How to Walk through a Doorway

You have been there before. You start to walk through a doorway, and someone else is walking towards you, trying to pass through the doorway at the same time. So, we have two people, face to face in the doorway with two possible actions, Left or Right (think of it as stage right and stage left or port and starboard). We call this a simultaneous move game because each person will make the decision at the same time. (You could argue that one could wait, but in the US the best time is now) Subtleties determine which direction you choose, but the main goal is to pass through the door quickly. There are four possibilities (LL, RR, LR, and RL). If LR or RL is played, then we have successful passing (or a Nash Equilibrium). If, however, we have LL or RR, then a new game erupts. Here's why. At LL (or RR), both players are on one side of the doorway, with a perfectly wide open space, large enough for passing, on the right. The best response, given the move L of your rival, is R, and likewise for your rival. In fact, that is the only response. So, we now have RR, and no passing. The best response the new outcome is left, again no passing, and so on.

  

 This is an example of an Adaptive Dynamic Game (related to Evolutionary stable Equilibrium) where we keep playing the game and best responding, but never settling. (This concept was introduced to me by Game Theory professor, Roy Gardner) The neat thing about it is that, theoretically, the game could go on forever, but it doesn't. So we need to add something to the model. This is the crucial step of game theory, defining the game. 

Let's go back, and again we could argue that waiting is a possibility. So, there is something idiosyncratic about the players that might make them more likely to wait. You could even have a case, where they both wait. So there might be a polite type and a rude type. We can go further by adding more moves. Here are a few strategies observed in the field: 1) the "moth" where they flatten themselves against the side; 2) the "bulldozer" where there is no turning; and 3) "committed" where the individual chooses a direction and looks down. Another addition to the model is patience. Some are more in a hurry than others, so maybe after three tries, you reset the game. But the number is arbitrary to each player. You want to announce that you want to reset the game with a perfect stranger. In the US, we like to be blunt, but no one ever says "Hey, this isn't working, let's start over." Another field observation, almost always, and usually between men, someone smugly states "age before beauty" or "shall we dance?"

A similar concept is the Four-Way Stop problem, but with more complexity.