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Players on the TV game show The Weakest Link should either take no chances at all or cast caution to the winds. A team does best if it banks its winnings either after every right answer or only after a run of six successive right answers.

Eight players compete on the show, answering questions in turn and accumulating money for the group when they answer correctly. At the end of each round the players vote off one player – the “weakest link” – amid sarcastic comments from the show’s famously vitriolic host, Anne Robinson. The last player takes home the entire team’s winnings.

Money accumulated in a round is only carried over to the next if one of the players banks it before hearing their question. Once a player banks the money, the stakes go back to the bottom of the scale. You risk losing the money if you don’t bank it but there are compensations, because the next question after each correct answer is worth more money.

The winner usually goes home with only a small fraction of the money up for grabs, reported Paul Coe, a mathematician at Dominican University in River Forest, Illinois, to the Joint Mathematics Meetings in San Diego, California, last week. “I am always struck by how little money is actually given away,” he says.

Losing their nerve

Although banking after, say, three right answers accumulates much more money than banking after one – $5000 compared to $1000 – strings of three are harder to come by. In a 21-question round, Coe’s team found that players would be better off banking after each question than after three unless their success rate was over 67 per cent, which is rarely achieved.

On the other hand, the reward for answering six questions right – $50,000 – is so high that waiting for a chain of six before banking is the best strategy for all but the weakest teams.

Players often elect to bank after only a few questions – even though it’s never the best strategy. “You usually see them lose their nerve and bank after three or four questions,” Coe says.

Psychological tension

The psychological tension of the game may make waiting for six right answers unrealistic, even if it is the best strategy on paper, cautions Edward Aboufadel, a mathematician at Grand Valley State University in Allendale, Michigan.

“Banking after six questions might give the better pay-off, but it could take several rounds for that pay-off to happen,” he said. “With Anne Robinson harassing the team for not banking any money, I think people would quickly back out.”

It could be hard to preserve a coalition under such circumstances, agrees Coe’s colleague William Butterworth of Barat College in Lake Forest, Illinois. “The venomous quality of the host does not come into our mathematical analysis,” he says.