Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over bundles of items.
Goal: Make everyone happy. 🙌
📦 →
🙋♀️
💐 →
🤦♀️
🍰 →
🤩
🚗 →
🤷♀️
Task:
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop: Agents have preferences over bundles of items.
Goal: Make everyone happy. 🙌
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.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop: Agents have preferences over bundles of items.
Goal: Make everyone happy. 🙌
Compute an allocation
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow \mathbb{Z} $$
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
0/1 Valuations
$$ \{0,1\} $$
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
Additive Valuations
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
For all subsets \( S \subseteq R \), \( \nu_\ell(S) = \sum_{o \in S} \nu_\ell(\{o\}) \).
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
If 🙋♀️ values the items 🍰 and 💐 at 1 each, then she values {🍰, 💐} at 2.
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🙌
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😀 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
👀 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😬 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😠 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: Make everyone happy. 🤔
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: The allocation \( \pi \) creates no envy
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: The allocation \( \pi \) creates no envy
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Agent a envies b in \( \pi \) if \( v_a(\pi(b)) > v_a(\pi(a)) \)
and a knows b.
Fair Division
Distribute m items among n agents.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: The allocation \( \pi \) creates no envy
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Agent a envies 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.
Backdrop: Agents have preferences over items.
Goal: Make everyone happy. 🙌
Task:
$$ \pi: A \rightarrow 2^R $$
Fair Division
Backdrop:
Goal: The allocation \( \pi \) creates no envy
Compute an allocation
$$ \nu_\ell: 2^R \rightarrow $$
$$ \{0,1\} $$
😭 → {🍰,🍵}
😍 → {🍔,🍟,☕️}
Agent a envies 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
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.
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)