The Housemates’ Problem
Marc Kilgour
Wilfrid Laurier University
Waterloo, Canada
The Original Housemates’ Problem
Francis Su, American Mathematical Monthly, 1999
My friend and his housemates were
moving into a house with rooms of various
sizes and features. He asked “Do you
think there’s always a way to partition the
rent so that each person will prefer a
different room?”
Housemates’ Problem: Fair Division
• Allocate n + 1 goods over n persons.
• Goods 1, 2, …, n, called rooms, are
discrete.
• Good n + 1, representing money, is
continuous.
• Each person is to be allocated one room
and an amount of money, called rent.
• All rents must sum to H > 0, the rent for
the house.
Su’s Solution: Sperner’s Lemma
• The rent for the house, H, can be
partitioned so that each housemate
prefers a different room, provided that
– No matter how the rent is partitioned, each
person finds some room acceptable.
– (“Miserly tenants”) Each person always
prefers a zero-rent room to any positive-rent
room.
– Preference sets are closed.
Su’s Solution: Comments
• Su found an algorithm that uses repeated
partitioning to approximate an envy-free
assignment.
• Miserly Tenants assumption can be weakened:
No-one ever chooses the highest-rent room
when there is a zero-rent room.
• Fair Division Calculator:
http://www.math.hmc.edu/~su/fairdivision/calc/
“Which do you prefer – Room R1 at rent p1 or Room R2
at rent p2?”
Competitive Fair Division
• Brams & Kilgour, 2001 (Journal of Political
Economy) introduced bids by each
housemate for each room.
• Let bi,j be the amount that Housemate i is
willing to pay for room Rj.
• Problem: Given n X n matrix B = (bi,j) and
H > 0, find an assignment σ of Rooms to
Housemates, plus a vector (p1, p2, …, pn)
of rents, satisfying Solution Conditions.
Solution Conditions (1)
• Room j assigned to housemate σ(j) at rent pj
• Conditions without a model of preference
- σ a permutation: one room per housemate
n
p H
- Budget:
j 1 j
- Feasibility: Housemate assigned room j must
be willing to pay at least pj for it, i.e.,
pj ≤ bσ(j),j for all j.

Feasible Bids

If, for every permutation, σ,
then the problem is infeasible.
n
b
j 1  ( j ), j
H
Brams and Kilgour (2001) ensure feasibility by
supposing that all housemates are committed. A
housemate is committed iff her bids for all rooms
sum to H.
Solution Conditions (2)
• Further conditions without model of preference
- No negative rents: pj ≥ 0 for all j
- Monotonicity: The assignment is unchanged if
a player raises her bid for the room she is
assigned or reduces her bids for other rooms.
- Independence: The rent for a room does not
depend on the (exact) amount of the winning
bid.
Solution Conditions (3)
• Preference Model (quasi-linear utilities): Utility =
Bid – Rent, so housemate i gains utility bi,j − pj if
assigned room j.
• Conditions incorporating Preference Model:
- Pareto-Optimality: The sum of all housemates’
utilities is a maximum
- Envy-Freeness: No housemate strictly prefers
a different room:
bσ(j),j − pj ≥ bσ(j,)h − ph for all j, h
Example 1
 80 40
• H = 100, n = 2, B  

 70 50
If σ = 12, utilities are 80 – p1 to 1 and 50 – p2 to 2.
If σ = 21, utilities are 40 – p2 to 1 and 70 – p1 to 2.
Total utility is 130 – (p1 + p2) for σ = 12
or
110 – (p1 + p2) for σ = 21
Since p1 + p2 = 100, choose σ = 12 to maximize total utility.
Maxsum Assignment
• A permutation σ* is a maxsum (or utilitarian)
assignment iff

n
b
j 1  *( j ), j


n
b
j 1  ( j ), j
for all permutations σ of {1, 2, …, n}.
• A problem is feasible iff

n
b
j 1  *( j ), j
H
where σ* is any maxsum assignment.
Maxsum Assignments
• Theorem: A solution of a Housemates Problem is
Pareto-Optimal iff it is based on a maxsum
assignment.
• Proof:  i ui   j b ( j ), j  p j   j b ( j ), j  H


Example 1, continued
 80 40
• H = 100, n = 2, B  
 , σ = 12
 70 50
• 1 has no envy if 80 – p1 ≥ 40 – p2
2 has no envy if 50 – p2 ≥ 70 – p1
These conditions, combined with p1 + p2 = 100,
are satisfied by (p1, p2) iff
60 ≤ p1 ≤ 70, p2 = 100 – p1
• E.g.: 1 gets R1 for p1 = 70, 2 gets R2 for p2 = 30
Gap Procedure
• Brams and Kilgour, 2001
• Given any feasible maxsum assignment to
committed housemates, GAP partitions
the rent for the house into feasible nonnegative rents. The solution is monotonic
and sometimes independent, but may not
be envy-free.
• O(n3)
Adjusting the Gap Procedure
• Potthoff (2002) showed that, if the rents
produced by GAP are not envy-free, then
linear programming can be used to find
the nearest efficient envy-free solution.
• Distance metric: Potthoff used sum of
absolute differences to measure distance,
but many other metrics are possible.
Compensation Procedure (HRS)
• Haake, Raith, and Su (Social Choice & Welfare,
2002). For any maxsum assignment of rooms in
a feasible Housemates’ Problem, HRS finds an
envy-free partition of the rent for the house.
• Housemates need not be committed.
• Solution is feasible (rents never exceed bids),
but rents may be negative.
• Procedure applies at most n – 1 compensation
rounds.
• O(n3)
Example 2
 150 0 
• H = 100, n = 2, B  

 160 10
• Both σ = 12 and σ = 21 are maxsum
assignments.
• The only envy-free rents that sum to 100 are p1
= 125 and p2 = –25.
• Therefore, there may be no envy-free
assignment of rooms at non-negative rents.
Envy-freeness and Negativity
• Even when all housemates are committed (each
row of B sums to H), there may be no feasible
envy-free solution with non-negative rents.
• Example (Brams and Kilgour, 2001)
 36 34

 31 36
B 
34 30

 32 33
30
33
36
35
0

0
0

0
Song & Vlach Algorithm (SV)
• Shao Chin Sung and Milan Vlach (Social
Choice & Welfare, 2004)
• For any maxsum assignment of rooms in a
feasible Housemates’ Problem, SV finds
rents that are feasible, non-negative, and
envy-free if they exist; if not, it assigns any
envious housemate rent zero.
• O(n3)
Non-negative Envy-free Rents for
Committed Housemates
• n = 2, clear
• n = 3, proven by Song and Vlach
• n ≥ 4, fails, by example
An Auction Approach (ASU)
• Abdulkadiroǧlu, Sönmez, and Ünver (Social Choice &
Welfare, 2004) propose an auction approach that yields
an efficient, envy-free solution at non-negative prices
whenever one exists.
• Adapts Demange, Gale & Sotomayor’s (1986) approach
based on Hall’s Theorem in solution of distribution
problems.
• Begin with equal rents; then incrementally raise rents for
overdemanded rooms and decrease rents for others
• Continuous or discrete
• Converges, but no complexity estimate
References
•
1999, Francis Edward Su, “Rental Harmony: Sperner’s Lemma in Fair Division,”
American Mathematical Monthly 106, 930-942.
•
2001, Steven J. Brams and D. Marc Kilgour, “Competitive Fair Division,” Journal of
Political Economy 109, 2, 418-443.
•
2002, Claus-Jochen Haake, Matthias G. Raith, and Francis Edward Su, “Bidding for
envy-freeness: A procedural approach to n-layer fair division problems,” Social
Choice and Welfare 19, 723-749.
•
2002, Richard Potthoff, “Use of Linear Programming to Find an Envy-Free Solution
Closest to the Brams-Kilgour Gap Solution to the Housemates Problem,” Group
Decision and Negotiation 11, 405-414.
•
2004, Atila Abdulkadiroǧlu, Tayfun Sönmez, and M. Utku Ünver, “Room Assignment –
Rent Division: A Market Approach,” Social Choice and Welfare, 22, 515-538.
•
2004, Shao Chin Sung and Milan Vlach, “Competitive Envy-Free Division,” Social
Choice and Welfare 23, 103-111.
References on the WWW
•
•
1999 - Francis Edward Su
http://www.math.hmc.edu/~su/papers.dir/rent.pdf
•
•
2001- Steven J. Brams and D. Marc Kilgour
http://www.jstor.org/view/00223808/sp030005/03x0050x/0
•
•
2002 - Claus-Jochen Haake, Matthias G. Raith, and Francis Edward Su
http://www.math.hmc.edu/~su/papers.dir/bfe.pdf
•
•
2002 - Richard Potthoff
http://www.springerlink.com/content/l452142657372855/fulltext.pdf
•
•
2004 - Atila Abdulkadiroǧlu, Tayfun Sönmez, and M. Utku Ünver
http://www.springerlink.com/content/xje7v7b758xghe1u/fulltext.pdf
•
•
2004 - Shao Chin Sung and Milan Vlach,
http://ideas.repec.org/a/spr/sochwe/v23y2004i1p103-111.html