Space of cooperate-defect games
March 14, 2012 25 Comments
A general two player, two strategy symmetric game between Alice and Bob can be represented by its payoff matrix for Alice:
Where is the payoff for Alice if both players do action 1, 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 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:
By relabeling what we mean by strategy 1 and 2, we can assume that (we will consider the case of 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 . This gives us:
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 and to give:
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 and 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 ? I refer to these as coordination games, instead of cooperate-defect games. For these games, the matrix looks like:
By the same relabeling of strategy 1 and strategy 2 trick as before, we can assume that . Now we have two cases to consider before we can proceed, is ? If that is the case, then we can divide by in the same normalizing argument as before to arrive at:
Setting we get:
This game has 3 distinct regions depending on if , , .
A remaining case is if . We can’t normalize by a negative number (since it flips signs), so I will divide by $P – T$ and set to get:
Note that and so there is only one qualitatively distinct game for this matrix. This leaves us with one last games, the zero game:
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!
[Kaz2010] Artem Kaznatcheev. Robustness of ethnocentrism to changes in inter-personal interactions. Complex Adaptive Systems – AAAI Fall Symposium, 2010.