In this paper, we study the social optima for a kind of time-inconsistent linear-quadratic (LQ) mean-field problem. We first propose the concept of \(\varepsilon\) -social equilibrium strategy for our problem within the game theoretic framework. Next, an auxiliary time-inconsistent control problem is constructed by the scheme of person-by-person optimality and a set of linear feedback equilibrium controls for this auxiliary problem is presented explicitly via several ordinary differential equations (ODEs). Then we can derive a new kind of consistency condition (CC) system consisting of a forward stochastic differential equation (SDE) and three flows of backward stochastic differential equations (BSDEs), and obtain a set of decentralized admissible controls for the original problem. The wellposedness of this kind of CC system is also discussed in this paper. We prove that the set of admissible controls given above is indeed an \(\varepsilon\) -social equilibrium strategy for our original problem. Finally, a numerical example is presented to illustrate our theoretic results visually.
