  Cooperation is an important research object in economics, sociology, and
biology, and the evolution of cooperation in structured populations is a
interesting research topic. We mainly focus on the evolution of cooperation
with two-order and higher-order game in two-layer network. We introduce a bias
coefficient of utility function and study the influence of bias coefficient on
the evolution of cooperation in two-layer network. We firstly provide
theoretical analysis of fixation probabilities of two-order and higher-order
game under weak selection in two-layer network.Secondly,based on the expression
of fixation probability, we obtain the critical value of the two different
games by comparing the size relationship of fixation probability under weak
selection condition and neutral selection condition. Finally, by comparing the
relationship between the critical value of single-layer and two-layer network
in two-order game and higher-order game, when the nonlinear factor satisfies
certain conditions, it is concluded that when the optimal bias coefficient
tends towards 0 is met, some two-layer networks promote the evolution of
cooperative behavior more than some single-layer networks.
