  Fictitious Play (FP) is a simple and natural dynamic for repeated play in
zero-sum games. Proposed by Brown in 1949, FP was shown to converge to a Nash
Equilibrium by Robinson in 1951, albeit at a slow rate that may depend on the
dimension of the problem. In 1959, Karlin conjectured that FP converges at the
more natural rate of $O(1/\sqrt{t})$. However, Daskalakis and Pan disproved a
version of this conjecture in 2014, showing that a slow rate can occur,
although their result relies on adversarial tie-breaking. In this paper, we
show that Karlin's conjecture is indeed correct for the class of diagonal
payoff matrices, as long as ties are broken lexicographically. Specifically, we
show that FP converges at a $O(1/\sqrt{t})$ rate in the case when the payoff
matrix is diagonal. We also prove this bound is tight by showing a matching
lower bound in the identity payoff case under the lexicographic tie-breaking
assumption.
