1. Mathematical Foundations of AI
Lecture 9 – Fair Allocation
(cake cutting)

2. Content Proportional and Envy-free cutting Dividing a cake to 2 diners
Dividing a cake to n diners
Other issues

3. Fair division

4. Fair division: definitions
The cake is represented by the unit interval [0,1]
The preferences of the i’th Player are represented by a function Vi : [0,1]  R+
Vi has the following properties:
Additivity: For 2 distinct portions A,B [0,1] Vi(A U B) = Vi(A) + Vi(B)
The value of the portion [x,x] (size 0) is 0
Vi([0, 1]) = 1

5. Divisions
A division S={S1,S2,…,SN} is a partition of the cake + allocation to the N agents
Each Si is the union of a finite number of intervals
The union of all N parts is [0,1]
For i≠j, Si,Sj are distinct

6. Fairness We use two different concepts of “fairness”
A division is proportional, if Vi(Si) ≥ 1/n ,i.e. every agent thinks he gets at least 1/n
A division is envy-free, if Vi(Si) ≥ Vi(Sj) for all j≠i, i.e. no agent wants to trade portions with anyone else

7. Fairness (example) The valuation functions: 2/3 1/3 1/3 1/3 1/3 1/3
[0… ⅓]
[⅓…⅔]
[⅔ …1]
2/3
1/3
1/3
1/3
1/3
1/3
2/3

8. Fairness (example) Non-proportional division 2/3 1/3 1/3 1/3 1/3 1/3
[0… ⅓]
[⅓…⅔]
[⅔ …1]
2/3
1/3
1/3
1/3
1/3
1/3
2/3

9. Fairness (example) Division is proportional, but not envy-free ≥ ≤ 2/3
[0… ⅓]
[⅓…⅔]
[⅔ …1] ≥
2/3
1/3
1/3
1/3
1/3 ≤
1/3
2/3

10. Fairness (example) Division is proportional, and envy-free 2/3 1/3 1/3
[0… ⅓]
[⅓…⅔]
[⅔ …1]
2/3
1/3
1/3
1/3
1/3
1/3
2/3

11. Fairness (example) Division is proportional, and envy-free 7/12 1/4
[0…¼]
[¼…½]
[½ …1]
7/12
1/4
1/6
1/4
1/4
1/2
1/4
5/12
1/3

12. Proportional vs. Envy-free
Lemma 1: If a division is Envy free, it is also proportional
Proof: For every agent i, there is at least one portion j, for which Vi(Sj) ≥ 1/n, Thus
Vi(Si) ≥ Vi(Sj) ≥ 1/n
Lemma 2: for n=2, a proportional division is also envy-free

13. Division protocols Example: N=2 The Cut-and-choose protocol:
Agent 1 divides the cake to 2 equal parts, i.e. V1(A)=V1(B)
Agent 2 selects the larger portion
Claim: The protocol guarantees an envy-free division (and also proportional)

14. Our protocol is perfect! (but only for 2 agents…)
Division Protocols (2)
Other properties of a protocol:
How many cuts do we need?
In Cut-and-choose: just one cut  optimal
Are portions contiguous?
Yes
Do we need any other tools/assumptions? No
Our protocol is perfect! (but only for 2 agents…)

15. N=3 Protocol for N = 3 (Steinhaus, 1943)
Agent 1 cuts the cake to three even pieces
If there are two parts that worth at least ⅓ to agent 2, then:
Agents select pieces in the order 3,2,1
Otherwise, agent 2 marks the two small pieces as “bad”
1/3
1/3
1/3
3/12
2/12
7/12

16. n=3 (cont.)
(similarly) If there are two pieces that worth at least ⅓ to agent 3, then
Agents select pieces in the order 2,3,1
Otherwise agent 3 marks two small pieces as “bad”
(if players 2+3 marked bad pieces) There is one piece marked twice
Agent 1 takes that piece
Other two pieces are merged to one smaller cake, and agents 2,3 split it using cut-and-choose.

17. Protocol properties Cut and choose 2 1 V Steinhaus 3 X name N #cuts
Proportional
Envy-free
contiguous
Cut and choose 2 1 V
Steinhaus 3 X

18. N=3 + envy free Envy-Free Protocol for N = 3 (Selfridge-Conway, 1960)
A pre stage:
(a) agent 1 divides the cake to three equal parts
(b) agent 2 trims the largest part, so that now the two largest parts are equal
We now have two cakes: A and B

19. N=3 + envy free
1/3
1/3
1/3
2/10
3/10
3/10
5/10
2/10 A B

20. Stage One (Dividing cake A)NT T
Agent 3 picks first, then 2
If 3 picked the trimmed part, 2 picks the other part of same size.
Otherwise 2 picks the trimmed part
Agent 1 takes the piece that is left
The player that got the trimmed piece is denoted by T.
The other one (from {2,3}), is denoted by NT

21. Stage two (dividing cake B)NT
Agent NT divides the cake to three equal parts
Agents select pieces in this order: T 1 NT
Claim: the protocol is envy free

22. Protocol properties Cut and choose 2 1 V Steinhaus 3 X
name N
#cuts
Proportional
Envy-free
contiguous
Cut and choose 2 1 V
Steinhaus 3 X
Selfridge-Conway 5

23. What if N>3 ! ! ! ! Protocol for arbitrary N (Dubins-Spanier, 1961)
Each agent stops the knife when it passes 1/N of its value function ! !
V3(S3)=1/5
V5(S5)=1/5
V2(S2)
=1/5
V4(S4)
=1/5 ! !
Claim: the protocol is proportional

24. Protocol properties Cut and choose 2 1 V Steinhaus 3 X
name N
#cuts
Proportional
Envy-free
contiguous
other
Cut and choose 2 1 V
Steinhaus 3 X
Selfridge-Conway 5
Dubins-Spanier
(Banach-Knaster)
N-1
Moving knife

25. Envy free for N>3
Is there an envy free allocation for general N?
Yes
Is there a protocol that guarantees such a partition?
Yes, but with an unbounded number of cuts ( Brams &Taylor ‘95)
Open question: can we do it with a bounded number of cuts?

26. More… Optimality Dishonesty
Allocation of indivisible goods (next class)
Other topologies
Independent / correlated value functions