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## Fair Allocations of Indivisible Goods Part I: Envy-freeness

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**Fair Allocations of Indivisible Goods Part I: Envy-freeness**Richard Lipton Vangelis Markakis Elchanan Mossel Amin Saberi Georgia Tech CWI U. C. Berkeley Stanford**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, Varian ’74] Infinitely divisible cakes: Envy-free partitions exist Cake-cutting procedures: minimize # cuts, achieve additional fairness criteria [Brams, Taylor ’96, Robertson, Webb ’98]**Discrete version**Set of indivisible goods M = {1, 2, …, m} Set of agents N = {1, 2, …, n}**Model**For agent p: utility function : (monotone) • Special cases: • Additive utilities (e.g. probability measures) • Same utility for every agent.**What is fair?**• Proportionality [Steinhaus - Banach - Knaster ’48] • Envy-freeness [Foley ’67, Varian ‘74] • Max-min fairness [Dubins - Spanier ’61] • Equitability • …..**Fairness Concept**Given an allocation A = (A1,…,An): Envy of p for q: Envy of A: Envy-free allocations may not exist Goal:Polynomial time algorithms with upper bounds on the envy**Outline**• Existence of allocations with bounded envy • Optimization problems: positive and negative results • Incentive Compatibility**Outline**• Existence of allocations with bounded envy • Optimization problems: positive and negative results • Incentive Compatibility**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.**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) ≤.**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 ● ● ● ● ●**●**● 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**Algorithm**At step i: • Find and eliminate all the directed cycles from the envy graph. • Give good i to an agent that no-one envies (any node with in-degree = 0). □**Remarks**• Bound is tight • Nonadditive utilities maximum marginal utility • Cyclic swaps: used in finding theater sponsors in ancient Greece, (2-cycles)!**Outline**• Existence of allocations with bounded envy • Optimization problems: positive and negative results • Incentive Compatibility**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?**Hardness Results**Both problems are NP-hard. Proof: Partition; even if n = 2 and both players have the same utility function. Approximation algorithms? Definition: An algorithm A, for a minimization problem , achieves an approximation factor of ( 1), if for every instance I of , the solution returned by A satisfies: SOL(I) OPT(I) Envy: Also hard to approximate with better than exponential approximation factor; even for the above case.**Envy-ratio: Identical Additive Utilities**Assume agents have the same utility function Value of good Envy-ratio(A) =**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.**Polynomial Time Approximation Schemes**PTAS: A family of algorithms {A} s.t. >0 A returns a solution with error (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**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: Integer Programming with constant # of variables • Transform allocation of rounded instance to an allocation in I.**Proof Outline Cont’d**• Rounding step ( I IR ) (with respect to ) • Large goods: give each to some agent and remove these agents from I • Small goods: Merge together and divide into equal pieces • Medium goods: delete some least significant digits and round up • Solve IR optimally • New instance has constant number of different bundles an agent can have in an optimal solution • Integer programming formulation with constant number of variables Lenstra’s algorithm • Transform allocation of rounded instance to an allocation in I. • Rounding error incurs at most 1 + loss**More General Utilities**Additive non-identical utilities: O(m)-approximation Non-additive utilities: (assuming access to the utilities via queries) Theorem 3: Any deterministic algorithm needs an exponential number of queries to produce any finite approximation. Proof: Counting argument, similar to [Nisan-Segal ’03].Not dependent on any complexity theory assumption.**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.**A Related Problem**Problem 3 [max-min fairness]: Find an allocation A that maximizes the utility of the least happy person:**Why we need better Linear Programming Techniques**Consider instances with a good of very high value Fractionally: Everybody can get a piece Integrally: Somebody will be unhappy**Conclusions**• There exist allocations, in which the envy is bounded by the maximum marginal utility. • Envy and max-min fairness are computationally hard in general. • If all players have the same (additive) utility function both problems can be well approximated. • Any algorithm that computes a minimum envy allocation is not truthful.**Step 1: Rounding (I IR)**Let L be the average utility: Rounding parameter: integer constant • 3 types of goods: • Large: • Medium: • Small:**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: **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: **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) Integer program formulation with constant number of variables Lenstra’s algorithm**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