In this paper we investigate the game chromatic number for complete multipartite graphs. Our results consist of two findings: (1) building on Dunn's strategies for equal-sized parts, we determine the closed-form formulas for the game chromatic number for all complete multipartite graphs without singletons, with respective optimal strategies for Alice and Bob, (2) for an explicit list of part sizes, we give an exact algorithm that computes the game chromatic number in polynomial time and allows for reconstructing an optimal strategy for both players. Additionally, we provide counterexamples to earlier, partial formulas for complete multipartite graphs with singletons and large equal-sized parts.

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