Tie breaking for singular channels is worth 1 Nat

We consider the finite-blocklength performance of singular channels (e.g., the binary erasure channel). At least for symmetric singular channels, it is known that the next correction term after the channel dispersion is the “constant” term. We show that for such channels, the asymptotic significance of tie breaking (i.e., making a fair decision in the case of multiple codewords of equal likelihood) is greater than for non-singular channels. Specifically, for an ensemble of codebooks, where the codewords are independent with the same marginal distribution, the constant correction term is increased by exactly one nat.