When asked why he no longer frequented Ruggeri’s restaurant in St. Louis, Yogi Bera famously replied: “Nobody goes there any more. It’s too crowded.” As other Yogi Bera quotes, this one is silly, yet insightful. And it is related to an interesting problem at the interface of mathematics and economics.

The problem was inspired by the El Farol bar in Santa Fe, near the famous Santa Fe Institute for the study of complex systems. Scientists at the institute like to frequent El Farol on Thursdays to listen to live Irish music. However, the bar is too small for all of them. Let’s assume that if more than half of them decide to go, it is so packed that nobody has a good time. However, when the bar is not overcrowded, an evening of music is more fun than staying at home. Importantly, in our example everybody needs to make up their mind at the same time about whether to go or not. Nobody can call ahead to see how many people are at the bar, or coordinate with others.

How can a patron decide whether to go or not? If everybody acts the same, then everybody either stays at home, or everybody goes to the bar. Neither of these outcomes is optimal, since the bar is either completely empty or completely packed.

But these are scientists. They observe the bar and track how crowded it is from one week to the next. They look for patterns in attendance to help them decide whether to go or not. Maybe the bar was nearly empty the last couple of weeks, indicating that it might be empty again this week. Or maybe there is a cycle with the bar overcrowded one week, and nearly empty the next. Thus each scientist develops a strategy to translate these observations into a decision.

But here is the catch – there is no one strategy that guaran-tees success. If there was, everybody would be using it. But with the same strategy everybody would again be making the same decision each week, and the bar would be either overcrowded or empty. We would be where we started. The best hope is that the scientists choose different strategies. Some stay home, and each Thursday the bar is filled exactly to capacity.

This may seem like a frivolous problem. However, it has been studied extensively by mathematicians and economists as a simple model of a market. Indeed, the scientists compete for a resource – music at a bar. They are rational, as they monitor attendance and use this information strategically. But, as in a real market, they have limited information – they do not know the strategies of others, only how many show up each week. To do well, everybody needs to keep learning and adapting.

But the El Farol problem applies more widely. Suppose that instead of a bar and patrons we are think of the ocean and fisher-man. If all go out to fish, the stock will collapse. But if none do, many will go hungry. Or think of how we make use of our environment. And consider, that unlike the patrons of the El Farol bar as residents of planet Earth, we will not get a chance to try again, if we overcrowd it and overtax its resources.

Some Notes:

The original El Farol problem was proposed in a very readable paper by Brian Arthur. In the original paper, the assumption is that if more than 60% of potential patrons visit the bar, then it will be overcrowded. I have changed this to make the exposition a bit simpler, and kind of combined the El Farol problem and the Minority Game. These are similar, but not the same. You can find the origi-nal paper here.

As noted, the El Farol problem has an even simpler version – the Minority Game. The idea is that each agent plays in successive rounds of a game where there are only two choices A and B. Any agent in the minority wins a round. Everybody in the majority loses. For instance if during one round most agents chose A, then all those who chose B win a set amount. Each agent learns a strategy based on their previous choices. Here is an introduction, and an accessible overview which discusses implications for economics can be found here.