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

Abstract: 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.

• The paper demonstrates that cyclic motions in RPS deviate from Nash equilibrium predictions in finite populations.
• The paper introduces a win-lose-tie conditional response model that predicts players' moves based on previous outcomes and yields higher payoffs.
• The paper validates its model by matching calculated expected payoffs with experimental results from 300 rounds across 60 groups.

Analysis of Social Cycling and Conditional Responses in the Rock-Paper-Scissors Game

The study titled "Social cycling and conditional responses in the Rock-Paper-Scissors game" explores human decision-making in non-cooperative strategic interactions, using the Rock-Paper-Scissors (RPS) game as a model. This research tests the predictive capacity of classical game theory, which advocates for Nash equilibrium (NE) with infinite rationality, versus evolutionary game theory, which anticipates cyclic motions, particularly in finite populations with bounded rationality.

Experimental Overview

The authors conducted laboratory experiments with 360 students distributed into 60 populations, each of six individuals. These groups engaged in 300 rounds of the discrete-time iterated RPS game, under a random pairwise-matching protocol. The study aimed to observe real-world deviations from the predicted NE state, where every player's choices are randomized to maintain unpredictability and, thus, prevent exploitation.

Key Findings

1. Cyclic Motions: The study observes population-level cyclic motions inconsistent with the NE framework. Despite variations in the payoff parameter $a$, the cycling direction and frequency remained generally consistent. Such behavior indicates persistent deviations from the NE, hinting at a universal pattern of social state cycling.
2. Conditional Response Model: The authors introduce a win-lose-tie conditional response model that aligns with empirical observations of cyclic behavior without parameter adjustments. This model suggests players utilize a predictable response strategy based on their previous game's outcome (win, lose, or tie), yielding higher payoffs than the NE mixed strategy.
3. Theoretical and Empirical Concordance: The calculated values of expected payoffs under the proposed model showed close alignment with experimental results, underscoring the model's effectiveness in explaining player behavior in RPS games.
4. Strategic Implications: Interestingly, the action marginal distribution of individuals under the conditional response strategy is indistinguishable from NE mixed strategy yet results in higher payoff allocations. This suggests that players potentially maximize their returns through optimized conditional responses, a strategy that may outperform NE in a practical setting.

Implications and Future Directions

The research challenges the established perception of NE as the ultimate strategy for rational decision-making in competitive contexts, particularly in finite-population scenarios where bounded rationality is prevalent. It opens the door for exploring more complex behavioral strategies in similar game-theoretic models and emphasizes the significance of empirical validation in theoretical constructs.

In terms of future implications, this paper posits that increasing game repetition might further entrench these cycles, potentially refining conditional response strategies for better outcomes. Additionally, understanding the brain's decision-making circuitry could be explored, potentially bridging the gap between psychological behavior and strategic decision-making.

Ultimately, this work encourages further empirical and theoretical investigations into decision-making paradigms, particularly how collective versus individual strategies evolve in fundamental game systems. Such studies could refine our understanding of human strategic behavior, extending beyond idealized rational models to incorporate realistic, bounded rationality approaches.