Pareto Optimality in House Allocation Problems

1
Pareto Optimality in House Allocation Problems
David Abraham Computer Science Department Carnegie
-Mellon University
Katarína Cechlárová Institute of Mathematics PJ
Safárik University in Košice

• David Manlove
• Department of Computing Science
• University of Glasgow

Kurt Mehlhorn Max-Planck-Institut fur
Informatik Saarbrucken
Supported by Royal Society of Edinburgh/Scottish
Executive Personal Research Fellowship and
Engineering and Physical Sciences Research
Council grant GR/R84597/01
2
House Allocation problem (HA)
?

• Set of agents Aa1, a2, , ar
• Set of houses Hh1, 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 nrs and let mtotal 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
• Abdulkadiroglu 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

4
The underlying graph

• Weighted bipartite graph G(V,E)
• Vertex set VA?H
• Edge set ai, hj ?E if and only if ai finds
  hj acceptable
• Weight of edge ai, hj is rank of hj in ais
  preference list
• Example
• a1 h2 h1
• a2 h3 h4 h2
• a3 h4 h3
• a4 h1 h4

a1
?
h1
2
1
a2
h2
?
3
1
a3
2
h3
?
2
1
1
a4
h4
?
2
5
The underlying graph

• Weighted bipartite graph G(V,E)
• Vertex set VA?H
• Edge set ai, hj ?E if and only if ai finds
  hj acceptable
• Weight of edge ai, hj is rank of hj in ais
  preference list
• Example
• a1 h2 h1
• a2 h3 h4 h2
• a3 h4 h3
• a4 h1 h4

a1
?
h1
2
M(a1)h1
1
a2
h2
?
3
1
a3
2
h3
?
2
1
1
M(a1, h1), (a2, h4), (a3, h3)
a4
h4
?
2
6
The underlying graph

• Weighted bipartite graph G(V,E)
• Vertex set VA?H
• Edge set ai, hj ?E if and only if ai finds
  hj acceptable
• Weight of edge ai, hj is rank of hj in ais
  preference list
• Example
• a1 h2 h1
• a2 h3 h4 h2
• a3 h4 h3
• a4 h1 h4

a1
?
h1
2
1
a2
h2
?
3
1
a3
2
h3
?
2
1
1
M(a1, h2), (a2, h3), (a3, h4), (a4, h1)
a4
h4
?
2
7
Pareto optimal matchings

• A matching M1 is Pareto optimal if there is no
  matching M2 such that
• Some agent is better off in M2 than in M1
• 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(ai1) (0?i?r-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
9
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(ai1) (0?i?r-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
10
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(ai1) to
  M(ai) (0?i?r-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

a1
h1
M is not coalition-free due to ?a1, a2, a4?
a2
h2
a3
h3
a4
h4
11
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(ai1) 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

12
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
• Abdulkadiroglu and Sönmez, 1998
• Example
• a1 h1 h2 h3
• a2 h1 h2
• a3 h1 h2

M1(a1,h1), (a2,h2)

• a1 h1 h2 h3
• a2 h1 h2
• a3 h1 h2

M2(a1,h3), (a2,h2), (a3,h1)
13
Related work

• Rank maximal matchings
• Matching M is rank maximal if, in M
• Maximum number of agents obtain their
  first-choice house
• 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

14
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 Gales Top-Trading Cycles (TTC)
  algorithm
• Shapley and Scarf, 1974

15
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

16
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

17
Phase 1
M not coalition-free

• 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 trade-in-free
18
Phase 2 outline

• Repeatedly search for a matched agent a and an
  unmatched house h such that a prefers h to
  hM(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

19
Phase 2 outline

• Repeatedly search for a matched agent a and an
  unmatched house h such that a prefers h to
  hM(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

20
Phase 2 outline

• Repeatedly search for a matched agent a and an
  unmatched house h such that a prefers h to
  hM(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

21
Phase 2 outline

• Repeatedly search for a matched agent a and an
  unmatched house h such that a prefers h to
  hM(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

22
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

23
Phase 3 outline

• 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 as 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)

24
Phase 3 termination

• Once Phase 3 terminates, matching is
  coalition-free

• 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

25
Initial property rights

• Suppose A?A and each member of A owns a house
  initially
• For each agent a?A, denote this house by h(a)
• Truncate as list at h(a)
• Form matching M by pre-assigning a to h(a)
• Use Hopcroft-Karp algorithm to augment M to a
  maximum cardinality matching M in restricted HA
  instance
• Then proceed with Phases 2 and 3 as before
• Constructed matching M is individually rational
• If AA then we have a housing market
• TTC algorithm finds the unique matching that
  belongs to the core
• Shapley and Scarf, 1974
• Roth and Postlewaite, 1977

26
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

27
Interpolation of Pareto optimal matchings

• Given an HA instance I, p-(I) and p(I) denote
  the sizes of a minimum and maximum Pareto optimal
  matching
• Theorem I admits a Pareto optimal matching of
  size k, for each k such that p-(I) ? k ? p(I)
• Given a Pareto optimal matching of size k, O(m)
  algorithm constructs a Pareto optimal matching of
  size k1 or reports that kp(I)
• Based on assigning a vector ?r1, , rk? to an
  augmenting path P?a1, h1, ,
  ak, hk? where
  rirankai(hi)
• Examples ?1,3,2?
  ?1,2,2?
• Find a lexicographically smallest augmenting path

1
h1
a1
1
a2
h2
2
3
1
h3
a3
2
1
h4
a4
2
h5
28
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 gt1)
• Non-bipartite case
• Solvable in O(?(n?(m, n))mlog3/2 n) time
• D.J. Abraham, D.F. Manlove
• Pareto optimality in the Roommates problem
• Technical Report TR-2004-182 of the Computing
  Science Department of Glasgow University
• Solvable in O(?nm) time?