Which Is the Fairest (Rent Division)
of Them All?
“Mirror mirror on the wall, who is the fairest of them all?”
The Evil Queen
Ya’akov (Kobi) Gal, Moshe Mash, Ariel Proccacia, Yair ZIck
Reaching fair solutions in the real world
2
Example 1
r = $6000
Room: Verde
Room: Azul
$854
$1082
Alice
Bob
Room: Naranja
$2908
$2743
$2154
Alice
Bob
$1721
Claire
$2403
Alice
$2010
$2124
Bob
Claire
Claire
3
Example 1
r = $6000
Room: Verde
Room: Azul
$854
$1082
Alice
Bob
Room: Naranja
$2908
$2743
$2154
Alice
Bob
$1721
Claire
$2403
Alice
$2010
$2124
Bob
Claire
Claire
4
Example 1
r = $6000
Room: Verde
Room: Azul
$854
$1082
Alice
Bob
Room: Naranja
$2908
$2743
Alice
Bob
$1721
Claire
$2403
$2010
Alice
Bob
$2124 $2382
$896
Claire
$2154 $2722
Claire
5
The Problem
How to fairly distribute rooms and rent among roommates
Output – outcome σ, p
Input
•
•
•
Roommates (players): 𝑁 = 1, … , 𝑛.
r: rent-price for the apartment,
𝑣𝑖𝑗 : player 𝑖’s value of room 𝑗
such tha𝑡
𝑗 𝑣𝑖𝑗
=𝑟
•
Room allocation: 𝜎: 𝑁 → 𝑁
•
rent division 𝑝 = 𝑝1 , … , 𝑝𝑛
such that
𝑗 𝑝𝑗
=𝑟
6
Envy Free (EF) Outcome
𝑣𝑖𝜎
𝑖
Outcome 𝜎, 𝑝 such that
− 𝑝𝜎 𝑖 ≥ 𝑣𝑖𝑗 − 𝑝𝑗 for all 𝑖, 𝑗 ∈ 𝑁
7
8
Where EF goes wrong
Room: Verde
$1000
Room: Azul
$1000
Alice
$0
Bob
$0
Claire
$0
Alice
Bob
$0
Claire
Room: Naranja
$1000
$0
Alice
$0
Bob
Claire
9
Where EF goes wrong
r = $1000
Room: Verde
$1000
Room: Azul
$1000
Alice
$0
Bob
$0
Claire
$0
Alice
Bob
$0
Claire
Room: Naranja
$1000
$0
Alice
$0
Bob
Claire
10
Where EF goes wrong
r = $1000
Room: Verde
$1000
Room: Azul
$1000
$333.3
Alice
$0
Bob
$0
Claire
$0
Alice
Bob
$333.3
$0
Claire
Room: Naranja
$1000
$333.3
$0
Alice
$0
Bob
Claire
11
Where EF goes wrong
r = $1000
Room: Azul
$1000
Alice
$1000
$0
Bob
$0
Claire
Room: Verde
$1000
$0
Alice
Bob
$0
Claire
Room: Naranja
$1000
$0
Alice
$0
Bob
Claire
12
Solutions Concepts
• Equitability (under EF constraint)
• Maximin (under EF constraint)
13
Known facts about EF
• EF allocation always exists [Svensson ‘83]
• Usually there is more than one possible EF solution.
14
Solutions Concepts
• Equitability (under EF constraint)
• Maximin (under EF constraint)
15
Equitability
Room: Verde
Room: Azul
Alice
Bob
Claire
Alice
Bob
Claire
Room: Naranja
Alice
Bob
Claire
16
Equitability
• Maximal difference between players’ utilities for a given
outcome
𝐷 𝜎, 𝑝 = max 𝑢𝑖 𝜎, 𝑝 − 𝑢𝑗 𝜎, 𝑝
𝑖,𝑗
• An outcome 𝜎, 𝑝 is equitable if it minimizes the maximal
difference subject to EF
𝐷 ∗ 𝑉 = min{𝐷 𝜎, 𝑝 ∣ 𝜎, 𝑝 is EF for 𝑉}
17
Solutions Concepts
• Equitability (under EF constraint)
• Maximin (under EF constraint)
18
Maximin
• The minimal utility among the players
𝑈 𝜎, 𝑝 = min 𝑢𝑖 𝜎, 𝑝
𝑖
• An outcome 𝜎, 𝑝 is maximin
if it maximizes the minimal utility subject to EF
𝑈 ∗ (𝑉) = 𝑚𝑎𝑥 {𝑈(𝜎, 𝑝) ∣ 〈𝜎, 𝑝〉 𝑖𝑠 𝐸𝐹 𝑓𝑜𝑟 𝑉}
19
General
Algorithmic
Framework
20
EF Results
1st Welfare Thm
EF Outcome
2nd Welfare Thm
Socially
Optimal room
Allocation
21
EF Result
′
Lemma: if 𝜎, 𝑝 is EF, and 𝜎 is socially optimal,
′
then 𝑢𝑖 𝜎, 𝑝 = 𝑢𝑖 𝜎 , 𝑝 for all 𝑖 ∈ 𝑛
𝑛𝑑
(immediately follows from 2
Welfare theorem)
22
General Algorithmic Framework
• Start with any socially optimal allocation
• Find the price vector in the Maximin / Equitable
outcome for this room allocation
23
Relationship between Maximin and Equitability
Theorem:
There is a unique price vector that maximizes utility of the worst off player,
and it also achieves equitability.
Maximin
subject to EF
Equitability
subject to EF
24
Relationship between Maximin and Equitability
In contrast,
an equitable solution may not be Maximin.
Maximin
subject to EF
Equitability
subject to EF
25
Maximin is good in theory.
Is it also good in practice?
26
Fair Division is hard to study in the lab
“the goods in the lab are not really distributed
among participants, but serve as temporary
substitutes for money.”
[Herreiner and Puppe 09]
27
28
Empirical Studies
• Study the practical benefit of the maximin solution
• The database included more than 16,000 instances!
• Most common instances included 2, 3 and 4 rooms
29
Motivation
30
Theoretical
Benefits
Empirical
Benefits
Are people willing to accept such
solutions in practice?
31
Participants
• Spliddit users who participated in rent division
instances with 2, 3 or 4 players
• Invited over email
• Offered $10 fixed compensation
32
Within-Subject Design
• The subjects were shown the two solutions — maximin and arbitrary EF
for their own instance
• The two solution outcomes were shown in sequence, and in random order
33
Survey
The subjects asked to rate the following questions (between 1-5):
Individual
This question relates to your own allocation.
In other words, we would like you to pay attention only to your own benefit.
How happy are you with getting the room called Verde for $2,382?
Other
This question relates to the allocation for everyone else.
How fair do you rate the allocation for Bob and Claire?
34
Results
35
Conclusion
Computational fair division can be applied to the real world
Interplay between theory and practice
Lots of interesting open problems
36
DRAFTS
SLIDES
37
Results
38
How Common are Large Differences in Utility?
• Let 𝑐 ∗ 𝜖 be the class of 3 players that satisfy the following property:
there exist some room 𝑗 for which 𝑣𝑖𝑗 − 𝑣𝑘𝑗 ≤ 𝜖 for all 𝑖, 𝑘 ∈ 𝑛 , but
𝑣𝑖𝑙 − 𝑣𝑘𝑙 ≥ 2𝜖 for all 𝑙 ∈ \ 𝑗 and all 𝑖 ∈ 𝑛 , 𝑘 ∈ 𝑛 \ 𝑖 .
Room 1 [agreement]
Room 2 [disagreement] Room 3 [disagreement]
39
How Common are Large Differences in Utility?
• Lemma: if 𝑉 ∈ 𝑐 ∗ (𝜖), then 𝑉 allows for and extremely equitable EF
solution, where each player has utility of nearly ( 𝑖 𝑣𝑖𝑗 − 1) /3 ;
however, it also admits an EF solution where one of the players has
utility 0 — the worst possible outcome.
Room 1 [agreement]
Room 2 [disagreement] Room 3 [disagreement]
40
How Common are Large Differences in Utility?
Theorem: for two players, the expected difference between the least fair and
𝑟
most fair EF rent divisions is .
3
For more than two tenants, things get really annoying really fast (even for the 3
player case…).
41
The Benefit Is Significant in
Theory
42
Outline
1
The Problem
2
Motivation
3
Solutions concepts – EF , Equitability, Maximin
4
Theoretical results
5
Empirical Results
43
Maximin algorithm
𝑛
𝑖=1 𝑣𝑖𝜋 𝑖
∗
1. Let 𝜎 ∈ 𝑎𝑟𝑔𝑚𝑎𝑥𝜋
be a socially optimal allocation
2. Compute a price vector 𝑝 by solving the linear program
max 𝑈
𝑠. 𝑡: 𝑈 ≤ 𝑣𝑖𝜎
𝑣𝑖𝜎 𝑖 −𝑝𝜎
𝑛
𝑗=1 𝑝𝑗
− 𝑝𝜎
𝑖
𝑖
=𝑟
𝑖
≥ 𝑣𝑖𝑗 − 𝑝𝑗
∀𝑖 ∈ 𝑁
∀𝑖, 𝑗 ∈ 𝑁
44
Motivation
Number of Rooms
Number of Instances
2
698
3
445
4
160
All
1356
45
EF Result
• Lemma: if 𝜎, 𝑝 is EF, and 𝜎 ′ is socially optimal,
then 𝑢𝑖 𝜎, 𝑝 = 𝑢𝑖 𝜎 ′ , 𝑝 for all 𝑖 ∈ 𝑛
(immediately from 2𝑠𝑡 Welfare theory)
EF with the same
utilities
2st Welfare Thm
Allocation 1
P
Allocation 2
P
Allocation 3
P
50
An equitable solution may not be Maximin
Room: B
Room: A
Alice
Bob
Room: C
$0
Alice
Bob
Claire
Claire
Alice
Bob
Claire
An equitable solution may not be Maximin
Room: B
Room: A
Alice
Bob
Room: C
$0
Alice
Bob
Claire
Claire
Alice
Bob
Claire