I have been studying some instances of strategy-stealing. The issue is that I am not sure whether proving the non-existence of a winning strategy for player 2 necessarily proves a winning (or at least tying) strategy for player 1. Wikipedia says
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy.
The arguments that I have seen make sense, in terms of proving that player 2 cannot have a winning strategy. For example, the Wikipedia article on Chomp makes the modest claim
The second player therefore cannot have a winning strategy.
However, in Engel's Problem-Solving Strategies, Double Chess is analyzed in problem #18 of the games chapter and the following seemingly grander claim is made:
The rules of chess are changed as follows: Black and White make alternately two legal moves. Show that there exists a strategy for white which guarantees him at least a tie.
The proof is
Suppose $B$ can win no matter what $A$ does. On his first move, $A$ moves one of his knights to any one of the two possible squares and then back to its original position. Now all the pieces are in their original position, but $A$ has becmoe the second player and must win. Contradiction!
When does the negation of the existence of a winning strategy for player 2 imply the existence of a winning/tying strategy for player 1?
I might be confused because I am actually not sure of how to formally describe a winning strategy, so a rigorous explanation of that might be helpful as a preface. (I considered using alternating existential and universal quantifiers to describe winning strategies, but didn't know how to terminate the sequence.)
P.S. I have included set theory as a tag because, according to the Wikipedia article on determinacy, the subject is a subfield of set theory. However, I am not familiar with this part of set theory and I did not really understand the article.
The Simplest Case
Simple Conditions
For now, let's consider two-player games of perfect information where there is a pre-known bound on the number of moves a game might take, and one of the two players wins in the end, and the first player is fixed (e.g. the player who controls the white pieces must move first).
Strategies
As mentioned in the comments, there is a theorem/collection of related theorems called Zermelo's theorem* (in game theory), that is relevant here. Summarized, it says that in a game of this type, either the first player to move or the second player to move must have a "winning strategy" — a function that provides a move for every situation that player could possibly be in (a strategy) which guarantees a win no matter how the opponent plays.
(*As an aside, most discussions of "Zermelo's theorem" aren't quite in line with what Zermelo wrote about and how his proof went. See Zermelo and the Early History of Game Theory by Ulrich Schwalbe and Paul Walker for some clarification about the history.)
The Argument
A full formal setup and proof would make this post much longer, and might not add much more conceptual understanding. Briefly, the idea is to use induction on, say, the maximum length of a game. A game with only one turn either has a winning move for the first player, in which case the winning strategy for them is "play one of the winning moves". Or there isn't, in which case the winning strategy for the second player is an empty function since they have no choices to make. For the induction step: if there are any moves to a "non-next player has a winning strategy" situation, the first player has a winning strategy to make one of those moves and then follow the existing strategy. If there are no such moves, then all moves are to a "next player has a winning strategy" position, and the second player can follow that strategy no matter which move the first player makes. Another, less brief, account can be found in Lecture 15 from the Open Yale Course ECON 159: Game Theory, which has accompanying lecture notes and transcript, etc.
Variants
There are many variants on the conditions that go beyond the simplest case, and I won't cover them all here, but I wanted to address the comments about Chess and give a picture of how this stuff can be extended.
Ties
Some games, like "Tic Tac Toe"/"Noughts and Crosses" don't satisfy the condition where someone must win in the end. The game can end in a "tie" (also known colloquially as a "draw", but we make a distinction in combinatorial game theory).
From a game $G$ like this, we can build a new game $G_1$ by breaking ties in favor for first player, and another game $G_2$ by breaking ties in favor of the second player. For $G_1$, Zermelo's Theorem tells us that either the first player or the second player has a winning strategy in $G_1$. This means that the first player can force at least a tie in $G$ or the second player can force a win in $G$. Analogously, Zermelo's Theorem applied to $G_2$ tells us that either the second player can force at least a tie in $G$ or the first player can force a win.
Now it's just a matter of listing the possibilities:
Repeated positions (like Chess)
Some games like chess have things like repeated board positions, which would seem to throw a wrench in the induction proof of Zermelo's Theorem. However, this is sort of an illusion because of things like the Chess's repetition rules. Chess doesn't allow real repetitions if you include a list of all previous positions with their frequency counts as part of the "position". With this clarified, Chess and similar games are basically just games with ties (though you could complicate the earlier story if you want to distinguish stalemates from repetition draws, etc.)
Repeated positions (for real)
Some other games (much less likely to be actually played) really can repeat positions, and hence can go on forever. For example, maybe Nim-but-you-can-return-objects-too, or "Fair Shares and Varied Pairs" (a description can be found in this mathstrek blog post or Winning Ways). For these so-called "loopy" games, we must consider "drawn" games which last forever, in addition to the ones that end.
It takes some extra work (see IV.4 of Combinatorial Game Theory, for instance), but it turns out there's a tidy description of things with just one more category in addition to first or second player having a winning strategy: those positions where both players can force the game to be drawn. So it's not obvious, but these drawn games basically work out the same as ties.
Partizan Games
If you study more combinatorial game theory, you'll encounter partizan games where players have an extra distinction aside from who moves first. For example, maybe you could consider a position on a chessboard and the "moves for Black" even if it would be White's turn in the original position. For the purposes of things like Zermelo's Theorem (ignoring a lot of good theory), this doesn't really matter. Given a partizan game, you could basically consider $G_1$ where Black moves first (and the color whose turn it is is part of the data of the position) and $G_2$ where White moves first and then apply a previous theorem to each of those.