Non-convergence of proportions of types in a preferential attachment graph with three co-existing types

We consider the preferential attachment model with multiple vertex types introduced by Antunovi ́c, Mossel and R ́acz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunovi ́c, Mossel and R ́acz. We also consider another family of examples where we show that the conjecture does hold.