Rules.of.Play.Game.Design.Fundamentals [Electronic resources] نسخه متنی

Strategies in Game Theory

Decision trees help us understand how players move through the space of possibility of a game. To see how this works, think back to the Tic-Tac-Toe decision tree. The tree contains every conceivable move, in every possible iteration of the game. This is actually more information than we need. Most players will not randomly pick their next square, but will actively try and score three in a row while keeping an opponent from doing the same. With this in mind, we can start to trim all of the "stupid move" branches from our tree. Poundstone describes what this process of "trimming" would be like:

Go through the diagram and carefully backtrack from every leaf. Each leaf is someone's last move, a move that creates a victory or a tie. For instance, at Point A, it is X's move, and there is only one empty cell. X has no choice but to fill it in and create a tie.

Now look at Point B, a move earlier in the game. It is O's turn, and he has two choices. Putting an O in one of the two open cells leads to the aforementioned Point A and a sure tie. Putting an O in the other cell, however, leads to a win for X. A rational O player prefers a tie to an X victory. Consequently, the right branch leading upward from Point B can never occur in rational play. Snip this branch from the diagram. Once the play gets to Point B, a tie is a forgone conclusion.

But look: X could have won earlier, at Point C. A rational X would have chosen an immediate win at Point C. So actually, we can snip off the entire left branch of the diagram.

Keep pruning the tree down to the root, and you will discover that ties are the only possible outcomes of rational play.(There is more than one rational way of playing, though.) The second player can and will veto any attempt at an X victory, and vice-versa.[3]

From this pruned-down version of Tic-Tac-Toe, it is possible to create what game theory calls a strategy. A strategy in game theory parlance offers a more precise meaning than what is commonly meant by "strategy." A common understanding of a strategy in Starcraft might be: "If you're playing the Zergs, create a lot of Zerglings at the beginning of the game and rush your opponent's central structures before they have time to build power." A strategy in this casual sense is a set of general heuristics or rules of thumb that will help guide you as you play. However, a strategy in game theory means a complete description of how you should act at every moment of the game. Once you select a strategy in the game theory sense of the word, you do not make any other choices, because the strategy already dictates how you should act for the rest of the game, regardless of what the other player does. This can make game theory strategies quite intricate. Poundstone lists a sample strategy for Tic-Tac-Toe for the first player X.

Put X in the center square. O can respond two ways:

As you can see, the strategy for even a simple game such as Tic-Tac-Toe is somewhat complex. A complete strategy is ultimately a methodology for navigating the branches of a decision tree. A strategy proscribes exact actions for the player utilizing the strategy, but it also has to take into account all of the possible branches that an opposing player could select. In Poundstone's example, the strategy dictates the way that the first player would move from the root of the tree to the first of nine possible points. From there the opposing player could move to any of the other eight points, a move that the strategy has to take into account.

A complete strategy for a game such as Chess would be mindbogglingly huge. However, game theory does not study games as strategically complicated as Chess. In fact, the games that game theory studies are remarkably simple. But as we already know, even very simple games can play out in quite complex ways.

[3]Ibid. p. 46.

[4]Ibid. p. 48.