The Two-by-Two Matrix
The most common game theory tool is the two-by-two matrix. This uncomplicated rectangle illustrates the simplest of games, involving two players, each having a single choice between two options. It is a basic tool that Schelling has frequently used in his strategic work. He has protested that his use of the matrix does not make him a game theorist, arguing, “That’s like saying because I use an equal sign, I’m a mathematician,”1 but the Nobel committee and the academic world has thought otherwise. This simple way of illustrating a game has been central to game theory since the subject’s formal inception and it remains a surprisingly versatile device for strategic decision making. This is a grid with one player’s choices represented running north-south and the other’s running east-west. The possible outcomes of their combined decisions appear in the intersecting cells of the grid. The matrix shows four cells that represent the four possible intersecting outcomes of the player’s choices (Table 5.1).
The two-by-two, or 2X2, matrix is illustrated and a number of basic game theory elements are presented with the “banknote game,” developed by Robert Sugden in The Economics of Rights, Cooperation, and Welfare.2 In this exercise Players A and B are in different rooms and the organizer says, “I have donated a $10 bill and a $5 bill for this game to be played. You say which bill you claim. If you both claim the same bill, you both get nothing, but if you claim different bills, you each get the bill you claim.”