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The Housemates’ Problem

The Housemates’ Problem. Marc Kilgour Wilfrid Laurier University Waterloo, Canada. The Original Housemates’ Problem. Francis Su, American Mathematical Monthly , 1999

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The Housemates’ Problem

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  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

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