1 On Approximately Fair Allocations of Indivisible Goods Elchanan Mossel Amin Saberi Richard Lipton Vangelis Markakis Georgia Tech AUEB U. C. Berkeley Stanford
Dec 21, 2015
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On Approximately Fair Allocations of Indivisible Goods
Elchanan Mossel Amin Saberi
Richard Lipton Vangelis Markakis Georgia Tech AUEB
U. C. Berkeley Stanford
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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)
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Set of agents N = {1, 2, …, n}
Set of indivisible goods M = {1, 2, …, m}
Discrete version
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Model
For agent p: utility function :
Special cases:
Additive utilities (e.g. probability measures)
Same utility for every agent.
(monotone)
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What is fair?
– Proportionality [Steinhaus - Banach - Knaster ’48]
– Envy-freeness [Foley ’67, Varian ‘74]
– Max-min fairness [Dubins - Spanier ’61]
– Equitability
– …..
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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
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Outline
• Existence of allocations with bounded envy
• Optimization problems: positive and negative results
• Incentive Compatibility
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Outline
• Existence of allocations with bounded envy
• Optimization problems: positive and negative results
• Incentive Compatibility
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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.
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A Tight Bound
Theorem: We can compute in time O(mn3) an allocation A, such that e(A) ≤ .
[Dall’Aglio - Hill ’03]: e(A) ≤ (2n)3/2
1 good, 2 players e(A)
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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.
●
●
● ●
●
●
● ●
●
A1
A5
A4
A3A2
A = (A1, A2,…,A5,…)
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●
●
● ●
●
●
● ●
●
# of edges decreases
Envy does not increase
A1
A5
A4
A3A2 ●
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● ●
●
● ●
A2
A1
A5
A4A3
●
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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).
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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).
□
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Remarks
• Bound is tight
• Nonadditive utilities
maximum marginal utility
• Cyclic swaps: used in finding theater sponsors in ancient Greece, (2-cycles)!
€
ντιδοσις
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Outline
• Existence of allocations with bounded envy
• Optimization problems: positive and negative results
• Incentive Compatibility
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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:
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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.
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Additive Utilities
Assume agents have the same utility function
Value of good
Envy-ratio(A) =
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Relations with Job Scheduling
People ProcessorsGoods 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.
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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
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A PTAS for the envy-ratio problem
Theorem: The envy-ratio problem admits a Polynomial Time Approximation Scheme.
Proof outline:
1. Rounding step ( I IR ).
2. Solve IR optimally: IP with constant # of variables
3. Transform allocation of rounded instance to an
allocation in I.
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Step 1: Rounding (I IR)
Let L be the average utility:
3 types of goods:
1. Large:
2. Medium:
3. Small:
Rounding parameter: integer constant
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Step 1: Rounding (I IR)
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.
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Step 1: Rounding (I IR)
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:
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Step 1: Rounding (I IR)
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:
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Step 2: Solve IR optimally
Constant number of distinct values for the goods in IR :
Claim: optimal allocation A in IR s.t.
# distinct bundles with 2λ goods is constant(exp(λ) but still constant)
# goods in
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Step 2: Solve IR optimally
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.
Integer variable XS: # agents with bundle S, for each S with 2λ goods
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Step 3 (IR I)
Lemma 1: Given an optimal solution of IR, we can find an allocation in I, B = (B1,…,Bn), such that:
OPTR: Optimal solution of the rounded instance.
Lemma 2: OPTR OPT
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Finally…
Which turns out to be:
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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)
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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
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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.
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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.
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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
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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]
……
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The End!
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Proof of Theorem 4Proof: Construction of an example in which any such algorithm will fail.
biscuit muffin k eggs
0.45 0.35 0.2/k each
0.35 0.45 0.2/k each
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Proof of Theorem 4Proof: Construction of an example in which any such algorithm will fail.
biscuit muffin k eggs
0.45 - δ 0.35 + δ 0.2/k each
0.35 0.45 0.2/k each
By misreporting Homer will receive the biscuit and more eggs than before.