1. Which Is the Fairest (Rent Division) of Them All?
“Mirror mirror on the wall, who is the fairest of them all?”
The Evil Queen
Good afternoon to you all, my name is Moshik Mash. I am a Ph.D student at BGU university. My advisor is Kobi Gal and I am very glad to present to you a joint work with Ariel Proccacia and Yair Zick from CMU university.
Ya’akov (Kobi) Gal, Moshe Mash, Ariel Proccacia, Yair ZIck

2. Reaching fair solutions in the real world
The research focuses on how to apply fair division algorithms in the real world. It is motivated by a free online service called Splidit developed by Ariel Procaccia and Jonathan Goldman at CMU. Spliddit allows people to upload their real fair division problems to the website and get in return a possible solution to their problem.
This paper deals with the share rent domain. Imagine that you and 2 other friends wish to rent an apartment together.
Roommates may have different values for each room. For example, imagine that there is a room which includes many windows; you like this room very much but your friend thinks it is a horrible room because the traffic noise for example.
The main question is how to divide the rooms and the rent between the roommates?

3. Example 1 r = $6000 Room: Azul Room: Naranja Room: Verde Alice Bob
Claire
$854
$1721
$1082
$2743
$2154
$2908
$2403
$2124
$2010
Here is an example of a real instance from Spliddit’s database.
The input is the total rent, the rooms, the roomates and the roomates’ valuations of the rooms. The valuations of each roommate over the rooms have to add up to the rent price.
For example Alice values the Azul room at $852, Verde room at $2403 and Narandja room at $2154.
Her valuations added up to $6000.

4. Example 1 r = $6000 Room: Azul Room: Naranja Room: Verde Alice Bob
Claire
$854
$1721
$1082
$2743
$2154
$2908
$2403
$2124
$2010
The output is a room allocation. For example Claire will gets the Azul room.

5. Example 1 r = $6000 Room: Azul Room: Naranja Room: Verde Alice Bob
Claire
$854
$1721
$1082
$2743
$2908
$2403
$2124
$2010
$2382
$896
$2722
$2154
and a price vector - the rent for each room. For example, Claire will pay $896 for the Azul room. Of course, the rooms’ rent have to add up to the total rent. For example equals the total rent.
This solution meets a special constraint– it is an EF solution. EF is an allocation such that each player prefers his OWN allocation to the allocations of each of the other roommates. For example, Claire’s utility is the difference between her valuations for her room minus the room’s price ( ). Suppose Claire should get Alice’s room, verdure room. Her utility would the rent her utility for the room (2124) minus the rent of Verder room (2382). We can see that her utility would be negative, therefore of course she prefers her own allocation.

6. How to fairly distribute rooms and rent among roommates
The Problem
How to fairly distribute rooms and rent among roommates
Roommates (players): 𝑁=1,…,𝑛.
r: rent-price for the apartment,
𝑣 𝑖𝑗 : player 𝑖’s value of room 𝑗
such tha𝑡 𝑗 𝑣 𝑖𝑗 =𝑟
Input
Output – outcome σ, p
Room allocation: 𝜎:𝑁→𝑁
rent division 𝑝 =𝑝 1 ,…, 𝑝 𝑛
such that 𝑗 𝑝 𝑗 =𝑟
So, lets formalize the problem.
The input is: the players, the total rent and the players’ values of the rooms. For example v_ij is the value of player I for room j. In addition the values of each player have to add up to the total rent
The output is an outcome which is represented by a pair of the room allocation (sigma) and the rooms’ prices (P). Of course the price of the rooms have to add up to the total rent.

7. Outcome 𝜎, 𝑝 such that 𝑣 𝑖𝜎 𝑖 − 𝑝 𝜎 𝑖 ≥ 𝑣 𝑖𝑗 − 𝑝 𝑗 for all 𝑖,𝑗∈𝑁
Envy Free (EF) Outcome
Outcome 𝜎, 𝑝 such that 𝑣 𝑖𝜎 𝑖 − 𝑝 𝜎 𝑖 ≥ 𝑣 𝑖𝑗 − 𝑝 𝑗 for all 𝑖,𝑗∈𝑁
Here is the formalization of the EF constraint.
The valuation of each player i for his room minus the room’s price is greater or equal then the valuation of the player for any other room minus the price of the specific room.

8. Here is a snapshot from Spliddit where the envy-freeness is explained to the users.
The power of the envy-free allocation is that it easy to explain to the users why it is a good solution.
For example, each roommate feels that he got the best deal (he prefers his own room, among all the other possible rooms). In other words, he doesn’t want to swap rooms with any other roommate.
But the problem is that EF alone is sometimes not sufficient. Let’s see an example.

9. Where EF goes wrong Room: Azul Room: Naranja Room: Verde Alice Bob
Claire
$1000 $0
In this example, each player has a strong preference for a different room. For example Alice values the Azul room at $1000 and the other rooms at zero.
How should we allocate the rooms among the players?

10. Where EF goes wrong r = $1000 Room: Azul Room: Naranja Room: Verde
Alice
Bob
Claire
$1000 $0
Of course. Each one will get the room he wants.
How should we divide the rent?

11. Where EF goes wrong r = $1000 Room: Azul Room: Naranja Room: Verde
Alice
Bob
Claire
$1000 $0
$333.3
$333.3
$333.3
Of course, equally!

12. Where EF goes wrong r = $1000 Room: Azul Room: Naranja Room: Verde
Alice
Bob
Claire
$1000 $0
$1000
I have another suggestion. Alice will pay all the rent. Of course, Alice will not like this solution! But why not? it is an EF solution.
Lets see why Alice doesn’t envy Bob and Claire?
Alice’s utility is 0 ( ). if for example she would get Bob’s room, Verde room, her utility would be 0 as well. Why? Her valuation for the room is zero and room’s price is zero.

13. Solutions Concepts Equitability (under EF constraint)
Maximin (under EF constraint)
So, we noticed that EF outcomes can be a non-intuitive. Therefore we wish to explore criteria to distinguish between EF outcomes. Equitability (under EF constraint) and Maximin (under EF constraint)

14. Known facts about EF EF allocation always exists [Svensson ‘83]
Usually there is more than one possible EF solution.
Firstly, I want to emphasize that EF outcome always exists (for all instances) as I mentioned before.
But, usually there is more than one possible EF outcome.

15. Solutions Concepts Equitability (under EF constraint)
Maximin (under EF constraint)
Lets start with the equitability under EF solution.

16. Equitability Room: Azul Room: Naranja Room: Verde Alice Bob Claire
Firstly, let me define three definitions. The worst off player is the player with the lowest utility from a specific outcome. The best off player is the player with the HIGHEST utility from a specific outcome.
The last definition is disparity, which describes the difference between the utility of the best off player and the utility of the worst off player.
In the equitable solution, we minimizes the disparity under the EF constraint.
Who is the worst off player in this example? All of them, because the utilities of all the players are the same. who is the best off player? Again, All of them as well, because the same reason. Therefore the disparity is zero.
We cannot promise we could achieve EF solution such that all the players will have the same utility for all instances, but we will choose the EF solution with the minimal disparity.

17. Equitability
Maximal difference between players’ utilities for a given outcome
𝐷 𝜎, 𝑝 = max 𝑖,𝑗 𝑢 𝑖 𝜎, 𝑝 − 𝑢 𝑗 𝜎, 𝑝
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 }
Lets formalize the Equitability under envy-free constraint.
The first formula describes the disparity (the maximal utilities’ difference).
The second one chooses the EF outcome with the minimal disparity among all the possible EF solutions.

18. Solutions Concepts Equitability (under EF constraint)
Maximin (under EF constraint)
The second criterion is the Maximin under EF

19. Maximin The minimal utility among the players 𝑈 𝜎, 𝑝 = min 𝑖 𝑢 𝑖 𝜎, 𝑝
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}
In the Maximin solution we choose the EF solution among all the possible EF solutions with maximal utility for the worst off player. We choose the EF solution with the maximal utility of the worst off player among all the possible EF solutions.
The first formula describes the utility of the worst off player
In the second formula, we choose the EF solution with the maximal utility off the worst player among all the possible EF solutions.

20. General Algorithmic Framework
Now, I want to present to you a general algorithm to find Maximin outcome or Equitability outcome or others objective functions subject EF.

21. Socially Optimal room Allocation
EF Results
Socially Optimal room Allocation
1st Welfare Thm
2nd Welfare Thm
EF Outcome
Firstly, I want to define the socially optimal room allocation term.
The socially optimal room allocation is a room allocation which maximizes the roommates’ values.
For example, lets see this room allocation (we maximize the sum of these values). We cannot allocate the rooms in another way such that the values will add up to a greater value than this sum. Therefore this is a socially optimal room allocation.
Now In will present to you a theorem about EF outcomes.
The room allocation of EF outcome must be a socially optimal room allocation. It derives directly from the first welfare theorem.
And for every socially optimal room allocation, there is a price vector for the rooms’ rent such the outcome will be EF. It derives directly from the second welfare theorem.

22. EF Result
Lemma: if 𝜎, 𝑝 is EF, and 𝜎 ′ is socially optimal, then 𝑢 𝑖 𝜎, 𝑝 = 𝑢 𝑖 𝜎 ′ , 𝑝 for all 𝑖∈ 𝑛 (immediately follows from 2 𝑛𝑑 Welfare theorem)
In addition, for every EF outcome - sigma and p (of course, sigma is an optimal room allocation because it is EF). If we will replace the room allocation, sigma, with any other socially optimal room allocation (without changing the price vector), we will get EF outcome and the utilities of all the players will remained the same). It also derives directly from the second Welfare theorem.
Therefore it doesn’t matter which room allocation we use for the EF outcome, the most important thing is to use socially optimal. Because, the utilities of the players will remain the same for all socially optimal rooms allocations for a given price vector.

23. General Algorithmic Framework
Start with any socially optimal allocation
Find the price vector in the Maximin / Equitable outcome for this room allocation
So, let’s see the algorithm to find maximn or equitable outcome.
We start with any socially optimal room allocation (it doesn't matter which one, as I explained before).
Then, we use linear programming to find a price vector which satisfies the EF constraint and the Maximin constraint, or equitability.
Therefore we can find Maximin or equitability in polynomial time.

24. 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.
Equitability
subject to EF
Maximin
I presented to you two interesting criteria under EF (Maximin and Equitability) and both of them can be computed in polynomial time.
The big question is which of we of we need to adopt? Which of them is more important? So, I want to present to you a very interesting result, which shows us that we can adopt Maximin and Equitability as the same time!
Firstly, there is a unique price vector which achieves maximin solution. In addition, the outcome with any socially optimal room allocation will achieves also Equitable solution.
Therefore, whether you find a maximin outcome under EF, you ge equitable outcome for free

25. Relationship between Maximin and Equitability
In contrast,
an equitable solution may not be Maximin.
Equitability
subject to EF
Maximin
In contrast, not every equitable outcome is Maximin.
So, it maybe the case that there are several possible price vectors for equitable solutions under EF but only one of them will be the price vector of the Maximin.
Therefore, if we think about our algorithm. It makes sense to add the Maximin constraint to the LP and not the equitability constraint because Maximin is also equitable (and not vise versa).

26. Maximin is good in theory. Is it also good in practice?
Therefore, it seems that Maximin is the best solution. So, based on the theoretical it is deployed to Spliddit.
But the question is- people are willing to accept the solutions in the real world?

27. 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]
Imagine that you are participating in a fair division experiment in the lab. The experimenter describes an apartment, rooms and your valuations for the rooms.
The problem is that the valuations in the lab are primed by the experimenter, they do not represent your real preferences. 
It is just an interpretation of money.

28. A main contribution of this work is that the valuations submitted by Spliddit users, represent their real preferences.

29. 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
We wanted to validate that we can actually improve the disparity and the utility of the worst off player in real world instances, therefore we performed a data analysis.
The database of Spliddit included 16,000 instances. The most common instances included 2, 3 and 4 rooms.

30. Motivation
We compared the disparity and the utility of the worst off player which were achieved by the Maximin algorithm and by an arbitrary EF.
For example, for n=2 we could decrease the disparity by 11% of the rent on average. So whenever the rent was $1000, it means we could decrease the disparity by $110 on average.
In addition, we also present in the paper theoretical analysis which support these improvements.

31. Are people willing to accept such solutions in practice?
Theoretical
Benefits
Empirical
Benefits
Are people willing to accept such solutions in practice?
We noticed that we can improve the disparity and the utility of the worst off player in theory and in the real world.
The big question is do people actually care about the disparity and the utility of the worst off player?
To answer this question, we conducted a user study.

32. Participants Spliddit users who participated in rent division
instances with 2, 3 or 4 players
Invited over
Offered $10 fixed compensation
We sent s to users of Spliddit and we offered them $10 for participating in a short user study.

33. 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
We showed them the instance which they uploaded to Spliddit in the past ,and we presented to them two possible solutions - Maximin under EF and arbitrary EF.

34. 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?
For each solution type (maximin and EF), we asked them to rate their satisfaction from thier own allocation as well as the allocations of the others.

35. Results Lets see the results.
The participants indeed prefer the maximin solution for them and for the others and this result is valid for each number of rooms.
All the results are statistically significant by Wilcoxon test.
So people indeed prefer the maximin solution, therefore if you use Spliddit today you will get in return a Maximin outcome.

36. Conclusion
Computational fair division can be applied to the real world
Interplay between theory and practice
Lots of interesting open problems
In conclusion I want to emphasize several important points.
First of all, fair division problems can be applied to the real world! Spliddit and this research are strong indications.
There is a clear interplay between theory and practice in fair division domains. For example, the findings in theory directly impact on the solutions in Spliddit On the other hand, Spliddit allows us to examine the algorithms in real world instances. In addition it allows us to get feedback from users.
3. Finally, we believe that this research is part of a broader agenda of research in many other real world fair division domains such as indivisible goods’ distribution, taxi share, tasks distribution and many others. In addition, we have also been learning from people about new real world domains and Spliddit allows users to suggest new domains.
Thank you very much for your kind attention.
//For example, Nisarg Shah presented in EC last year a research which deals with classrooms allocations among district charter schools. This research suggest one //of the largest schools in California. est new domains. Joun shing

37. DRAFTS SLIDES

38. Results

39. 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 𝑣 𝑖𝑗 − 𝑣 𝑘𝑗 ≤𝜖 for all 𝑖,𝑘∈ 𝑛 , but 𝑣 𝑖𝑙 − 𝑣 𝑘𝑙 ≥2𝜖 for all 𝑙∈\ 𝑗 and all 𝑖∈ 𝑛 , 𝑘∈ 𝑛 \ 𝑖 .
Room 1 [agreement]
Room 2 [disagreement]
Room 3 [disagreement]
כמו-כן, ניתחנו את ההפרש עבור מקרים פרטיים. לדוגמא:
נניח וכל השחקנים מערכים חדר מסוים בערך אותו דבר אבל את שני החדרים האחרים כל אחד מעריך באופן מאוד שונה.

40. 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 𝑣 𝑖𝑗 −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]
ה- equitable outcome יספק לנו D=0. לעומת זאת, קיים פתרון EF אשר יניב תועלת של 0 עבור אחד מהשחקנים.

41. How Common are Large Differences in Utility?
Theorem: for two players, the expected difference between the least fair and most fair EF rent divisions is 𝑟 3 .
For more than two tenants, things get really annoying really fast (even for the 3 player case…).
עבור שני שחקנים ההפרש של ה- D בין שני סוגי הפתרונות הינו שכר הדירה חלקי 3. עבור שלושה שחקנים הבעיה הופכת לקשה לחישוב (אפילו עבור 3 שחקנים)

42. The Benefit Is Significant in Theory
בשקופיות הבאות השווה בין EF outcome אשר מביא להפרש הגדול ביותר בין התועלות של שחקנים (המצב הגרוע ביותר( לבין equitable outcome.
אנו נחשב את ה- expected difference בין ה- D (ההפרש המקסימלי) של שני ה- outcomes עבור התפלגויות אחידות

43. Outline The Problem Motivation1
Motivation 2
Solutions concepts – EF , Equitability, Maximin 3
Empirical Results 4 5
Theoretical results

44. Maximin algorithm
Let 𝜎 ∗ ∈𝑎𝑟𝑔𝑚𝑎 𝑥 𝜋 𝑖=1 𝑛 𝑣 𝑖𝜋 𝑖 be a socially optimal allocation
Compute a price vector 𝑝 by solving the linear program
max 𝑈
𝑠.𝑡:𝑈≤ 𝑣 𝑖𝜎 𝑖 − 𝑝 𝜎 𝑖 ∀𝑖∈𝑁
𝑣 𝑖𝜎 𝑖 − 𝑝 𝜎 𝑖 ≥ 𝑣 𝑖𝑗 − 𝑝 𝑗 ∀𝑖,𝑗∈𝑁
𝑗=1 𝑛 𝑝 𝑗 =𝑟
כפי שהראנו בשקופית הקודמת מספיק למצוא מקסימין תחת EF על מנת להשיג equitability תחת EF. להלן האלגוריתם שלנו:
תחילה נמצא socially optimal room allocation בעזרת מציאת זיווג מקסימלי ב- weighted bipartite graph. קבוצה אחת של קודקודים תייצג את השחקנים והקבוצה השנייה את החדרים. ה- valuations של השחקנים לחדרים יתואר ע"י הקשתות בין הקודקודים בין שני הקבוצות.
ע"מ למצוא את וקטור המחירים נפתור linear programming הבא:
פונקציית המטרה הינה מיקסום U, ונדאג שהתועלת של כל אחד מהשחקנים הינה מעל U (בעצם מקסממנו את השחקן בעל התועלת הנמוכה ביותר).
כמו-כן, נוסיף את אילוץ ה- EF ואת האילוץ שידאג לכך שמחירי החדרים יסתכמו לשכר הדירה.
החישוביות של האלגוריתם הינה פולינומאליות למרות שהוספנו אילוצים של equitability ו- maxiimin

45. Motivation Number of Instances Number of Rooms 698 2 445 3 160 4 1356
All
we chose a subset of these instances conservatively (no need to be specific in the talk). 

46. EF with the same utilities
EF Result
Lemma: if 𝜎, 𝑝 is EF, and 𝜎 ′ is socially optimal,
then 𝑢 𝑖 𝜎, 𝑝 = 𝑢 𝑖 𝜎 ′ , 𝑝 for all 𝑖∈ 𝑛
(immediately from 2 𝑠𝑡 Welfare theory)
Allocation 1 P
We showed any room allocation can achieve an EF outcome. The question is, will any room allocation also be maximin and equitable? (i.e. minimize D, maximize U). The answer is yes!!
This implies an algorithm for achieving a maximin and Equitability under EF outcome: start with a socially optimal allocation, and find the price vector that is EF and minimize D. The same algorithm can also compute the equitable solution by adding a constraint for equitability instead of Maximin. But the Maximin is also equitablilty but equitability is not Maximin.
we actually give a general algorithmic framework that can also compute the equitable solution, and others. 
2st Welfare Thm
EF with the same utilities
Allocation 2 P
Allocation 3 P

47. An equitable solution may not be Maximin
Room: A
Room: C
Room: B
Alice
Bob
Claire $0
Visualization of the problem and a possible solution

48. An equitable solution may not be Maximin
Room: A
Room: C
Room: B
Alice
Bob
Claire $0
Visualization of the problem and a possible solution