Strategy-stealing argument

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 strategy-stealing 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.

Strategy-Stealing was invented by John Nash in the 1940s to show that the game of hex is always a first-player win, as ties are not possible in this game.^[1] However, Nash did not publish this method, and Beck (2008) credits its first publication to Alfred W. Hales and Robert I. Jewett, in the 1963 paper on tic-tac-toe in which they also proved the Hales–Jewett theorem.^[2] Other examples of games to which the argument applies include the m,n,k-games such as gomoku. In the game of Sylver coinage, strategy stealing has been used to show that the first player wins, rather than that the game ends in a tie.^[3]

Example

A strategy-stealing argument can be used on the example of the game of tic-tac-toe, for a board and winning rows of any size.^[1][2] Suppose that the second player is using a strategy, S, which guarantees them a win. The first player places an X in a arbitrary 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 tic-tac-toe 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 strategy-stealing argument cannot be applied to chess.^[4] 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 high-level games have White's winning 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,^[5] the second player is typically awarded some compensation points, which makes the starting position asymmetrical, and the strategy-stealing argument will no longer work. See also mirror go.

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. However, it does not provide an explicit strategy for the first player, and because of this it has been called non-constructive.^[4] This usage of the word "constructive" is informal, not matching the definitions of constructive mathematics. According to the BHK interpretation, the most widely used basis for constructive interpretation of logical formulae, the fact that the second player has no winning strategy 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 non-constructive, 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 games with a finite number of reachable positions, and games where the appropriate instance of Markov's rule can be constructively established by means of bar induction, then the non-constructive proof of a winning strategy for the first player can be converted into a winning strategy.

References