How to play the one-lie Rényi–Ulam game

The one-lie Rényi–Ulam liar game is a two-player perfect information zero-sum game, lasting qq rounds, on the set [n]≔{1,…,n}[n]≔{1,…,n}. In each round Paul chooses a subset A⊆[n]A⊆[n] and Carole either assigns one lie to each element of AA or to each element of [n]∖A[n]∖A. Paul wins the original (resp. pathological) game if after qq rounds there is at most one (resp. at least one) element with one or fewer lies. We exhibit a simple, unified, optimal strategy for Paul to follow in both games, and use this to determine which player can win for all q,nq,n and for both games.