 The fact that one of the players wins a single play of game \(G(\phi , M)\) does not yet tell us anything about the truth-value of the sentence \(\phi\). Truth and falsity are characterized in terms of the notion of winning strategy. The sentence \((\forall x)(\exists y)(\forall z)(\exists w) R(x, y, z, w)\) is true in \(M\) if there is a winning strategy for player 2 in the game just described: a recipe telling player 2 what to do, when (a specified amount of) information about the opponent’s earlier moves is given. Technically, the requirement is that there be strategy functions \(f\) and \(g\), such that for any choices \(a\) and \(c\) by player \(1, R(a, f(a), c, g(a, c))\) holds in \(M\). Observe that strategy function \(f\) is a Skolem function for the quantifier \(\exists y\) in \(\phi\), and similarly \(g\) is a Skolem function for the quantifier \(\exists w\). The sentence \(\phi\) is false in \(M\) if there is a winning strategy (set of strategy functions) for player 1 in the corresponding game: a constant \(c\) and a function \(h\) such that for any choices \(b\) and \(d\) by player \(2, R(c, b, h(b), d)\) fails to hold in \(M\).