Better Ways to Cut a Cake

1
Better Ways to Cut a Cake

• Steven Brams NYU
• Mike Jones Montclair State University
• Christian Klamler Graz University
• Paris, October 2006

2
Fair Division 2

• Division of a heterogeneous divisible good among
  various players
• land division,
• service used over time by different players

• Various Procedures (Brams Taylor 1996)
• Comparisons on the basis of
• Complexity of the rules
• Properties satisfied
• Manipulability

3
Desirable Properties 3

• Efficiency
• There is no other allocation that is better for
  one player and at least as good for all others.
• Envy-freeness
• Each player thinks it receives at least a
  tied-for-largest portion, so it does not envy
  another player.
• Equitability
• Each players valuation of the portion that it
  receives is the same as every other players
  valuation of the portion it receives.

4
Assumptions 4

• CAKE as the unit interval X0,1
• Cuts divide the cake into subintervals
• Every player i has a continuous value function vi
  on 0,1 with the following properties
• For all x ? X, vi(x) ? 0
• vi(?) 0 i.e. measure is non-atomic
• For any disjoint x,y ? X, vi(xy) vi(x)
  vi(y), i.e. measure is finitely additive
• vi(X) 1
• Players are ignorant about other players value
  functions.
• Goal of each player is to maximize the value of
  the minimum-size piece that it can guarantee for
  itself, regardless of what the other players do
  (maximin value), i.e. players are risk-averse
  they never choose strategies that entail the
  possibility of giving them less than their
  maximin values.

5
Cut and Choose 5
Satisfies Efficiency, Envy-Freeness but NOT
Equitability
6
Does a perfect cut exist? 6

• Efficient, envy-free and equitable solution at x
  3/7
• (see Jones (2002))
• However, (even risk averse) players have no
  incentive to state their true value functions!

7
The Surplus Procedure 7

• RULES
• Independently, A and B report their value
  functions fA(x) and fB(x) to a referee.
• Referee determines the 50-50 points a and b.
• 0---------------------a-------------b---------1
• If a and b coincide, the cake is cut at that
  point and the pieces are randomly assigned.
• Let a be to the left of b. Then A gets 0,a and
  B gets b,1.

8
The Surplus Procedure 8

• Let c be the point in a,b at which the players
  receive the same proportion p of the cake in this
  interval as each values it.
• 0---------------------a-----c--------b---------1
  
• A receives portion a,c and B c,b for a total
  of 0,c for A and (c,1 for B.

9
The Surplus Procedure 9

• To solve for c we set

For the previous example we get
Which yields c 7/16. This does not ensure
pure equitability as they value the interval
a,b differently only proportional
equitability!
10
The Surplus Procedure 10
For pure equitability we need to cut the cake
at point e such that
for e 3/7 (which is further to the left than c).
There are conflicting arguments for cutting at c
(proportional equitability) and e (equitability).
Property A procedure is strategy-vulnerable if a
maximin player can, by misrepresenting its value
function, assuredly do better, whatever the value
function of the other players. A procedure that
is not strategy-vulnerable is called
strategy-proof.
11
Theorem 1 11
Theorem 1 SP is strategy-proof, whereas any
procedure that makes e the cut-point is
strategy-vulnerable.

• Proof
• Misrepresenting a and/or b.
• 0-----------a-----b---a------------1
• Misrepresenting their value functions over a,b.
• 0-----------a-----c?----b----------1
• Shift of c to the right for A possible if it
  either
• decreases
• increases
• But therefore A would have to know fB(x) which
  it does not!

12
Theorem 1 12

• If A knew the location of b manipulation was
  possible
• concentrate the value just to the left of b,
  what moves c rightward

• Manipulation is possible when cake is cut at e!
• submit fA(x) with the same 50-50 point
• if a is to the left of b, then decrease
• if a is to the right of b, then decrease

13
Extensions to Three or More Players 13

• Consider the following value functions for 3
  players

14
Extensions to Three or More Players 14

• It is not always possible to divide a cake among
  three players into envy-free and equitable
  portions using two cuts!

15
Extensions to Three or More Players 15
However, an envy-free allocation that uses n-1
parallel, vertical cuts is always efficient.
(Gale, 1993 Brams and Taylor, 1996)

• There are 2 envy-free procedures for 3-person,
  2-cut cake division
• Stromquist (1980) requires 4 simultaneously
  moving knifes
• Barbanel Brams (2004) requires 2
  simultaneously moving knifes

Beyond 4 players, no procedure is known that
yields an envy-free division unless an unbounded
number of cuts is allowed.
16
Equitability Procedure (EP) 16
It is always possible to find an equitable
division of a cake among three or more players
that is efficient.

• The rules of EP are
• Independently, A,B,C, report their (possibly
  false) value functions fA(x), fB(x), fC(x),
  over 0,1 to a referee.
• The referee determines the cutpoints that
  equalize the common value that all players
  receive (for the n! possible assignments of
  pieces)
• Choose the assignment that gives the players
  their maximum common value.

17
Equitability Procedure (EP) 17
Using the above 3-player example, the cutpoints
e1 and e2 have to be such that
giving e1 0.269 and e2 0.662 with a value of
0.393 for each player.
18
Theorems 2 and 3 18
Theorem 2 EP is strategy-proof.
In order to misrepresent, a player would have to
know the borders of the pieces. As it does not do
so it cannot ensure itself a more valuable piece.
Theorem 3 If a player is truthful under EP, it
will receive at least 1/n of the cake regardless
of whether or not the other players are truthful
otherwise, it may not.
We know that there is a division where each
player receives at least 1/n (e.g. Dubins-Spanier
moving knife procedure). As vi(X)1, undervaluing
the cake at one part will overvalue it at some
other part, but an ignorant player might get the
latter.
19
Example 19
In the previous example assume C knows the value
functions of A and B. Let c1 and c2 be the
cutpoints. Then C should undervalue the middle
portion between those points so that
It is maximal if B is indifferent between
receiving the right portion and the middle
portion, i.e.
This leads to c1 0.230 and c2 0.707 where A
and B receive a value of 0.354 and C receives a
value of 0.477 (compared to the 0.393
before). However, a bit more undervaluation ? C
gets a value less than 0.393.
20
Conclusion 20

• We have described a new 2-person, 1-cut cake
  cutting procedure (SP).
• Like cut-and-choose it induces players to be
  truthful, but produces a proportionally equitable
  division.
• SP is more information demanding.
• For three persons, there may be no envy-free
  division that is also equitable. For four
  persons, there is no known minimal-cut envy-free
  procedure. However, EP ensures equitability and
  efficiency.