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

Abstract

h i g h l i g h t s• Cycles are measured in a standard Rock-Paper-Scissors human experiment.• The existence of persistent cycles is confirmed from analyzing the evolutionary trajectories.• The mean frequency of cycles is quantitatively measured. • The observed highly stochastic but weak cyclic motions are quantitatively understood by a discrete-time logit dynamics model. a b s t r a c tThe 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 socialstate 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.

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

“…Here and in Ref. 28 we find that even discrete-time updating of actions will lead to collective cyclic motions in the RPS game, and such a population-level behavior is not affected by the particular value of a .…”

“…Each population leaves a trajectory on this plane as the RPS game proceeds. To detect rotational flows, we assign for every social state transition s ( t ) → s ( t + 1) a rotation angle θ ( t ), which measures the angle this transition rotates with respect to the centroid c 0 ≡ ( N /3, N /3, N /3) of the social state plane (see Methods)28. Positive and negative θ values signify counter-clockwise and clockwise rotations, respectively, while θ = 0 means the transition is not a rotation around c 0 .…”

“…Two of the present authors partially overcame these difficulties by using social state velocity vectors26 and forward and backward transition vectors27 to visualize violation of detailed balance in game evolution trajectories, but a simple way of quantitatively measuring persistent cyclic behavoiors in a highly stochastic trajectory was still lacking. The cycling frequency of directional flows in the neutral RPS game ( a = 2) was later quantitatively measured in28 using a coarse-grained counting technique. Cason and co-workers29 using another cycle rotation index as the order parameter also obtained evidence of persistent cycling in some evolutionarily stable RPS-like games, if players were allowed to update actions asynchronously in continuous time and were informed about the social states of the whole population by some sophisticated ‘heat maps'.…”

“…We adopt an improved cycle counting method on the basis of our earlier experiences28 and study directional flows in evolutionarily stable ( a > 2) and unstable ( a < 2) discrete-time RPS games. We show strong evidence that the RPS game is an intrinsic non-equilibrium system, which cannot be fully described by the NE concept even in the evolutionarily stable region but rather exhibits persistent population-level cyclic motions.…”

A Behavioral Study of 'Noise' in Coordination Games

“…There is empirical research on evolutionary dynamics related to replicator dynamics (Taylor and Jonker, 1978;Weibull, 1995). Examples include the evolutionary interpretation by Crawford (1991) of the seminal study by Van Huyck et al (1990), as well as more recent experiments by Friedman and Ostrov (2013), Xu et al (2013), Cason et al (2014), Hoffman et al (2015). These studies show that replication makes macro-predictions that correspond to observed system behavior such as cyclical patterns in rock-paper-scissor games.…”