Feynman’s forgotten ‘restaurant problem’ finally solved after 50 years

Researchers have finally deciphered a mathematical puzzle scribbled in the notes of former Manhattan Project physicist Richard Feynman more than half a century ago. The solution reveals how diners should decide when to stop trying new restaurants and commit to a favourite.

A set of handwritten notes left behind by physicist Richard Feynman has been decoded more than 50 years after they were first drafted, revealing an elegant solution to what researchers have dubbed the “restaurant problem”.

Published in the Proceedings of the National Academy of Sciences (PNAS), the study reconstructs a decision-making problem Feynman developed during a lunch conversation with his friend Ralph Leighton at a Thai restaurant in California during the 1970s.

According to the paper, Leighton was debating whether to order his favourite dish or try something new. Feynman responded by turning the dilemma into a mathematical optimisation problem and deriving a solution by hand.

The notes survived, but the meaning behind them remained largely indecipherable for decades.

The explore versus exploit dilemma

The problem centres on a familiar question: when should someone stop exploring new options and stick with the best one they have found?

In Feynman’s formulation, each restaurant has a fixed but unknown quality score. A diner visiting a city for a limited number of nights can either continue trying new restaurants or return to the best restaurant already discovered.

The objective is to maximise the total quality of meals consumed over the entire stay.

Researchers Brian Christian of the University of Oxford, Evan Russek of Hunter College and Thomas Griffiths of Princeton University reconstructed the problem from Feynman’s original notes and proved that his solution was mathematically optimal.

Their analysis showed that the optimal strategy is to continue exploring until a restaurant exceeds a specific quality threshold, with that threshold gradually decreasing as the number of nights remaining falls.

As departure approaches, diners should become increasingly willing to settle for a restaurant that is good enough rather than continuing the search for something better.

Beyond restaurants

While inspired by lunch, the researchers note that the same principle applies to many aspects of life.

The paper links the restaurant problem to decisions about jobs, housing, relationships, parking spaces and other situations where people must balance exploration against exploitation.

The study places Feynman’s puzzle alongside classic decision-making challenges such as the secretary problem and multi-armed bandit problems, both of which have long been studied by economists and psychologists.

Unlike those models, however, Feynman’s version assumes that once a restaurant has been visited, its quality is known with certainty.

Humans are surprisingly good at it

The researchers did not stop at deciphering Feynman’s mathematics, they also tested how closely people follow the optimal strategy. In a preregistered experiment involving 2,520 participants, volunteers were asked to make repeated restaurant choices under different conditions. The results showed that people generally use a simple rule in which their willingness to settle decreases in a roughly linear fashion as opportunities run out.

Although not perfectly optimal, the strategy performed remarkably close to Feynman’s mathematically ideal solution. According to the researchers, participants consistently adjusted their behaviour depending on the distribution of restaurant quality available to them, demonstrating an intuitive understanding of the trade-off between exploration and exploitation.

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