Dominating set is a subset of nodes called dominators in a graph such that every non-dominator nodes (called dominatee) is adjacent to at least one dominator. This paper considers a more general multi-dominating problem where each node$i$, dominator or dominatee, is required to have at least$k_i$neighboring dominators, and different node can have different$k_i$value. We first propose a game design toward this problem. This game is self-stabilizing (i.e., it always ends up with a legitimate state regardless of its initial configuration). The obtained result is guaranteed minimal (i.e., it contains no proper subset that is also a multi-dominating set) and Pareto optimal (we cannot increase the payoff of some player without sacrificing the payoff of any other). We then point out challenges when turning the design into a distributed algorithm using guarded commands. We present an algorithm that is proved weakly stabilizing. Simulation results show that the proposed game and algorithm produce smaller dominating sets,$k$-dominating sets, and multi-dominating sets in various network topologies when compared with prior approaches.