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

Abstract

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.

Figure 1. The Rock-Paper-Scissors game.

(A) Each matrix entry specifies the row action's payoff. (B) Non-transitive dominance relations (R beats S, P beats R, S beats P) among the three actions. (C) The social state plane for a population of size N = 6. Each filled circle denotes a social state (n_R, n_P, n_S); the star marks the centroid c₀; the arrows indicate three social state transitions at game rounds t = 1, 2, 3.

Figure 2. Action shift probability conditional on a player's current action.

If a player adopts action R at one game round, this player's probability of repeating the same action at the next game round is denoted as R₀, while the probability of performing a counter-clockwise or clockwise action shift is denoted, respectively, as R₊ and R₋. The conditional probabilities P₀, P₊, P₋ and S₀, S₊, S₋ are defined similarly. (A–E) The mean (vertical bin) and the SEM (standard error of the mean, error bar) of each conditional probability obtained by averaging over the different populations of the same payoff parameter a = 1.1, 2, 4, 9, and 100 (from left to right). (F–J) The corresponding action shift probability values predicted by the conditional response model using the parameters of Fig. 3F–3J as inputs.

Figure 3. Social cycling explained by conditional response.

The payoff parameter is a = 1.1, 2, 4, 9 and 100 from left-most column to right-most column. (A–E) Accumulated cycle numbers C_1,t of 59 populations. (F–J) Empirically determined CR parameters, with the mean (vertical bin) and the SEM (error bar) of each CR parameter obtained by considering all the populations of the same a value. (K–O) Comparison between the empirical cycling frequency (vertical axis) of each population and the theoretical frequency (horizontal axis) obtained by using the empirical CR parameters of this population as inputs.

Figure 4. Probability distribution of payoff difference g_cr − g₀ at population size N = 6.

We assume a > 2 and set the unit of the horizontal axis to be (a − 2). The solid line is obtained by sampling 2.4 × 10⁹ CR strategies uniformly at random; the filled circle denotes the maximal value of g_cr among these samples.