Has every sentence in first-order logic a gaming metaphor? In Terence Tao's book Analysis 1 (see sample chapters), he explains quantifiers through so-called gaming metaphors – here an example (for $\forall\exists$)

To continue the gaming metaphor, suppose you play a game
  where your opponent first picks a positive number x, and then you
  pick a positive number y. You win the game if y
  2 = x. If you can
  always win the game regardless of what your opponent does, then
  you have proven that for every positive x, there exists a positive
  y such that y
  2 = x.

I figured out that one can imagine a $\exists\forall$ sentence, say $\exists x\forall y: P(x, y)$, also a gaming metaphor: I choose an $x$, then my oponent chooses any $y$; I have won iff $P(x, y)$. If I can always win this game with a fixed $x$, then the sentence $\exists x\forall y: P(x, y)$ is true.
Has every sentence in first-order logic a gaming metaphor?
I gave two examples: $\forall\exists$ and $\exists\forall$ – but there are so many other combinations of these quantifiers, for example $\exists\forall \exists\forall\exists$. It's hard for me to find a corresponding gaming metaphor for these types of sentences.
 A: Yes, imagine there are two players — call them $\exists$ and $\forall.$
Put the sentence into prenex normal form: first all the quantifiers (the prefix part), followed by a part without any quantifiers at all (the matrix part).
The two players make moves in the order specified by the quantifiers.  Each move is simply a value for the indicated variable.  So, for instance, if the first quantifier is $\exists x,$ then the first move is made by player $\exists$ and the move itself is a choice of a value for $x.$
After all the moves are made, player $\exists$ wins if the matrix (the non-quantifier part, after all the quantifiers) is true for the chosen values of the variables; otherwise, player $\forall$ wins.
Then the sentence is true iff there's a winning strategy for player $\exists,$ and the sentence is false iff there's a winning strategy for player $\forall.$
A: Yes, though I'd call it an interpretation rather than a metaphor. See Jaakko Hintikka, Game-theoretical semantics: insights and prospects, Notre Dame J. Formal Logic 23 (1982), 219–241.
