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A co-evolving and adaptive Rock (R)-Paper (P)-Scissors (S) game (ARPS) inwhich an agent uses one of three cyclically dominating strategies is proposedand studied numerically and analytically. An agent takes adaptive actions toachieve a neighborhood to his advantage by rewiring a dissatisfying link with aprobability $p$ or switching strategy with a probability $1-p$. Numericalresults revealed two phases in the steady state. An active phase for$p<p_{\text{cri}}$ has one connected network of agents using differentstrategies who are continually interacting and taking adaptive actions. Afrozen phase for $p>p_{\text{cri}}$ has three separate clusters of agents usingonly R, P, and S, respectively with terminated adaptive actions. A mean-fieldtheory of link densities in co-evolving network is formulated in a general waythat can be readily modified to other co-evolving network problems of multiplestrategies. The analytic results agree with simulation results on ARPS well. Wepoint out the different probabilities of winning, losing, and drawing a gameamong the agents as the origin of the small discrepancy between analytic andsimulation results. As a result of the adaptive actions, agents of higherdegrees are often those being taken advantage of. Agents with a smaller(larger) degree than the mean degree have a higher (smaller) probability ofwinning than losing. The results are useful in future attempts on formulatingmore accurate theories.

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