El Farol Bar problem

The El Farol bar problem is a problem in game theory. Based on a bar in Santa Fe, New Mexico, it was created in 1994 by W. Brian Arthur.

The problem is as follows: There is a particular, finite population of people. Every Thursday night, all of these people want to go to the El Farol Bar. However, the El Farol is quite small, and it's no fun to go there if it's too crowded. So much so, in fact, that the preferences of the population can be described as follows:

• If less than 60% of the population go to the bar, they'll all have a better time than if they stayed at home.
• If more than 60% of the population go to the bar, they'll all have a worse time than if they stayed at home.

Unfortunately, it is necessary for everyone to decide at the same time whether they will go to the bar or not. They cannot wait and see how many others go on a particular Thursday before deciding to go themselves on that Thursday.

One aspect of the problem is that, no matter what method each person uses to decide if they will go to the bar or not, if everyone uses the same pure strategy it is guaranteed to fail. If everyone uses the same deterministic method, then if that method suggests that the bar will not be crowded, everyone will go, and thus it will be crowded; likewise, if that method suggests that the bar will be crowded, nobody will go, and thus it will not be crowded. Often the solution to such problems in game theory is to permit each player to use a mixed strategy, where a choice is...
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