 When Hintikka used Skolem functions as strategies in his games for first-order logic, he made the strategies for a player depend only on the previous moves of the other player. (A Skolem function for \(\exists\) depends only on universally quantified variables.) Because the games were games of perfect information, there was no loss in this, by the second comment above. But when he moved to IF logic, the requirement that strategies depend only on moves of the other player really did make a difference. Hodges 1997 showed this by revising the notation, so that for example \((\exists y/x)\) means: “There is \(y\) independent of \(x\), regardless of which player chose \(x\)”.