Asymptotic Neyman-Pearson games for converse to the channel coding theorem

Upper bounds have recently been derived on the maximum volume of length-n codes for memoryless channels subject to either a maximum or an average decoding error probability ε. These bounds are expressed in terms of a minmax game whose variables are n-dimensional probability distributions and whose payoff function is the power of a Neyman-Pearson test at significance level 1 - ε. We derive the exact asymptotics (as n → ∞) of this game by relating it to a problem that admits an asymptotic saddlepoint with an equalizer property.