The Housemates’ Problem

1. The Housemates’ Problem Marc Kilgour Wilfrid Laurier University Waterloo, Canada

2. 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?”

3. 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.

4. 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.

5. 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?”

6. 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.

7. 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 - Budget: - Feasibility: Housemate assigned room j must be willing to pay at least pj for it, i.e., pj ≤ bσ(j),j for all j.

8. Feasible Bids If, for every permutation, σ, then the problem is infeasible. 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.

9. 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.

10. 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

11. Example 1 • H = 100, n = 2, 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.

12. Maxsum Assignment • A permutation σ* is a maxsum (or utilitarian) assignment iff for all permutations σ of {1, 2, …, n}. • A problem is feasible iff where σ* is any maxsum assignment.

13. Maxsum Assignments • Theorem: A solution of a Housemates Problem is Pareto-Optimal iff it is based on a maxsum assignment. • Proof:

14. Example 1, continued • H = 100, n = 2, , σ = 12 • 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

15. Gap Procedure • Brams and Kilgour, 2001 • Given any feasible maxsum assignment to committed housemates, GAP partitions the rent for the house into feasible non-negative rents. The solution is monotonic and sometimes independent, but may not be envy-free. • O(n3)

16. 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.

17. 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)

18. Example 2 • H = 100, n = 2, • 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.

19. 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)

20. 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)

21. Non-negative Envy-free Rents for Committed Housemates • n = 2, clear • n = 3, proven by Song and Vlach • n≥ 4, fails, by example

22. 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

23. 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.

24. 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