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?

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?

College admission → one-to-many matching

Task scheduling → many-to-many matchingRoommate allocation → 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

Generalizations

How can stability be used?

College admission → one-to-many matchingTask scheduling → many-to-many matching

Roommate allocation → 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

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

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