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

of 67 /67

Embed Size (px)

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?

    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