  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.
