Every symmetric 3×3 global game of strategic complementarities has noise-independent selection

We prove that the global game selection in all 3×3 payoff-symmetric supermodular games is independent of the noise structure. As far as we know, all other proofs of noise-independent selection in such games rely on the existence of a so-called monotone potential (MP) maximiser. Our result is more general, since some 3×3 symmetric supermodular games do not admit an MP maximiser. As a corollary, noise-independent selection does not imply the existence of an MP maximiser, nor the existence of an equilibrium robust to incomplete information.