By Edoardo Ballico, Ciro Ciliberto

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8: Arrangement of the 2 × 2 games by indices, g rc rows, columns and layers beginning with the (symmetric) Prisoner’s Dilemma because it is the best known game of all. With six column patterns, six row patterns, and four layers, we have room for exactly 144 2 × 2 games. 8, as an array that is six games wide, six games high and four layers deep. The three-digit index locates each game in the array. 16 show the order graphs for all 144 games. 14 contains the 36 games in layer 1. 16 show layers 2, 3 and 4.

Notice first that any game with an order graph that is symmetric about the positive diagonal is its own reflection, and second, that there can be only 12 such games. A symmetric game must have two payoffs on the positive diagonal {(1, 1), (2, 2), (3, 3), (4, 4)}. There are 4C2 = 6 ways to achieve this. Having chosen two symmetric points there are only two symmetric ways to join them to the remaining two points. 2 ×4 C2 = 12. Therefore there are 144 − 12 = 132 asymmetric games. Every asymmetric game has a reflection.

6 produces the desired orientation. 4 Counting the 2 × 2 games Rapoport and Guyer [23] established a commonly accepted count for the 2 × 2 games. They began with the observation that there are 576 ways to arrange two sequences of four numbers in a bi-matrix. 4 COUNTING THE 2 × 2 GAMES 17 Convention for constructing standard payoff matrices: Apply the first rule that the payoff matrix allows. If the game has a symmetric pair, 1. (4, 4) −→ upper right cell 2. (1, 1) −→ lower left cell 3. (3, 3) −→ upper right cell 4.

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