 In a back-and-forth game there are two structures \(A\) and \(B\), and two players who are commonly called Spoiler and Duplicator. (The names are due to Joel Spencer in the early 1990s. More recently Neil Immerman suggested Samson and Delilah, using the same initials; this places Spoiler as the male player \(\forall\) and Duplicator as the female \(\exists\).) Each step in the game consists of a move of Spoiler, followed by a move of Duplicator. Spoiler chooses an element of one of the two structures, and Duplicator must then choose an element of the other structure. So after \(n\) steps, two sequences have been chosen, one from \(A\) and one from \(B\):