Adaptive cyclically dominating game on co-evolving networks: Numerical and analytic results

A co-evolving and adaptive Rock (R)-Paper (P)-Scissors (S) game (ARPS) in which an agent uses one of three cyclically dominating strategies is proposed and studied numerically and analytically. An agent takes adaptive actions to achieve a neighborhood to his advantage by rewiring a dissatisfying link with a probability $p$ or switching strategy with a probability $1-p$. Numerical results revealed two phases in the steady state. An active phase for $p<p_{\text{cri}}$ has one connected network of agents using different strategies who are continually interacting and taking adaptive actions. A frozen phase for $p>p_{\text{cri}}$ has three separate clusters of agents using only R, P, and S, respectively with terminated adaptive actions. A mean-field theory of link densities in co-evolving network is formulated in a general way that can be readily modified to other co-evolving network problems of multiple strategies. The analytic results agree with simulation results on ARPS well. We point out the different probabilities of winning, losing, and drawing a game among the agents as the origin of the small discrepancy between analytic and simulation results. As a result of the adaptive actions, agents of higher degrees are often those being taken advantage of. Agents with a smaller (larger) degree than the mean degree have a higher (smaller) probability of winning than losing. The results are useful in future attempts on formulating more accurate theories.