Exacting Princess

Dr Fedor Duzhin,Nanyang Technological University,

School of Physical and Mathematical Sciences

About me

• High School:

• 1994, Russian Mathematics Olympiad – 2nd prize• 1995, Chinese Mathematics Olympiad – 2nd prize• 1995, Russian Mathematics Olympiad – 3rd prize

• 1994, Russian Informatics Olympiad – 1st prize• 1995, Russian Informatics Olympiad – 2nd prize

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About me

• University Education:

• M.S. in mathematics, 2000, Moscow State University

• Major: Pure Mathematics (Topology)

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About me

• Graduate Education:

• Ph.D. in mathematics, 2005, Royal Institute of Technology, Stockholm

• Major: Pure Mathematics (Topology and Dynamical Systems)

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Characters

Martin Gardner (b. 1914)Famous American science writer specializing in recreational mathematicsHe stated the problem in 1960

Sabir Gusein-Zade (b. 1950)Russian mathematicianHe gave a general solution to the problem in 1966

Boris Berezovsky (b. 1946)One of Russia's first billionairesOnce he was an applied mathematician. His doctoral thesis is devoted to optimal stopping of stochastic processes, which is a generalization of the problem.

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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

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 Band prince B is better than prince Cthen A is better than C

Therefore, indeed, there is the bestIf she could speak with each of the princes, then she would be able to chose the best of them

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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 conventb. If he is the best, they get married and live

happily

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Problem Statement

The procedure is as followsShe 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?

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Some thoughts

If she decides to pick up the 1st oneor the 3rd one,or the 100th one

the chance of success is just 1%

QUESTION:How can she make the chance of

success reasonable, for example at least 25%?

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Some thoughts

Instead, she could do as follows:First, reject half of them, that is, 50 princesAnd pick up the first one who exceeds these 50

QUESTION:What is the chance of success under

this strategy?

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Some thoughts

QUESTION:What is the chance of success under

this strategy?

Let S be the chance in % to get the best fiancéIf 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%

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Idea!

The princess’s strategy can be like:First, reject R% of the candidatesAnd pick up the first one who exceeds all the rejected

ones

QUESTION: What R should be taken to make the chance of success maximal?

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General facts

In mathematics, the probability is measured not in %

If there is one chance of two, the probability is 1/2=0.5If 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 100The probability is between 0 and 1

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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 nth audience?

Let us introduce some parameters describing the process

Suppose that she already rejected the first n-1 candidates and the nth 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)

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Rigorous discussion

Recall that A(n) is the probability to winif she rejects the first n-1 candidates,if the nth prince is better than any of the first n-1,and if she picks the nth prince

Obviously, A(100)=1:if she rejected 99 princes the 100th turned out to be better than all of themthen he is the best automatically

Further, A(99)=1-0.01=0.99:if she rejected 98 princes the 99th turned out to be better then all of themthen the 100th can be the best (probability is 0.01)otherwise the 99th is the best

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Challenge

Let A(n) be the probability to winif she rejects the first n-1 candidates,if the nth prince is better than any of the first n-

1,and if she picks the nth prince

Prove (by mathematical induction) that 100

)(n

nA

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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

Rigorous discussion

Recall that B(n) is the probability to win ifshe rejects the first n candidates anduses the optimal strategy starting the (n+1)st prince

Let’s think about properties B(n) may have

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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.

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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.

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Rigorous discussion

Recall that B(n) be the probability to win ifshe rejects the first n candidatesuses the optimal strategy starting the (n+1)st prince

Obviously, B(100)=0if 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 100th candidate isto accept him.

Let’s try to calculate B(n)

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Complete Probability Formula

Imagine that a knight came to a crossroads.To choose the way, he throws a diceBut the dice is broken so that

the probability to go to the left is 0.2the probability to go straight is 0.3the probability to go to the right is 0.5

A fairy lives on the rightChance of survival = 1

0.2 0.3

0.5

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A warlock lives straight aheadChance of survival = 0.1

A monster lives on the leftChance of survival = 0.5

Complete Probability Formula

is the knight’s chance to survive

To calculate the knight’s chance to survive, we do as follows

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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

Rigorous discussion

Assume that 98 princes are rejected.

99th one is better than all of themProbability of this is 1/99Chance of success is 99/100

B(100)=0, B(99)=1/100 Let’s calculate B(98)

There are two possibilities:

99th one is not better than all of themProbability of this is 98/99Chance of success is 1/100

Complete probability to win is

10099

9998

100

1

99

98

100

99

99

1

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Rigorous discussion

n

)(nA

10099

9998

Let us fill the following table

1009998

)(nB

)(

)(

nA

nB99

1

0

100

98

100

991

99

1

98

1

100

1

0

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Challenge

Recall that A(n) is the probability to win ifshe rejects the first n-1 candidatesthe nth prince is better than any of the first n-1and she picks the nth prince

Prove by mathematical induction that

(already known to be true for n=100, 99, 98)

99

1

1

11

)(

)(

nnnA

nB

Recall that B(n) is the probability to win ifshe rejects the first n candidatesuses the optimal strategy starting the (n+1)st prince

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Rigorous discussion

1)(

)(

0

0 nA

nB1

)1(

)1(

0

0

nA

nB

1)(

)(

0

0 nA

nB

Now it’s clear what to do

We must find n0 such that , but

That is

In other words, 199

1

1

11

00

nn

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Some calculus

99

1

1

11)(

nn

nfConsider the expression

Obviously, f(n) is the area of the union of the strips on the picture

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Some calculus

Area does not change if we stretch the figure 100 times vertically and squeeze 100 times horizontally

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Some calculus

Thus f(n) is approximately the area under the graph

1

100

)( nx

dxnf

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Rigorous discussion

11

100

n x

dx

1100

log n

e

n 1

100

Now we have the equation

Calculating the integral,we see that

Multiplying by -1, we have

Thus the solution is

1100

log n

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Summary

Thus the optimal strategy for the princess isreject automatically 100/e≈36 candidatesand pick up the first one who exceeds all the rejected

ones

The probability to get the best guy is about 1/e≈0.37

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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.

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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 best400 points is the second

best350 points is the third

bestetc (she has her own criteria)

QUESTION: How should the princess act to maximize the expected value of her husband?

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Super-Challenge

Let V1 be the value of the best princeV2 be the value of the second princeV3 be the value of the third prince

etc.

Let P1 be the probability to get the best princeP2 be the probability to get the second prince P3 be the probability to get the third prince

etc.DEFINITION: Vexp=P1V1+P2V2+P3V3+…+PNVN

is the average expected value of the husbandQUESTION: How should the princess act to maximize Vexp?

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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)=0.98b)Second, try to show by induction that A(n)=n/100

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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

99

1

1

11

)(

)(

nnnA

nB

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?

Thanks for your attention!

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