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.

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.

Zero-determinant (ZD) strategies, as discovered by Press and Dyson, can enforce a linear relationship between a pair of players' scores in the iterated prisoner's dilemma. Particularly, the extortionate ZD strategies can enforce and exploit cooperation, providing a player with a score advantage, and consequently higher scores than those from either mutual cooperation or generous ZD strategies. In laboratory experiments in which human subjects were paired with computer co-players, we demonstrate that both the generous and the extortionate ZD strategies indeed enforce a unilateral control of the reward. When the experimental setting is sufficiently long and the computerized nature of the opponent is known to human subjects, the extortionate strategy outperforms the generous strategy. Human subjects' cooperation rates when playing against extortionate and generous ZD strategies are similar after learning has occurred. More than half of extortionate strategists finally obtain an average score higher than that from mutual cooperation.

Abstract. Evolutionary dynamics provides an iconic relationship-the periodic frequency of a game is determined by the payoff matrix of the game. This paper reports the first experimental evidence to demonstrate this relationship. Evidence comes from two populations randomly-matched 2 × 2 games with 12 different payoff matrix parameters. The directions, frequencies and changes in the radius of the cycles are measured definitively. The main finding is that the observed periodic frequencies of the persistent cycles are significantly different in games with different parameters. Two replicator dynamics, standard and adjusted, are employed as predictors for the periodic frequency. Interestingly, both of the models could infer the difference of the observed frequencies well. The experimental frequencies linearly, positively and significantly relate to the theoretical frequencies, but the adjusted model performs slightly better.

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In a two-person Rock-Paper-Scissors (RPS) game, if we set a loss worth nothing and a tie worth 1, and the payoff of winning (the incentive a) as a variable, this game is called as generalized RPS game. The generalized RPS game is a representative mathematical model to illustrate the game dynamics, appearing widely in textbook. However, how actual motions in these games depend on the incentive has never been report quantitatively. Using the data from 7 games with different incentives, including 84 groups of 6 subjects playing the game in 300-round, with random-pair tournaments and local information recorded, we find that, both on social and individual level, the actual motions are changing continuously with the incentive. More expressively, some representative findings are, (1) in social collective strategy transit views, the forward transition vector field is more and more centripetal as the stability of the system increasing; (2) In the individual behavior of strategy transit view, there exists a phase transformation as the stability of the systems increasing, and the phase transformation point being near the standard RPS;(3) Conditional response behaviors are structurally changing accompanied by the controlled incentive. As a whole, the best response behavior increases and the win-stay lose-shift (WSLS) behavior declines with the incentive. Further, the outcome of win, tie, and lose influence the best response behavior and WSLS behavior. Both as the best response behavior, the win-stay behavior declines with the incentive while the lose-left-shift behavior increase with the incentive. And both as the WSLS behavior, the lose-left-shift behavior increase with the incentive, but the lose-right-shift behaviors declines with the incentive. We hope to learn which one in tens of learning models can interpret the empirical observation above.

Abstract:We examine the effectiveness of the individual-punishment mechanism in larger groups, comparing groups of four to groups of 40 participants. We find that the individual punishment mechanism is remarkably robust when the marginal per capita return (MPCR), i.e. the return to each participant from each dollar that is contributed, is held constant. Moreover, the efficiency gains from the punishment mechanism are significantly higher in the 40-participant than in the four-participant treatment. This is true despite the coordination problems inherent in an institution relying on decentralized individual punishment decisions in the context of a larger group. It reflects increased per capita expenditures on punishment that offset the greater coordination difficulties in the larger group. However, if the marginal group return (MGR), i.e. the return to the entire group of participants, stays constant, resulting in an MPCR that shrinks with group size, no such offset occurs and punishment loses much but not all of its effectiveness at encouraging OPEN ACCESSGames 2013, 4 90 voluntary contributions to a public good. Efficiency is not significantly different from the small-group treatment.

Using laboratory experimental data, we test the uncertainty of social state transitions in various competing environments of fixed paired two-person constant sum 2 × 2 games. It firstly shows that, the distributions of social strategy transitions are not erratic but obey the principle of the maximum entropy (MaxEnt). This finding indicates that human subject social systems and natural systems could share wider common backgrounds.