How many two-player two-action simultaneous move games are there?

Question

A two-player two-action simultaneous move game is represented by the following matrix:

┌───┬───┬───┐
│1\2│ C │ D │
├───┼───┼───┤
│ A │a\b│c\d│
├───┼───┼───┤
│ B │e\f│g\h│
└───┴───┴───┘

Player 1 has two actions, $A$ and $B$. Player 2 has two actions, $C$ and $D$. If player 1 plays action $A$ and player 2 plays action $C$ then player 1 gets utility $a$ and player 2 gets utility $b$. Similarly, for every other combination of actions. The objective of each player is to maximize their expected utility.

An example of such a game is Prisoner's dilemma for which $d = e < a = b < g = h < c = f$ given that $A$ and $C$ are betray and $B$ and $D$ are silent. From this it can be seen that the best action for player 1 is $A$ since $e < a$ and $g < c$. Similarly, the best action for player 2 is $C$. Hence, $AC$ is a state of Nash equilibrium.

One way to represent such games is to use the notation $a \leq b \leq c \leq d \leq e \leq f \leq g \leq h$ where the variables $a, \ldots, g$ may be arranged in any permutation and the intermediate $\leq$ operators may either be $<$ or $=$ since when instantiated the values of the varaibles $a, \ldots, g$ are known. Hence, the upper bound on the number of possible games is $8! \times 2^7 = 5,160,960$.

Unfortunately, this notation is not unique since equality is symmetric. Furthermore, the same game can be represented in upto 4 different ways depending upon what actions we assign to $A$ and $C$.

Another representation of such games is to assign to each state (i.e. $AC$, $AD$, $BC$ or $BD$) the values $0$, $1$, $2$ and $3$ representing which players would like to switch their action. Hence, $0$ denotes Nash equilibrium and $3$ denotes that both players would like to switch. Hence, the upper bound on the number of possible games is $4^4 = 256$.

Again, a lot of games might be repeated because the same game can be represented in upto 4 different ways depending upon what actions we assign to $A$ and $C$. However, this representation has the advantage of being unique with respect to notation. That being said, the difference between $5,160,960$ and $256$ makes me wonder whether there are some games that I'm missing.

So, my question is how do you calculate the number of unique two-player two-action simultaneous move games? Two games are considered the same iff the only difference between them is the choice of actions assigned to $A$ and $C$.

Answer 1

There are several classification of $2 \times 2$ games in the literature. I believe that the ones most useful to you are the following two.

1) Rapoport, A. and M. J. Guyer: 1966, ‘A taxonomy of 2 × 2 games’, General Systems 11, 203–214. (This classification assumes no ties in players' payoffs and derives 78 classes.)

2) Kilgour, D.M. & Fraser, N.M.: 1988, 'A taxonomy of all ordinal 2 × 2 games". Theor Decis 24: 99-117. (This classification extends the former one by allowing for ties and derives 726 classes.)