In this paper, we consider cooperative games arising from total domination problem on graphs. We introduce two games, rigid total dominating set game and relaxed total dominating set game, and focus on their cores. First, a common necessary and sufficient condition for the balancedness of the two total dominating set games is obtained. Next, the computational issues on cores are discussed. We prove that in general the problems of testing the balancedness and testing the membership of the core are all $\cal {NP}$-hard for both total dominating set games.