On Approximately Fair Allocations of Indivisible Goods

1. On Approximately Fair Allocations of Indivisible Goods Richard Lipton Vangelis Markakis Elchanan Mossel Amin Saberi Georgia Tech AUEB U. C. Berkeley Stanford

2. Cake-cutting problems Divide the cake among a set of people in a fair manner Empirically: since Pharaoh times (land division) Mathematical approaches: [Steinhaus, Banach, Knaster ’48] Fairness measure: Envy[Foley ’67] Infinitely divisible cakes: Envy-free partitions exist Cake-cutting procedures: minimize # cuts, achieve additional fairness criteria [Brams, Taylor ’96, Robertson, Webb ‘98]

3. Discrete version Set of indivisible goods M = {1, 2, …, m} Set of agents N = {1, 2, …, n}

4. Model For agent p: utility function : (monotone) • Special cases: • Additive utilities (e.g. probability measures) • Same utility for every agent.

5. What is fair? • Proportionality [Steinhaus - Banach - Knaster ’48] • Envy-freeness [Foley ’67, Varian ‘74] • Max-min fairness [Dubins - Spanier ’61] • Equitability • …..

6. Fairness Concept Given an allocation A = (A1,…,An): Envy of p for q: Envy of A: Envy-free allocations may not exist Goal:Algorithms with upper bounds on the envy

7. Outline • Existence of allocations with bounded envy • Optimization problems: positive and negative results • Incentive Compatibility

8. Outline • Existence of allocations with bounded envy • Optimization problems: positive and negative results • Incentive Compatibility

9. Additive Utilities Theorem[Dall’Aglio - Hill ’03]: There exists an allocation A with e(A) ≤(2n)3/2. Proof: probability measure on [0,1], Tools: convexity arguments, envy seen as the distance between a certain space and its convex hull.

10. A Tight Bound [Dall’Aglio - Hill ’03]:e(A) ≤(2n)3/2 1 good, 2 players e(A)   Theorem: We can compute in time O(mn3) an allocation A, such that e(A) ≤.

11. Proof A: allocation of a subset of the goods S  M. G(A) = (V, E) : envy graph of A • V = {agents} • pq  E iff p envies q in A. ● ● A5 ● A1 ● A4 A = (A1, A2,…,A5,…)  A2 A3 ● ● ● ● ●

12. ● ● A1 ● ● A2 A5 A3 A4 ● ● ● ● ● • Claim: For any allocation A, there exists an allocation B s.t.: • e(B) ≤ e(A). • envy-graph of B is acyclic ( i with in-degree = 0). ● A5 ● ● A1 ● A4 A2 A3 ● ● ● ● ● # of edges decreases Envy does not increase

13. Algorithm At step i: • Eliminate all the cycles from the envy graph. • Give good i to an agent that no-one envies (any node with in-degree = 0). □

14. Remarks • Bound is tight • Nonadditive utilities maximum marginal utility • Cyclic swaps: used in finding theater sponsors in ancient Greece, (2-cycles)!

15. Outline • Existence of allocations with bounded envy • Optimization problems: positive and negative results • Incentive Compatibility

16. Optimization Problem 1 [envy]: Find an allocation A that minimizes the envy: Problem 2[envy-ratio]: Find an allocation A that minimizes the ratio: Polynomial time algorithms?

17. Hardness Results Both problems are NP-hard. Proof: Partition; even if n = 2 and both players have the same utility function. Envy: Also hard to approximate; even for the above case.

18. Additive Utilities Assume agents have the same utility function Value of good Envy-ratio(A) =

19. Relations with Job Scheduling People  Processors Goods  Jobs • [Graham ’69]: • Order the goods in decreasing value. • Give next good to the person with the minimum current bundle. [Coffman-Langston ’84]: Graham’s algorithm achieves an approximation factor of 1.4 for the envy-ratio problem.

20. Polynomial Time Approximation Schemes PTAS:   > 0,  algorithm A with cost  (1 + )OPT in time poly(| I |),  instance I PTAS’s in job scheduling: [Hochbaum, Shmoys ’87]: Makespan [Woeginger ’97]: Maximize min. completion time [Alon, Azar, Woeginger, Yadid ’98]: Generalizations

21. A PTAS for the envy-ratio problem Theorem: The envy-ratio problem admits a Polynomial Time Approximation Scheme. Proof outline: • Rounding step ( I  IR ). • Solve IR optimally: IP with constant # of variables • Transform allocation of rounded instance to an allocation in I.

22. Step 1: Rounding (I  IR) Let L be the average utility: Rounding parameter: integer constant • 3 types of goods: • Large: • Medium: • Small:

23. Step 1: Rounding (I  IR) Large: give to some agent, remove agent We may assume there are no large goods in I Claim: There exists an optimal solution in which every large good is assigned to a person with no other goods in her bundle.

24. Step 1: Rounding (I  IR) • Large: WLOG no large goods in I • Medium: round to next integer multiple of • (ignore some of the least significant digits) • Small: merge together and round:     

25. Step 1: Rounding (I  IR) • Large: WLOG no large goods in I • Medium: round to next integer multiple of • (ignore some of the least significant digits) • Small: merge together and round:          

26. Step 2: Solve IR optimally Constant number of distinct values for the goods in IR : Claim: optimal allocation A in IR s.t.  # goods in #distinct bundles with  2λ goods is constant (exp(λ) but still constant)

27. Step 2: Solve IR optimally Integer variable XS: # agents with bundle S, for each S with  2λ goods • For , solve the decision problem: • Is there an allocation A = (A1,…,An) with ? • Integer program, constant number of variables  Lenstra’s algorithm • Repeat only for a constant number of pairs (t1, t2). • Pick solution with best envy-ratio.

28. Step 3 (IR I) OPTR: Optimal solution of the rounded instance. Lemma 1: Given an optimal solution of IR, we can find an allocation in I, B = (B1,…,Bn), such that: Lemma 2: OPTR OPT

29. Finally… Which turns out to be:

30. Non-additive utilities Input: exponential in size Use only polynomial amount of input? (query model) Theorem 3: Any deterministic algorithm that computes a finite approximation to minimum envy or minimum envy-ratio needs an exponential number of queries for the players’ utilities. Proof: Counting argument, similar to [Nisan-Segal ’03] Note: Not dependent on any complexity theory assumption.

31. Related Work and Extensions • Envy-ratio • Additive non-identical utilities: O(m)-approximation • Nonadditive (e.g. submodular) ? • Max-min fairness: • [Bezakova, Dani ’05, Saberi, Asadpour ’07]: new approximations + hardness results

32. Incentive Compatibility So far we have assumed that players report their true utilities. Definition: An algorithm is truthful if being honest is always a dominant strategy for every player. Theorem 4: An algorithm that outputs a minimum envy allocation is not truthful.

33. Conclusions • There exist allocations, in which the envy is bounded by the maximum marginal utility. • Minimizing the envy is hard in general. • If all players have the same (additive) utility function the envy ratio can be well approximated. • Any algorithm that computes a minimum envy allocation is not truthful.

34. Economic Theory: models and solution concepts Rationality, fairness, incentive compatibility,… Mathematically rich; however mostly non-constructive Discrete math and theory of algorithms: Dealing with indivisibilities Computational complexity Post-mortem

35. Post-mortem Finding efficient algorithms for computing / approximating economic solution concepts: • Fair division (partially here, [DH ’88, BD ’05, AS ’07]) • Nash Equilibria [P ’94, LMM ’03, LM ’04, PT ‘04, DMP ’06, BBM ’07] • Market Equilibria [DPS ’02, DPSV ’02, JMS ’03, DV ’03] • Cost Sharing [MS ’97, FPS ’00, JV ‘01] • ……

36. The End!

37. Proof of Theorem 4 Proof: Construction of an example in which any such algorithm will fail.

38. Proof of Theorem 4 Proof: Construction of an example in which any such algorithm will fail. By misreporting Homer will receive the biscuit and more eggs than before.