# Bumping into people: the awkward dance

You know when you open the door, and you find someone else is trying to get in at the same time, and so you both end up right in each other’s faces? And then you each try to get out of the other’s way, only to go in the same direction and still be in each other’s face? And then you do a sort of weird dance?

I’ve been trying to find some good Youtube clips of this, and while I know they’re out there, I can’t find them in a brief search. But I think you know what I’m talking about.

Well, surprise surprise, we can model this interaction as a game! Each person has two strategies: move left (L), or move right (R).(1)  If they both move in the same direction, then they are still stuck doing the awkward dance, and get payoff -1. Otherwise, they move out of each other’s way, and so they happily go along their way, getting payoff 1.

 1\2 Left Right Left (-1,-1) (1,1) Right (1,1) (-1,-1) Fig. 1: Awkward dance game

There are a couple of pure-strategy Nash equilibria: one player goes left, and the other goes right. But which equilibrium is going to be chosen? A priori, there is no way to tell. Here, social conventions can be useful, such as always moving forward on the right side (for a similar post, see Marli’s post, “When in New York, do as the New Yorkers do“). The problem is when some people didn’t get the memo (*sigh*).

There is a third Nash equilibrium in mixed strategies, where each person chooses to go in one direction with a 50-50 chance. This means that they will have a 50-50 chance each time they play that they will bump into each other, but eventually, after perhaps dancing for a while, they will get it right.

(1) This will all be from the perspective of player 1.