Hypergame Paradox

I've recently hear about this paradox which I didn't knew before. It surprised me, because I'm a fan of paradoxes, and this one is quite neat. I also wanted to post about it, because when I tried to google for it, the first few hits weren't really relevant, or seemed to have an incorrect explanation of the paradox.

So, this paradox is about games. In particular, we are interested in 2-player games which are played in turns. We say that one of such games is a finite game if, no matter what the players do, the game eventually ends after a finite number of turns. For example, Chess is not a finite game since, after playing for a while, the players might decide to pick one of their pieces, say a rook, and forever move it back and forth between two squares on the board. On the other hand Tic-Tac-Toe is a finite game, since it can only take at most 9 turns, when the board will be full and the game will end either in a tie, or with one of the two players winning.

Now, hypergame, is a really fun game whose rules are as follows: the first player starts by choosing a finite game to play; then the second player has the first turn in the game selected, and then they keep playing that game as usual. Whoever wins the game selected by the first player, is also the winner of hypergame.

It turns out, interestingly, that hypergame is also a finite game! Observe that, no matter which game the first player chooses to play, we know that the selected game is a finite game, and must also terminate after a finite number of turns.

However... if hypergame is a finite game, the first player then might also decide “let's play hypergame” as his first move. The second player, which has to start now playing hypergame, might choose as well “let's play hypergame!”, ... and so on, forever, ...

So, we are left with the paradoxical question, is hypergame a finite game or not?