The optimal strategy when playing The Weakest Link
The Weakest Link, a British TV quiz show, ran its last episode in 2017. Did players miss their chance to make the most money possible?
The Weakest Link has a group of people answer questions in a circle. For each correct answer, the team goes up the totem pole of values they can “bank.” After one correct answer they can bank £250, after two correct answers they can bank £500, and so on. Once a player gets a question wrong, however, the amount returns to £0. What is the optimal stopping strategy? Should players hedge their bets after reaching £500? £4000? What if they have low accuracy?
A Monte Carlo simulation allowed me to calculate average earnings when comparing stopping strategy to percent accuracy.
Here is the link to the Jupyter notebook. I estimated 100 questions per game. I averaged over 500 simulated games. The payment ladder is: £0, £250, £500, £1000, £1500, £2250, £3000, £4000, £5000.*
The heat map and contour map below show how much confidence and risk-aversion play a role in the game’s outcome. It is easy for players of high skill to make dramatically less money than if they took risks. And conversely, players with low skill can make just as much as a perfect player if they deploy the right strategy. For example, a player with 65 percent accuracy deciding to wait for 5 or 6 turns (£1500 and £2250 respectively) would make roughly the same as a conservative player with perfect accuracy.
While the contour map shows that many players will roughly make the most money if they take risks all the time, the optimal strategy reveals that most players should bank whenever possible. If a player has less than 77 percent accuracy, the player should always bank. There is a grey area between 77 and 85 percent, and then above 85 percent accuracy, a player should always shoot for top payout.
For those interested in a more rigorous study check out Barmish and Boston, U. of Michigan.
*On the TV show’s graphic, it labels the bottom rung of the ladder as £100, but you can only bank after reaching the £250 rung. It’s a strange setup.