This is a story about two boys repeatedly trying to meet, how they often failed, and how they eventually learned to do so. It is also a story about non-equilibrium behavior, the evolution of language (at a rudimentary level), and the gradual emergence of equilibrium.

When I was about 10 years old, I passed the test to get a bike-riding license. One of my best friends lived about a 30–40-minute walk or a 15–20-minute bike ride away from where I lived at the time. With our newfound independence (he also had a bike-riding license), we were now able to visit each other without any parental help. So, this is how things usually unfolded. One of us would call the other sometime after school (we didn’t go to the same school). We then decided who would visit whom. But, regardless of what we decided, we always agreed that we would both leave our homes and meet somewhere halfway. But there were many paths that we could take. So, this was our game-theoretic problem: it was one of coordination. How could we make sure that we both picked the same path so that we would be able to meet?

Our problem was, in fact, a bit more complicated than you would think from what I have sketched so far. I need to provide a bit more context. First, we didn’t always want to bike; sometimes we preferred to walk. And either way, we didn’t always want to take the same route. This could be because of the weather, because one of our bikes didn’t work, or because we just felt like having some variation. Second, we both didn’t really look at maps. Especially in the beginning, we probably didn’t even have a good sense of all the options we had. We also never looked at or knew any street names (apart from the streets we lived on). We both knew ways to get from one of us to the other, but we had no full mental picture of the urban geography between us. Third, there were no mobile phones yet. All we had was a landline phone, which we would use before we left, but once one of us left, there was no possibility of further communication.

So, the game we had was one of coordination, yes, but with varying preferences and, at least initially, an incomplete understanding of the possibilities (the strategies). It also had (only) pre-play communication and was repeated. But I will not write down a formal model of this game. Instead, I will just use some game-theoretic language to express what actually happened.

In the beginning, my friend and I missed each other astonishingly often. Sometimes I would go all the way to his place only to find that he had gone all the way to mine, and then we would have to call each other to start the process again. Sometimes I would go all the way to his place only to find that he hadn’t even left yet, often because his mother had not let him go yet, as he hadn’t finished all his household or homework duties. [Another complication for the game-theoretic modeling.] I don’t think this ever happened on my end; I always left the moment I put the phone down. Why I was allowed to do so is a question I will not address here. Sometimes we did manage to meet. In any case, it became clear that we didn’t always leave at the same time, so when we met, where we met varied.

When we missed each other, we, of course, discussed afterwards how that happened. I think that, at the beginning, we didn’t even always know how we could have missed each other. But slowly (surprisingly slowly, I now think), we started to understand the geography of the land between us, and we also started to be able to articulate (without ever using street names, however) where we missed each other. We also started to give certain routes names. We had very unimaginative names, in fact. One thing we realized quickly is that the routes can be partitioned into those that go over and those that go around the hill that sat between our two homes. If I recall this correctly, we weren’t so keen to go over the hill in winter, especially not by bike, nor when it was very hot, as it wasn’t very shady. Although sometimes, in milder weather, we liked biking over the hill; especially the going-down bit was fun. So, the vocabulary that we developed fairly quickly, and that helped us distinguish between going over and going around the hill, was this: we said that we would either use the “up” or the “down” path.

But then we slowly realized that “up” and “down” were not enough to fully pin down the path after all. Somewhat annoyingly, both the “up” and “down” paths had multiple options, especially somewhere around the midpoint between our two homes. But after some time, we refined our vocabulary to allow us to further distinguish the various routes. We also settled on only four routes that we would ever take, and we finally called them “up up,” “up down,” “down up,” and “down down.” A sufficiently rich language had evolved. While I no longer know which is which, at the time this vocabulary was enough for us to use in our pre-game phone conversations to allow us to coordinate with an almost 100% success rate. [Except for those cases where my friend was not allowed to leave. This also got better, because I started to ask my friend if he was sure that he was allowed to leave. But it was never fully solved, as my friend tended to be a bit too optimistic in his assessment.]

Game-theoretically speaking, then, when we started to play this game, we had a fair amount of miscoordination, that is, non-equilibrium behavior. Over time, a language evolved (or was developed by us), a language that gradually became sufficiently rich (and no richer) to allow us to coordinate almost perfectly, so that, ultimately, equilibrium behavior emerged. A nice example of “evolution” leading to equilibrium behavior, I feel.