1. Exacting Princess Dr Fedor Duzhin, Nanyang Technological University, School of Physical and Mathematical Sciences

2. About me High School: 1994, Russian Mathematics Olympiad – 2 nd prize 1995, Chinese Mathematics Olympiad – 2 nd prize 1995, Russian Mathematics Olympiad – 3 rd prize 1994, Russian Informatics Olympiad – 1 st prize 1995, Russian Informatics Olympiad – 2 nd prize 2

3. About me University Education: M.S. in mathematics, 2000, Moscow State University Major: Pure Mathematics (Topology) 3

4. About me Graduate Education: Ph.D. in mathematics, 2005, Royal Institute of Technology, Stockholm Major: Pure Mathematics (Topology and Dynamical Systems) 4

5. Characters Martin Gardner (b. 1914) Famous American science writer specializing in recreational mathematics He stated the problem in 1960 Sabir Gusein-Zade (b. 1950) Russian mathematician He gave a general solution to the problem in 1966 Boris Berezovsky (b. 1946) One of Russia's first billionaires Once he was an applied mathematician. His doctoral thesis is devoted to optimal stopping of stochastic processes, which is a generalization of the problem. 5

6. Problem Statement Once upon a time in the land of Fantasia a princess decided to get married. 100 princes came to seek for her hand; and she intends to choose the best of them. 6

7. Problem Statement She can compare princes Once she spoke to any two of them, she can decide which one is better ? <><> > ♥ The princes form an ordered set: If prince A is better than prince B and prince B is better than prince C then A is better than C Therefore, indeed, there is the best If she could speak with each of the princes, then she would be able to chose the best of them 7

8. Problem Statement But! Once she’s spoken to a prince, she has either to accept or reject him ♥ If she rejects him, the proud prince leaves the country immediately and never comes back. Once she accepts the offer, there are two possibilities: a.If the candidate is not the best, she goes into a convent b.If he is the best, they get married and live happily 8

9. Problem Statement The procedure is as follows She faces the princes appearing in a random order. On each audience she decides whether to accept the current candidate. PROBLEM: Find the optimal strategy for the princess: Which prince must be accepted to make the chance of success as high as possible? 9

10. Some thoughts If she decides to pick up the 1 st one or the 3 rd one, or the 100 th one the chance of success is just 1% QUESTION: How can she make the chance of success reasonable, for example at least 25%? 10

11. Some thoughts Instead, she could do as follows: First, reject half of them, that is, 50 princes And pick up the first one who exceeds these 50 QUESTION: What is the chance of success under this strategy? 11

12. Some thoughts QUESTION: What is the chance of success under this strategy? Let S be the chance in % to get the best fiancé If the best prince is among the first half, then she loses automatically S<50% But if the best prince is among the second half and the second best is among the first half, then she wins automatically 25%<S<50% 12

13. Idea! The princess’s strategy can be like: First, reject R % of the candidates And pick up the first one who exceeds all the rejected ones QUESTION: What R should be taken to make the chance of success maximal? 13

14. General facts In mathematics, the probability is measured not in % If there is one chance of two, the probability is 1/2=0.5 If there are three chances of eight, the probability is 3/8=0.375 Thus if the chance of success is S %, then the probability is P = S /100. The chance of success in % lies between 0 and 100 The probability is between 0 and 1 14

15. Rigorous discussion Let’s try to think backwards. If there is only the last prince, the situation is clear. Assume that the princess knows what to do with (n+1) st prince. What should she do on the n th audience? Let us introduce some parameters describing the process Suppose that she already rejected the first n-1 candidates and the n th one is better than any of them (otherwise accepting him does not make sense). Let A(n) be the probability to win if she accepts him QUESTION : Calculate A(n) 15

16. Rigorous discussion Recall that A(n) is the probability to win if she rejects the first n-1 candidates, if the n th prince is better than any of the first n-1, and if she picks the n th prince Obviously, A(100)=1 : if she rejected 99 princes the 100 th turned out to be better than all of them then he is the best automatically Further, A(99)=1-0.01=0.99 : if she rejected 98 princes the 99 th turned out to be better then all of them then the 100 th can be the best (probability is 0.01 ) otherwise the 99 th is the best 16

17. Challenge Let A(n) be the probability to win if she rejects the first n-1 candidates, if the n th prince is better than any of the first n-1, and if she picks the n th prince Prove (by mathematical induction) that 17

18. Rigorous discussion Assume that she just rejects n candidates and then uses the optimal strategy. Let B(n) be the probability to win in this case. Now the optimal strategy is obvious: The princess rejects the first, the second etc. candidates while B(n)>A(n) Once A(n) becomes larger than B(n), she accepts the first one who is better than all the rejected guys. QUESTION : Calculate B(n) 18

19. Rigorous discussion Recall that B(n) is the probability to win if she rejects the first n candidates and uses the optimal strategy starting the ( n+1 )st prince Let’s think about properties B(n) may have 19

20. Rigorous discussion Let’s think about properties B(n) may have First, B(n) is a decreasing function: B(n)≥B(n+1) for any n Indeed, the earlier the princess starts using the optimal strategy, the greater chance of success is. 20

21. Rigorous discussion Second, B(n) must be constant in the beginning of the process Indeed, in the beginning the princess just skips guys, it doesn’t affect the probability of success. 21

22. Rigorous discussion Recall that B(n) be the probability to win if she rejects the first n candidates uses the optimal strategy starting the (n+1) st prince Obviously, B(100)=0 if she rejected all the 100 princes, she loses automatically Further, B(99)=0.01 : if she rejected 99 princes, the only way to deal with the 100 th candidate is to accept him. Let’s try to calculate B(n) 22

23. Complete Probability Formula Imagine that a knight came to a crossroads. To choose the way, he throws a dice But the dice is broken so that the probability to go to the left is 0.2 the probability to go straight is 0.3 the probability to go to the right is 0.5 A fairy lives on the right Chance of survival = 1 0.2 0.3 0.5 23 A warlock lives straight ahead Chance of survival = 0.1 A monster lives on the left Chance of survival = 0.5

24. Complete Probability Formula is the knight’s chance to survive To calculate the knight’s chance to survive, we do as follows 24 Chance of survival = 1 0.2 0.3 0.5 Chance of survival = 0.1 Chance of survival = 0.5 0.2x0.50.3x0.10.5x1++=0.63

25. Rigorous discussion Assume that 98 princes are rejected. 99 th one is better than all of them Probability of this is 1/99 Chance of success is 99/100 B(100)=0, B(99)=1/100 Let’s calculate B(98) There are two possibilities: 99 th one is not better than all of them Probability of this is 98/99 Chance of success is 1/100 Complete probability to win is 25

26. Rigorous discussion Let us fill the following table 26

27. Challenge Recall that A(n) is the probability to win if she rejects the first n-1 candidates the n th prince is better than any of the first n-1 and she picks the n th prince Prove by mathematical induction that (already known to be true for n=100, 99, 98 ) Recall that B(n) is the probability to win if she rejects the first n candidates uses the optimal strategy starting the ( n+1 )st prince 27

28. Rigorous discussion Now it’s clear what to do We must find n 0 such that, but That is In other words, 28

29. Some calculus Consider the expression Obviously, f(n) is the area of the union of the strips on the picture 29

30. Some calculus Area does not change if we stretch the figure 100 times vertically and squeeze 100 times horizontally 30

31. Some calculus Thus f(n) is approximately the area under the graph 31

32. Rigorous discussion Now we have the equation Calculating the integral, we see that Multiplying by -1, we have Thus the solution is 32

33. Summary Thus the optimal strategy for the princess is reject automatically 100/e≈36 candidates and pick up the first one who exceeds all the rejected ones The probability to get the best guy is about 1/e≈0.37 33

34. Generalization What to do if there are more applicants? 1000 princes or N princes? This is easy – in the same way the princess rejects N/e of them and accepts the first one who is better than all the rejected guys. The probability to win approaches 1/e as N grows. 34

35. Generalization Imagine that the princess is not so exacting. She does not want to go into a convent. Instead, she ranks princes, for example: 500 points is the best 400 points is the second best 350 points is the third best etc (she has her own criteria) QUESTION : How should the princess act to maximize the expected value of her husband? 35

36. Super-Challenge Let V 1 be the value of the best prince V 2 be the value of the second prince V 3 be the value of the third prince etc. Let P 1 be the probability to get the best prince P 2 be the probability to get the second prince P 3 be the probability to get the third prince etc. DEFINITION: V exp =P 1 V 1 +P 2 V 2 +P 3 V 3 +…+P N V N is the average expected value of the husband QUESTION : How should the princess act to maximize V exp ? 36

37. Homework Question 1 (slide 17): Recall that we proved that A(100)=1 and A(99)=0.99. a)First, try to prove that A(98)=0.98 b)Second, try to show by induction that A(n)=n/100 37 Question 2 (slide 27): Recall that we found B(100), B(99), B(98). a)First, try to calculate B(97) b)Second, try to show by induction that Question 3 (slide 36): Recall that we worked out the optimal strategy for a princess who aims to get only the best possible husband. a)First, try to figure out how the princess should act if she wants to get either the best or the second best one. b)What should she do if she wants to get any of the first 3 candidates? c)What if she would be satisfied with any of best k among N princes? d)What if she ranks the candidates and tries to maximize the expected value?

38. Thanks for your attention! 38