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

2 Cake-cutting problems Divide the cake among a set of people in a fair manner 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] Mathematical approaches: [Steinhaus, Banach, Knaster ’48] Empirically: since Pharaoh times (land division)

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

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

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

11 Proof G(A) = (V, E) : envy graph of A  V = {agents}  pq  E iff p envies q in A. A: allocation of a subset of the goods S  M. ● ● ●● ● ● ● ● ● A1A1 A5A5 A4A4 A3A3 A2A2 A = (A 1, A 2,…,A 5,…) 

12 ● ● ●● ● ● ● ● ● # of edges decreases Envy does not increase A1A1 A5A5 A4A4 A3A3 A2A2 ● ● ●● ● ● ● A2A2 A1A1 A5A5 A4A4 A3A3 ● ● 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).

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

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 [Coffman-Langston ’84]: Graham’s algorithm achieves an approximation factor of 1.4 for the envy-ratio problem. [Graham ’69]:  Order the goods in decreasing value.  Give next good to the person with the minimum current bundle.

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: 1.Rounding step ( I  I R ). 2.Solve I R optimally: IP with constant # of variables 3.Transform allocation of rounded instance to an allocation in I.

22 Step 1: Rounding (I  I R ) Let L be the average utility: 3 types of goods: 1.Large: 2.Medium: 3.Small: Rounding parameter: integer constant

23 Step 1: Rounding (I  I R ) 1.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  I R ) 1.Large: WLOG no large goods in I 2.Medium: round to next integer multiple of (ignore some of the least significant digits) 3.Small: merge together and round:         

25 Step 1: Rounding (I  I R ) 1.Large: WLOG no large goods in I 2.Medium: round to next integer multiple of (ignore some of the least significant digits) 3.Small: merge together and round:                  

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

27 Step 2: Solve I R optimally  For, solve the decision problem:  Is there an allocation A = (A 1, …,A n ) with ?  Integer program, constant number of variables  Lenstra ’ s algorithm  Repeat only for a constant number of pairs (t 1, t 2 ).  Pick solution with best envy-ratio. Integer variable X S : # agents with bundle S, for each S with  2 λ goods

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

29 Finally … Which turns out to be:

30 Non-additive utilities 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. Input: exponential in size Use only polynomial amount of input? (query model)

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 Definition: An algorithm is truthful if being honest is always a dominant strategy for every player. So far we have assumed that players report their true utilities. 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 Post-mortem 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

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. biscuitmuffink eggs /k each /k each

38 Proof of Theorem 4 Proof: Construction of an example in which any such algorithm will fail. biscuitmuffink eggs   0.2/k each /k each By misreporting Homer will receive the biscuit and more eggs than before.