In combinatorial game theory, the strategystealing argument is a general argument that shows, for many twoplayer games, that the second player cannot have a guaranteed winning strategy. The strategystealing argument applies to any symmetric 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.
The argument works by obtaining a contradiction. A winning strategy is assumed to exist for the second player, who is using it. But then, roughly speaking, after making their 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.
Examples of games to which the argument applies are hex and the m,n,kgames such as gomoku. In hex ties are not possible, so the argument shows that it is a firstplayer win.
Contents

Example 1

Chess 2

Go 3

Constructivity 4

References 5
Example
A strategystealing argument can be used on the example of the game of tictactoe, for a board and winning rows of any size. Suppose that the second player is using a strategy, S, which guarantees them a win. The first player places an X in a random position, and the second player then responds by placing an O according to S. But if they ignore the first random X that they placed, the first player finds themselves in the same situation that the second player faced on their first move; a single enemy piece on the board. The first player may therefore make their moves according to S  that is, unless S calls for another X to be placed where the ignored X is already placed. But in this case, the player may simply place his X in some other random position on the board, the net effect of which will be that one X is in the position demanded by S, 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 ignored X of no consequence). But then the second player has lost  contradicting the supposition that they had a guaranteed winning strategy. Such a winning strategy for the second player, therefore, does not exist, and tictactoe is either a forced win for the first player or a tie. Further analysis shows it is in fact a tie.
Chess
There is a class of chess positions called Zugzwang in which the player obligated to move would prefer to "pass" if this were allowed. Because of this, the strategystealing argument cannot be applied to chess. 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 statistics from modern highlevel games have White winning's percentage about 10% higher than Black's.
Go
In Go 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,^{[1]} the second player is typically awarded some compensation points, which makes the starting position asymmetrical, and the strategystealing argument will no longer work.
Constructivity
The argument shows that the second player cannot win, by means of deriving a contradiction from any purported winning strategy for the second player. According to the BHK interpretation, the most widely used basis for constructive interpretation of logical formulae, this is constructive.
The argument is commonly employed in games where there can be no draw to show that the first player has a winning strategy, such as in Hex. This application of the argument is usually nonconstructive, where the inference from the absence of a strategy and the impossibility of a draw is made by means of the law of the excluded middle. For finite games, and games where the appropriate instance of Markov's rule can be constructively established by means of bar induction, then the nonconstructive proof of a winning strategy for the first player can be converted into a winning strategy.
References

^ Fairbairn, John, History of Komi, retrieved 20100409
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