Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and...

94
Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Transcript of Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and...

Page 1: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular Matchings: Structure and CheatingStrategies

Meghana Nasre

April 19, 2013.

Page 2: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• Set of agents A and set of posts P.

• Agents rank posts according to individual preferences.

Goal

• Compute a good assignment of agents to posts.

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p

• Relevant in matching students to housing, NETflix, ...

• Preferences may be strict..

Page 3: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• Set of agents A and set of posts P.

• Agents rank posts according to individual preferences.

Goal

• Compute a good assignment of agents to posts.

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p

• Relevant in matching students to housing, NETflix, ...

• Preferences may be strict..

Page 4: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• Set of agents A and set of posts P.

• Agents rank posts according to individual preferences.

Goal

• Compute a good assignment of agents to posts.

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p

• Relevant in matching students to housing, NETflix, ...

• Preferences may be strict..

Page 5: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• Set of agents A and set of posts P.

• Agents rank posts according to individual preferences.

Goal

• Compute a good assignment of agents to posts.

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p

• Relevant in matching students to housing, NETflix, ...

• Preferences may be strict..

Page 6: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• Set of agents A and set of posts P.

• Agents rank posts according to individual preferences.

Goal

• Compute a good assignment of agents to posts.

a

a

a

1

2

3

1 2 3 4 5

1

1

2

2

1

3

2

p p p p p

1

22

3 4

• Relevant in matching students to housing, NETflix, ...

• Preferences may be strict or can contain ties.

Page 7: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• G = (A ∪ P,E ); edge set E is ranked.

A P

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p a

a

a

1

2

3

p

p

p

p

p

1

2

3

4

5

Page 8: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• G = (A ∪ P,E ); edge set E is ranked.

Output

• A matching M in G .

A P

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p a

a

a

1

2

3

p

p

p

p

p

1

2

3

4

5

• Several notions of optimality – rank-maximality, popularity,fairness...

Page 9: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Setup

Input

• G = (A ∪ P,E ); edge set E is ranked.

Output

• A matching M in G .

A P

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

p p p p p a

a

a

1

2

3

p

p

p

p

p

1

2

3

4

5

• Several notions of optimality – rank-maximality, popularity,fairness...

Page 10: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matching – definition

• An agent compares two matchings M1 and M2 using votes.

• Matching that gets more votes is the better of the two.

M2

M1

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

a

a

a

1

2

3

1 2 3 5

1

1

2

23 4 5

1

3

2 3

4p p p p p p p p p p

• M1 � M2.

• M is popular if no matching M ′ is more popular than M.

• Instance may admit no popular matching.

• Instance may admit many popular matchings.

Page 11: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matching – definition

• An agent compares two matchings M1 and M2 using votes.

• Matching that gets more votes is the better of the two.

M2

M1

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

a

a

a

1

2

3

1 2 3 5

1

1

2

23 4 5

1

3

2 3

4p p p p p p p p p p

• M1 � M2.

• M is popular if no matching M ′ is more popular than M.

• Instance may admit no popular matching.

• Instance may admit many popular matchings.

Page 12: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matching – definition

• An agent compares two matchings M1 and M2 using votes.

• Matching that gets more votes is the better of the two.

M2

M1

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

a

a

a

1

2

3

1 2 3 5

1

1

2

23 4 5

1

3

2 3

4p p p p p p p p p p

• M1 � M2.

• M is popular if no matching M ′ is more popular than M.

• Instance may admit no popular matching.

• Instance may admit many popular matchings.

Page 13: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matching – definition

• An agent compares two matchings M1 and M2 using votes.

• Matching that gets more votes is the better of the two.

M2

M1

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

a

a

a

1

2

3

1 2 3 5

1

1

2

23 4 5

1

3

2 3

4p p p p p p p p p p

• M1 � M2.

• M is popular if no matching M ′ is more popular than M.

• Instance may admit no popular matching.

• Instance may admit many popular matchings.

Page 14: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matching – definition

• An agent compares two matchings M1 and M2 using votes.

• Matching that gets more votes is the better of the two.

M2

M1

a

a

a

1

2

3

1 2 3 4 5

1

1

2

23 4 5

1

3

2 3

a

a

a

1

2

3

1 2 3 5

1

1

2

23 4 5

1

3

2 3

4p p p p p p p p p p

• M1 � M2.

• M is popular if no matching M ′ is more popular than M.

• Instance may admit no popular matching.

• Instance may admit many popular matchings.

Page 15: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

This talk..

A centralized matching market.

• Assume: Input instance admits a popular matching.

• A central authority chooses any popular matching.

• A manipulative agent – a1.• a1 is aware of the true preferences of everybody.

• Goal: a1 wants to manipulate to get better always.• Other agents do not change preferences.

• What is a1’s best strategy?

• What if all agents are manipulative?

Page 16: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

This talk..

A centralized matching market.

• Assume: Input instance admits a popular matching.

• A central authority chooses any popular matching.

• A manipulative agent – a1.• a1 is aware of the true preferences of everybody.

• Goal: a1 wants to manipulate to get better always.• Other agents do not change preferences.

• What is a1’s best strategy?

• What if all agents are manipulative?

Page 17: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

This talk..

A centralized matching market.

• Assume: Input instance admits a popular matching.

• A central authority chooses any popular matching.

• A manipulative agent – a1.• a1 is aware of the true preferences of everybody.

• Goal: a1 wants to manipulate to get better always.• Other agents do not change preferences.

• What is a1’s best strategy?

• What if all agents are manipulative?

Page 18: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Background

• Notion of popularity – introduced by Gardenfors in 1975.• Initially defined in the context of stable marriage problem.

• Abraham et al. – popularity in one-sided preferences (SODA 05).

• In terms of strategic issues• Strategic issues of stable marriage/ roommates well

understood.Teo et al. (Mgmt Sci 01), Huang (STACS 07).

• Our study is in similar spirit for popularity.

Page 19: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Background

• Notion of popularity – introduced by Gardenfors in 1975.• Initially defined in the context of stable marriage problem.

• Abraham et al. – popularity in one-sided preferences (SODA 05).

• In terms of strategic issues• Strategic issues of stable marriage/ roommates well

understood.Teo et al. (Mgmt Sci 01), Huang (STACS 07).

• Our study is in similar spirit for popularity.

Page 20: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Background

• Notion of popularity – introduced by Gardenfors in 1975.• Initially defined in the context of stable marriage problem.

• Abraham et al. – popularity in one-sided preferences (SODA 05).

• In terms of strategic issues• Strategic issues of stable marriage/ roommates well

understood.Teo et al. (Mgmt Sci 01), Huang (STACS 07).

• Our study is in similar spirit for popularity.

Page 21: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Background

• Notion of popularity – introduced by Gardenfors in 1975.• Initially defined in the context of stable marriage problem.

• Abraham et al. – popularity in one-sided preferences (SODA 05).

• In terms of strategic issues• Strategic issues of stable marriage/ roommates well

understood.Teo et al. (Mgmt Sci 01), Huang (STACS 07).

• Our study is in similar spirit for popularity.

Page 22: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Outline

• Strict preferences.• Characterization of popular matchings.• Switching graph.• A single manipulative agent.• All agents are manipulative.

• Ties• Characterization of popular matchings.• Switching graph.

prior to this, switching graph for ties was open

• Summary.

Page 23: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Strict preferences

Page 24: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• Uses notion of f -post and s-post for an agent.

• f -post: rank-1 post for an agent.

• s-post: most preferred non-f -post.

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

• s-post may be a dummy last-resort post in some cases.

Page 25: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• Uses notion of f -post and s-post for an agent.

• f -post: rank-1 post for an agent.

• s-post: most preferred non-f -post.

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

• s-post may be a dummy last-resort post in some cases.

Page 26: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• Uses notion of f -post and s-post for an agent.

• f -post: rank-1 post for an agent.

• s-post: most preferred non-f -post.

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

• s-post may be a dummy last-resort post in some cases.

Page 27: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• Uses notion of f -post and s-post for an agent.

• f -post: rank-1 post for an agent.

• s-post: most preferred non-f -post.

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

a

a

a

a

p

p

p

1 1

2 2

33

4

4

reduced graph G’

25

p

p5

• s-post may be a dummy last-resort post in some cases.

Page 28: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• A matching M is popular iff:

1. M matches all f -posts (the red posts).2. M matches each agent a to either f (a) or s(a).

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a popular matching

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

Page 29: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• A matching M is popular iff:

1. M matches all f -posts (the red posts).2. M matches each agent a to either f (a) or s(a).

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a popular matching

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

Page 30: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• A matching M is popular iff:

1. M matches all f -posts (the red posts).2. M matches each agent a to either f (a) or s(a).

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a popular matching

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

Page 31: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• A matching M is popular iff:

1. M matches all f -posts (the red posts).2. M matches each agent a to either f (a) or s(a).

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

25

another pop. matching

• An O(m + n) time algorithm for strict lists.

Page 32: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: characterization Abraham et al. SICOMP 07

• A matching M is popular iff:

1. M matches all f -posts (the red posts).2. M matches each agent a to either f (a) or s(a).

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

25

another pop. matching

• An O(m + n) time algorithm for strict lists.

Page 33: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Recall.. our problem

• A centralized matching market.

• Central authority chooses any popular matching.

• A manipulative agent – a1.• a1 is aware of the true preferences of everybody.• Other agents do not change preferences.

What does a1 get when every one is truthful?

• since M is popular, M(a1) ∈ {f (a1), s(a1)}.• identify edges (a1, p) such that (a1, p) belongs to some pop.

mat.

Page 34: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Recall.. our problem

• A centralized matching market.

• Central authority chooses any popular matching.

• A manipulative agent – a1.• a1 is aware of the true preferences of everybody.• Other agents do not change preferences.

What does a1 get when every one is truthful?

• since M is popular, M(a1) ∈ {f (a1), s(a1)}.• identify edges (a1, p) such that (a1, p) belongs to some pop.

mat.

Page 35: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph

Page 36: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

Start with a pop. mat. M⇓ switch

to another pop. mat. M ′

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

Page 37: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

Start with a pop. mat. M⇓ switch

to another pop. mat. M ′

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

Page 38: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

a popular matching

• For a, define M(a) and OM(a).

Page 39: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

a popular matching

• For a, define M(a) and OM(a).

Page 40: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

p2

p3

p1

p4

p5

a popular matchingswitching graph

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

Page 41: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

p2

p3

p1

p4

p5

a popular matchingswitching graph

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

Page 42: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

a popular matchingswitching graph

p2

p3

p4

p1

p5

a1

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

Page 43: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

a popular matchingswitching graph

p2

p3

p4

p1

p5

a2

a1

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

Page 44: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

a popular matchingswitching graph

p2

p3

p4

p1

p5

a2

a1 a

3a

4

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

Page 45: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

a popular matchingswitching graph

p2

p3

p4

p1

p5

a2

a1 a

3a

4

• Switching path p5, p1, p4.

Page 46: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

a

p p p p p1 2 3 4 5

a

a

a

1

2

3

4

2

2

1

1

1

1

3

3 2 4

34

43

5

25

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

a

a

a

a

p

p

p

p

1 1

2 2

33

4

4

p5

reduced graph

switching graph

p2

p3

p4

p5

a

aa

3a

4

another popular matching

p1

1

2

• Switching path p5, p1, p4.

Page 47: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

• Reduced graph has exactly two edges per agent.

• Switching graph is extremely simple!

cycle componenttree component

Page 48: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

• Reduced graph has exactly two edges per agent.

• Switching graph is extremely simple!

cycle componenttree component

Page 49: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph Mc Dermid and Irving. JOCO 11

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t a popular matching M in G .

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

• Reduced graph has exactly two edges per agent.

• Switching graph is extremely simple!

cycle componenttree component

Page 50: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t. a popular matching M in G .

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

• Reduced graph has exactly two edges.

• Switching graph is extremely simple!

cycle componenttree component

agents stuck with their posts

Page 51: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Switching graph

• GM = (P,EM); vertices: posts in G ; edges: agents in G .

• Edges of GM : defined w.r.t. a popular matching M in G .

• For a, define M(a) and OM(a). Edge from M(a) to OM(a).

• Reduced graph has exactly two edges.

• Switching graph is extremely simple!

cycle component

agents stuck with their posts

tree component

agents can switch

Page 52: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Partition set of agents

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• Af ⇒ agents that get matched to f -post always.

• As ⇒ agents that get matched to s-post always.

• Af /s ⇒ agents that get matched to f -post sometime, s-postat others.

Page 53: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Partition set of agents

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• Af ⇒ agents that get matched to f -post always.

• As ⇒ agents that get matched to s-post always.

• Af /s ⇒ agents that get matched to f -post sometime, s-postat others.

Page 54: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Manipulative agent

Page 55: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Manipulative agent a1

What partition does a1 belong when every one is truthful?

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• If a1 ∈ Af , every pop. mat in G matches a1 to f (a1).

• Recall other agents remain truthful.

• Hence, true preference list is optimal for a1.

What if a1 belongs to As or Af /s?

Page 56: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Manipulative agent a1

What partition does a1 belong when every one is truthful?

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• If a1 ∈ Af , every pop. mat in G matches a1 to f (a1).

• Recall other agents remain truthful.

• Hence, true preference list is optimal for a1.

What if a1 belongs to As or Af /s?

Page 57: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Manipulative agent a1

What partition does a1 belong when every one is truthful?

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• If a1 ∈ Af , every pop. mat in G matches a1 to f (a1).

• Recall other agents remain truthful.

• Hence, true preference list is optimal for a1.

What if a1 belongs to As or Af /s?

Page 58: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Manipulative agent a1

What partition does a1 belong when every one is truthful?

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• If a1 ∈ Af , every pop. mat in G matches a1 to f (a1).

• Recall other agents remain truthful.

• Hence, true preference list is optimal for a1.

What if a1 belongs to As or Af /s?

Page 59: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Manipulative agent a1

What partition does a1 belong when every one is truthful?

cycle component

agents stuck with their posts

tree component

agents can switch

Af

Ab

A = Af ∪ As ∪ Af /s

• If a1 ∈ Af , every pop. mat in G matches a1 to f (a1).

• Recall other agents remain truthful.

• Hence, true preference list is optimal for a1.

What if a1 belongs to As or Af /s?

Page 60: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Always gets matched to s1 by being truthful.

• Recall a1 belongs to the tree part of a cycle component in GM .

s1

a1

f1

tree component cycle component

• Can a1 get matched to f1 by manipulation?No! There is no way to switch other agents.

Page 61: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Always gets matched to s1 by being truthful.

• Recall a1 belongs to the tree part of a cycle component in GM .

s1

a1

f1

tree component cycle component

• Can a1 get matched to f1 by manipulation?No! There is no way to switch other agents.

Page 62: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Always gets matched to s1 by being truthful.

• Recall a1 belongs to the tree part of a cycle component in GM .

s1

a1

f1

tree component cycle component

• Can a1 get matched to f1 by manipulation?

No! There is no way to switch other agents.

Page 63: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Always gets matched to s1 by being truthful.

• Recall a1 belongs to the tree part of a cycle component in GM .

s1

a1

f1

tree component cycle component

• Can a1 get matched to f1 by manipulation?No! There is no way to switch other agents.

Page 64: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Always gets matched to s1 by being truthful.

• Recall a1 belongs to the tree part of a cycle component in GM .

s1

a1

f1

tree component cycle component

pi

• Can a1 get matched to one of p1, . . . , pk by manipulation?Yes! Provided one of pi belongs to a tree component.

Page 65: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Always gets matched to s1 by being truthful.

• Recall a1 belongs to the tree part of a cycle component in GM .

s1

a1

f1

tree component cycle component

pi

Theorem: a1 can cheat if and only if one of pi belongs to a treecomponent.

Page 66: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

a1

f1

tree component cycle component

pi

s

Note: This change does not affect s(a) for others.

Page 67: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

a1

f1

tree component cycle component

pi

s

Note: This change does not affect s(a) for others.

Page 68: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

a1

f1

tree component cycle component

pi

s

Note: This change does not affect s(a) for others.

Page 69: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

a1

f1

tree component cycle component

pi

s

• Apply the switching path starting at pi .

• This makes pi unmatched.

Page 70: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

a1

f1

tree component cycle component

pi

s

• Apply the switching path starting at pi .

• This makes pi unmatched.

Page 71: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

f1

tree component cycle component

a1

pi

s

• Assign pi to a1.

• Ensures that every pop. mat. assigns a1 to pi .

Page 72: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

As agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : pi , s

If part: If one of pi belongs to a tree component, a1 can getmatched to it always.

s1

f1

tree component cycle component

a1

pi

s

• Assign pi to a1.

• Ensures that every pop. mat. assigns a1 to pi .

Page 73: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Gets matched to f1 sometimes, and to s1 at other times.

• Recall a1 belongs to a cycle or a tree component in GM .

tree component cycle component

f1

a1

s1

s

• Can she get matched to f1 always by cheating?Yes, provided there exists another cycle component in GM .

Theorem: a1 can cheat iff there exists a cycle component in GM

not containing a1.

Page 74: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Gets matched to f1 sometimes, and to s1 at other times.

• Recall a1 belongs to a cycle or a tree component in GM .

tree component cycle component

f1

a1

s1

s

• Can she get matched to f1 always by cheating?Yes, provided there exists another cycle component in GM .

Theorem: a1 can cheat iff there exists a cycle component in GM

not containing a1.

Page 75: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Gets matched to f1 sometimes, and to s1 at other times.

• Recall a1 belongs to a cycle or a tree component in GM .

tree component cycle component

f1

a1

s1

s

• Can she get matched to f1 always by cheating?

Yes, provided there exists another cycle component in GM .

Theorem: a1 can cheat iff there exists a cycle component in GM

not containing a1.

Page 76: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Gets matched to f1 sometimes, and to s1 at other times.

• Recall a1 belongs to a cycle or a tree component in GM .

tree component cycle component

f1

a1

s1

s

• Can she get matched to f1 always by cheating?Yes, provided there exists another cycle component in GM .

Theorem: a1 can cheat iff there exists a cycle component in GM

not containing a1.

Page 77: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .

Gets matched to f1 sometimes, and to s1 at other times.

• Recall a1 belongs to a cycle or a tree component in GM .

tree component cycle component

f1

a1

s1

s

• Can she get matched to f1 always by cheating?Yes, provided there exists another cycle component in GM .

Theorem: a1 can cheat iff there exists a cycle component in GM

not containing a1.

Page 78: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : f1, s

If part: If there exists a cycle component in GM not containing a1,then a1 can get matched to f1 always.

tree component cycle component

f1

a1

s1

s

Page 79: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : f1, s

If part: If there exists a cycle component in GM not containing a1,then a1 can get matched to f1 always.

tree component cycle component

f1

a1

s1

s

Page 80: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Af /s agent

a1 : f1, p1, p2, . . . , pk , s1, . . .a1 : f1, s

If part: If there exists a cycle component in GM not containing a1,then a1 can get matched to f1 always.

tree component cycle component

f1

s1

s

a1

Page 81: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Complexity of computing strategy

• Strategy requires simple checks which are linear in size ofswitching graph.

Theorem: Optimal cheating strategy for a1 can be computed inO(m + n) time for strict lists.

Page 82: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Complexity of computing strategy

• Strategy requires simple checks which are linear in size ofswitching graph.

Theorem: Optimal cheating strategy for a1 can be computed inO(m + n) time for strict lists.

Page 83: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

A non co-operative game

• All agents are manipulative.

• All agents have complete information.

• Do I manipulate if all other agents remain truthful?

• If answer is NO for every agent, then true preferences form aNash equilibrium.

• Using our strategy, possible to decide in O(m + n) time if truepreferences form an equilibrium.uses simple checks on switching graph.

Page 84: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

A non co-operative game

• All agents are manipulative.

• All agents have complete information.

• Do I manipulate if all other agents remain truthful?

• If answer is NO for every agent, then true preferences form aNash equilibrium.

• Using our strategy, possible to decide in O(m + n) time if truepreferences form an equilibrium.uses simple checks on switching graph.

Page 85: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

A non co-operative game

• All agents are manipulative.

• All agents have complete information.

• Do I manipulate if all other agents remain truthful?

• If answer is NO for every agent, then true preferences form aNash equilibrium.

• Using our strategy, possible to decide in O(m + n) time if truepreferences form an equilibrium.uses simple checks on switching graph.

Page 86: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

A non co-operative game

• All agents are manipulative.

• All agents have complete information.

• Do I manipulate if all other agents remain truthful?

• If answer is NO for every agent, then true preferences form aNash equilibrium.

• Using our strategy, possible to decide in O(m + n) time if truepreferences form an equilibrium.uses simple checks on switching graph.

Page 87: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Ties

Page 88: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Preference lists with ties

Challenges

• The characterization of popular matchings is more involved.Abraham et al. SICOMP 07

• Uses well known Dulmage Mendelson decomposition.

• Prior to this, no switching graph was known for ties.

• Switch. graph is a collection of sink and non-sink components.

S NS

Page 89: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Preference lists with ties

Challenges

• The characterization of popular matchings is more involved.Abraham et al. SICOMP 07

• Uses well known Dulmage Mendelson decomposition.

• Prior to this, no switching graph was known for ties.

• Switch. graph is a collection of sink and non-sink components.

S NS

Page 90: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Strategic issues – ties

• Using switching graph, develop a cheating strategy formanipulative agent.

• Strategies remain similar in spirit – require more work forproofs.

Theorem: Optimal cheating strategy for a1 can be computed inO(√nm) time for the case of ties.

Page 91: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Strategic issues – ties

• Using switching graph, develop a cheating strategy formanipulative agent.

• Strategies remain similar in spirit – require more work forproofs.

Theorem: Optimal cheating strategy for a1 can be computed inO(√nm) time for the case of ties.

Page 92: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Summary:

• Study of strategic issues of popular matching.

• A switching graph characterization for ties.

• Counting number of pop. mat. for ties is #P − complete.This is in contrast with strict lists.

Extensions:

• How can agents co-operate to get better always?

• Other ways to define improvement rather than better always?

• Cheating strategies for other measures – like rank-maximality,fairness?

Page 93: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Thank You.

Page 94: Popular Matchings: Structure and Cheating Strategies€¦ · Popular Matchings: Structure and Cheating Strategies Meghana Nasre April 19, 2013.

Popular matchings: need not exist

a

a

a

p p p

1

2

3

1 2 3

1 2 3

1

1

2

2

3

3

a

a

a

p p p

1

2

3

1 2 3

1 2 3

1

1

2

2

3

3

a

a

a

p p p

1

2

3

1 2 3

1 2 3

1

1

2

2

3

3

MM1 2

M3

More popular than relation is not transitive.