Transcript Computational Aspects of Game Theory and Microeconomics

On Approximately Fair
Allocations of Indivisible Goods
Richard Lipton
Georgia Tech
Elchanan Mossel
U. C. Berkeley
Vangelis Markakis
AUEB
Amin Saberi
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]
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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Set of agents
N = {1, 2, …, n}
Discrete version
Set of indivisible goods
M = {1, 2, …, m}
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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: 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
[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) ≤ .
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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
A = (A1, A2,…,A5,…) 
A2
●
●
A4 ●
A3
●
●
●
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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).
●
●
A1
●
A5
●A1
A2
●
●
●A2
A4 ●
A3
●
A3
●
●
●
●
A5●
A4
●
●
●
# of edges decreases
Envy does not increase
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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)!

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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.
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  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.
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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:
Rounding parameter: integer constant
3 types of goods:
1. Large:
2. Medium:
3. Small:
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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:
2. Medium:
WLOG no large goods in I
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:
2. Medium:
WLOG no large goods in I
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.

# goods in
# distinct bundles with  2λ goods is constant
(exp(λ) but still constant)
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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.
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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
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Finally…
Which turns out to be:
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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.
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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
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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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 nonconstructive
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 4
Proof: 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 4
Proof: 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
By misreporting Homer will receive the biscuit and
more eggs than before.
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