Suppose we have a game between two players in which they take alternating turns. The game can have finite length, length $\omega$ or any transfinite number of steps (however, I'm not concerning games which are continuous). Every game has a *winning condition*, which can be interpreted as the set of all game histories possible which result in a win for player 1, and other such set which specifies winning for player 2. Here we are only interested in games which always end (possibly after transfinite time) and which have no ties (if player 1 doesn't win, then player 2 wins).

We define a *strategy* for player 1 as a function from the possible partial (i.e. up to some move of 1) history of the game to an allowed move which 1 can make. We say that strategy for player 1 is *winning* if it guarantees him a win in the game provided that he follows the strategy. Same for strategy and winning strategy for player 2. The game is *determined* if one of the players has a winning strategy. Otherwise, the game is *undetermined*.

It's a well-known result that, if we assume axiom of choice, then there is an undetermined game of length $\omega$ with allowed moves being elements of $\Bbb N$. This is necessarilly a non-constructive proof. It's also known that there is an undetermined game in which players play for time $\omega_1$ with moves being bits, or they play countable ordinals for time $\omega$, or they play elements of $\mathcal{P}(\Bbb R)$ for time $\omega$, without assuming AC. However, only proofs of these facts which I know of are by contradiction.

A game which is somewhat closer to being constructive example is the following game which uses non-principal ultrafilter on $\Bbb N$: players alternatingly form increasing sequence of numbers $a_1,a_2,...\in\Bbb N$. Then we partition $\Bbb N$ into two sets, $[0,a_1)\cup[a_2,a_3)\cup...$ which belongs to player 1, and $[a_1,a_2)\cup[a_3,a_4)\cup...$ which belongs to player 2. The winner is the player whose set is in the non-principal ultrafilter. Simple strategy-stealing shows that neither player has a winning strategy.

All other examples of games which are undetermined are heavily non-constructive, which made me ask the following question:

Are there any explicit examples of games which are not determined?

With "explicit" I mean that we can provide the example without assuming existence of any sets beyond these which ZF can prove are existing.