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
Georgia Tech
CWI

• Elchanan Mossel Amin
  Saberi

U. C. Berkeley
Stanford
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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
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Discrete version

• Set of indivisible goods M 1, 2, , m

• Set of agents N 1, 2, , n

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Model

• For agent p utility function

(monotone)

• Special cases
• Additive utilities (e.g. probability measures)
• Same utility for every agent.

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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 Polynomial time 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 DallAglio - 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
DallAglio - 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) ?.
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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,) ?
A3
A2
?
?
?
?
?
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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).

?
A5
?
?
?
A1
A4
A3
A2
?
?
?
?
?
of edges decreases Envy does not increase
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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).
• 
  ?

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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)!

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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
Problem 2 envy-ratio Find an allocation A that
minimizes the ratio
Polynomial time algorithms?
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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.
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.
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Envy-ratio Identical Additive Utilities
Assume agents have the same utility
function Value of good
Envy-ratio(A)
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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 Grahams algorithm
achieves an approximation factor of 1.4 for the
envy-ratio problem.
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Polynomial Time Approximation Schemes
PTAS A family of algorithms A? s.t. ? ?gt0 A?
returns a solution with error ? (1 ?)OPT in
time poly( I ), ? instance I
PTASs 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
• Rounding step ( I ? IR ).
• Solve IR optimally Integer Programming with
  constant of variables
• Transform allocation of rounded instance to an
  allocation in I.

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Proof Outline Contd

• 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 ? Lenstras algorithm
• Transform allocation of rounded instance to an
  allocation in I.
• Rounding error incurs at most 1 ? loss

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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.
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Summary of Approximability
Positive Negative
Additive Identical PTAS NP-hard
General Additive O(m) NP-hard
General Non-additive Hard for any f(n, m)
Other (e.g. submodular) ? ?
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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.
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A Related Problem
Problem 3 max-min fairness Find an allocation
A that maximizes the utility of the least happy
person
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Comparisons with envy-ratio
Envy-ratio Envy-ratio Max-min fairness Max-min fairness
Positive Negative Positive Negative
Additive Identical PTAS NP-hard PTAS NP-hard
General Additive O(m) NP-hard O(m) BD 05 O(k) G 05 ?k AS 07 2-hard
General Non-additive Hard for any f(k, m) Hard for any f(k,m)
Other (e.g. submodular) ? ? O(k) G 05, KP 07 ?
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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
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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.

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Thank You!
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Step 1 Rounding (I ? IR)
Let L be the average utility
Rounding parameter integer constant

• 3 types of goods
• Large
• Medium
• Small

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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

? ? ? ? ?
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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

? ? ? ? ?
? ? ? ? ?
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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.


?
goods in
distinct bundles with ? 2? goods is
constant (exp(?) but still constant)
Integer program formulation with constant number
of variables ? Lenstras algorithm
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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