Two-Person Game Theory: The Basic Concepts
Game theory is about choices. In a game such as Advanced Dungeons and Dragons, there may be infinitely many choices. Such a game is not applicable to the laws of game theory. A game such as chess, checkers, or tic-tac-toe can be analyzed by game theory. Game theory is concerned not with games of imagination, but of games of strategy. Just as laws of probability prescribe how games of complete chance operate, laws of game theory apply to games of strategy.
While game theory cannot often determine the best possible strategy, it can determine whether there one exists. Two ideas that can outline games are the game tree and the game matrix. A game tree is a diagram of possible choices that can outline every single possibility within the game. In a game of rock-paper-scissors, there are nine possible outcomes. Player A can choose rock, paper, or scissors. Player B has the same set of choices. Charted as a game tree, this is how the game appears:
Through this analysis, it is clear that any strategy that player A takes will not give him a higher chance of winning this game. This is assumed, of course, that each player makes his choice independent of knowing what the other player has chosen. If player A knew what player B chose, he would be able to win automatically. Because he does not know this information, however, there does not exist a best strategy for either him or player B.
In order to more clearly present the idea of a game tree; allow for the modification of rock-paper-scissors. Let it be assumed that before player A makes his choice, player B tells him one of the two choices he has not made. That is, if player B has chosen "scissors" he may tell player A either that he has not chosen "rock" or that he has not chosen "paper". Given that player B tells player A that he has not chosen "rock", I will analyze player A's options. In this situation, player A could choose scissors and have a one-half chance of winning (a win or a tie). Alternatively, player A could choose rock and also have a one-half chance of winning (a win or a loss). Player A's last option would be to choose paper and have no chance of winning (a tie or loss).
Obviously, player A's best option would then be to choose scissors. Clearly there is a best strategy in this game. The best strategy is to choose the item that would be beaten by the disclosed item that B did not guess (in this case, scissors). Fifty percent of the time player A will win with this strategy, and fifty percent he will tie. Had player B told player A "no paper" instead of "no rock", this strategy would lead to a tie instead of a win as in this situation. Unless he does not use this strategy, player A cannot lose. Now examine this example in game tree form:
By choosing scissors, player A can maximize his winnings in this game. This is a very simple example. To add some complexity, allow the assignment of payoffs. Perhaps player A would be receiving a higher payoff if he chose rock instead of scissors. To compensate for his lower chance of winning, he would need to determine whether the payoffs were fair. Before adding payoffs, a game matrix is a method that needs to be introduced. A matrix is similar to a grid in where all possibilities can be charted. As used in the previous example, here is the game in matrix form:
|Player B "no rock"||Player B - Paper||Player B - Scissors|
|Player A - Rock||-1||1|
|Player A - Paper||0||-1|
|Player A - Scissors||1||0|
The payoffs are from player A's point of view. A negative score for player A is equal to a positive score for player B. By totaling the columns and finding an average payoff, it is clear that scissors is the best choice for player A. Player B's strategy can also be outlined by this matrix. All columns add up to the same amount, and have the same three quantities. However, it is apparent that player A will never choose paper. With that row deleted, there is a 2x2 matrix remaining. If player B chooses scissors he can either draw or lose. If player B chooses paper he can either win or lose. Therefore paper should be the logical choice for player B. Combined with player A's choice of scissors, it is apparent that player A will win and player B will lose this game.
However, it should seem logical that player A will choose scissors, so the best player B can do is draw by choosing scissors. If player A sees that player B is going to choose scissors because of this, however, he may choose rock and take the win. However, player B can anticipate further and choose paper. Since this psychology can go on forever, we must anticipate the regular odds and stick with them for analytical purposes.
There are a few ways to determine the best choice. While the third column averages 0.5, the second negative 0.5, and the third 0, had the payoffs been different, a different approach might have been better. Another effective strategy is trying to minimize one's loss. That is, you may not have as much to win, but your chances of losing are less. Take this game payoff matrix, for example:
|Player B "no rock"||Player B - Paper||Player B - Scissors|
|Player A - Rock||-10||20|
|Player A - Paper||0||-5|
|Player A - Scissors||5||0|
In this case, player A still would not want to choose paper because he cannot win anything. However, his choice is now not as clear. With rock, he has a chance of either winning big or losing a lot. With scissors, he can win but the payoff is not great. In minimizing his loss, player A's best choice is A, since he will never lose anything. However, the average payoff for rock is better. Even though he may lose 10, he has the opportunity to win 20. The average payoff for rock is then 5 (both items added and then halved). The average payoff for scissors is only 2.5. If player A only has the chance to play one game, minimizing his loss will be the effective strategy. If player A can play many games, choosing the best average payoff will be more effective.
From player B's standpoint in this game matrix, paper is the choice if he wants to minimize immediate loss. Since player A will never choose paper, player B should never choose scissors since he cannot win anything. Therefore paper is the overall best choice for player B. He will either win 10 or lose 5 depending on player A's decision of either short-term or long-term strategy.
While rock-paper-scissors is a very simple childhood game, it presents a very ideal example for the demonstration of game trees and matrices. Now that these ideas are clear, they can be used to analyze the decision making process necessary in strategy games such as Magic: The Gathering. The effects of probability will also become apparent, but first I will explain the game theory aspect of this game.