Three of us have just been to Emirates Stadium to see a football game. We witnessed and participated in some interesting rational herding walking to the stadium, as I described in my previous post. But once inside, we had another game-theoretic problem. The problem was caused by the fact that we didn’t all have seats together. After a quick bite to eat at one of the food stalls inside the stadium before kickoff, we went to our separate seats without communicating how we would meet again at halftime. It didn’t occur to us at the time that maybe we should have talked about this. When we then got to halftime, when people started flooding back from their seats into the food stall area, I realized that we hadn’t arranged where we would meet, and I was wondering for a moment how we would manage to do so. The food stall area is huge. It goes all the way around the stadium, I believe. So where was I supposed to go?  

I thought about it a bit, and realized that there is really only one place that sticked out, most likely not only in my mind, but quite possibly also in the mind of my family members I was trying to meet: the stand-up table where we had our food together and that was also the last place at which we were together before the game started. And this is indeed where we found each other, pretty quickly, and without having to resort to communicating with our (Austrian) phones (that don’t work so very reliably in London).

In principle, we had a difficult coordination problem. We could have met anywhere in the stadium. And the stadium is huge and full of people. If I had thought for some reason that my family members would go to, say, Stand-Up-Table 157 (counting from the entrance, say), I should have gone there as well. If I had thought that my family members would all come to my seat, I should have waited for them at my seat. If I had thought that my family members would go to the entrance we came in through, I should have done the same. Game-theoretically, this situation is well described by a large pure coordination game in which all three of us have the same large strategy space (the set of all possible places we could meet) with payoffs such that we all get the highest possible payoff, say 1, if we all choose the same strategy, and, for simplicity, 0 otherwise. Such a game has as many (pure strategy) Nash equilibria as it has strategies. A Nash equilibrium is a strategy for each of us such that if the others follow it, I also want to do so (and the same is true for the others). So, for every place that we could have met at, the strategy of going there is a Nash equilibrium strategy.

Contrary to popular belief, game theory does not generally predict (Nash) equilibrium play, even if the players are assumed to be extremely rational. In fact, even common knowledge of rationality does not imply equilibrium play. Common knowledge of rationality means that everyone involved is rational (which in turn means that everyone has clear goals and chooses actions that best achieve these goals) and that everyone knows that everyone is rational and that everyone knows that everyone knows that everyone is rational, and so on ad infinitum (as we like to say). We probably rarely have common knowledge of rationality in actual real-life situations of strategic interaction (as in our case here). But, it would, in any case, not imply that we would play an equilibrium. Common knowledge of rationality only implies that the outcome will be, what is called rationalizable. Without telling you what rationalizable strategies are, I can tell you that there are some games in which there is only one rationalizable strategy profile, and that would have to be a Nash equilibrium. But in coordination games the assumption of common knowledge of rationality yields the following prediction: anything is possible.

So, how did we manage to meet after all in these difficult circumstances? In some situations of strategic interaction there are strategies that Thomas Schelling called “focal points,” see the Wikipedia entry for a starting point. As you will notice when you read this entry, there is no generally workable definition of a focal point given. I have once attempted to provide such a definition of a focal point in a paper with Carlos Alós-Ferrer, but while I like it a lot, it is perhaps only partially satisfying. The idea is relatively straightforward, though. A focal point is a strategy (profile) that, among all other strategies, jumps out at all of you: it is specially earmarked, relative to all other options, in the minds of all the people involved. In our case, the table at which we had food together, and that was also the last place before we went our different ways to our seats, was so earmarked in all our minds. And given this earmarking, we all followed the strategy of “go to the place you have collectively earmarked.” This strategy only works if you indeed earmark the same thing. In reality, many situations lack such a clear focal point, with the implication that you do not manage to meet, or at least not quickly, or not without further communication. One of the problems with the theory of focal points, is that it is a bit difficult to state when and when not it should work. But it did work in our case, and I was happy about that. One could say that because of Newton we did not float into space, and because of Schelling we managed to meet.