Summary: Given a graph \(G\) and an integer \(k\), two players take turns coloring the vertices of \(G\) one by one using \(k\) colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of \(G\) are colored. The game chromatic number \(\chi_g(G)\) is the minimum \(k\) for which the first player has a winning strategy. The paper of [T. Bohman et al., Random Struct. Algorithms 32, No. 2, 223–235 (2008; Zbl 1139.05017)] began the analysis of the asymptotic behavior of this parameter for a random graph \(G_{n,p}\). This paper provides some further analysis for graphs with constant average degree, i.e., \(np=O(1)\), and for random regular graphs. We show that with high probability (w.h.p.) \(c_1\chi(G_{n,p})\leq \chi_g(G_{n,p})\leq c_2\chi(G_{n,p})\) for some absolute constants \(1<c_1< c_2\). We also prove that if \(G_{n,3}\) denotes a random \(n\)-vertex cubic graph, then w.h.p. \(\chi_g(G_{n,3})=4\).