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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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