Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists....
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Transcript of Paths to stable allocations · Theorem (Gale, Shapley, 1962) A stable matching always exists....

Stable marriages Stable allocations
Paths to stable allocations
Ágnes Cseh, Martin Skutella
The 7th International Symposium on Algorithmic Game Theory,30 September 2014

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → onetomany matchingTask scheduling → manytomany matchingRoommate allocation → nonbipartite graph

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission
→ onetomany matchingTask scheduling → manytomany matchingRoommate allocation → nonbipartite graph

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → onetomany matching
Task scheduling → manytomany matchingRoommate allocation → nonbipartite graph

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → onetomany matchingTask scheduling
→ manytomany matchingRoommate allocation → nonbipartite graph

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → onetomany matchingTask scheduling → manytomany matching
Roommate allocation → nonbipartite graph

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → onetomany matchingTask scheduling → manytomany matchingRoommate allocation
→ nonbipartite graph

Stable marriages Stable allocations
Generalizations
How can stability be used?
College admission → onetomany matchingTask scheduling → manytomany matchingRoommate allocation → nonbipartite graph

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

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

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

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

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

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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
Twophase bestresponse algorithm (matchings)1 married men2 unmarried men
Twophase algorithm (allocations)1 freeze the quota! → improve along better edges2 free quota

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.

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.

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.

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.

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.
Stable marriagesBasic notionsMyopic changesAlgorithmic results
Stable allocationsGeneralizationsBasic notionsMyopic changesSummary