Exacting Princess

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Exacting Princess

• Dr Fedor Duzhin,
• Nanyang Technological University,
• School of Physical and Mathematical Sciences

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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 mathematics
He stated the problem in 1960
Sabir Gusein-Zade (b. 1950) Russian mathematician
He gave a general solution to the problem in 196
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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.
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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.
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Problem Statement
She can compare princes
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, ther
e is the best
?
Once she spoke to any two of them, she can decide
which one is better
If 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 shes 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
  
• If the candidate is not the best, she goes into a
  convent
  
• If he is the best, they get married and live
  happily

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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 t
he chance of success as high as possible?
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Some thoughts
If she decides to pick up the 1st one or 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 princes
And pick up the first one who exceeds these 50
QUESTION What is the chance of success under th
is strategy?
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Some thoughts
Let S be the chance in to get the best fiancé
If the best prince is among the first half, then
she loses automatically St prince is among the second half and the second
best is among the first half, then she wins
automatically 25QUESTION What is the chance of success under th
is strategy?
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Idea!
The princesss 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?
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General facts
In mathematics, the probability is measured not
in If there is one chance of two, the probabi
lity is 1/20.5 If there are three chances of eig
ht, the probability is 3/80.375
Thus if the chance of success is S, then the pr
obability is PS/100. The chance of success in
lies between 0 and 100 The probability is betwe
en 0 and 1
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Rigorous discussion
Lets try to think backwards. If there is only
the last prince, the situation is clear.
Assume that the princess knows what to do with (
n1)st prince. What should she do on the nth audi
ence? Let us introduce some parameters describi
ng the process Suppose that she already rejecte
d 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 probabil
ity to win if she accepts him QUESTION Calcula
te A(n)
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Rigorous discussion
Recall that A(n) is the probability to win
if 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 t
hem
then he is the best automatically
Further, A(99)1-0.010.99 if she rejected 98 p
rinces the 99th turned out to be better then al
l of them then the 100th can be the best (probab
ility is 0.01) otherwise the 99th is the best
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Challenge
Let A(n) be the probability to win
if 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
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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 la
rger than B(n), she accepts the first one who is
better than all the rejected guys.
QUESTION Calculate B(n)
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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 (n1)st
prince
Lets think about properties B(n) may have
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Rigorous discussion
First, B(n) is a decreasing function
B(n)B(n1) for any n Indeed, the earlier the
princess starts using the optimal strategy, the
greater chance of success is.
Lets think about properties B(n) may have
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Rigorous discussion
Second, B(n) must be constant in the beginning of
the process Indeed, in the beginning the prince
ss just skips guys, it doesnt affect the
probability of success.
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Rigorous discussion
Lets try to calculate B(n)
Recall that B(n) be the probability to win if
she rejects the first n candidates
uses the optimal strategy starting the (n1)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 prince
s, the only way to deal with the 100th candidat
e is to accept him.
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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 warlock lives straight ahead
Chance of survival 0.1
A monster lives on the left Chance of survival
0.5
0.2
0.3
A fairy lives on the right Chance of survival 1
0.5
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Complete Probability Formula
To calculate the knights chance to survive, we
do as follows
is the knights chance to survive
0.2x0.5
0.3x0.1
0.5x1


0.63
Chance of survival 0.1
Chance of survival 0.5
0.2
0.3
Chance of survival 1
0.5
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Rigorous discussion
B(100)0, B(99)1/100
Lets calculate B(98)
Assume that 98 princes are rejected.
99th one is better than all of them Probability
of this is 1/99
Chance of success is 99/100
There are two possibilities 99th one is not be
tter than all of them Probability of this is 98/9
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Chance of success is 1/100
Complete probability to win is
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Rigorous discussion
Let us fill the following table
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Challenge
Prove by mathematical induction that

(already known to be true for n100, 99, 98)
Recall that A(n) is the probability to win if
she rejects the first n-1 candidates
the nth prince is better than any of the first
n-1 and she picks the nth prince
Recall that B(n) is the probability to win if
she rejects the first n candidates
uses the optimal strategy starting the (n1)st
prince
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Rigorous discussion
Now its clear what to do We must find n0 such
that , but

That is In other words,
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Some calculus
Consider 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
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Rigorous discussion
Now we have the equation Calculating the inte
gral, we see that Multiplying by -1, we ha
ve
Thus the solution is
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Summary
Thus the optimal strategy for the princess is
reject automatically 100/e36 candidates
and pick up the first one who exceeds all the
rejected ones The probability to get the best g
uy is about 1/e0.37
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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 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?
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Super-Challenge
Let V1 be the value of the best prince V2 be t
he value of the second prince V3 be the value of
the third prince
etc.
Let P1 be the probability to get the best princ
e P2 be the probability to get the second prince
P3 be the probability to get the third prince
etc.
DEFINITION VexpP1V1P2V2P3V3PNVN
is the average expected value of the husband
QUESTION 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.
  
• First, try to prove that A(98)0.98
• Second, try to show by induction that A(n)n/100

• Question 2 (slide 27) Recall that we found
  B(100), B(99), B(98).
  
• First, try to calculate B(97)
• 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.
  
• First, try to figure out how the princess should
  act if she wants to get either the best or the
  second best one.
  
• What should she do if she wants to get any of the
  first 3 candidates?
  
• What if she would be satisfied with any of best k
  among N princes?
  
• What if she ranks the candidates and tries to
  maximize the expected value?

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Thanks for your attention!
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