Better Ways to Cut a Cake

Steven Brams – NYUMike Jones – Montclair State UniversityChristian Klamler – Graz UniversityParis, October 2006

Fair Division

2

Various Procedures (Brams & Taylor 1996) Comparisons on the basis of:

Complexity of the rules Properties satisfied Manipulability

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

Desirable Properties

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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 player’s valuation of the portion that it

receives is the same as every other player’s valuation of the portion it receives.

Assumptions 4

CAKE as the unit interval X=[0,1] 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(x+y) = 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.

0 1

Cut and Choose 5

0 1/2 1

4 10,3 2

2 1 ,13 2A

xv x

x

1 0,1Bv x x

Satisfies Efficiency, Envy-Freeness but NOT Equitability

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!

1/2

1/2

2/3

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The Surplus Procedure 7

RULES:

1. Independently, A and B report their value functions fA(x) and fB(x) to a referee.

2. Referee determines the 50-50 points a and b.

0---------------------a-------------b---------1

3. If a and b coincide, the cake is cut at that point and the pieces are randomly assigned.

4. Let a be to the left of b. Then A gets [0,a] and B gets [b,1].

The Surplus Procedure 85. 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.

The Surplus Procedure 9To 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!

The Surplus Procedure

10For “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.

Theorem 1 11

Proof:1. Misrepresenting a and/or b.

0-----------a-----b---a’------------1

2. Misrepresenting their value functions over [a,b].

0-----------a-----c----b----------1Shift 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!

Theorem 1: SP is strategy-proof, whereas any procedure that makes e the cut-point is strategy-vulnerable.

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

Extensions to Three or More Players 13

Consider the following value functions for 3 players:

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!

Extensions to Three or More Players 15

There are 2 envy-free procedures for 3-person, 2-cut cake division:

o Stromquist (1980): requires 4 simultaneously moving knifes

o 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.

However, an envy-free allocation that uses n-1 parallel, vertical cuts is always efficient. (Gale, 1993; Brams and Taylor, 1996)

Equitability Procedure (EP)

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The rules of EP are:1. Independently, A,B,C, … report their (possibly false) value

functions fA(x), fB(x), fC(x), … over [0,1] to a referee.

2. The referee determines the cutpoints that equalize the common value that all players receive (for the n! possible assignments of pieces)

3. Choose the assignment that gives the players their maximum common value.

It is always possible to find an equitable division of a cake among three or more players that is efficient.

Equitability Procedure (EP)

17Using 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.

Theorems 2 and 3

18Theorem 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.

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.

Conclusion

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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.