of trust and interaction dynamics differ from those with human opponents. Our findings enhance the understanding of algorithm aversion and reliance on AI, suggesting that increasing familiarity with algorithms can improve their integration into decision‐making processes.

Enhancement of persistence in the rock-paper-scissors dynamics through higher-order interactions

Higher-order interactions (HOIs) are ubiquitous in real systems, and HOIs among individuals complicate the exploration of species diversity for populations in ecosystems. Therefore, understanding how higher-order interactions affect biodiversity is essential but poses significant challenges. In this paper, we consider higher-order interactions in the competitive processes of spatial rock-paper-scissors (RPS) dynamics, specifically within spatially embedded hyper-lattices, and investigate what biodiversity can be driven by HOIs. Considering hyperedges, species in hyperedges can be different, and different relationships are considered, which can eventually demonstrate the strength of competition. With the competition intensity modulated by the sensitivity of HOIs, extensive simulations have shown that higher-order competitive processes can promote coexistence. We also found that stratification of species densities occurs when three species coexist due to HOIs, which are closely related to the increase of empty sites and disordered spiral entanglement. From investigating the extinction probability in a broad aspect, it has been found that, even if HOIs are working, the coexistence in the system is eventually broken at high mobility regimes. The intensified sensitivity to HOIs can somewhat alleviate extinction, although no striking change has occurred. Our findings may address that modeling predator-prey interactions incorporating HOIs is important for gaining insights into biodiversity, and we hope our study may provide a broad aspect for exploring complex ecosystems.

Coalescence model of rock-paper-scissors particles

The rock–paper–scissors game, commonly played in East Asia, gives a simple model to understand physical, biological, psychological, and other problems. The interacting rock–paper–scissors particle system connects two models: the collision model (Maxwell and Boltzmann’s kinetic theory of gas) and the coalescence model (Smoluchowski’s coagulation theory). A 2s+1 type rock–paper–scissors collision model naturally introduces a nonlinear integrable Lotka–Volterra system. The time evolution of the coalescence model is obtained from the logarithmic time change of the collision model. We also discuss the behavior of a discrete rock–paper–scissors coalescence model.

Momentum accelerates evolutionary dynamics

We combine momentum from machine learning with evolutionary dynamics, where momentum can be viewed as a simple mechanism of intergenerational memory similar to epigenetic mechanisms. Using information divergences as Lyapunov functions, we show that momentum accelerates the convergence of evolutionary dynamics including the continuous and discrete replicator equations and Euclidean gradient descent on populations. When evolutionarily stable states are present, these methods prove convergence for small learning rates or small momentum, and yield an analytic determination of the relative decrease in time to converge that agrees well with computations. The main results apply even when the evolutionary dynamic is not a gradient flow. We also show that momentum can alter the convergence properties of these dynamics, for example by breaking the cycling associated to the rock–paper–scissors landscape, leading to either convergence to the ordinarily non-absorbing equilibrium, or divergence, depending on the value and mechanism of momentum.

Exploring cyclic dominance in rock-paper-scissor strategies within the Optional Prisoner's Dilemma: A quantitative analysis with environmental feedback

This research investigation delves into the concept of cyclic dominance and its interplay with environmental feedback within the framework of the Optional Prisoner's Dilemma (OPD) game. A quantitative model has been meticulously constructed to encompass both the OPD and Rock-Paper-Scissors (RPS) games, wherein the outcomes are contingent upon environmental feedback. The intricate relationship between player populations and their surrounding environment is elucidated through the utilization of replicator dynamics. The empirical findings of this study shed light on the existence of distinct equilibria contingent upon the prevailing environmental circumstances. Under scenarios characterized by resource depletion, an equilibrium state emerges wherein only solitary individuals (referred to as "loners") persist. Conversely, under conditions of resource replenishment, an equilibrium is established where various strategic approaches coexist in equal proportions, akin to the dynamics of the Rock-Paper-Scissors game. Notably, cyclic dominance manifests itself during periods characterized by intermediate environmental conditions. Several pivotal factors have been identified as determinants of these dynamic phenomena, including the rate of environmental replenishment, the strength of the dilemma presented, the payoff for loners, and the reward for a draw. It has been empirically demonstrated that the presence of extreme values in the replenishment rate, irrespective of whether exceptionally high or exceptionally low, serves as an impediment to the emergence of cyclic dominance. This model underscores the substantial influence of environmental feedback on social dilemmas, particularly in the context of the OPD game, ultimately leading to the emergence of competitive cyclic dominance games reminiscent of the Rock-Paper-Scissors game. In doing so, it elucidates the intricate interplay between evolutionary game incentives and the dynamics of the surrounding environment.

Ambush strategy enhances organisms’ performance in rock–paper–scissors games

We study a five-species cyclic system wherein individuals of one species strategically adapt their movements to enhance their performance in the spatial rock–paper–scissors game. Environmental cues enable the awareness of the presence of organisms targeted for elimination in the cyclic game. If the local density of target organisms is sufficiently high, individuals move towards concentrated areas for direct attack; otherwise, they employ an ambush tactic, maximising the chances of success by targeting regions likely to be dominated by opponents. Running stochastic simulations, we discover that the ambush strategy enhances the likelihood of individual success compared to direct attacks alone, leading to uneven spatial patterns characterised by spiral waves. We compute the autocorrelation function and measure how the ambush tactic unbalances the organisms’ spatial organisation by calculating the characteristic length scale of typical spatial domains of each species. We demonstrate that the threshold for local species density influences the ambush strategy’s effectiveness, while the neighbourhood perception range significantly impacts decision-making accuracy. The outcomes show that long-range perception improves performance by over 60%, although there is potential interference in decision-making under high attack triggers. Understanding how organisms’ adaptation their environment enhances their performance may be helpful not only for ecologists, but also for data scientists, aiming to improve artificial intelligence systems.

Prevalence of impaired foot function in baseball players with and without disabled throwing shoulder/elbow: a case–control study

Prevalence of impaired foot function among baseball players with and without a disabled throwing shoulder/elbow was investigated. The study included 138 male players. Players who had previously complained of shoulder/elbow pain during throwing motion were defined as the players with a history, and those who experienced shoulder/elbow pain during the examination were defined as having the injury. Foot function was evaluated by foot “rock paper scissors” movements and floating toes. Their prevalence was assessed and the relationships between players with and without the injuries were statistically analyzed. The prevalence of players with a history and injury was 27% and 7%, respectively. The prevalence of impaired foot function on the non-throwing side among players with injury was significantly higher than those without (60% vs. 28%, P < 0.001) and higher tendency on the throwing side than those without (60% vs. 32%). Regarding floating toes, players with a relevant history showed a significantly higher prevalence on the throwing side than those without (49% vs 28%, P < 0.001) and higher tendency on the non-throwing side than those without (49% vs 32%). Players with disabled throwing shoulder/elbow have a significantly higher prevalence of impaired foot function and floating toes than players without it.

Spatial dynamics of higher order rock-paper-scissors and generalisations

We introduce and study the spatial replicator equation with higher order interactions and both infinite (spatially homogeneous) populations and finite (spatially inhomogeneous) populations. We show that in the special case of three strategies (rock–paper–scissors) higher order interaction terms allow travelling waves to emerge in non-declining finite populations. We show that these travelling waves arise from diffusion stabilisation of an unstable interior equilibrium point that is present in the aspatial dynamics. Based on these observations and prior results, we offer two conjectures whose proofs would fully generalise our results to all odd cyclic games, both with and without higher order interactions, assuming a spatial replicator dynamic. Intriguingly, these generalisations for N⩾5 strategies seem to require declining populations, as we show in our discussion.

Principal trade-off analysis

How are the advantage relations between a set of agents playing a game organized and how do they reflect the structure of the game? In this paper, we illustrate ‘Principal Trade-off Analysis’ (PTA), a decomposition method that embeds games into a low-dimensional feature space. We argue that the embeddings are more revealing than previously demonstrated by developing an analogy to Principal Component Analysis (PCA). PTA represents an arbitrary two-player zero-sum game as linear combination of simple games via the projection of policy profiles into orthogonal 2D feature planes. We show that the feature planes represent unique strategic trade-offs and truncation of the sequence provides insightful model reduction and visualization. We demonstrate the validity of PTA on a quartet of games (Kuhn poker, RPS + 2, Blotto and Pokemon). In Kuhn poker, PTA clearly identifies the trade-off between bluffing and calling. In Blotto, PTA identifies game symmetries and specifies strategic trade-offs associated with distinct win conditions. These symmetries reveal limitations of PTA unaddressed in previous work. For Pokemon, PTA recovers clusters that naturally correspond to Pokemon types, correctly identifies the designed trade-off between those types, and discovers a rock-paper-scissor (RPS) cycle in the Pokemon generation type – all absent any specific information except game outcomes.

Handling sensory disabilities in a smart society

Billions of people live with visual and/or hearing impairments. Regrettably, their access to systems remains delayed, leaving them socially excluded. The need for universal access of next-generation systems and users’ inclusion is paramount. We pose that a smart society should respond to this crucial need. Following ability-based design principles, we introduce a simulated social robot that adapts to users’ sensory abilities. Its working was assessed via a Rock–Paper–Scissors game in an Intelligent Environment (IE), using three modes: where the user is able to see and hear, only see, or only hear. With this game, two user-studies were conducted using the UMUX-LITE usability score, an expectation rating, and the gap between experience and expectation, complemented with two open questions. A repeated measures Multivariate ANalysis Of VAriance (MANOVA) on the data from study 1 unveiled an overall difference between the three modes, F ( 6 , 6 ) = 6.823, η p 2 = .872, p = .017. Users expected applications to be harder to use with a disability, especially a visual impairment. All modes were considered accessible, with the experience exceeding expectations for the mode with a hearing impairment. In parallel, substantial variance was observed across participants and the results from the open questions suggested improvements. To reduce this variance and increase system stability, study 2 was run with an enhanced design. A repeated measures MANOVA on the data from study 2 confirmed study 1’s findings, F ( 6 , 6 ) = 12.801, η p 2 = .928, p = .003. Moreover, experiences exceeded expectations in all modes and the variance among participants was substantially decreased. We conclude that IE applications managed by a social robot can be adapted to user’s sensory abilities, improving smart society’s accessibility, and, hence, reducing social exclusion.

The effect of lookahead on phase transition in migration of three species with cyclic predator–prey relations

A three-species predator–prey system with cyclic predator–prey relations (also called the rock–paper–scissors game) on a one-dimensional lattice where all individuals migrate in the same direction is studied. Each individual can look ahead within a certain range and can stop its migration when too many predators occur within its lookahead range. Simulation experiments revealed that the three species can coexist within a wide range of model parameter values, providing insights into the dynamical phase transition between coexisting and single phases.

Stability of cycles and survival in a jungle game with four species

The Jungle Game is used in population dynamics to describe cyclic competition among species that interact via a food chain. The dynamics of the Jungle Game supports a heteroclinic network whose cycles represent coexisting species. The stability of all heteroclinic cycles in the network for the Jungle Game with four species determines that only three species coexist in the long-run, interacting under cyclic dominance as a Rock–Paper–Scissors Game. This is in stark contrast with other interactions involving four species, such as cyclic interaction and intraguild predation. We use the Jungle Game with four species to determine the success of a fourth species invading a population of Rock–Paper–Scissors players.

Adaptive rock-paper-scissors game enhances eco-evolutionary performance at cost of dynamic stability

The rock-paper-scissors (RPS) game is a classic model for exploring the performance of how multiple strategies interact and evolve over time. The classic RPS game assumes a fixed benefit and cost for each strategy against another one when two players meet, while its evolutionary game considers the frequency dynamics of the three strategies with each's fitness influenced by its net payoff. This may not reflect the complexity of real-world scenarios as strategies can co-adapt with each other. We introduce an adaptive RPS game that captures the dynamics of strategy densities with trait-mediated payoffs, and the adaptive dynamics of coevolving traits via incremental mutations leading to adaptively evolving payoffs. Results show that the adaptive RPS game approaches a steady state of strategy density, if any, faster than the evolutionary RPS game. The stable coexistence of all strategies in the evolutionary game can be easily destabilized in the adaptive game. Strategies that are allowed to adaptively change their traits also performed better and achieved greater strategy densities than those fixed strategies in the adaptive game. The coevolving strategies in the adaptive RPS game exhibit complex and diverse attractors in the trait space, sensitive to both initial conditions and model parameters, but exhibiting positive payoffs with greater benefits than costs. These findings highlight how adaptive games enhance strategy performance by sacrificing eco-evolutionary dynamic stability.

Semi-Restricted Rock, Paper, Scissors

Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman. The game consists of $3n$ rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly $n$ times during the $3n$ rounds, while Norman is allowed to play normally without any restrictions. Answering a question of Spiro, we show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is $\Theta(\sqrt{n})$. Moreover, we study semi-restricted versions of general zero sum games and prove a number of results concerning their optimal strategies and expected scores, which in particular implies our results for semi-restricted RPS.

Historic Behavior in Nonautonomous Rock–Paper–Scissors Dynamics

Historic Behavior in Nonautonomous Rock–Paper–Scissors Dynamics

Gene drives for the extinction of wild metapopulations

Population-suppressing gene drives may be capable of extinguishing wild populations, with proposed applications in conservation, agriculture, and public health. However, unintended and potentially disastrous consequences of release of drive-engineered individuals are extremely difficult to predict. We propose a model for the dynamics of a sex ratio-biasing drive, and using simulations, we show that failure of the suppression drive is often a natural outcome due to stochastic and spatial effects. We further demonstrate rock–paper–scissors dynamics among wild-type, drive-infected, and extinct populations that can persist for arbitrarily long times. Gene drive-mediated extinction of wild populations entails critical complications that lurk far beyond the reach of laboratory-based studies. Our findings help in addressing these challenges.

Cyclic pairwise interaction representing a rock-paper-scissors game maintains the population of the vulnerable yeast Saccharomyces cerevisiae within a multispecies sourdough microbiome.

Sourdough starters are spontaneously generated multispecies communities consisting of lactic acid bacteria (LAB) and yeasts. Traditionally used to make baked goods, these communities of microorganisms can be propagated through successive transfers to other containers. Successive transfers result in microbial community dynamics; however, the mechanisms guiding the dynamics are not fully elucidated. This study tracked five species of sourdough LAB-yeast communities (three LAB species and two yeast species) during successive transfers in vitro to explore the interspecies interactions driving the multispecies community dynamics. The microbial dynamics were modeled with a generalized Lotka-Volterra (gLV) equation, which can mathematically identify ecological types of interspecies interactions. The gLV model simulated the population dynamics of each species with Pearson's correlation coefficient values of ≥0.817. Pairwise experiments revealed that competition (negative/negative) predominated among all pairwise interactions (70%), followed by amensalism (negative/neutral) (30%). In the pairwise community, the LAB species Limosilactobacillus fermentum suppressed Saccharomyces cerevisiae growth through amensalism; however, S. cerevisiae population was maintained when the community comprised 3-5 species, including the LAB species L. fermentum and Pediococcus pentosaceus. The key interspecies interactions for maintaining the S. cerevisiae population were theoretically determined using the gLV model; the three identified species interacted non-transitively with S. cerevisiae in a cyclic pairwise interaction, metaphorically representing a rock-paper-scissors relationship, which systemically supported the S. cerevisiae population in the multispecies community. These theoretical insights indicate that cyclic pairwise interaction is the main driver of LAB-yeast population dynamics, which helps sustain a vulnerable yeast species in a sourdough multispecies community.IMPORTANCETraditionally, multispecies consisting of lactic acid bacteria and yeasts collaboratively engage sourdough fermentation, which determines the quality of the resulting baked goods. Nonetheless, the successive transfer of these microbial communities can result in undesirable community dynamics that prevent the formation of high-quality sourdough bread. Thus, a mechanistic understanding of the community dynamics is fundamental to engineer sourdough complex fermentation. This study describes the population dynamics of five species of lactic acid bacteria-yeast communities in vitro using a generalized Lotka-Volterra model that examines interspecies interactions. A vulnerable yeast species was maintained within up to five species community dynamics by obtaining support with a cyclic interspecies interaction. Metaphorically, it involves a rock-paper-scissors game between two lactic acid bacteria species. Application of the generalized Lotka-Volterra model to real food microbiomes including sourdoughs will increase the reliability of the model prediction and help identify key microbial interactions that drive microbiome dynamics.

Predicting rock-paper-scissors choices based on single-trial EEG signals.

Decision prediction based on neurophysiological signals is of great application value in many real-life situations, especially in human-AI collaboration or counteraction. Single-trial analysis of electroencephalogram (EEG) signals is a very valuable step in the development of an online decision-prediction system. However, previous EEG-based decision-prediction methods focused mainly on averaged EEG signals of all decision-making trials to predict an individual's general decision tendency (e.g., risk seeking or aversion) over a period rather than on a specific decision response in a single trial. In the present study, we used a rock-paper-scissors game, which is a common multichoice decision-making task, to explore how to predict participants' single-trial choice with EEG signals. Forty participants, comprising 20 females and 20 males, played the game with a computer player for 330 trials. Considering that the decision-making process of this game involves multiple brain regions and neural networks, we proposed a new algorithm named common spatial pattern-attractor metagene (CSP-AM) to extract CSP features from different frequency bands of EEG signals that occurred during decision making. The results showed that a multilayer perceptron classifier achieved an accuracy significantly exceeding the chance level among 88.57% (31 of 35) of participants, verifying the classification ability of CSP features in multichoice decision-making prediction. We believe that the CSP-AM algorithm could be used in the development of proactive AI systems.

Evolutionary dynamics in the cyclic competition system of seven species: Common cascading dynamics in biodiversity

Complex systems in ecological science can be generally defined by either the number of different species or the structure among species having many relations, and understanding the given interaction structure is essential to predict the evolution of ecosystems. In this paper, we propose a multi-species system whose competition can occur cyclically. By exploiting the generalized system of cyclic competition among seven species, we explore how species biodiversity can appear when the generalized system is established by possessing the underlying mechanism of rock–paper–scissors (RPS) and rock–paper–scissors–lizard–spock (RPSLS) games. Through Monte-Carlo simulations, similar to the RPSLS system having the phase transition in biodiversity from five to one containing the three species survival in the middle, the model for seven species also exhibits similar cascading features in the biodiversity as mobility increases, validated by measuring the survival probability. We also found that not every cyclic structured system among seven species exhibits a common cascading feature in the transition in biodiversity. It is revealed that such a characteristic may require sufficient structures of RPS-like subgroups. Our findings may provide insights into the biodiversity of cyclically competing species and the link to predict biodiversity associated with the interaction structure in the microscopic framework.

Historic Behavior in Rock–Paper–Scissor Dynamics

Historic Behavior in Rock–Paper–Scissor Dynamics