Scientists uncover Feynman’s formula for finding best holiday restaurant | Mathematics

When it comes to exploring a new city, it can be tricky to know when to stop searching for a different restaurant to try every night, or to visit the first place you love on repeat.

Now researchers have found that the late physicist and Nobel laureate Richard Feynman devised a mathematical equation that can tackle the conundrum – at least when the range of options is known – and they believe the approach is similar to tactics people use intuitively.

“The essence of the problem is that the value of exploring, of looking around and trying something new, decreases the opportunities you’re going to have to make use of that information,” said Prof Tom Griffiths of Princeton University, a co-author of the study.

The team note the dilemma is a form of “stopping problem” – a situation where it is necessary to decide when to stop one action and start another.

Griffiths noted the restaurant problem has specific features – including the chance to go back to a venue.

Writing in the Proceedings of the National Academy of Sciences, researchers describe how Feynman’s interest was sparked by a lunch with his friend Ralph Leighton at a Thai restaurant in California in the 1970s. Leighton, the researchers note, was debating whether to stick with his favourite meal of ginger chicken or try a new dish.

Feynman turned the issue into a mathematical problem, but his work remained concealed in handwritten notes.

“The notes remained inscrutable for decades, until we managed to decipher them and reconstruct Feynman’s original problem and solution,” the team wrote.

Rather than focusing on which dish to choose, using Feynman’s solution, the researchers reframed the conundrum in terms of choosing which restaurant to dine at when visiting a city for a certain number of nights.

According to Feynman’s approach, in this context, people should try a different restaurant each night until they find one that exceeds a particular threshold that reflects a desired quality.

In Feynman’s equations this threshold is not fixed. Instead it declines more and more rapidly as the number of days left in the city reduces. In other words, as the days go by there is increasingly less motivation to hunt for an amazing dining spot, because the time you will have to enjoy it has decreased.

“The thresholds are being guided by the best thing you might be able to find if you kept looking,” said Griffiths. “If you have a long time to look, finding something amazing has a lot of value because you can go back many times.”

Feynman’s approach assumed there is equal possibility of finding any restaurant within a fixed range of quality. However the researchers also explored other scenarios.

“We showed that if the distribution of restaurants varies, then the strategy you should follow will change too,” said Griffiths.

If a place has several awful restaurants with one or two gems, for example, the threshold starts much higher – meaning it is worth exploring for longer. By contrast if most are of similar – above average – quality, the threshold is lower, meaning it is not worth exploring for so long.

While Griffiths and another author, Brian Christian from the University of Oxford, first encountered and tackled Feynman’s conundrum more than a decade ago, their new work also tests how people behave.

The team recruited 2,520 participants to take part in an online task, where they were asked to imagine being in a city for different periods of time. The variation in the quality of restaurants available was indicated.

The participants were then presented with a grid where each square represented a restaurant and were asked to pick one for each day of their stay. Once a square – or restaurant – had been selected, its quality was revealed.

The team found that rather than the threshold decreasing more and more rapidly as days left reduced, it fell linearly with the proportion of nights remaining.

“It’s a little bit simpler than Feynman’s solution, but it actually turns out to be quite good,” said Griffiths. “The trick is having a threshold and then decreasing that threshold as you get closer to the end [of a trip]. And as long as you are doing something like that, that’ll actually work pretty well.”