CMU 15-896 Fair division 4: Indivisible goods Teacher: Ariel - - PowerPoint PPT Presentation

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CMU 15-896 Fair division 4: Indivisible goods Teacher: Ariel - - PowerPoint PPT Presentation

Dec 11, 2022 404 likes •642 views

CMU 15-896 Fair division 4: Indivisible goods Teacher: Ariel Procaccia Indivisible goods Set of goods Each good is indivisible Players have arbitrary valuations for bundles of goods Envy-freeness and proportionality are

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CMU 15-896

Fair division 4: Indivisible goods

Teacher: Ariel Procaccia

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15896 Spring 2016: Lecture 9

Indivisible goods

  • Set
  • f

goods

  • Each good is indivisible
  • Players

have arbitrary valuations

for bundles of goods

  • Envy-freeness and

proportionality are infeasible!

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15896 Spring 2016: Lecture 9

Minimizing envy

  • Given allocation , denote
  • Theorem [Nisan and Segal 2002]: Every

protocol that finds an allocation minimizing must use an exponential number of bits of communication in the worst case

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15896 Spring 2016: Lecture 9

Communication complexity

  • Protocol defined by a

binary tree

  • Complexity is the

height of the tree

  • Complexity of a

problem is the height

  • f the shortest tree

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1 2 1 1 1 1 1 0 1 1 1 2 1

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15896 Spring 2016: Lecture 9

Proof of theorem

  • Let
  • is a set of functions s.t. for all

,

  • 5
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15896 Spring 2016: Lecture 9

Proof of theorem

  • Suppose

, and denote a valuation profile by

  • Lemma: Suppose

, then the sequence of bits transmitted on input is different from the sequence transmitted on

  • Assume the lemma is true, then there must

be at least sequences, and the height of the tree must be at least

  • 6
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15896 Spring 2016: Lecture 9

Proof of lemma

  • Assume not; then

and generate the same sequence

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1 2 1 1 1 1 1 0 1 1 1 2 1 1 2 1 1 1 1 1 0 1 1 1 2 1 , ,

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15896 Spring 2016: Lecture 9

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1 2 1 1 1 1 1 0 1 1 1 2 1 1 2 1 1 1 1 1 0 1 1 1 2 1 , , 1 2 1 1 1 1 1 0 1 1 1 2 1 ,

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15896 Spring 2016: Lecture 9

Proof of lemma

  • If

, such that ,

  • The allocation

is EF for , is EF for

  • Given

, protocol produces an EF ,

  • is also returned on

, but is not EF

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15896 Spring 2016: Lecture 9

Approximate EF

  • Define the maximum marginal utility
  • Theorem [Lipton et al. 2004]: An

allocation with can be found in polynomial time

  • Note: we are still not assuming anything

about the valuation functions!

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15896 Spring 2016: Lecture 9

Proof of Theorem

  • Given allocation , we have an edge

in its envy graph if envies

  • Lemma: Given partial allocation

with envy graph , can find allocation with acyclic envy graph s.t.

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15896 Spring 2016: Lecture 9

Proof of lemma

  • If

has a cycle , shift allocations along to obtain ; clearly

  • #edges in envy graph of

decreased:

  • Same edges between ∖
  • Edges from ∖ to shifted
  • Edges from to ∖ can only

decrease

  • Edges inside C decreased
  • Iteratively remove cycles

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15896 Spring 2016: Lecture 9

Proof of theorem

  • Maintain envy

and acyclic graph

  • In round 1, allocate good to arbitrary

agent

  • are allocated in acyclic
  • Derive

by allocating to source

  • Use lemma to eliminate cycles

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15896 Spring 2016: Lecture 9

EF cake cutting, revisited

  • Want to get -EF cake division
  • Agent makes

marks

  • ⁄
  • such

that for every

  • If intervals between consecutive marks are

indivisible goods then

  • Now we can apply the theorem
  • Need

cut queries and

  • eval

queries

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15896 Spring 2016: Lecture 9

An even simpler solution

  • Relies on additive valuations
  • Create the “indivisible goods” like before
  • Agents choose pieces in a round-robin

fashion:

  • Each good chosen by agent is preferred

to the next good chosen by agent

  • This may not account for the first good

chosen by , but

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15896 Spring 2016: Lecture 9

Maximin share guarantee

  • Let us focus on indivisible goods and

additive valuations

  • Communication complexity is not an issue
  • But computational complexity is
  • Observation: Deciding whether there exists

an EF allocation is NP-hard, even for two players with identical additive valuations

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15896 Spring 2016: Lecture 9 Total: $30

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$30 $50 $2 $5 $5 $3 $5 Total: $50 Total: $20 $2 $10 $5 $20 $20 $3 $40 Total: $40 Total: $30 Total: $30

Maximin share guarantee

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15896 Spring 2016: Lecture 9

  • Maximin share (MMS) guarantee [Budish,

2011] of player :

,…,

  • Theorem [P & Wang, 2014]:

there exist additive valuation functions that do not admit an MMS allocation

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15896 Spring 2016: Lecture 9

Counterexample for

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17 25 12 1 2 22 3 28 11 0 21 2 22 3 28 23 17 12 1 2 22 3 28 11 0 21 23 1 25 12 25 17 11 0 21 23 17 25 12 1 2 22 3 28 11 0 21 23 17 25 12 1 2 22 3 28 11 0 21 23

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15896 Spring 2016: Lecture 9

Counterexample for

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1 1 1 1 1 1 1 1 1 1 1 1 17 25 12 1 2 22 3 28 11 0 21 23 3 -1 -1 -1 0 0 0 0 0 0 0 0 3 -1 0 0

  • 1 0 0 0
  • 1 0 0 0

3 0 -1 0 0 0 -1 0 0 0 0 -1

Player 1 Player 2 Player 3

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15896 Spring 2016: Lecture 9

  • Maximin share (MMS) guarantee [Budish,

2011] of player :

,…,

  • Theorem [P & Wang, 2014]:

there exist additive valuation functions that do not admit an MMS allocation

  • Theorem [P & Wang, 2014]: It is always

possible to guarantee each player

  • f

his MMS guarantee

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15896 Spring 2016: Lecture 9