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.”

Games as graphs

Let me introduce a slightly more abstract graph-theoretic way for thinking about games, a perspective that will assist in thinking about tactics as opposed to strategies. For a given game with a given concept of “board position,” we will think of the game at bottom as constituted by a certain labeled directed graph. Namely, we have a node in the graph for each possible board position, all the various possible states of the game. We place an edge from board position p to board position q, labeled for a player, if that player can make a legal move in the game transforming the game state from p to q. If the turn indicator is part of the board-position information, the label information is redundant, since only one player would have a legal move from that board position, and so it can be omitted. Some of the board positions are terminal, with no outgoing edges, and they are labeled as wins for one player or the other or as draws in the game. One of the board positions is labeled as the starting position.

Any such labeled directed graph can be conceived as a game. A play in such a game is simply a path in the graph, starting at the start node and alternately following edges for one player and then the other. We play in such a game by jumping from one node to another, following the edges, just as we conceive of playing a game on the game tree by stepping successively from nodes to child nodes. The winner of such a play is determined by the label on the terminal node, if such a node is reached. The game is a finite-play game if all plays of the game are finite.

A tactic for a player is a function on the graph mapping each board state to an outgoing edge for that player (when there is one). Such a tactic is winning for that player, if all plays of the game in which that player has played in accordance with the tactic lead eventually to a win for that player.

For a game given by a game tree, a board position conception of the game consists of a labeled graph of the type I have mentioned and a mapping of the nodes of the tree to nodes in the graph, mapping each game position to a corresponding board position, in such a way that the move structure is respected. That is, every possible move in the game tree corresponds to a move for that player in the graph from the corresponding node, and conversely, every possible move in the graph corresponds to a move in the game tree for any position having that board state.

Meanwhile, this graph-theoretic conception is not necessarily the most general way to think about tactics, since with some concepts of board position, the moves of the game are not actually well-defined on the board position. For example, in the game of chess, if one's conception of board position consists just of the state of pieces on the chessboard, then one cannot tell from this whether a certain move such as castling or en passant is legal. Or even if that information is taken to be part of the board position, one cannot tell whether a given move will cause the end of the game by the 50-move rule, and so these conceptions of board position will not respect the move structure as required.

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.