 There are three important comments to make on the distinction between perfect and imperfect information. The first is that the Gale-Stewart theorem holds only for games of perfect information. Suppose for example that \(\forall\) and \(\exists\) play the following game. First, \(\forall\) chooses one of the numbers 0 and 1. Then \(\exists\) chooses one of these two numbers. Player \(\exists\) wins if the two numbers chosen are the same, and otherwise player \(\forall\) wins. We require that \(\exists\), when she makes her choice, doesn’t know what \(\forall\) chose; so a Skolem function for her will be a constant. (This game corresponds to the IF sentence above with \(R\) read as equality, in a structure with a domain consisting of 0 and 1.) It’s clear that player \(\exists\) doesn’t have a constant winning strategy, and also that player \(\forall\) doesn’t have a winning strategy at all. So this game is undetermined, although its length is only 2.