Tactics versus strategies in the theory of games

How do tactics differ from strategies? Does the fundamental theorem of finite games hold for tactics? Must every finite game have a winning tactic for one player or drawing tactics for both?

This is part one of a series of essays on tactics and strategies in the theory of games.

Tactics

In the theory of games, let us explore the difference between tactics and strategies. Game-theoretically, what is a tactic?

In informal usage, the word “tactic” commonly refers to a local plan of action concerned with specific, limited aims, while strategies in contrast have global ambitions; one thinks of the difference between a battle and the larger war of which it is a part. In chess, we have the various familiar tactics—forks, pins, skewers, sacrifices, discovered attacks, removal of the guard, and so forth. In each case, the tactic is a specific kind of move or sequence of moves characterized by certain local goals and local arrangements of pieces, rather than a consideration of the whole game board or the history of the game play that had been undertaken to reach that position. Chess tactics are often vitally important techniques to consider, for they often provide the key manner of play to achieve a favorable outcome.

In this essay, however, I shall be interested in the abstract game-theoretic notion of tactic, using the word with its specific technical meaning in the theory of games, a meaning which is not exactly the same as the common usage of this term in chess, say, although it does retain something of the local-character aspect.

Let me explain. We have defined in the theory of games that a strategy is a function that tells a player exactly how to move in any given game situation—a strategy is a function on the full game tree, mapping the nodes for which it is that player's turn to the child node arising from the strategy's recommended move. Since nodes in the game tree in effect encode the entire history of how play got to that position, a strategy thus in effect has access to that game-history information.

A tactic, in contrast, tells a player exactly how to move given only the current board position, that is, the current state of affairs on the playing board, without knowing the game-playing history of how we got to that state. So the difference between a strategy and a tactic is that a strategy makes its recommendation knowing the whole history of play to the current state of affairs, while a tactic must make its recommendation based solely on the current board position.

The notion of tactic therefore makes sense only for games that do indeed have something that might be called the game board, and the concept of tactic is sensitive to which information exactly is taken to be included as part of the board position. For many games, to be sure, including Hex, Connect Four, Othello, tic-tac-toe, and Nim, there is a clear and natural notion of board position—one need only look at the playing board (plus perhaps the turn indicator saying whose turn it is) and know everything about the complete space of how the game could possibly continue. For these games, therefore, we have a corresponding clear and natural notion of tactic.

With chess, in contrast, the standard conception of “board position” is a little more unsettled and perhaps even ambiguous. If we think of the board position as consisting only of the information visible by looking at the chessboard—a photograph of the board—then we wouldn't necessarily know whether castling was possible, since perhaps we had already moved the rook or king; we wouldn't necessarily know whether en passant was possible, since applicability of this right depends on the previous move; we wouldn't necessarily know if our opponent has castling privileges or not, and we wouldn't necessarily know the extent to which we are subject to draw by three-fold repetition or the fifty-move rule. We just can't always determine this sometimes crucial information by looking at the playing board.

In regard to most common games the notion of “board position” that we are talking about here is actually closer to how people use the term “position” in discussions of the game. In a newspaper chess puzzle, for example, we generally find the state of the game on the board, without knowing exactly the history of play how it was arrived at, even if there are conventions for part of this information, concerning the possibility of castling and en passant.

Meanwhile, for any given game one can present a strategically equivalent form of it by imagining ourselves to be playing the game literally on the game tree itself. We imagine the entire game tree spread before us on a great lawn, and making a move in the game consists of stepping from the current node to a child node in the tree. This way of playing is strategically isomorphic to the original game, since it has the same game tree, and the key observation is that with this game-tree concept of game board, strategies and tactics are the same thing.

The general point I am trying to make here is that when speaking of tactics, we must specify exactly which information is available on the game board for the tactics to consult when making their recommendation. The same game might have several different natural conceptions of what constitutes the “game board.”

Winning tactics

For which games should we expect winning tactics for one player or the other as opposed merely to winning strategies? Is there a winning tactic in chess, for example, or drawing tactics for both players? Indeed, which of our familiar common games admit winning and drawing tactics, as opposed to winning and drawing strategies? More generally:

Question. Does the fundamental theorem of finite games hold generally for tactics?

In other words, in every finite game, must one player have a winning tactic or both players have drawing tactics?

Interlude

We shall see the answers momentarily, first with a variety of examples, including some showing a surprising aspect of the question. In the end we shall prove a very general result, and eventually consider the case of tactics in chess.