The optimal marriage
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Theory Applications Experiments
The optimal marriage
Ferenc Huszár
Computational and Biological Learning LabDepartment of Engineering, University of Cambridge
May 14, 2010
optimal marriage - tea talk CBL
![Page 2: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/2.jpg)
Theory Applications Experiments
The standard marriage problema.k.a. the standard secretary problem
Marriage as an optimal stopping problem:
1. you have to choose one partner to marry2. The number of potential partners, N, is finite and known3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept orreject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back6. you are satisfied by nothing but the best (0-1 loss)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 3: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/3.jpg)
Theory Applications Experiments
The standard marriage problema.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 known3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept orreject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back6. you are satisfied by nothing but the best (0-1 loss)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 4: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/4.jpg)
Theory Applications Experiments
The standard marriage problema.k.a. the standard secretary problem
Marriage as an optimal stopping problem:1. you have to choose one partner to marry2. 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 orreject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back6. you are satisfied by nothing but the best (0-1 loss)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 5: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/5.jpg)
Theory Applications Experiments
The standard marriage problema.k.a. the standard secretary problem
Marriage as an optimal stopping problem:1. you have to choose one partner to marry2. The number of potential partners, N, is finite and known3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept orreject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back6. you are satisfied by nothing but the best (0-1 loss)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 6: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/6.jpg)
Theory Applications Experiments
The standard marriage problema.k.a. the standard secretary problem
Marriage as an optimal stopping problem:1. you have to choose one partner to marry2. The number of potential partners, N, is finite and known3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept orreject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back6. you are satisfied by nothing but the best (0-1 loss)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 7: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/7.jpg)
Theory Applications Experiments
The standard marriage problema.k.a. the standard secretary problem
Marriage as an optimal stopping problem:1. you have to choose one partner to marry2. The number of potential partners, N, is finite and known3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept orreject 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)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 8: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/8.jpg)
Theory Applications Experiments
The standard marriage problema.k.a. the standard secretary problem
Marriage as an optimal stopping problem:1. you have to choose one partner to marry2. The number of potential partners, N, is finite and known3. the N partners are “tried” sequentially in a random order1
4. There is a clear ranking of partners, the decision is either accept orreject based only on the relative ranking of partners “tried’ ’ so far
5. once rejected a partner cannot be called back6. you are satisfied by nothing but the best (0-1 loss)
1uniform distribution over permutationsoptimal marriage - tea talk CBL
![Page 9: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/9.jpg)
Theory Applications Experiments
The optimal strategyin the standard marriage problem
I there is no point of accepting anyone who is not the best so farI P[#r is the best |#r is the best in first r ] = 1/N
1/r = rN
I the optimal strategy is a cutoff rule with threshold r∗:reject first r∗ − 1, then accept the first, that is best-so-far
I determining r∗:
φN(r∗) = P[you win with threshold r∗]
=N∑
j=r∗P[#j is the best and you select it]
=N∑
j=r∗
1N
r∗ − 1j − 1 =
r∗ − 1N
N∑j=r∗
1j − 1
I r∗(N) = argmaxr φN(r)
optimal marriage - tea talk CBL
![Page 10: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/10.jpg)
Theory Applications Experiments
The optimal strategyin the standard marriage problem
I there is no point of accepting anyone who is not the best so far
I P[#r is the best |#r is the best in first r ] = 1/N1/r = r
NI the optimal strategy is a cutoff rule with threshold r∗:
reject first r∗ − 1, then accept the first, that is best-so-farI determining r∗:
φN(r∗) = P[you win with threshold r∗]
=N∑
j=r∗P[#j is the best and you select it]
=N∑
j=r∗
1N
r∗ − 1j − 1 =
r∗ − 1N
N∑j=r∗
1j − 1
I r∗(N) = argmaxr φN(r)
optimal marriage - tea talk CBL
![Page 11: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/11.jpg)
Theory Applications Experiments
The optimal strategyin the standard marriage problem
I there is no point of accepting anyone who is not the best so farI P[#r is the best |#r is the best in first r ] = 1/N
1/r = rN
I the optimal strategy is a cutoff rule with threshold r∗:reject first r∗ − 1, then accept the first, that is best-so-far
I determining r∗:
φN(r∗) = P[you win with threshold r∗]
=N∑
j=r∗P[#j is the best and you select it]
=N∑
j=r∗
1N
r∗ − 1j − 1 =
r∗ − 1N
N∑j=r∗
1j − 1
I r∗(N) = argmaxr φN(r)
optimal marriage - tea talk CBL
![Page 12: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/12.jpg)
Theory Applications Experiments
The optimal strategyin the standard marriage problem
I there is no point of accepting anyone who is not the best so farI P[#r is the best |#r is the best in first r ] = 1/N
1/r = rN
I the optimal strategy is a cutoff rule with threshold r∗:reject first r∗ − 1, then accept the first, that is best-so-far
I determining r∗:
φN(r∗) = P[you win with threshold r∗]
=N∑
j=r∗P[#j is the best and you select it]
=N∑
j=r∗
1N
r∗ − 1j − 1 =
r∗ − 1N
N∑j=r∗
1j − 1
I r∗(N) = argmaxr φN(r)
optimal marriage - tea talk CBL
![Page 13: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/13.jpg)
Theory Applications Experiments
The optimal strategyin the standard marriage problem
I there is no point of accepting anyone who is not the best so farI P[#r is the best |#r is the best in first r ] = 1/N
1/r = rN
I the optimal strategy is a cutoff rule with threshold r∗:reject first r∗ − 1, then accept the first, that is best-so-far
I determining r∗:
φN(r∗) = P[you win with threshold r∗]
=N∑
j=r∗P[#j is the best and you select it]
=N∑
j=r∗
1N
r∗ − 1j − 1 =
r∗ − 1N
N∑j=r∗
1j − 1
I r∗(N) = argmaxr φN(r)
optimal marriage - tea talk CBL
![Page 14: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/14.jpg)
Theory Applications Experiments
The optimal strategyin the standard marriage problem
I there is no point of accepting anyone who is not the best so farI P[#r is the best |#r is the best in first r ] = 1/N
1/r = rN
I the optimal strategy is a cutoff rule with threshold r∗:reject first r∗ − 1, then accept the first, that is best-so-far
I determining r∗:
φN(r∗) = P[you win with threshold r∗]
=N∑
j=r∗P[#j is the best and you select it]
=N∑
j=r∗
1N
r∗ − 1j − 1 =
r∗ − 1N
N∑j=r∗
1j − 1
I r∗(N) = argmaxr φN(r)
optimal marriage - tea talk CBL
![Page 15: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/15.jpg)
Theory Applications Experiments
Assymptotic behaviourin the standard marriage problem
I introduce x = limN→∞rN
φN(r) =r − 1
N
N∑j=r
(N
j − 1
)(1N
)
→ x∫ 1
x
1t dt = −x log x =: φ∞(x)
I this is maximised by x∗ = 1e ≈ 0.37
I probability of winning is also φ∞(x∗) = 1e
optimal marriage - tea talk CBL
![Page 16: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/16.jpg)
Theory Applications Experiments
Assymptotic behaviourin the standard marriage problem
I introduce x = limN→∞rN
φN(r) =r − 1
N
N∑j=r
(N
j − 1
)(1N
)
→ x∫ 1
x
1t dt = −x log x =: φ∞(x)
I this is maximised by x∗ = 1e ≈ 0.37
I probability of winning is also φ∞(x∗) = 1e
optimal marriage - tea talk CBL
![Page 17: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/17.jpg)
Theory Applications Experiments
Assymptotic behaviourin the standard marriage problem
I introduce x = limN→∞rN
φN(r) =r − 1
N
N∑j=r
(N
j − 1
)(1N
)
→ x∫ 1
x
1t dt = −x log x =: φ∞(x)
I this is maximised by x∗ = 1e ≈ 0.37
I probability of winning is also φ∞(x∗) = 1e
optimal marriage - tea talk CBL
![Page 18: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/18.jpg)
Theory Applications Experiments
Assymptotic behaviourin the standard marriage problem
I introduce x = limN→∞rN
φN(r) =r − 1
N
N∑j=r
(N
j − 1
)(1N
)
→ x∫ 1
x
1t dt = −x log x =: φ∞(x)
I this is maximised by x∗ = 1e ≈ 0.37
I probability of winning is also φ∞(x∗) = 1e
optimal marriage - tea talk CBL
![Page 19: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/19.jpg)
Theory Applications Experiments
Real-world applicationfinding a long-term relationship in Hungary
I total population of Hungary: 10,090,330I single/widowed/divorced women,aged 20-29: 533,142 = NI r∗(533, 142) ≈ 196, 132I probability of finding the best is around 0.37I “try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL
![Page 20: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/20.jpg)
Theory Applications Experiments
Real-world applicationfinding a long-term relationship in Hungary
I total population of Hungary: 10,090,330
I single/widowed/divorced women,aged 20-29: 533,142 = NI r∗(533, 142) ≈ 196, 132I probability of finding the best is around 0.37I “try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL
![Page 21: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/21.jpg)
Theory Applications Experiments
Real-world applicationfinding a long-term relationship in Hungary
I total population of Hungary: 10,090,330I single/widowed/divorced women,aged 20-29: 533,142 = N
I r∗(533, 142) ≈ 196, 132I probability of finding the best is around 0.37I “try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL
![Page 22: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/22.jpg)
Theory Applications Experiments
Real-world applicationfinding a long-term relationship in Hungary
I total population of Hungary: 10,090,330I single/widowed/divorced women,aged 20-29: 533,142 = NI r∗(533, 142) ≈ 196, 132
I probability of finding the best is around 0.37I “try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL
![Page 23: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/23.jpg)
Theory Applications Experiments
Real-world applicationfinding a long-term relationship in Hungary
I total population of Hungary: 10,090,330I single/widowed/divorced women,aged 20-29: 533,142 = NI r∗(533, 142) ≈ 196, 132I probability of finding the best is around 0.37
I “try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL
![Page 24: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/24.jpg)
Theory Applications Experiments
Real-world applicationfinding a long-term relationship in Hungary
I total population of Hungary: 10,090,330I single/widowed/divorced women,aged 20-29: 533,142 = NI r∗(533, 142) ≈ 196, 132I probability of finding the best is around 0.37I “try” and reject 200,000 partners before even thinking of marriage
optimal marriage - tea talk CBL
![Page 25: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/25.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of rankingI Rapoport and Tversky (1970): absolute values drawn Gaussian
valuesI Kogut (1999): lowest price of an item with known price distributionI Seale and Rapoport (1997): the standard marriage problemI all studies found that subjects stopped earlier than optimalI explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL
![Page 26: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/26.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of ranking
I Rapoport and Tversky (1970): absolute values drawn Gaussianvalues
I Kogut (1999): lowest price of an item with known price distributionI Seale and Rapoport (1997): the standard marriage problemI all studies found that subjects stopped earlier than optimalI explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL
![Page 27: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/27.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of rankingI Rapoport and Tversky (1970): absolute values drawn Gaussian
values
I Kogut (1999): lowest price of an item with known price distributionI Seale and Rapoport (1997): the standard marriage problemI all studies found that subjects stopped earlier than optimalI explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL
![Page 28: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/28.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of rankingI Rapoport and Tversky (1970): absolute values drawn Gaussian
valuesI Kogut (1999): lowest price of an item with known price distribution
I Seale and Rapoport (1997): the standard marriage problemI all studies found that subjects stopped earlier than optimalI explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL
![Page 29: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/29.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of rankingI Rapoport and Tversky (1970): absolute values drawn Gaussian
valuesI Kogut (1999): lowest price of an item with known price distributionI Seale and Rapoport (1997): the standard marriage problem
I all studies found that subjects stopped earlier than optimalI explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL
![Page 30: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/30.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of rankingI Rapoport and Tversky (1970): absolute values drawn Gaussian
valuesI Kogut (1999): lowest price of an item with known price distributionI Seale and Rapoport (1997): the standard marriage problemI all studies found that subjects stopped earlier than optimal
I explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL
![Page 31: The optimal marriage](https://reader034.fdocuments.in/reader034/viewer/2022051513/547e9fec5806b5c75e8b46e4/html5/thumbnails/31.jpg)
Theory Applications Experiments
Human experiments
I Kahan et al (1967): absolute value instead of rankingI Rapoport and Tversky (1970): absolute values drawn Gaussian
valuesI Kogut (1999): lowest price of an item with known price distributionI Seale and Rapoport (1997): the standard marriage problemI all studies found that subjects stopped earlier than optimalI explained with a constant cost of evaluaing an option
optimal marriage - tea talk CBL