The Computation of Nash Equilibrium in Fashion Games via Semi-Tensor Product Method

Using the semi-tensor product of matrices, this paper investigates the computation of pure-strategy Nash equilibrium (PNE) for fashion games, and presents several new results. First, a formal fashion game model on a social network is given. Second, the utility function of each player is converted into an algebraic form via the semi-tensor product of matrices, based on which the case of two-strategy fashion game is studied and two methods are obtained for the case to verify the existence of PNE. Third, the multi-strategy fashion game model is investigated and an algorithm is established to find all the PNEs for the general case. Finally, two kinds of optimization problems, that is, the so-called social welfare and normalized satisfaction degree optimization problems are investigated and two useful results are given. The study of several illustrative examples shows that the new results obtained in this paper are effective.