Room 1 $10 Room 2 Room 3 E NVY -F REE R ENT D IVISION Theorem - - PowerPoint PPT Presentation

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Room 1 $10 Room 2 Room 3 E NVY -F REE R ENT D IVISION Theorem - - PowerPoint PPT Presentation

Aug 29, 2023 360 likes •649 views

O VERVIEW Rent Division Computationally efficient algorithms for assigning rooms and dividing rent Indivisible Goods A new, approximate notion of fairness and its application in Spliddit Classroom Allocation Leximin in the real world:

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OVERVIEW

Rent Division

Computationally efficient algorithms for assigning rooms and dividing rent

Indivisible Goods

A new, approximate notion of fairness and its application in Spliddit Leximin in the real world: properties,

  • ptimization, and implementation

Classroom Allocation

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OVERVIEW

Rent Division

Computationally efficient algorithms for assigning rooms and dividing rent

Indivisible Goods

A new “computational” notion of fairness and its application in Spliddit Leximin in the real world: properties,

  • ptimization, implementation

Classroom Allocation

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THE

HE WHIN INING PHILOSOPHERS PROBLEM

Nir Ben Moshe Naomi Sender Ariel Procaccia

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Total rent: $10 Room 1 Room 2 Room 3

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ENVY-FREE RENT DIVISION

  • Theorem [Sevensson 1983]: An envy-free

solution always exists

  • Theorem [Aragones 1995]: An envy-free

solution can be computed in polynomial time

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Total rent: $3 Room 1 Room 2 Room 3

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ENVY-FREE RENT DIVISION

  • Theorem [Sevensson 1983]: An envy-free

solution always exists

  • Theorem [Aragones 1995]: An envy-free

solution can be computed in polynomial time

  • Theorem [Gal, Mash, P, Zick 2015]:

A solution that maximizes the minimum utility subject to envy-freeness can be found in polynomial time

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OVERVIEW

Rent Division

Computationally efficient algorithms for assigning rooms and dividing rent

Indivisible Goods

A new “computational” notion of fairness and its application in Spliddit Leximin in the real world: properties,

  • ptimization, implementation

Classroom Allocation

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INDIVISIBLE GOODS

Assume: additive valuations

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  • EF is infeasible ⇒

random values?

  • For each good 𝑕, draw

values 𝑊

1 𝑕 , … , 𝑊 𝑜(𝑕)

from a distribution

  • ver 0,1 𝑜
  • Theorem [Dickerson et

al., 2014]: Under mild technical assumptions, if 𝑛 = Ω(𝑜 ⋅ log𝑜) then an EF allocation exists w.h.p. as 𝑛 → ∞

Min value of 𝑛 such that 99% of instances admit an EF allocation

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MAXIMIN SHARE GUARANTEE

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

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  • Maximin share (MMS) guarantee [Budish

2011] of player 𝑗: max

𝑌1,…,𝑌𝑜 min 𝑘

𝑊

𝑗(𝑌 𝑘)

  • Theorem [P & Wang 2014]: ∀𝑜 ≥ 3 there

exist additive valuation functions that do not admit an MMS allocation

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COUNTEREXAMPLE FOR 𝑜 = 3

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COUNTEREXAMPLE FOR 𝑜 = 3 × 106 × 103 + +

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  • Maximin share (MMS) guarantee [Budish

2011] of player 𝑗: max

𝑌1,…,𝑌𝑜 min 𝑘

𝑊

𝑗(𝑌 𝑘)

  • Theorem [P & Wang 2014]: ∀𝑜 ≥ 3 there

exist additive valuation functions that do not admit an MMS allocation

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

possible to guarantee each player 2/3 of his MMS guarantee (in poly time for constant 𝑜)

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OVERVIEW

Rent Division

Computationally efficient algorithms for assigning rooms and dividing rent

Indivisible Goods

A new “computational” notion of fairness and its application in Spliddit Leximin in the real world: properties,

  • ptimization, implementation

Classroom Allocation

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“… the reward of helping people who have a real fair division problem by explaining our solutions, is that they in return pose interesting and difficult new questions, food for our thoughts. … It could be a goldmine of ideas, as well as a costly proposition if there are too many questions!”

Hervé Moulin

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“… public school facilities should be shared fairly among all public school pupils, including those in charter schools.”

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I object, your honor — the method is provably fair!

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OUR

UR APPROACH

  • Facilities have capacities
  • Players have demands
  • Preferences are dichotomous
  • Starting point: the Leximin

Mechanism [Bogomolnaia and Moulin 2004]

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THE

HE LEXIMIN MECHANISM

1 2 1 2

1 2

:

1 2 1 2

1

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1 2 1 4 1 4

1 2

:

1 2 3 4 3 4

THE

HE LEXIMIN MECHANISM

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  • Theorem [Kurokawa et al. 2015]:

The leximin mechanism satisfies proportionality, envy-freeness, Pareto efficiency, and group strategyproofness

  • We actually prove this in a much

more general framework

  • Theorem [Kurokawa et al. 2015]:

The expected number of units allocated by the leximin mechanism 1/4-approximates the maximum number of units that can be allocated simultaneously

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BIBLIOGRAPHY

  • Procaccia. Cake Cutting: Not Just Child’s
  • Play. Communications of the ACM 2013.
  • Dickerson, Goldman, Karp, Procaccia, and
  • Sandholm. The Computational Rise and Fall
  • f Fairness. AAAI 2014.
  • Procaccia and Wang. Fair Enough:

Guaranteeing Approximate Maximin Shares. EC 2014.

  • Kurokawa, Procaccia, and Shah. Leximin

Allocations in the Real World. EC 2015.