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

Zhijian Wang, Bin Xu, Hai-Jun Zhou

arXiv:1404.5199v1physics.soc-phcs.GT

TL;DR

Population cycling in human RPS is not explained by the mixed-strategy Nash equilibrium or independent action choices. The paper models win-, tie-, and loss-conditioned responses and finds that the model reproduces observed cycling; optimized conditional responses can also yield higher payoffs than NE.

  • Problem

    Population cycling in human RPS is not explained by mixed-strategy NE or independent decisions based only on a player's previous action.

    Source: Empirical conditional response patterns · p. 3

  • Method

    The authors model each player's next action with probabilities conditioned on whether the previous outcome was a win, tie, or loss.

    Source: The conditional response model · p. 4

  • Results

    Using empirical response parameters, the model predicted mean cycling frequencies of 0.035 (a = 1.1), 0.026 (a = 2), 0.030 (a = 4), 0.018 (a = 9), and 0.017 (a = 100), agreeing well with measurements.

    Source: The conditional response model · p. 4

  • Takeaways & Limitations

    Optimized conditional responses can yield higher payoffs than NE despite having the same uniform action marginals.

    Source: Discussion · p. 5

    The study leaves CR-parameter optimization out; effects of memory length and population size on optimal strategies remain to be studied.

    Source: Discussion · p. 5

Abstract

from arXiv · show

How humans make decisions in non-cooperative strategic interactions is a challenging question. For the fundamental model system of Rock-Paper-Scissors (RPS) game, classic game theory of infinite rationality predicts the Nash equilibrium (NE) state with every player randomizing her choices to avoid being exploited, while evolutionary game theory of bounded rationality in general predicts persistent cyclic motions, especially for finite populations. However, as empirical studies on human subjects 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 cyclic motions in a laboratory experiment of the discrete-time iterated RPS game under the traditional random pairwise-matching protocol. The cycling direction and frequency are not sensitive to the payoff parameter a. This collective behavior contradicts with the NE theory but it is quantitatively explained by a microscopic model of win-lose-tie conditional response without any adjustable parameter. Our theoretical calculations reveal that this new strategy may offer higher payoffs to individual players in comparison with the NE mixed strategy, suggesting that high social efficiency is achievable through optimized conditional response.

Results

In discrete-time, pairwise-matched RPS, populations cycled persistently across payoff settings, even though individual action frequencies resembled the Nash equilibrium. Win-, tie-, and loss-conditioned responses revealed patterns beyond independent action choice.

Experimental system

Collective behaviors of the whole population

  • Mean cycling frequencies were 0.031 ± 0.006 (a = 1.1), 0.027 ± 0.008 (a = 2), 0.031 ± 0.008 (a = 4), 0.022 ± 0.008 (a = 9), and 0.018 ± 0.007 (a = 100), all significantly positive.Frequency did not significantly correlate with a, and cycling did not die out over time.Source: Collective behaviors of the whole… · p. 3, Figure 2 · p. 4

Empirical conditional response patterns

  • Winners were more likely to repeat their action, while for a ≥ 2 losers were more likely to shift clockwise; independent-choice predictions were near zero and differed from observed cycling.The independent model uses each player's previous action alone, not the opponent's outcome.Source: Empirical conditional response patterns · p. 3

The conditional response model

The model conditions each player's next action on whether the previous play was a win, tie, or loss. Its predicted cycling frequencies agree with experiment, while its payoff analysis shows that carefully optimized responses can outperform the NE mixed strategy.

The conditional response model

  • The model assigns each player six shift probabilities, conditioning clockwise, counter-clockwise, or repeat choices on whether the prior outcome was a win, tie, or loss.These six parameters are W−, W+, T−, T+, L−, and L+; the repeat probability is the remainder.Source: The conditional response model · p. 4
  • The model derives social-state transition probabilities from pair outcomes, then computes the steady-state distribution and mean cycling frequency.Cycling frequency is computed from the rotation angle associated with each social-state transition.Source: The conditional response model · p. 4
  • Using empirical response parameters, the model predicted mean cycling frequencies of 0.035 (a = 1.1), 0.026 (a = 2), 0.030 (a = 4), 0.018 (a = 9), and 0.017 (a = 100), agreeing well with measurements.Agreement also held across the 59 individual populations.Source: The conditional response model · p. 4
  • For a > 2, sampled CR strategies had mean gcr − g0 = −0.0085(a − 2), while empirical-parameter payoffs all weakly exceeded g0.Here g0 is the expected payoff under the NE mixed strategy.Source: The conditional response model · p. 5
  • For large a, three optimized CR examples outperformed NE by about 10%; one had fcr = 0.003, so high payoff did not require population-level cycling.A different example, with W0 ≈ 1 and L0 ≈ 0, extends win-stay lose-shift.Source: The conditional response model · p. 5

Discussion

The discussion argues that conditional response can explain collective cycling even when individual action marginals resemble the Nash equilibrium, while optimized parameters can yield higher payoffs. It also identifies open questions about optimization and the neural basis of conditional response.

Discussion

  • Optimized conditional-response strategies can yield higher payoffs than the Nash mixed strategy, although their optimization remains an open question.The authors also describe conditional response as psychologically plausible and identify win-stay lose-shift as a special case.Source: Discussion · p. 5
  • Whether conditional response is a basic human-brain decision mechanism or a consequence of deeper neural mechanisms remains unresolved.Source: Discussion · p. 6

Materials and Methods

  • The experiment used 360 Zhejiang University students in 60 populations of six, playing 300 rounds with a fixed payoff parameter under random pairwise matching.Rewards were paid privately, with the exchange rate set so the Nash-equilibrium expected reward was 50 RMB across populations.Source: Materials and Methods · p. 6, S1. EXPERIMENTAL SETUP · p. 14

Supplementary Figure 1

  • The supplementary figures describe conditional action-shift probabilities across five payoff values and sampled conditional-response strategy payoffs and cycling frequencies.They also show cycling frequency by population size and parameter choices, including symmetric strategies.Source: Figure S1 · p. 10, Figure S2 · p. 11, Figure S3 · p. 12

S1. EXPERIMENTAL SETUP

  • Each session involved six players repeatedly playing 300 rounds, with three randomly matched pairs formed by computer in each round.The study ran five sets of sessions at a = 1.1, 2, 4, 9, and 100.Source: S1. EXPERIMENTAL SETUP · p. 14

S2. THE MIXED-STRATEGY NASH EQUILIBRIUM

The section establishes the equal-probability mixed-strategy Nash equilibrium and discusses other equilibria across payoff values. It reports that observed action marginals generally fit the mixed strategy, with one excluded cooperative exception.

S2. THE MIXED-STRATEGY NASH EQUILIBRIUM

S3. EVOLUTIONARY STABILITY OF THE NASH EQUILIBRIUM

The mixed-strategy Nash equilibrium is evolutionarily stable when a > 2, unstable when a < 2, and neutral at a = 2. Mutant-strategy payoffs are compared with the NE payoff to establish these conditions.

S3. EVOLUTIONARY STABILITY OF THE NASH EQUILIBRIUM

S4. CYCLING FREQUENCIES PREDICTED BY TWO SIMPLE MODELS

The observed persistent cycling is inconsistent with the mixed-strategy NE and is not explained by the independent decision model.

S4. CYCLING FREQUENCIES PREDICTED BY TWO SIMPLE MODELS

A. Assuming the mixed-strategy Nash equilibrium

Under the mixed-strategy NE, action choices are uniform and the social-state transition is history-independent, so detailed balance rules out persistent cycling. A model based on each player’s own current action also predicts near-zero cycling frequencies, unlike the empirical values.

A. Assuming the mixed-strategy Nash equilibrium

B. Assuming the independent decision model

  • In the independent decision model, each player’s next action depends probabilistically only on that player’s current action, through action-specific transition probabilities.Source: B. Assuming the independent decision… · p. 16
  • Using empirically determined action-transition probabilities, the model predicts cycling frequencies near zero and significantly different from empirical values across the listed payoff parameters.Predictions are 0.0050 (a = 1.1), −0.0005 (a = 2), −0.0024 (a = 4), −0.0075 (a = 9), and −0.0081 (a = 100).Source: B. Assuming the independent decision… · p. 17

S5. DETAILS OF THE CONDITIONAL RESPONSE MODEL

The conditional-response model specifies action shifts based on whether a player won, tied, or lost. It combines these responses with random pairwise matching to calculate transitions between population social states.

A. Social state transition matrix

  • The win-lose-tie conditional response strategy uses action-shift probabilities conditioned on the current outcome, with three probabilities for each outcome summing to one.W, T, and L denote responses after a win, tie, and loss; the six shift probabilities W−, W+, T−, T+, L−, and L+ specify a strategy.Source: A. Social state transition matrix · p. 17
  • A social state records the numbers of players choosing R, P, and S, while the current matching is described by counts of the six possible pair types.The action counts are constrained by the pair counts, and multiple pairing patterns can correspond to the same social state.Source: A. Social state transition matrix · p. 17
  • For each pair type, the model assigns probabilities to possible next-round action pairs according to the two players’ outcome-conditioned responses.For an R–P pair, for example, the players respond as loser and winner, respectively, producing six possible pair outcomes.Source: A. Social state transition matrix · p. 18
  • The transition probability from one social state to the next combines random pairwise matching with the possible outcomes across pairs.Source: A. Social state transition matrix · pp. 18–19

Steady-state properties

The conditional-response model’s predicted mean payoff is slightly above the Nash-equilibrium benchmark for a ≠ 2, and empirical mean payoffs are also above that benchmark in the four reported population sets. Its steady-state distribution is calculated numerically to obtain cycling frequency and mean payoff.

Steady-state properties

  • The CR model’s social-state steady-state distribution is determined numerically, then used to compute cycling frequency and per-round mean payoff.Source: Steady-state properties · p. 19
  • For a ≠ 2, predicted g_cr is slightly above g0, and empirical mean payoff g is above g0 in all four reported population sets.Here g0 = (1 + a)/3 is the expected per-round payoff under the NE mixed strategy.Source: Steady-state properties · p. 20
  • The conditional response strategy may yield higher individual-player payoffs than the NE mixed strategy, with empirical g−g0 estimates reported across five a values.Empirical g−g0 equals 0.009±0.004 at a=1.1 (mean±SEM), 0.000 ± 0.006 at a=2, 0.004 ± 0.012 at a=4, 0.01 ± 0.02 at a=9, and 0.05 ± 0.37 at a=100.Source: Steady-state properties · p. 20

B. The symmetric case

Cycling persists in the CR model even when its parameters are symmetric with respect to clockwise and counter-clockwise action shifts. The direction depends on whether winners or losers are more likely to change actions.

B. The symmetric case

  • With symmetric clockwise and counter-clockwise parameters, cycling remains nonzero when W0 ≠ L0; its magnitude rises with |W0 − L0| and falls with T0.Source: B. The symmetric case · p. 20
  • When W0 > L0, cycling is counter-clockwise; when L0 > W0, it is clockwise, and symmetrized empirical parameters yield nonzero frequencies for all five payoff values.The reported frequencies are 0.024 (a = 1.1), 0.017 (a = 2.0), 0.017 (a = 4.0), 0.015 (a = 9.0), and 0.017 (a = 100.0).Source: B. The symmetric case · p. 20

C. Sampling the conditional response parameters

Sampling CR parameters reveals both the cycling patterns they favor and the payoff gap between random and optimized strategies. At steady state, however, a population using the same CR strategy has individual action marginals identical to the NE mixed strategy.

C. Sampling the conditional response parameters

D. Action marginal distribution of a single player

E. Computer simulations

Computer simulations confirm the theoretical predictions of the CR model. The paper also extends the model to allow responses to depend on both a player’s and an opponent’s previous actions, with the original win-lose-tie model as a restricted case.

E. Computer simulations

  • Computer simulations of populations using fixed CR parameters under random pairwise matching confirmed all theoretical predictions of the model.Source: E. Computer simulations · p. 21

S6. THE GENERALIZED CONDITIONAL RESPONSE MODEL

  • The generalized CR model lets a player’s next action depend on both players’ previous actions and has 27 transition parameters subject to nine normalization conditions.Source: S6. THE GENERALIZED CONDITIONAL… · p. 21
  • The win-lose-tie CR model is a limiting case of the generalized model, whose additional assumptions are considered reasonable only for the simplest payoff matrix.Source: S6. THE GENERALIZED CONDITIONAL… · p. 21
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