My tea-time talk about

1. Theory Applications Experiments
The optimal marriage
Ferenc Huszár
Computational and Biological Learning Lab
Department of Engineering, University of Cambridge
May 14, 2010
optimal marriage - tea talk CBL

2. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1 uniform distribution over permutations
optimal marriage - tea talk CBL

3. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry
1 uniform distribution over permutations
optimal marriage - tea talk CBL

4. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry
2. The number of potential partners, N, is finite and known
1 uniform distribution over permutations
optimal marriage - tea talk CBL

5. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry
2. The number of potential partners, N, is finite and known
3. the N partners are “tried” sequentially in a random order1
1 uniform distribution over permutations
optimal marriage - tea talk CBL

6. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry
2. The number of potential partners, N, is finite and known
3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept or
reject based only on the relative ranking of partners “tried’ ’ so far
1 uniform distribution over permutations
optimal marriage - tea talk CBL

7. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry
2. The number of potential partners, N, is finite and known
3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept or
reject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back
1 uniform distribution over permutations
optimal marriage - tea talk CBL

8. Theory Applications Experiments
The standard marriage problem
a.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry
2. The number of potential partners, N, is finite and known
3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept or
reject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back
6. you are satisfied by nothing but the best (0-1 loss)
1 uniform distribution over permutations
optimal marriage - tea talk CBL

9. Theory Applications Experiments
The optimal strategy
in the standard marriage problem
optimal marriage - tea talk CBL

10. Theory Applications Experiments
The optimal strategy
in the standard marriage problem
there is no point of accepting anyone who is not the best so far
optimal marriage - tea talk CBL

11. Theory Applications Experiments
The optimal strategy
in the standard marriage problem
there is no point of accepting anyone who is not the best so far
P[#r is the best |#r is the best in first r ] = 1/N = N
1/r
r
optimal marriage - tea talk CBL

12. Theory Applications Experiments
The optimal strategy
in the standard marriage problem
there is no point of accepting anyone who is not the best so far
P[#r is the best |#r is the best in first r ] = 1/N = N
1/r
r
∗
the optimal strategy is a cutoff rule with threshold r :
reject first r ∗ − 1, then accept the first, that is best-so-far
optimal marriage - tea talk CBL

13. Theory Applications Experiments
The optimal strategy
in the standard marriage problem
there is no point of accepting anyone who is not the best so far
P[#r is the best |#r is the best in first r ] = 1/N = N
1/r
r
∗
the optimal strategy is a cutoff rule with threshold r :
reject first r ∗ − 1, then accept the first, that is best-so-far
determining r ∗ :
φN (r ∗ ) = P[you win with threshold r ∗ ]
N
= P[#j is the best and you select it]
j=r ∗
N N
1 r∗ − 1 r∗ − 1 1
= =
j=r ∗
N j −1 N j=r ∗ j − 1
optimal marriage - tea talk CBL

14. Theory Applications Experiments
The optimal strategy
in the standard marriage problem
there is no point of accepting anyone who is not the best so far
P[#r is the best |#r is the best in first r ] = 1/N = N
1/r
r
∗
the optimal strategy is a cutoff rule with threshold r :
reject first r ∗ − 1, then accept the first, that is best-so-far
determining r ∗ :
φN (r ∗ ) = P[you win with threshold r ∗ ]
N
= P[#j is the best and you select it]
j=r ∗
N N
1 r∗ − 1 r∗ − 1 1
= =
j=r ∗
N j −1 N j=r ∗ j − 1
r ∗ (N) = argmaxr φN (r )
optimal marriage - tea talk CBL

15. Theory Applications Experiments
Assymptotic behaviour
in the standard marriage problem
optimal marriage - tea talk CBL

16. Theory Applications Experiments
Assymptotic behaviour
in the standard marriage problem
r
introduce x = limN→∞ N
N
r −1 N 1
φN (r ) =
N j=r
j −1 N
1
1
→x dt = −x log x =: φ∞ (x )
x t
optimal marriage - tea talk CBL

17. Theory Applications Experiments
Assymptotic behaviour
in the standard marriage problem
r
introduce x = limN→∞ N
N
r −1 N 1
φN (r ) =
N j=r
j −1 N
1
1
→x dt = −x log x =: φ∞ (x )
x t
this is maximised by x ∗ = 1
e ≈ 0.37
optimal marriage - tea talk CBL

18. Theory Applications Experiments
Assymptotic behaviour
in the standard marriage problem
r
introduce x = limN→∞ N
N
r −1 N 1
φN (r ) =
N j=r
j −1 N
1
1
→x dt = −x log x =: φ∞ (x )
x t
this is maximised by x ∗ = 1
e ≈ 0.37
probability of winning is also φ∞ (x ∗ ) = 1
e
optimal marriage - tea talk CBL

19. Theory Applications Experiments
Real-world application
finding a long-term relationship in Hungary
optimal marriage - tea talk CBL

20. Theory Applications Experiments
Real-world application
finding a long-term relationship in Hungary
total population of Hungary: 10,090,330
optimal marriage - tea talk CBL

21. Theory Applications Experiments
Real-world application
finding a long-term relationship in Hungary
total population of Hungary: 10,090,330
single/widowed/divorced women,aged 20-29: 533,142 = N
optimal marriage - tea talk CBL

22. Theory Applications Experiments
Real-world application
finding a long-term relationship in Hungary
total population of Hungary: 10,090,330
single/widowed/divorced women,aged 20-29: 533,142 = N
r ∗ (533, 142) ≈ 196, 132
optimal marriage - tea talk CBL

23. Theory Applications Experiments
Real-world application
finding a long-term relationship in Hungary
total population of Hungary: 10,090,330
single/widowed/divorced women,aged 20-29: 533,142 = N
r ∗ (533, 142) ≈ 196, 132
probability of finding the best is around 0.37
optimal marriage - tea talk CBL

24. Theory Applications Experiments
Real-world application
finding a long-term relationship in Hungary
total population of Hungary: 10,090,330
single/widowed/divorced women,aged 20-29: 533,142 = N
r ∗ (533, 142) ≈ 196, 132
probability of finding the best is around 0.37
“try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL

25. Theory Applications Experiments
Human experiments
optimal marriage - tea talk CBL

26. Theory Applications Experiments
Human experiments
Kahan et al (1967): absolute value instead of ranking
optimal marriage - tea talk CBL

27. Theory Applications Experiments
Human experiments
Kahan et al (1967): absolute value instead of ranking
Rapoport and Tversky (1970): absolute values drawn Gaussian
values
optimal marriage - tea talk CBL

28. Theory Applications Experiments
Human experiments
Kahan et al (1967): absolute value instead of ranking
Rapoport and Tversky (1970): absolute values drawn Gaussian
values
Kogut (1999): lowest price of an item with known price distribution
optimal marriage - tea talk CBL

29. Theory Applications Experiments
Human experiments
Kahan et al (1967): absolute value instead of ranking
Rapoport and Tversky (1970): absolute values drawn Gaussian
values
Kogut (1999): lowest price of an item with known price distribution
Seale and Rapoport (1997): the standard marriage problem
optimal marriage - tea talk CBL

30. Theory Applications Experiments
Human experiments
Kahan et al (1967): absolute value instead of ranking
Rapoport and Tversky (1970): absolute values drawn Gaussian
values
Kogut (1999): lowest price of an item with known price distribution
Seale and Rapoport (1997): the standard marriage problem
all studies found that subjects stopped earlier than optimal
optimal marriage - tea talk CBL

31. Theory Applications Experiments
Human experiments
Kahan et al (1967): absolute value instead of ranking
Rapoport and Tversky (1970): absolute values drawn Gaussian
values
Kogut (1999): lowest price of an item with known price distribution
Seale and Rapoport (1997): the standard marriage problem
all studies found that subjects stopped earlier than optimal
explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL