  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.
