Table Of ContentCake-Cutting Algorithms
Cake-Cutting Algorithms
Be Fair If You Can
Jack Robertson
William Webb
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~CRC Press
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Contents
Preface ix
1 Fairly Dividing a Cake 1
101 First Things 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
102 On to Three Persons - A False Start 5
103 On to Three Persons - Successfully 7
1.4 Envy Rears Its Ugly Head 11
1.5 EXERCISES 0 0 o o 0 0 o o 0 0 o o 0 14
2 Pieces or Crumbs - How Many Cuts Are Needed? 17
201 First Things - Some Agreements on Assumptions and Notation 17
202 One Cut Suffices 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22
203 On to the Count 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23
2.4 Using Fewer Cuts with the Divide and Conquer Algorithm 25
2.5 Two Cuts Are Never Enough 0 0 0 0 0 0 0 0 0 0 0 28
206 What is the Best We Can Do for Four Players? 0 29
207 Other Information on the Number of Cuts 0 31
2o8 EXERCISES 0 o 0 0 0 o 0 0 0 o 0 0 o 0 0 0 0 0 0 32
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3 Unequal Shares 35
3.1 First Things . . . . . . . . . . . . . . . . . . . . 35
3.2 A Better Solution Leading to Ramsey Partitions 36
3.3 Cut Near-Halves Algorithm for Unequal Shares . 42
3.4 More Than Two Players with Unequal Shares . 45
3.5 Unequal Shares in an Irrational Ratio 46
3.6 EXERCISES . . . . . . . . . . . . . . . . . . 48
4 The Serendipity of Disagreement 49
4.1 First Things . . . . . . 49
4.2 Some Other Examples 50
4.3 EXERCISES . . . . . 57
5 Some Variations on the Theme of ''Fair'' Division 61
5.1 Other Interpretations of "Fair" 61
5.2 Strong Fair Division 62
5.3 Envy-Free Division ..... . 65
5.4 Exact Division . . . . . . . . 68
5.5 Less is More-The Dirty Work Problem. 73
5.6 Existence Theorems .. 75
5.7 Classes of Algorithms 76
5.8 EXERCISES . . . . . 77
6 Some Combinatorial Observations 83
6.1 Applying Graph Theory to Fair Division 83
6.2 What Can Be Done with n Arbitrary Pieces? 88
6.3 EXERCISES . . . . . . . . . . . . . . . . . 91
7 Interlude: An Inventory of Results 93
7.1 Algorithms for Fair Division . . 93
7.2 Number of Cuts for Fair Division 95
8 Impossibility Theorems 99
8.1 Resetting the Stage ..................... . 99
8.2 More Than n- 1 Cuts Are Required for n Players, n 2: 3 . . 102
8.3 No Finite Algorithm Can Accomplish Exact Division . 103
8.4 PROJECTS ......................... . . 105
CONTENTS vii
9 Attempting Fair Division with a Limited Number of Cuts 107
9.1 The Problem . . . . . . . . . . . . . . . . . . . . . . 107
9.2 Simple Fair Division for a Small Number of Players . 108
9.3 What Can Be Done with k Cuts? . 114
9.4 Divide and Conquer Revisited . 120
9.5 EXERCISES . 122
9.6 PROJECTS . . . . . . . . . . . 123
10 Exact and Envy-Free Algorithms 125
10.1 Resetting the Stage . . 125
10.2 Near-Exact Division . . . . 126
10.3 Envy-Free Division . . . . . 128
10.4 Super Envy-Free Division . 136
10.5 EXERCISES . 138
10.6 PROJECTS . . . . . . . . . 138
11 A Return to Division for Unequal Shares 141
11.1 Resetting the Stage . . . . . . . . . . . 141
11.2 Ramsey Partitions and Fair Division . . . 141
11.3 Cut and Choose for Unequal Shares . . . 145
11.4 Envy-Free Division for Three Players with Unequal Shares . 149
11.5 EXERCISES . 154
11.6 PROJECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Solutions 155
References 173
Index 179
Preface
Why a Book on Cake-Cutting Algorithms?
Since the famous Polish school of mathematicians (Steinhaus, Banach, and
Knaster) introduced the problem of fair division and presented the first algo
rithms in the 1940's, the problem has blossomed and been widely popularized.
There are a large number of results scattered in research journals. Different
interpretations of "fair" have been proposed, and various classes of algorithms
employed. More recently, some impossibility and optimality results address the
question raised by Steinhaus at the outset: How many cuts are required for fair
division? Although serious mathematical attention to the problem is fairly re
cent, it is becoming widely exposed as results in the area are presented in popular
periodicals, elementary courses, and textbooks. In the numerous opportunities
we have had to discuss this material in a variety of settings, it has been our ex
perience that people quickly take up the challenge of searching for fair division
methods. It seems to us that the time has come to have the algorithms for fair
division collected in one readable, inclusive source.
We have attempted to give as comprehensive an answer as we could to the
question: What is known about cake-cutting algorithms using various definitions
of fair? The first six chapters provide a leisurely survey of the problem, chapter
7 is a quick reference which summarizes known results, while the last four
chapters contain technical proofs for previously introduced problems.
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