Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over bundles of items.Β 

Goal: Make everyone happy. πŸ™Œ

πŸ“¦ β†’

Β πŸ™‹β€β™€οΈ

πŸ’ β†’

πŸ€¦β€β™€οΈ

πŸ°Β β†’

🀩

πŸš—Β β†’

πŸ€·β€β™€οΈ

Task:

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop: Agents have preferences over bundles of items.

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \pi(a_i) \cap \pi(a_j) = \emptyset $$

No sharing...

Complete

$$ \bigcup_{a \in A} \pi(a) = R $$

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop: Agents have preferences over bundles of items.

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow \mathbb{Z} $$

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

0/1 Valuations

$$ \{0,1\} $$

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Additive Valuations

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

For all subsets \( S \subseteq R \),Β \( \nu_\ell(S) = \sum_{o \in S} \nu_\ell(\{o\}) \).

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

If πŸ™‹β€β™€οΈ values the items 🍰 and πŸ’ at 1 each, then she values {🍰, πŸ’} at 2.

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ™Œ

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

πŸ˜€ β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

πŸ‘€ β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😬 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😠 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Β Agent a enviesΒ b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: Make everyone happy. πŸ€”

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Β Agent a enviesΒ b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: The allocation \( \pi \) creates no envy

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Β Agent a enviesΒ b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: The allocation \( \pi \) creates no envy

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Β Agent a enviesΒ b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

and a knows b.

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: The allocation \( \pi \) creates no envy

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Β Agent a enviesΒ b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

and a knows b.

(between friends)

Fair Division

Distribute m items among n agents.Β 

Backdrop: Agents have preferences over items.Β 

Goal: Make everyone happy. πŸ™Œ

Task:

$$ \pi: A \rightarrow 2^R $$

Fair Division

Backdrop:

Goal: The allocation \( \pi \) creates no envy

Compute an allocation

$$ \nu_\ell: 2^R \rightarrow $$

$$ \{0,1\} $$

😭 β†’ {🍰,🍡}

😍 β†’ {πŸ”,🍟,β˜•οΈ}

Β Agent a enviesΒ b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)

and a knows b.

(between friends)

and a social network on \( A \)

Fair Division via Social Comparison

AAMAS 2017

Rediet Abebe,
Jon M. Kleinberg, and David C. Parkes

Networked fairness in cake cutting

IJCAI 2017

Xiaohui Bei.

Youming Qiao, and Shengyu Zhang.

Preprint (2022)
Ganesh Ghalme, Xin Huang, Nidhi Rathi

Envy-Free Cake Cutting with Graph Constraints

Β Local envy-freeness in house allocation problems.

AAMAS 2018

Aurélie Beynier, Yann Chevaleyre, Laurent Gourvès, Julien Lesca, Nicolas Maudet, and Anaëlle Wilczynski.

Envy-Free Allocations Respecting Social Networks
Journal of Artifical Intelligence, 2022

&

AAMAS 2018

Robert Bredereck,

Andrzej Kaczmarczyk,

and Rolf Niedermeier.

Β 

Parameterized Complexity of Envy-Free

Resource Allocation

in Social Networks

AAAI 2020

Eduard Eiben,
Robert Ganian,
Thekla Hamm,
and Sebastian Ordyniak

Preprint (2022)
Justin Payan, Rik Sengupta, Vignesh Viswanathan

Locally EFX Allocations Over a Graph

Survey (2021)

Fair Division

By Neeldhara Misra