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 image rounds, on the set image. In each round Paul chooses a subset image and Carole either assigns one lie to each element of image or to each element of image. Paul wins the original (resp. pathological) game if after image 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 image and for both games.