Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists....
Transcript of Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists....
![Page 1: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/1.jpg)
Stable marriages Stable allocations
Paths to stable allocations
Ágnes Cseh, Martin Skutella
The 7th International Symposium on Algorithmic Game Theory,30 September 2014
![Page 2: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/2.jpg)
Stable marriages Stable allocations
Basic notions
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 3: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/3.jpg)
Stable marriages Stable allocations
Basic notions
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 4: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/4.jpg)
Stable marriages Stable allocations
Basic notions
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 5: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/5.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 6: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/6.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 7: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/7.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 8: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/8.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 9: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/9.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 10: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/10.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 11: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/11.jpg)
Stable marriages Stable allocations
Basic notions
1 2 3 4
2 1 3 42 1 4 3
4 2 1 3
3 1 4 2
3 4 2 1
3 4 2 1
2 1 3 4
De�nition
Edge uv is blocking if
1 it is not in the matching and
2 u prefers v to his wife and
3 v prefers u to her husband.
Theorem (Gale, Shapley, 1962)
A stable matching always exists.
![Page 12: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/12.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 13: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/13.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 14: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/14.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 15: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/15.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 16: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/16.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 17: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/17.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 18: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/18.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 19: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/19.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 20: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/20.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 21: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/21.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 22: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/22.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 23: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/23.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 24: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/24.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 25: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/25.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 26: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/26.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 27: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/27.jpg)
Stable marriages Stable allocations
Myopic changes
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
Theorem (Knuth, 1976)
Uncoordinated processes may cycle.
Theorem (Roth, Vande Vate, 1990)
Uncoordinated processes terminate with probability one.
![Page 28: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/28.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 29: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/29.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?
1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 30: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/30.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 31: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/31.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 32: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/32.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 33: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/33.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 34: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/34.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 35: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/35.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 36: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/36.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 37: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/37.jpg)
Stable marriages Stable allocations
Algorithmic results
Theorem (Ackermann et al., 2011)
There is a best response strategy leading to a stable matching in
polynomial time.
How does it work?1 married men2 unmarried men
1
3
2
1
1
2
3
1
1
2
2
3
2
2
3
3
3
1
![Page 38: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/38.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → one-to-many matching
Task scheduling → many-to-many matching
Roommate allocation → non-bipartite graph
![Page 39: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/39.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission
→ one-to-many matching
Task scheduling → many-to-many matching
Roommate allocation → non-bipartite graph
![Page 40: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/40.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → one-to-many matching
Task scheduling → many-to-many matching
Roommate allocation → non-bipartite graph
![Page 41: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/41.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → one-to-many matching
Task scheduling
→ many-to-many matching
Roommate allocation → non-bipartite graph
![Page 42: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/42.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → one-to-many matching
Task scheduling → many-to-many matching
Roommate allocation → non-bipartite graph
![Page 43: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/43.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → one-to-many matching
Task scheduling → many-to-many matching
Roommate allocation
→ non-bipartite graph
![Page 44: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/44.jpg)
Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → one-to-many matching
Task scheduling → many-to-many matching
Roommate allocation → non-bipartite graph
![Page 45: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/45.jpg)
Stable marriages Stable allocations
Basic notions
jobs
machines
2 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
De�nition
An edge jm is blocking if
1 it is unsaturated and
2 j prefers m to its least preferred machine or j is incomplete and
3 m prefers j to his worst job or m has free time
![Page 46: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/46.jpg)
Stable marriages Stable allocations
Basic notions
jobs
machines
2 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
De�nition
An edge jm is blocking if
1 it is unsaturated and
2 j prefers m to its least preferred machine or j is incomplete and
3 m prefers j to his worst job or m has free time
![Page 47: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/47.jpg)
Stable marriages Stable allocations
Basic notions
jobs
machines
2 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
De�nition
An edge jm is blocking if
1 it is unsaturated and
2 j prefers m to its least preferred machine or j is incomplete and
3 m prefers j to his worst job or m has free time
![Page 48: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/48.jpg)
Stable marriages Stable allocations
Basic notions
jobs
machines
2 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
De�nition
An edge jm is blocking if
1 it is unsaturated and
2 j prefers m to its least preferred machine or j is incomplete and
3 m prefers j to his worst job or m has free time
![Page 49: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/49.jpg)
Stable marriages Stable allocations
Basic notions
jobs
machines
2 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
De�nition
An edge jm is blocking if
1 it is unsaturated and
2 j prefers m to its least preferred machine or j is incomplete and
3 m prefers j to his worst job or m has free time
![Page 50: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/50.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 51: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/51.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 52: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/52.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 53: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/53.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 54: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/54.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 55: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/55.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 56: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/56.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 57: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/57.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 58: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/58.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 59: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/59.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 60: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/60.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 61: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/61.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 62: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/62.jpg)
Stable marriages Stable allocations
Myopic changes
jobs
machines
3 3 1
3 1 1
1
3
1
1
1
2
3
1
2
2
2
3
2
2
3
3
3
1
Two-phase best-response algorithm (matchings)1 married men2 unmarried men
Two-phase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota
![Page 63: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/63.jpg)
Stable marriages Stable allocations
Summary
shortest path to stability random path to stabilitybest response dynamics exponential length converges with probability 1better response dynamics polynomial length converges with probability 1
What did we see?
1 Stable marriage problem with capacities2 Myopic procedures leading to a stable solution
Thank you for your attention.
![Page 64: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/64.jpg)
Stable marriages Stable allocations
Summary
shortest path to stability random path to stabilitybest response dynamics exponential length converges with probability 1better response dynamics polynomial length converges with probability 1
What did we see?
1 Stable marriage problem with capacities2 Myopic procedures leading to a stable solution
Thank you for your attention.
![Page 65: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/65.jpg)
Stable marriages Stable allocations
Summary
shortest path to stability random path to stabilitybest response dynamics exponential length converges with probability 1better response dynamics polynomial length converges with probability 1
What did we see?
1 Stable marriage problem with capacities
2 Myopic procedures leading to a stable solution
Thank you for your attention.
![Page 66: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/66.jpg)
Stable marriages Stable allocations
Summary
shortest path to stability random path to stabilitybest response dynamics exponential length converges with probability 1better response dynamics polynomial length converges with probability 1
What did we see?
1 Stable marriage problem with capacities2 Myopic procedures leading to a stable solution
Thank you for your attention.
![Page 67: Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists. Stable marriages Stable allocations Basic notions De nition Edge uv isblockingif 1 it](https://reader034.fdocuments.net/reader034/viewer/2022042622/5f8b74163a8dfc5d2b0a33c3/html5/thumbnails/67.jpg)
Stable marriages Stable allocations
Summary
shortest path to stability random path to stabilitybest response dynamics exponential length converges with probability 1better response dynamics polynomial length converges with probability 1
What did we see?
1 Stable marriage problem with capacities2 Myopic procedures leading to a stable solution
Thank you for your attention.