When we are at a restaurant, we should order the usual flatwhat we already know is good, or order something new risking it disappointing us? This is the “restaurant problem”solved at the end of the Seventies by the Nobel Prize-winning physicist Richard Feynman, who derived a formula to establish when it is appropriate continue to experiment and when is best stop and choose what we already know. Fifty years later, a study published in the “Proceedings of the National Academy of Sciences” demonstrated not only that Feynman’s solution was mathematically the best, but discovered that, when they have to make decisions of this type, people adopt a similar strategy to that proposed by Feynman, although simpler.
What is the restaurant dilemma
The restaurant dilemma is a mathematical problem in which we have to decide between an option that we already know and know is good, like our favorite dish, and a completely new one, which could turn out to be better but also disappoint us.
This dilemma, typical of our everyday life, was formalized and solved by the Nobel Prize-winning physicist Richard Feynman. In the late 1970s, Feynman was having lunch at a Thai restaurant in Glendale, California, with his friend Ralph Leighton. Leighton was torn between ordering the ginger chicken, his favorite dish, or trying something different that might have been even better. To help him, Feynman transformed the situation into a mathematical problem and derived the formula that allowed one to best decide when to experiment and when to choose one’s favorite dish.
The restaurant dilemma is part of a category of mathematical problems called “optimal stopping problems” which consist of trying to understand if the option we have in front of us is valid enough or if it is worth continuing to look. Optimal stopping problems often arise in everyday life, not only in choosing what to eat, but also in finding a home, choosing a parking space, and knowing when to quit a job.
Feynman’s solution to the mathematical problem
The solution identified by Feynman is this: continue to try new dishes until one is found that exceeds a certain “goodness score”, and from then on always order that. The score depends on how many meals are left and it is calculated like this:
√number of meals remaining / (√number of meals remaining + 1)
To understand how this works, let’s imagine we have the opportunity to have just one more meal at our favorite restaurant. In this case, the number of meals remaining is 1 and the score to pass is 1/(1+1) = 1/2i.e. 0.5. So, if our favorite dish in that restaurant has a “goodness score” higher than 0.5 for us (on a scale of 0 to 1), we should order it. Otherwise, we’d better change.
If, however, we still have many meals ahead of us, we can afford to be much more demanding. With 81 meals remaining, for example, the score to beat becomes 9/10i.e. 0.9. In this case it is better for us to choose something we know only if it is truly exceptional: if our favorite dish does not exceed a score of 0.9, it is better to continue exploring the menu.
In other words, the more time we have to search, the more we can afford to risk. However, when there are few remaining opportunities, it is worth it settle of something we already know is pretty good.
How people behave: the study that confirms the effectiveness of the mathematical formula
Fifty years later, a study published in “Proceedings of the National Academy of Sciences” took up Feynman’s problem to understand if people, without knowing the formula, spontaneously arrived at a similar strategy. The researchers involved 2,520 participants and they proposed them a similar problem: imagine being in a new city for 7, 14 or 28 nights and having to choose which one restaurant have dinner every night.
Each restaurant was associated with a score from 0 to 100, which participants could only discover after trying it and which represented the quality of the restaurant. The goal was to be able to eat at the best restaurants, that is accumulate more points possible during your stay. The participants, therefore, had to decide every evening whether to return to an already known restaurant or take a risk and try a new one.
Even though the participants did not know Feynman’s formula, their behavior followed one very similar logic. At the beginning of their stay they were more likely to explore new restaurants, while as the end approached they became more likely to return to those they already knew.
Their strategy, however, did not exactly coincide with the mathematically best one. The “goodness scores” people used decreased more simple and linear compared to Feynman’s formula and participants tended to explore longer. Despite this, their strategy allowed them to obtain almost as good results.
The mathematical model, however, remains one simplification of real life. As observed by the authors of the article themselves, in everyday life we do not make these decisions only based on how many opportunities we have left, but other factors also come into play, such as how hungry we are, the boredom of always eating the same thing or the influence that the people we are at the table with have on us.
