Incombinatorial game theory, thestrategy-stealing argument is a generalargument that shows, for manytwo-player games, that the second player cannot have a guaranteedwinning strategy. The strategy-stealing argument applies to anysymmetric game (one in which either player has the same set of available moves with the same results, so that the first player can "use" the second player's strategy) in which an extra move can never be a disadvantage.[1] A key property of a strategy-stealing argument is that it proves that the first player can win (or possibly draw) the game without actually constructing such a strategy. So, although it might prove the existence of a winning strategy, the proof gives no information about what that strategy is.
The argument works by obtaining acontradiction. A winning strategy is assumed to exist for the second player, who is using it. But then, roughly speaking, after making an arbitrary first move – which by the conditions above is not a disadvantage – the first player may then also play according to this winning strategy. The result is that both players are guaranteed to win – which is absurd, thus contradicting the assumption that such a strategy exists.
Strategy-stealing was invented byJohn Nash in the 1940s to show that the game ofhex is always a first-player win, as ties are not possible in this game.[2] However, Nash did not publish this method, andJózsef Beck credits its first publication toAlfred W. Hales and Robert I. Jewett, in the 1963 paper ontic-tac-toe in which they also proved theHales–Jewett theorem.[2][3] Other examples of games to which the argument applies include them,n,k-games such asgomoku. In the game ofChomp strategy stealing shows that the first player has a winning strategy in any rectangular board (other than 1x1). In the game ofSylver coinage, strategy stealing has been used to show that the first player can win in certain positions called "enders".[4] In all of these examples the proof reveals nothing about the actual strategy.
A strategy-stealing argument can be used on the example of the game oftic-tac-toe, for a board and winning rows of any size.[2][3] Suppose that the second player (P2) is using a strategyS which guarantees a win. The first player (P1) places anX in an arbitrary position. P2 responds by placing anO according toS. But if P1 ignores the first randomX, P1 is now in the same situation as P2 on P2's first move: a single enemy piece on the board. P1 may therefore make a move according toS – that is, unlessS calls for anotherX to be placed where the ignoredX is already placed. But in this case, P1 may simply place anX in some other random position on the board, the net effect of which will be that oneX is in the position demanded byS, while another is in a random position, and becomes the new ignored piece, leaving the situation as before. Continuing in this way,S is, by hypothesis, guaranteed to produce a winning position (with an additional ignoredX of no consequence). But then P2 has lost – contradicting the supposition that P2 had a guaranteed winning strategy. Such a winning strategy for P2, therefore, does not exist, and tic-tac-toe is either a forced win for P1 or a tie. (Further analysis shows it is in fact a tie.)
The same proof holds for anystrong positional game.
| a | b | c | d | e | f | g | h | ||
| 8 |      | 8 | |||||||
| 7 | 7 | ||||||||
| 6 | 6 | ||||||||
| 5 | 5 | ||||||||
| 4 | 4 | ||||||||
| 3 | 3 | ||||||||
| 2 | 2 | ||||||||
| 1 | 1 | ||||||||
| a | b | c | d | e | f | g | h | ||
There is a class ofchess positions calledZugzwang in which the player obligated to move would prefer to "pass" if this were allowed. Because of this, the strategy-stealing argument cannot be applied to chess.[5] It is not currently known whether White or Black can force a win with optimal play, or if both players can force a draw. However, virtually all students of chess consider White's first move to be an advantage and White wins more often than black in high-level games.
InGo passing is allowed. When the starting position is symmetrical (empty board, neither player has any points), this means that the first player could steal the second player's winning strategy simply by giving up the first move. Since the 1930s, however,[6] the second player is typically awarded somecompensation points, which makes the starting position asymmetrical, and the strategy-stealing argument will no longer work.
An elementary strategy in the game is "mirror go", where the second player performs moves which are diagonally opposite those of this opponent. This approach may be defeated usingladder tactics,ko fights, or successfully competing for control of the board's central point.
The strategy-stealing argument shows that the second player cannot win, by means of deriving a contradiction from any hypothetical winning strategy for the second player. The argument is commonly employed in games where there can be no draw, by means of thelaw of the excluded middle. However, it does not provide an explicit strategy for the first player, and because of this it has been called non-constructive.[5] This raises the question of how to actually compute a winning strategy.
For games with a finite number of reachable positions, such aschomp, a winning strategy can be found by exhaustive search.[7] However, this might be impractical if the number of positions is large.
In 2019, Greg Bodwin and Ofer Grossman proved that the problem of finding a winning strategy isPSPACE-hard in two kinds of games in which strategy-stealing arguments were used: theminimum poset game and the symmetricMaker-Maker game.[8]