Gamepiece Gambit

Alice and Bob are playing a board game. The board is divided into positions labeled a, b, c, d, . . . and the players use a gamepiece to mark the current position. Each round of the game consists of two steps: 1. Alice makes a choice. Depending on the current position, she has different options, where each option is a set of positions. Alice chooses one set S among the available sets of positions.