Mistakes can stabilise the dynamics of rock-paper-scissors games

Author summary A game of rock-paper-scissors is more than just a children's game. This type of interactions is often used to describe competition among animals or humans. A special feature of such an interaction is that none of the pure strategies dominates, resulting in a cyclic pattern. However, in wild communities such interactions are rarely observed by biologists. Our results suggest that this lack of cyclicity may stem from imperfectness of interacting individuals. In other words, we show analytically that heterogeneity in behavioural patterns may break a cyclic relationship and lead to a stable equilibrium in pure or mixed strategies. A game of rock-paper-scissors is an interesting example of an interaction where none of the pure strategies strictly dominates all others, leading to a cyclic pattern. In this work, we consider an unstable version of rock-paper-scissors dynamics and allow individuals to make behavioural mistakes during the strategy execution. We show that such an assumption can break a cyclic relationship leading to a stable equilibrium emerging with only one strategy surviving. We consider two cases: completely random mistakes when individuals have no bias towards any strategy and a general form of mistakes. Then, we determine conditions for a strategy to dominate all other strategies. However, given that individuals who adopt a dominating strategy are still prone to behavioural mistakes in the observed behaviour, we may still observe extinct strategies. That is, behavioural mistakes in strategy execution stabilise evolutionary dynamics leading to an evolutionary stable and, potentially, mixed co-existence equilibrium.

Automated summary

The game of rock-paper-scissors is used as a model to understand competition among animals or humans, where none of the pure strategies strictly dominates, resulting in a cyclic pattern. However, behavioral mistakes in strategy execution can break this cyclic relationship and lead to a stable equilibrium with only one strategy surviving. This study shows that imperfectness in interacting individuals may stabilize evolutionary dynamics and potentially result in mixed co-existence equilibrium.