Space of cooperate-defect games

A general two player, two strategy symmetric game between Alice and Bob can be represented by its payoff matrix for Alice:

\begin{pmatrix}R & S \\ T & P \end{pmatrix}

Where R is the payoff for Alice if both players do action 1, S is the payoff for Alice if she does action 1 and Bob does action 2, etc. Note that every time Alice and Bob play we could give each $10 and that would not change their strategies (since they get the money regardless of what they do). Similarily, we can subtract P from each payoff and not change the structure of the game (note that in this settings players can’t chose NOT to play). This reduces the matrix to:

\begin{pmatrix}R - P & S - P \\ T - P & 0 \end{pmatrix}

By relabeling what we mean by strategy 1 and 2, we can assume that R > P (we will consider the case of R = P later). What are the payoff measured in? It could be dollars, tens-of-dollars, or number-of-children; the key point is that the payoffs have no natural unit of measure. Thus, we can re-scale them by any positive number. The easiest choice is to re-scale by R - P. This gives us:

\begin{pmatrix} 1 & \frac{S - P}{R - P} \\ \frac{T - P}{R - P} & 0 \end{pmatrix}

I will usually refer to strategy 1 as “cooperate” and strategy 2 as “defect”. The intuition is that cooperation is mutually beneficial (a payoff of 1) while mutual defection is not (a payoff of 0). To simplify the matrix, I will relabel by setting U = \frac{S - P}{R - P} and V = \frac{T - P}{R - P} to give:

\begin{pmatrix} 1 & U \\ V & 0 \end{pmatrix}

Regular readers might remember me using this payoff matrix without justification. The big upside is that it lets us look at games by plotting them in two dimensions; I do this in the intro of [Kaz2010].

What makes a game qualitatively “different” is the possible orderings of U and V compared to each other and 0 and 1. There are 12 possible orderings, and hence 12 different types of games. I label some of them with names. Of course, some regions have multiple names for example the Stag Hunt game. On wikipedia it is defined the same way as game 5 in my figure, but in some settings it is defined to include both regions 1 and 5. Also, I don’t remember why I called game 4 as Battle of the Sexes since that game is usually only studied in the asymmetric case.

What about the case with R = P? I refer to these as coordination games, instead of cooperate-defect games. For these games, the matrix looks like:

\begin{pmatrix}0 & S - P \\ T - P & 0 \end{pmatrix}

By the same relabeling of strategy 1 and strategy 2 trick as before, we can assume that S - P \geq T - P. Now we have two cases to consider before we can proceed, is S - P > 0? If that is the case, then we can divide by S - P in the same normalizing argument as before to arrive at:

\begin{pmatrix}0 & 1 \\ \frac{T - P}{S - P} & 0 \end{pmatrix}

Setting X = \frac{T - P}{S - P} we get:

\begin{pmatrix}0 & 1 \\ X & 0 \end{pmatrix}

This game has 3 distinct regions depending on if X > 1, 1 > X > 0, 0 > X.

A remaining case is if 0 \geq S - P > T - P. We can’t normalize by a negative number (since it flips signs), so I will divide by $P – T$ and set Y = \frac{S - P}{P - T} to get:

\begin{pmatrix}0 & Y \\ -1 & 0 \end{pmatrix}

Note that - 1 \leq Y \leq 0 and so there is only one qualitatively distinct game for this matrix. This leaves us with one last games, the zero game:

\begin{pmatrix}0 & 0 \\ 0 & 0 \end{pmatrix}

For a total of 17 distinct games. Challenge for the reader: give a descriptive name to every game and give an example of it in the ‘real’ world!

References

[Kaz2010] Artem Kaznatcheev. Robustness of ethnocentrism to changes in inter-personal interactions. Complex Adaptive Systems – AAAI Fall Symposium, 2010.