the stable marriage problem · problem of stable marriage imagine you are a matchmaker, with n...
TRANSCRIPT
![Page 1: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/1.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
The Stable Marriage Problem
Mathias Lindemann
February 14, 2004
![Page 2: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/2.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 1 of 15
Problem of stable marriage
Imagine you are a matchmaker, with N female clients and N maleclients.
back to Overview
![Page 3: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/3.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 1 of 15
Problem of stable marriage
Imagine you are a matchmaker, with N female clients and N maleclients.
Each woman has given you a complete list of the N men, ordered byher preference: her first choice, her second choice, and so on.
back to Overview
![Page 4: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/4.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 1 of 15
Problem of stable marriage
Imagine you are a matchmaker, with N female clients and N maleclients.
Each woman has given you a complete list of the N men, ordered byher preference: her first choice, her second choice, and so on.
Each of the men has given you a complete list of the women, rankedsimilarly.
back to Overview
![Page 5: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/5.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 1 of 15
Problem of stable marriage
Imagine you are a matchmaker, with N female clients and N maleclients.
Each woman has given you a complete list of the N men, ordered byher preference: her first choice, her second choice, and so on.
Each of the men has given you a complete list of the women, rankedsimilarly.
Your job: Arrange N ”happy”(stable) marriages!
back to Overview
![Page 6: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/6.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 2 of 15
By stable we mean that once the matchmaker has arranged themarriages, there should be no man who says to another woman,
”You know, I love you more than the woman I was matched with - let’srun away together!”
where the woman agrees, because she loves the man more than herhusband.
back to Overview
![Page 7: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/7.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 2 of 15
By stable we mean that once the matchmaker has arranged themarriages, there should be no man who says to another woman,
”You know, I love you more than the woman I was matched with - let’srun away together!”
where the woman agrees, because she loves the man more than herhusband.In the spirit of equality, no woman should make such a successfulproposal to a man: should she so propose, we want the man to respond,
”Madam, I am flattered by your attention, but I am married to someoneI love more than you, so i am not interested.”
back to Overview
![Page 8: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/8.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 2 of 15
By stable we mean that once the matchmaker has arranged themarriages, there should be no man who says to another woman,
”You know, I love you more than the woman I was matched with - let’srun away together!”
where the woman agrees, because she loves the man more than herhusband.In the spirit of equality, no woman should make such a successfulproposal to a man: should she so propose, we want the man to respond,
”Madam, I am flattered by your attention, but I am married to someoneI love more than you, so i am not interested.”
Question: Is it always possible for the matchmaker to arrange such agroup of stable marriages, regardless of the preference lists of the menand women ?
back to Overview
![Page 9: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/9.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 3 of 15
Application
• Assignment of students to college places
• Medicine students where assigned to hospitals in USA until 1982(Students prepared a list of favourite hospitals and hospistals wrotea list of favourite students after interviews)
• Lists were sent to a central computer who computed the assignment
back to Overview
![Page 10: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/10.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 4 of 15
Setting
• M set of male clients, F set of female clients, |M | = |F |
back to Overview
![Page 11: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/11.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 4 of 15
Setting
• M set of male clients, F set of female clients, |M | = |F |
• each m ∈ M has an order <f of all f ∈ F
back to Overview
![Page 12: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/12.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 4 of 15
Setting
• M set of male clients, F set of female clients, |M | = |F |
• each m ∈ M has an order <f of all f ∈ F
• each f ∈ F has an order <m of all m ∈ M
back to Overview
![Page 13: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/13.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 4 of 15
Setting
• M set of male clients, F set of female clients, |M | = |F |
• each m ∈ M has an order <f of all f ∈ F
• each f ∈ F has an order <m of all m ∈ M
m1 <f m2 means f prefers m1
back to Overview
![Page 14: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/14.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 4 of 15
Setting
• M set of male clients, F set of female clients, |M | = |F |
• each m ∈ M has an order <f of all f ∈ F
• each f ∈ F has an order <m of all m ∈ M
m1 <f m2 means f prefers m1
A marriage is a bijective mapping H : M −→ F .We write
(m, f) ∈ H, H(m) = f, H−1(f) = m.
back to Overview
![Page 15: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/15.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 5 of 15
Definition 1. A marriage is instable, if there exists m ∈ M, f ∈ F suchthat
(1) (m, f) /∈ H, that is, m and f are not married
(2) m would prefer to be married with f instead of his wife H(m)
(3) f would prefer to be married with m instead of her husband H−1(f)
back to Overview
![Page 16: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/16.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 6 of 15
Example 2. M = {anton, bernd, christoph},F = {Anna,Bettina, Carola}
anton(a): Bettina, Anna, Carolabernd(b): Carola, Anna, Bettinachristoph(c): Carola, Bettina, Anna
Anna(A): anton, bernd, christophBettina(B): bernd, christoph, antonCarola(C): anton, christoph, bernd
One instable marriage(because of anton and Anna):
{aC, bA, cB}
back to Overview
![Page 17: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/17.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 7 of 15
The Algorithm of G ALE and S HAPLEY (1962)
while (@ a married man m ∈ M){m proposes to his favourite f on his list;if (f is not married)
marry m and f ;else if (f prefers m instead of her husband m′{
divorce (f,m′);marry f,m;m′ removes f from his list; }
else (f is happy with m′)m removes f from his list;
}
back to Overview
![Page 18: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/18.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}a B A Cb C A Bc C B AA a b cB b c aC a c b
back to Overview
![Page 19: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/19.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC} a B A C
b C A Bc C B AA a b cB b c aC a c b
back to Overview
![Page 20: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/20.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC}c proposes to C, C prefers c → divorce bC → {aB, cC}
a B A Cb C A Bc C B AA a b cB b c aC a c b
back to Overview
![Page 21: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/21.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC}c proposes to C, C prefers c → divorce bC → {aB, cC}b proposes to A → {aB, bA, cC} = stable marriage
a B A Cb C A Bc C B AA a b cB b c aC a c b
back to Overview
![Page 22: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/22.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC}c proposes to C, C prefers c → divorce bC → {aB, cC}b proposes to A → {aB, bA, cC} = stable marriage
Test:a : satisfied, b : prefers just C, c : satisfied
a B A Cb C A Bc C B AA a b cB b c aC a c b
back to Overview
![Page 23: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/23.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC}c proposes to C, C prefers c → divorce bC → {aB, cC}b proposes to A → {aB, bA, cC} = stable marriage
Test:a : satisfied, b : prefers just C, c : satisfiedA : prefers just a, B : prefers b and c, C : prefers just a
a B A Cb C A Bc C B AA a b cB b c aC a c b
back to Overview
![Page 24: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/24.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC}c proposes to C, C prefers c → divorce bC → {aB, cC}b proposes to A → {aB, bA, cC} = stable marriage
Test:a : satisfied, b : prefers just C, c : satisfiedA : prefers just a, B : prefers b and c, C : prefers just a
a B A Cb C A Bc C B AA a b cB b c aC a c b
Proposition 3. The algorithm stops after finite steps.
back to Overview
![Page 25: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/25.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 8 of 15
Test of the algorithm for our example
a proposes to B → {aB}b proposes to C → {aB, bC}c proposes to C, C prefers c → divorce bC → {aB, cC}b proposes to A → {aB, bA, cC} = stable marriage
Test:a : satisfied, b : prefers just C, c : satisfiedA : prefers just a, B : prefers b and c, C : prefers just a
a B A Cb C A Bc C B AA a b cB b c aC a c b
Proposition 3. The algorithm stops after finite steps.
In each step either a woman is married or an entry is removed from alist. A woman that is married once will be married forever.
back to Overview
![Page 26: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/26.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 9 of 15
Proposition 4. The algorithm gives a stable marriage H, that is, H iscomplete and H is stable.
back to Overview
![Page 27: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/27.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 9 of 15
Proposition 4. The algorithm gives a stable marriage H, that is, H iscomplete and H is stable.
1. H is complete. Assume, there remains one single woman f ∈ F .Then, there exists also one single man m ∈ M . But, m has alsoproposed to f .
back to Overview
![Page 28: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/28.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 9 of 15
Proposition 4. The algorithm gives a stable marriage H, that is, H iscomplete and H is stable.
1. H is complete. Assume, there remains one single woman f ∈ F .Then, there exists also one single man m ∈ M . But, m has alsoproposed to f .
2. The marriage is stable.Suppose, H is instable, i.e. ∃(m, f) which prefer each other morethan their partners H(m) and H−1(f).=⇒ m has proposed to f before H(m) but was rejected because ofH−1(f).Since the women can only improve their partnership this is acontradiction.
back to Overview
![Page 29: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/29.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 10 of 15
Remark 5. The algorithm seems to be symmetric, that is, both, M andF have the same success. This is not true!
back to Overview
![Page 30: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/30.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 10 of 15
Remark 5. The algorithm seems to be symmetric, that is, both, M andF have the same success. This is not true!
The active set (in this case the men) has significantly more successthan the passive (in this case the women). The algorithm can also bereversed.
back to Overview
![Page 31: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/31.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 10 of 15
Remark 5. The algorithm seems to be symmetric, that is, both, M andF have the same success. This is not true!
The active set (in this case the men) has significantly more successthan the passive (in this case the women). The algorithm can also bereversed.
Definition 6. A woman f is called unreachable for a man m if thereexists no stable marriage H ′ with (m, f) ∈ H ′.
back to Overview
![Page 32: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/32.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
back to Overview
![Page 33: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/33.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
We show: If m is rejected by f , then f is unreachable for m.
back to Overview
![Page 34: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/34.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
We show: If m is rejected by f , then f is unreachable for m.Assume, that m is rejected by f and there exists a stable marriage H ′
with (m, f) ∈ H ′.
back to Overview
![Page 35: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/35.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
We show: If m is rejected by f , then f is unreachable for m.Assume, that m is rejected by f and there exists a stable marriage H ′
with (m, f) ∈ H ′.Let m′ be the husband of f in H. With whom m′ is married in H ′ ?
back to Overview
![Page 36: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/36.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
We show: If m is rejected by f , then f is unreachable for m.Assume, that m is rejected by f and there exists a stable marriage H ′
with (m, f) ∈ H ′.Let m′ be the husband of f in H. With whom m′ is married in H ′ ?m is married with f 6= f ′
back to Overview
![Page 37: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/37.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
We show: If m is rejected by f , then f is unreachable for m.Assume, that m is rejected by f and there exists a stable marriage H ′
with (m, f) ∈ H ′.Let m′ be the husband of f in H. With whom m′ is married in H ′ ?m is married with f 6= f ′
Case1 m′ prefers f ′ rather than fImpossible !! m′ has already proposed to f ′ in H before f
back to Overview
![Page 38: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/38.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 11 of 15
Proposition 7. H is men–optimal, i.e. each man marries the bestreachable woman. That is, there exists no other stable marriage, wherem obtains a better woman.
We show: If m is rejected by f , then f is unreachable for m.Assume, that m is rejected by f and there exists a stable marriage H ′
with (m, f) ∈ H ′.Let m′ be the husband of f in H. With whom m′ is married in H ′ ?m is married with f 6= f ′
Case1 m′ prefers f ′ rather than fImpossible !! m′ has already proposed to f ′ in H before f
Case2 m′ prefers f rather than f ′
m′ : . . . <m′ f <m′ . . . <m′ f ′ <m′ . . .f : . . . <f m′ <f . . . <f m <f . . .
=⇒ contradiction to stability of H ′
back to Overview
![Page 39: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/39.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 12 of 15
Proposition 8. The algorithm provides a unique result.
back to Overview
![Page 40: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/40.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 12 of 15
Proposition 8. The algorithm provides a unique result.
Assume, that H1 and H2 are two different men–optimal solutions =⇒∃ m that comes off bad in H1 or H2. Contradiction to optimality.
back to Overview
![Page 41: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/41.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 12 of 15
Proposition 8. The algorithm provides a unique result.
Assume, that H1 and H2 are two different men–optimal solutions =⇒∃ m that comes off bad in H1 or H2. Contradiction to optimality.
Proposition 9. H is ”women–pessimal” (opposition of optimal).
back to Overview
![Page 42: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/42.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 12 of 15
Proposition 8. The algorithm provides a unique result.
Assume, that H1 and H2 are two different men–optimal solutions =⇒∃ m that comes off bad in H1 or H2. Contradiction to optimality.
Proposition 9. H is ”women–pessimal” (opposition of optimal).
Let the stable marriage H ′ 6= H be the worst possible arrangement forthe women.Let (f,m) ∈ H, (f,m′) ∈ H ′.m : unreachable women <m f <m . . .f : . . .m <f . . . <f m′ . . .Contradiction to stability of H ′!
back to Overview
![Page 43: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/43.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 13 of 15
Remarks
• Stable Marriage Problem is an assignment problem of matching–algorithms, transversal theory
back to Overview
![Page 44: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/44.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 13 of 15
Remarks
• Stable Marriage Problem is an assignment problem of matching–algorithms, transversal theory
• can be modelled as a graph problem −→ Perfect Matching Problem(A matching is a subset of edges where each edge is connected withat most one knot. A perfect matching is a subset of edges whereeach edge is connected with exactly one knot.)
back to Overview
![Page 45: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/45.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 13 of 15
Remarks
• Stable Marriage Problem is an assignment problem of matching–algorithms, transversal theory
• can be modelled as a graph problem −→ Perfect Matching Problem(A matching is a subset of edges where each edge is connected withat most one knot. A perfect matching is a subset of edges whereeach edge is connected with exactly one knot.)
• Marry to obtain the most happiness −→ Largest Perfect MatchingProblem
back to Overview
![Page 46: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/46.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 13 of 15
Remarks
• Stable Marriage Problem is an assignment problem of matching–algorithms, transversal theory
• can be modelled as a graph problem −→ Perfect Matching Problem(A matching is a subset of edges where each edge is connected withat most one knot. A perfect matching is a subset of edges whereeach edge is connected with exactly one knot.)
• Marry to obtain the most happiness −→ Largest Perfect MatchingProblem
• List uncomplete −→ yet another solution (Theorem of HALL)
back to Overview
![Page 47: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/47.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 14 of 15
Questions
• Gives the algorithm always a solution that optimizes the totalhappiness ?
back to Overview
![Page 48: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/48.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 14 of 15
Questions
• Gives the algorithm always a solution that optimizes the totalhappiness ?
• One can find an example where the men–optimal stable marriage isas much as possible worser than the optimal solution of the PerfectMatching Problem.
back to Overview
![Page 49: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/49.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 14 of 15
Questions
• Gives the algorithm always a solution that optimizes the totalhappiness ?
• One can find an example where the men–optimal stable marriage isas much as possible worser than the optimal solution of the PerfectMatching Problem.
• Is the solution of the Perfect Matching Problem always stable ?
back to Overview
![Page 50: The Stable Marriage Problem · Problem of stable marriage Imagine you are a matchmaker, with N female clients and N male clients. Each woman has given you a complete list of the N](https://reader035.vdocuments.us/reader035/viewer/2022071011/5fc93fe1d795e81b2c0e8380/html5/thumbnails/50.jpg)
Zentrum furTechnomathematik Fachbereich 3
Mathematik und Informatik
Zuruck page 15 of 15
Conclusion
Back to application: The algorithm prefers the hospitals. In 1981appeared an article in the New England Journal of Medicine about thisfact. Before that, people assumed that the assignment is fair for both.
back to Overview