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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 → one-to-many matchingTask scheduling → many-to-many matchingRoommate allocation → non-bipartite graph

• Stable marriages Stable allocations

Generalizations

How can stability be used?

→ one-to-many matchingTask scheduling → many-to-many matchingRoommate allocation → non-bipartite graph

• Stable marriages Stable allocations

Generalizations

How can stability be used?

Task scheduling → many-to-many matchingRoommate allocation → non-bipartite graph

• Stable marriages Stable allocations

Generalizations

How can stability be used?

→ many-to-many matchingRoommate allocation → non-bipartite graph

• Stable marriages Stable allocations

Generalizations

How can stability be used?

Roommate allocation → non-bipartite graph

• Stable marriages Stable allocations

Generalizations

How can stability be used?

→ non-bipartite graph

• Stable marriages Stable allocations

Generalizations

How can stability be used?

College admission → one-to-many matchingTask scheduling → many-to-many matchingRoommate allocation → non-bipartite 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

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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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