PLAYERS on the 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鈥攖he 鈥渨eakest link鈥濃�攁mid
sarcastic comments from the show鈥檚 famously vitriolic host, Anne Robinson. The
last player takes home the entire team鈥檚 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鈥檛 bank it but there are compensations, because the next question after each
correct answer is worth more money.
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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. 鈥淚 am always struck by how little money is actually given away,鈥� he
says.
Although banking after, say, three right answers accumulates much more money
than banking after one鈥�$5000 compared to $1000鈥攕trings
of three are harder to come by. In a 21-question round, Coe鈥檚 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鈥�$50,000鈥攊s
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鈥攅ven though it鈥檚
never the best strategy. 鈥淵ou usually see them lose their nerve and bank after
three or four questions,鈥� Coe says.
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. 鈥淏anking after six questions might give the better pay-off, but it
could take several rounds for that pay-off to happen,鈥� he said. 鈥淲ith Anne
Robinson harassing the team for not banking any money, I think people would
quickly back out.鈥�
It could be hard to preserve a coalition under such circumstances, agrees
Coe鈥檚 colleague William Butterworth of Barat College in Lake Forest, Illinois.
鈥淭he venomous quality of the host does not come into our mathematical analysis,鈥�
he says.