Wednesday, December 7, 2011

Game Theory: Part IX: Coordination and Anti-Coordination

In the last game, we assumed that the players are able to coordinate, in order to avoid a mutual mistake. But we cannot always confer with others before making decisions. And so, we must decide: how much is it worth to confer with someone else?
The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.
So, let us look at two classic games, the Coordination Game and the Anti-Coordination Game. These are very common; one example of the Coordination Game might be deciding which side of the road to drive on.
The Coordination Game:


X Y
X (A,A) (B,B)
Y(B,B)(A,A)

The Anti-Coordination Game:

X Y
X (B,B) (A,A)
Y(A,A)(B,B)

where in all cases A>B.

So, as you can see, in the coordination game, the players do better if they play the same move. In the anti-coordination game, the players do better if they play different moves. And in neither case is one move objectively better for one player than its alternative. So, how do the players resolve this?
The solution is obvious. The players must arrange ahead of time what move they will make.

(The players gain nothing from deviating from their arrangement, so they gain no benefit from the ability to make binding promises. In effect, any promise either player makes will punish them if they break it. This will be important later.)

Now, let us say that the players cannot get this for free. Maybe the players are playing via the postal service, and need to pay for stamps. How much should the players pay, maximum, for the ability to coordinate their moves? (As mentioned in Utility Functions, we are assuming that there is some conversion function between the payment and what the players are rewarded in. )

Let's look at what will happen if the players cannot coordinate. Since neither move is dominant, and in fact no move offers any advantage over the other, the players have no choice but to act randomly. Each move will be taken 50% of the time. (It occurs to me that perhaps I should explain how to calculate the odds with which you should decide your moves. Later. For now, just accept that the moves should be played with even odds.) So, the value of this game to each player is:

A/2+B/2.

So what is the value of the game where the players are allowed to communicate? Well, that should be obvious. Neither player stands to gain anything by going against their agreement, so the value of that game is A. So, the value of the ability to communicate is, in this case, A-(A/2+B/2)=A-A/2-B/2=A/2-B/2.

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