Pareto Optimality in House Allocation

Problems

David ManloveDepartment of Computing ScienceUniversity of Glasgow

David AbrahamComputer Science DepartmentCarnegie-Mellon University

Katarína CechlárováInstitute of MathematicsPJ Safárik University in Košice

Kurt MehlhornMax-Planck-Institut fűr InformatikSaarbrűcken

Supported by Royal Society of Edinburgh/Scottish Executive Personal Research Fellowshipand Engineering and Physical Sciences Research Council grant GR/R84597/01

2

House Allocation problem (HA)

Set of agents A={a1, a2, …, ar} Set of houses H={h1, h2, …, hs}

Each agent ai has an acceptable set of houses Ai H ai ranks Ai in strict order of preference

Example: a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

Let n=r+s and let m=total length of preference lists

a1 finds h1 and h2 acceptable

a3 prefers h4 to h3

3

Applications

House allocation context: Large-scale residence exchange in Chinese housing

markets Yuan, 1996

Allocation of campus housing in American universities, such as Carnegie-Mellon, Rochester and Stanford

Abdulkadiroğlu and Sönmez, 1998

Other matching problems: US Naval Academy: students to naval officer positions

Roth and Sotomayor, 1990 Scottish Executive Teacher Induction Scheme Assigning students to projects

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The underlying graph

Weighted bipartite graph G=(V,E) Vertex set V=AH Edge set: { ai, hj } E if and only if ai finds hj acceptable Weight of edge { ai, hj } is rank of hj in ai’s preference list

Example a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

a1

a2

a3

a4

h1

h2

h3

h4

2

1

1

2

3

2

11

2

5

Matchings in the underlying graph

A matching M in G is a subset of E such that each vertex of G is incident to at most one edge of M

Each agent is assigned to at most one house Each house is assigned at most one agent An agent is only ever assigned to an acceptable house

Example a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

a1

a2

a3

a4

h1

h2

h3

h4

2

1

1

2

3

2

11

2M={(a1, h1), (a2, h4), (a3, h3)}

M(a1)=h1

6

A matching M in G is a subset of E such that each vertex of G is incident to at most one edge of M

Each agent is assigned to at most one house Each house is assigned at most one agent An agent is only ever assigned to an acceptable house

Example a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

a1

a2

a3

a4

h1

h2

h3

h4

2

1

1

2

3

2

11

2M={(a1, h2), (a2, h3), (a3, h4), (a4, h1)}

Matchings in the underlying graph

7

Pareto optimal matchings

A matching M1 is Pareto optimal if there is no matching M2 such that:

1. Some agent is better off in M2 than in M1

2. No agent is worse off in M2 than in M1

Example

M1 is not Pareto optimal since a1 and a2 could swap houses – each would be better off

M2 is Pareto optimal

a1 : h2 h1

a2 : h1 h2

a3 : h3

a1 : h2 h1

a2 : h1 h2

a3 : h3

M1

M2

8

Testing for Pareto optimality

A matching M is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai) to M(ai+1) (0ir-1)

a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

Proposition: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

M is not maximal due to a3 and h3

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Testing for Pareto optimality

A matching M is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai) to M(ai+1) (0ir-1)

a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

Proposition: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

M is not trade-in-free due to a2 and h3

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Testing for Pareto optimality

A matching M is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai+1) to M(ai) (0ir-1)

a1 : h2 h1

a2 : h3 h4 h2

a3 : h4 h3

a4 : h1 h4

Proposition: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

M is not coalition-free due to a1, a2, a4

a1

a3

a4

h1

h3

h4

a2 h2

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Testing for Pareto optimality

A matching M is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai+1) to M(ai) (0 i r-1)

Lemma: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

Theorem: we may check whether a given matching M is Pareto optimal in O(m) time

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Finding a Pareto optimal matching

Simple greedy algorithm, referred to as the serial dictatorship mechanism by economists

for each agent a in turn if a has an unmatched house on his list

match a to the most-preferred such house; else report a as unmatched;

Theorem: The serial dictatorship mechanism constructs a Pareto optimal matching in O(m) time

Abdulkadiroğlu and Sönmez, 1998

Example a1 : h1 h2 h3 a2 : h1 h2 a3 : h1 h2

a1 : h1 h2 h3

a2 : h1 h2

a3 : h1 h2

M1={(a1,h1), (a2,h2)}

M2={(a1,h3), (a2,h2), (a3,h1)}

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Related work

Rank maximal matchings Matching M is rank maximal if, in M

1. Maximum number of agents obtain their first-choice house;2. Subject to (1), maximum number of agents obtain their second-choice

house;etc.

Irving, Kavitha, Mehlhorn, Michail, Paluch, SODA 04 A rank maximal matching is Pareto optimal, but need not be of

maximum size

Popular matchings Matching M is popular if there is no other matching M’ such that:

more agents prefer M’ to M than prefer M to M’ Abraham, Irving, Kavitha, Mehlhorn, SODA 05 A popular matching is Pareto optimal, but need not exist

Maximum cardinality minimum weight matchings Such a matching M may be found in G in O(nmlog n) time Gabow and Tarjan, 1989 M is a maximum Pareto optimal matching

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Faster algorithm for finding a maximum Pareto optimal matching

Three-phase algorithm with O(nm) overall complexity

Phase 1 – O(nm) time Find a maximum matching in G Classical O(nm) augmenting path algorithm

Hopcroft and Karp, 1973

Phase 2 – O(m) time Enforce trade-in-free property

Phase 3 – O(m) time Enforce coalition-free property Extension of Gale’s Top-Trading Cycles (TTC) algorithm

Shapley and Scarf, 1974

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Phase 1

a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

Maximum matching M in G has size 8 M must be maximal No guarantee that M is trade-in-free or coalition-free

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Phase 1

a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

Maximum matching M in G has size 9 M must be maximal No guarantee that M is trade-in-free or coalition-free

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Phase 1

a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

Maximum matching M in G has size 9 M must be maximal No guarantee that M is trade-in-free or coalition-free

M not coalition-free

M not trade-in-free

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Phase 2 outline

Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

Promote a to h h’ is now unmatched

Example a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

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Phase 2 outline

Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

Promote a to h h’ is now unmatched

Example a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

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Phase 2 outline

Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

Promote a to h h’ is now unmatched

Example a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

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Phase 2 outline

Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

Promote a to h h’ is now unmatched

Example a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

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Phase 2 termination

Once Phase 2 terminates, matching is trade-in-free With suitable data structures, Phase 2 is O(m) Coalitions may remain…

a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

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Build a path P of agents (represented by a stack) Each house is initially unlabelled Each agent a has a pointer p(a) pointing to M(a) or the first

unlabelled house on a’s preference list (whichever comes first)

Keep a counter c(a) for each agent a (initially c(a)=0) This represents the number of times a appears on the stack

Outer loop iterates over each matched agent a such that p(a)M(a)

Initialise P to contain agent a Inner loop iterates while P is nonempty

Pop an agent a’ from P If c(a’)=2 we have a coalition (CYCLE)

Remove by popping the stack and label the houses involved Else if p(a’)=M(a’) we reach a dead end (BACKTRACK)

Label M(a’) Else add a’’ where p(a’)=M(a’’) to the path (EXTEND)

Push a’ and a’’ onto the stack Increment c(a’’)

Phase 3 outline

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Once Phase 3 terminates, matching is coalition-free

Phase 3 termination

a1 : h4 h5 h3 h2 h1

a2 : h3 h4 h5 h9 h1 h2

a3 : h5 h4 h1 h2 h3

a4 : h3 h5 h4

a5 : h4 h3 h5

a6 : h2 h3 h5 h8 h6 h7 h1 h11 h4 h10

a7 : h1 h4 h3 h6 h7 h2 h10 h5 h11

a8 : h1 h5 h4 h3 h7 h6 h8

a9 : h4 h3 h5 h9

Phase 3 is O(m) Theorem: A maximum Pareto optimal matching

can be found in O(nm) time

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Minimum Pareto optimal matchings

Theorem: Problem of finding a minimum Pareto optimal matching is NP-hard

Result holds even if all preference lists have length 3 Reduction from Minimum Maximal Matching

Problem is approximable within a factor of 2 Follows since any Pareto optimal matching is a maximal

matching in the underlying graph G Any two maximal matchings differ in size by at most a factor

of 2 Korte and Hausmann, 1978

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Conclusions

Open problems - finding a maximum Pareto optimal matching Ties in the preference lists

Solvable in O(nmlog n) time Solvable in O(nm) time?

One-many case (houses may have capacity >1) Non-bipartite case

Solvable in O((n(m, n))mlog3/2 n) time D.J. Abraham, D.F. Manlove

Pareto optimality in the Roommates problemTechnical Report TR-2004-182 of the Computing Science Department of Glasgow University

Solvable in O(nm) time?

Further details D.J. Abraham, K. Cechlárová, D.F. Manlove and K.

Mehlhorn, Pareto Optimality in House Allocation Problems, In Proceedings of ISAAC 2004, vol 3341 of Lecture Notes in Computer Science