Tomorrow's Mathematicians Today, the first Undergraduate Mathematics Conference was held in Greenwich, London on 6th February 2010. To open the conference, Nic Mortimer spoke about 'How To Play "The Weakest Link" Optimally'. Nic Mortimer is a 2nd year Mathematics for Decision Science, BSc student at The University of Greenwich.

1. How To Play “The Weakest Link” Optimally An exploration into the mathematics behind the popular game show Nic Mortimer University of Greenwich mn836@greenwich.ac.uk

2. In solid geometry, what H is the name given to half a sphere? “Half a circle”

3. The Rules

4. The Rules 1000 800 600 450 300 200 100 50 20

5. The Rules 1000 800 600 450 300 200 100 50 20

6. The Rules 1000 800 600 450 300 200 100 50 20

7. The Rules 1000 800 600 450 300 200 100 50 20

8. The Rules 1000 800 600 450 300 200 100 50 20 BANK!

9. The Rules 1000 800 600 450 300 200 100 50 20 200

10. The Rules 1000 800 600 450 300 200 100 50 20 200

11. The Rules 1000 800 600 450 300 200 100 50 20 200

12. The Rules 1000 800 600 450 300 200 100 50 20 200

13. The Rules 1000 800 600 450 300 200 100 50 20 200

14. The Rules 1000 800 600 450 300 200 100 50 20 BANK! 200

15. The Rules 1000 800 600 450 300 200 100 50 20 220

16. The Rules 1000 800 600 450 300 200 100 50 20 1000

17. The Rules

18. The Rules Round 1: 9 players Round 2: 8 players Round 3: 7 players Round 4: 6 players Round 5: 5 players Round 6: 4 players Round 7: 3 players Round 8: 2 players (money tripled) Round 9: Head-to-head final

20. How important it is for you to eliminate strong opponents

22. My voting strategy Wait for “close” rounds then vote off the strongest of the players who tied This looks more innocent/honest, and gives a greater chance that you will not be the only one voting for them VOTE VOTE

23. Banking strategies Banking can be quantified mathematically Most basic strategy: bank after every correct question Suppose you get questions correct 50% of the time Answer 1 Correct: £20 0.5 Question 1 0.5 Incorrect: £0

24. Banking strategies Answer 1 Correct: £20 0.5 Question 1 0.5 Incorrect: £0 What are the average (expected) winnings? (0.5 x £20) + (0.5 x £0) = £10

25. Banking strategies What about after two questions? Answer 2 Amount Banked Answer 1 Correct: £20 £40 0.5 0.5 Correct: £20 Incorrect: £0 £20 0.5 Question 1 Correct: £20 £20 0.5 0.5 Incorrect: £0 Incorrect: £0 £0 0.5 What are the expected winnings? (0.5 x 0.5 x £40) + (0.5 x 0.5 x £20) + (0.5 x 0.5 x £20) + (0.5 x 0.5 x £0) = £20 ie. £10 per question

26. Banking strategies What about if we bank at £50 each time instead? Answer 2 Amount Banked Answer 1 Correct £50 0.5 0.5 Correct Incorrect £0 0.5 Question 1 Correct £20 0.5 0.5 Incorrect (last question of round so bank anything) Incorrect £0 0.5 What are the expected winnings? (0.5 x 0.5 x £50) + (0.5 x 0.5 x £0) + (0.5 x 0.5 x £20) + (0.5 x 0.5 x £0) = £17.50 ie. £8.75 per question

27. Banking strategies No matter when you bank, if you are getting 50% of questions correct, it is always more profitable to bank at £20 (after each correct question)

28. Banking strategies 70%, banking at £20 Answer 2 Amount Banked Answer 1 Correct £40 0.7 0.7 Correct Incorrect £20 0.3 Question 1 Correct £20 0.7 0.3 Incorrect Incorrect £0 0.3 What are the expected winnings? (0.7 x 0.7 x £40) + (0.7 x 0.3 x £20) + (0.3 x 0.7 x £20) + (0.3 x 0.3 x £0) = £28 ie. £14 per question

29. Banking strategies 70%, banking at £50 Answer 2 Amount Banked Answer 1 Correct £50 0.7 0.7 Correct Incorrect £0 0.3 Question 1 Correct £20 0.7 0.3 Incorrect Incorrect £0 0.3 What are the expected winnings? (0.7 x 0.7 x £50) + (0.7 x 0.3 x £0) + (0.3 x 0.7 x £20) + (0.3 x 0.3 x £0) = £28.70 ie. £14.35 per question

30. Banking strategies If you are getting 70% of questions correct, it turns out that £200 is the best banking level (after every 4 correct questions)

31. Banking strategies Used Microsoft Excel to simulate a 5-question decision tree for different %s of questions correct

32. Banking strategies Was able to create graphs for each % of questions correct

33. Banking strategies

34. Banking strategies

35. Banking strategies Was able to create graph for all possible % of questions correct

36. Banking strategies £300 better than £200 for 72%...but what about £450? Would this trend increase all the way to the £1000 banking level or decrease after £300?

37. Banking strategies

38. Banking strategies So what does all this mean? By estimating the % of questions you get correct, you can bank optimally If you get 90% of questions correct (ie. first round), only bank at £1000 If you only get 50% of questions correct, bank every time Anne gets to you and there is money in the pot

39. Banking strategies Notice that £50 and £100 are neveroptimal banking levels However, if you are getting 0-61% of questions correct, you should bank if it gets to you on £50 or £100. This is because you should be banking at £20 or more whenever possible. The same applies if you should be banking at £200 and it gets to you on £600, for example.

40. Banking strategies Your optimal banking level Easier to see in a table: How much has been earned when Anne reaches you

41. Other thoughts If Anne gets to you and there is <10 seconds on the timer, bank no matter what your optimal banking level is.

42. Conclusion Difficult to play all aspects of the game optimally Psychology of the game has to be factored in But in general, it is possible to bank optimally and vote intelligently

43. References “How long is a piece of string?” by Rob Eastaway and Jeremy Wyndham “Freakonomics: A Rogue Economist Explores the Hidden Side of Everything” by Steven D. Levitt and Stephen J. Dubner BBC iPlayer – The Weakest Link My e-mail: mn836@greenwich.ac.uk