FAT in Resource Allocation: Equilibrium to the Rescue Noam Nisan - - PowerPoint PPT Presentation

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FAT in Resource Allocation: Equilibrium to the Rescue Noam Nisan - - PowerPoint PPT Presentation

Jan 23, 2023 353 likes •667 views

FAT in Resource Allocation: Equilibrium to the Rescue Noam Nisan Hebrew U. Computer Science Moshe Babaioff Microsoft Research Inbal Talgam-Cohen Technion Computer Science (+ LL.B.) (Based on our paper Fair Allocation through

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SLIDE 1

FAT in Resource Allocation: Equilibrium to the Rescue

Noam Nisan – Hebrew U. Computer Science Moshe Babaioff – Microsoft Research → Inbal Talgam-Cohen – Technion Computer Science (+ LL.B.)

(Based on our paper “Fair Allocation through Competitive Equilibrium from Generic Incomes”)

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SLIDE 2

Resource Allocation

  • Who gets what
  • Fair allocation decisions should take into account:
  • 1. who wants what (preferences)
  • 2. who’s entitled to what
  • Increasingly driven by algorithms

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 2

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SLIDE 3

Motivating Scenario

  • Players: 2 food banks catering to populations with different needs
  • Resources: Donated food items to allocate among food banks

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen

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SLIDE 4

Motivating Scenario

  • Players: 2 food banks catering to populations with different needs
  • Resources: Donated food items to allocate among food banks
  • Properties:
  • 1. Using (real) money is inappropriate

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 4

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SLIDE 5

Motivating Scenario

  • Players: 2 food banks catering to populations with different needs
  • Resources: Donated food items to allocate among food banks
  • Properties:
  • 1. Using (real) money is inappropriate
  • 2. Preferences (roughly) additive
  • 3. Entitlements by population served

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 5

beans cereal

value 100 value 80 Food bank 1: 3K mouths to feed value 180

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SLIDE 6

Motivating Scenario

  • Players: 2 food banks catering to populations with different needs
  • Resources: Donated food items to allocate among food banks
  • Properties:
  • 1. Using (real) money is inappropriate
  • 2. Preferences (roughly) additive
  • 3. Entitlements by population served

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 6

beans cereal

value 60 value 140 value 200 Food bank 2: 7K mouths to feed

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SLIDE 7

Motivating Scenario

  • Players: 2 food banks catering to populations with different needs
  • Resources: Donated food items to allocate among food banks
  • Properties:
  • 1. Using (real) money is inappropriate
  • 2. Preferences (roughly) additive
  • 3. Entitlements by population served
  • Main question: What constitutes a fair allocation of the items?

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen

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SLIDE 8

Additional Examples

What constitutes a fair allocation of:

  • Courses to students?
  • Shifts to workers?
  • Computational resources across a university/company?
  • Heirloom items among family members?

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 8

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SLIDE 9

Fundamental Fairness Notion #1: Fair Share

  • Procedure of dividing a cake fairly:
  • first kid divides the cake
  • second kid picks a piece
  • First gets her fair share = ½ her value for the entire cake
  • Second gets at least ½

*Image from Spliddit 9

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SLIDE 10

Fundamental Fairness Notion #1: Fair Share

  • Procedure of dividing a cake fairly:
  • first kid divides the cake
  • second kid picks a piece
  • First gets her fair share = ½ her value for the entire cake
  • But we’re allocating indivisible items (truckloads…)!
  • Seek as fair an allocation as possible

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 10

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SLIDE 11

Fundamental Fairness Notion #2: Maximin Share

  • Procedure of dividing a cake fairly:
  • first kid divides the cake
  • second kid picks a piece

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 11

indi divisi isible ble items ms

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SLIDE 12

Fundamental Fairness Notion #2: Maximin Share

  • Procedure of dividing a cake fairly:
  • first kid divides the cake
  • second kid picks a piece

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 12

indi divisi isible ble items ms Items s into 2 piles

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SLIDE 13

Fundamental Fairness Notion #2: Maximin Share

  • Procedure of dividing a cake fairly:
  • first kid divides the cake
  • second kid picks a piece

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 13

indi divisi isible ble items ms Items s into 2 piles pile

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SLIDE 14

Fundamental Fairness Notion #2: Maximin Share

  • Procedure of dividing indivisible items fairly:
  • first kid divides items into 2 piles
  • second kid picks a pile
  • First gets her maximin share = as close to ½ as possible
  • A Rawlsian guarantee from behind a “thin veil of ignorance”

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 14

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SLIDE 15

Equilibrium to the Rescue?

  • In market equilibrium supply=demand
  • Equilibrium prices are:
  • sufficiently high (no over-demand)
  • sufficiently low (market clears – no excess supply)

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 15

Price Supply Demand Equil.

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SLIDE 16

Equilibrium to the Rescue?

  • Market equilibrium applicable to us via fake “money”
  • How it works:
  • Food banks get budgets (say 10K “points”) representing entitlements
  • Equilibrium item prices are such that
  • each food bank takes its best affordable set of items
  • market clears

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 16

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SLIDE 17

Equilibrium to the Rescue?

  • Amazing properties of equilibrium allocation (with equal budgets):
  • Each player gets at least her maximin share [Varian’74, Budish’11]
  • Transparency – everyone faces same prices
  • Efficiency

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 17

Price Supply Demand Equil.

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SLIDE 18

Too Good to Be True?

  • Equilibrium not guaranteed to exist!
  • Example:
  • Price above 10K → market doesn’t clear
  • Price at or below 10K → over-demand

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 18

cereal

Food bank 1: Budget 10K Food bank 2: Budget 10K

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SLIDE 19

Too Good to Be True?

  • Equilibrium not guaranteed to exist!
  • Example:
  • Price above 10K → market doesn’t clear
  • Price at or below 10K → over-demand
  • Reason for optimism: Issue is equal budgets – a knife’s edge case
  • Disappears when entitlements are slightly different (as in reality)

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 19

cereal

Food bank 1: Budget 10K Food bank 2: Budget 10K

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SLIDE 20

Equilibrium Exists Except on Knife’s Edge

  • Equilibrium price = 10K + 1
  • We show: With slightly different budgets an equilibrium exists

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 20

cereal

Food bank 1: Budget 10K + 1 Food bank 2: Budget 10K

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SLIDE 21

Equilibrium Exists Except on Knife’s Edge

  • Equilibrium price = 10K + 1
  • We show: With slightly different budgets an equilibrium exists
  • Theorem: For 2 additive players with equal budgets, after tiny random

perturbations to the budgets, a market equilibrium exists

  • [More existence theorems in paper]

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 21

cereal

Food bank 1: Budget 10K + 1 Food bank 2: Budget 10K

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SLIDE 22

Deriving New Fairness Notions from Equilibrium

  • With slightly perturbed budgets, is the equilibrium allocation still fair?
  • Intuition: fairness will hold approximately
  • relaxed to the extent necessary given indivisibilities

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 22

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SLIDE 23

Deriving New Fairness Notions from Equilibrium

  • With slightly perturbed budgets, is the equilibrium allocation still fair?
  • Intuition: fairness will hold approximately
  • relaxed to the extent necessary given indivisibilities
  • What if the budgets are very different?
  • What constitutes a fair allocation of indivisibles among unequals?

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 23

Food bank 1: 3K mouths to feed Food bank 2: 7K mouths to feed

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SLIDE 24

Deriving New Fairness Notions from Equilibrium

  • Theorem: In any equilibrium allocation among players with (possibly

very different) budgets, each player gets at least her proportional maximin share (or as close as possible given indivisibilities)

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 24

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SLIDE 25

Deriving New Fairness Notions from Equilibrium

  • Theorem: In any equilibrium allocation among players with (possibly

very different) budgets, each player gets at least her proportional maximin share (or as close as possible given indivisibilities)

  • Example:
  • first divides items into 10 piles
  • second picks 7 piles

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 25

Food bank 1: 3K mouths to feed Food bank 2: 7K mouths to feed

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SLIDE 26

Deriving New Fairness Notions from Equilibrium

  • Theorem: In any equilibrium allocation among players with (possibly

very different) budgets, each player gets at least her proportional maximin share (or as close as possible given indivisibilities)

  • Example:
  • first divides items into 10 piles
  • second picks 7 piles

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 26

Food bank 1: 3K mouths to feed Food bank 2: 7K mouths to feed 4 piles es 3 3 piles es

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SLIDE 27

Deriving New Fairness Notions from Equilibrium

  • Theorem: In any equilibrium allocation among players with (possibly

very different) budgets, each player gets at least her proportional maximin share (or as close as possible given indivisibilities)

  • Example:
  • first divides items into 10 piles
  • second picks 7 piles
  • First gets proportional maximin share = as close to 30% as possible

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 27

Food bank 1: 3K mouths to feed Food bank 2: 7K mouths to feed 4 piles es 3 3 piles es

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SLIDE 28

Deriving New Fairness Notions from Equilibrium

  • Theorem: In any equilibrium allocation among players with (possibly

very different) budgets, each player gets at least her proportional maximin share (or as close as possible given indivisibilities)

  • Example:
  • first divides items into 10 piles
  • second picks 7 piles
  • First gets proportional maximin share = as close to 30% as possible
  • [Another fairness notion from equilibrium in paper]

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 28

Food bank 1: 3K mouths to feed Food bank 2: 7K mouths to feed

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SLIDE 29

Take-Aways

  • Markets, prices, equilibrium: useful toolbox for defining and

implementing fairness + transparency

  • alongside “usual suspects” like statistics, ML
  • 1. Equil. helps define fair allocation of indivisibles among unequals
  • Open: Envy-freeness notions among unequals
  • 2. The existence of approx. fair allocations follows from equilibrium

existence after tiny budget perturbations

  • Open: Typical existence

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 29

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SLIDE 30

Thank you!

“Algorithms and Law” initiative – please contact me for details! italgam@cs.technion.ac.il

Fair Resource Allocation through Equilibrium Babaioff, Nisan, Talgam-Cohen 30