When exploring a new city, travelers often face a dilemma: should they try a different restaurant every night or return to a favorite spot? Researchers have now uncovered that the late physicist and Nobel laureate Richard Feynman devised a mathematical equation to tackle this conundrum, at least when the range of options is known. The approach, they believe, mirrors tactics people use intuitively.
The Stopping Problem
The team notes that the dilemma is a form of a 'stopping problem' – a situation where one must decide when to stop one action and start another. Prof Tom Griffiths of Princeton University, a co-author of the study, explained: '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.'
Unlike classic stopping problems, the restaurant dilemma includes the possibility of returning to a previously visited venue. This unique feature adds complexity to the decision-making process.
Feynman's Hidden Notes
Writing in the Proceedings of the National Academy of Sciences, the researchers describe how Feynman's interest was sparked during a lunch with his friend Ralph Leighton at a Thai restaurant in California in the 1970s. Leighton was debating whether to stick with his favorite 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, the researchers reframed the conundrum in terms of choosing which restaurant to dine at when visiting a city for a certain number of nights.
The Threshold Strategy
According to Feynman's approach, people should try a different restaurant each night until they find one that exceeds a particular threshold reflecting 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 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.'
Variations in Restaurant Quality
Feynman's approach assumed 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.
Testing Human Behavior
While Griffiths and co-author Brian Christian from the University of Oxford first tackled Feynman's conundrum over a decade ago, their new work also tests how people behave. The team recruited 2,520 participants for an online task, where they were asked to imagine being in a city for different periods of time, with variations in the quality of available restaurants indicated.
Participants were 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 was 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.'
