What is the simplest way to prove that each finite game is also determined? I know that a game is said to be determined if one of the players has a winning strategy. I was hoping to prove by contradiction that assuming in a finite game Player 1 does not have a winning strategy it needs to be the case that Player 2 has the winning strategy, but maybe there is an easier way. 



Determined, finite games
There are numerous proofs of what I call the fundamental theorem of finite
games.


**Theorem.** (Fundamental theorem of finite games)
 In any finite two-player game of perfect information, one of the players has a winning strategy.


**Proof 1.** Back-propagation through the game tree. Label the
nodes with the player who has a winning strategy from that
position. Every node will get a label, by recursively labeling from the terminal nodes of the tree (which amounts to working backwards from the won-game positions). What is the label on the root node? The winning strategy
is to stay on the nodes with that label.**QED**


**Proof 2.** Let $W$ be the nodes in the game tree for which player
I has a winning strategy in the game proceeding from that position.
If this includes the top node, then I wins. Otherwise, it is not difficult to see that player 
II can avoid the nodes in W, and therefore win. **QED**


**Proof 3.** The theorem amounts to the de Morgan law. The
assertion "player II has a winning strategy" is simply $$\forall
x\_1\exists x\_2\dots \vec x\text{ is a win for II}.$$ So if player II
does not have a winning strategy, we negate that assertion, and
push the negation through all the quantifiers, by de Morgan's law,
arriving at: $$\exists x\_1\forall x\_2\dots\vec x\text{ is a win for
I}.$$ And this is what it means for player I to have a winning
strategy. **QED**


**Proof 4.** First prove the Gale-Stewart theorem that every open
game is determined, which itself has several classic proofs. For example,
you can see some of them explained in the introduction of my recent
article,


* V. Gitman, J. D. Hamkins, [Open determinacy for class
games](http://jdh.hamkins.org/open-determinacy-for-class-games/),
to appear.


One of these proofs involves transfinite game values, but in the case of bounded-length finite games, all the game values will be finite. Finally, note that any finite game is an open game. **QED**


Most of these arguments (all except the de Morgan argument) generalize to prove the determinacy of clopen games, rather than merely uniformly bounded-length finite games. A clopen game is a game whose game tree is well-founded, or in other words, every play terminates in a win for one of the players. 


