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.