## The Geometry of Efficient Fair DivisionWhat is the best way to divide a 'cake' and allocate the pieces among some finite collection of players? In this book, the cake is a measure space, and each player uses a countably additive, non-atomic probability measure to evaluate the size of the pieces of cake, with different players generally using different measures. The author investigates efficiency properties (is there another partition that would make everyone at least as happy, and would make at least one player happier, than the present partition?) and fairness properties (do all players think that their piece is at least as large as every other player's piece?). He focuses exclusively on abstract existence results rather than algorithms, and on the geometric objects that arise naturally in this context. By examining the shape of these objects and the relationship between them, he demonstrates results concerning the existence of efficient and fair partitions. |

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### Contents

1 | |

7 | |

The Individual Pieces Set IPS | 16 |

What the IPS Tells Us About Fairness and Efficiency in | 25 |

The Individual Pieces Set IPS and the Full Individual | 56 |

What the IPS and the FIPS Tell Us About Fairness | 82 |

Introduction | 151 |

The IPS | 160 |

The RNS Wellers | 236 |

The Shape of the IPS | 286 |

The Relationship Between the IPS and the RNS | 298 |

Other Issues Involving Wellers Construction Partition | 352 |

Strong Pareto Optimality | 385 |

Characterizing Pareto Optimality Using Hyperreal | 416 |

The Multicake Individual Pieces | 444 |

451 | |

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### Common terms and phrases

absolutely continuous assume c-proportional Chapter characterization chores claim closed combination of measures concentrate condition consider consists contains context continuous with respect contradiction convex combination coordinates Corollary corresponding define Definition direction discussed distinct envy-free equal equivalent establishes example exists fact family of parallel fi(a Figure FIPS follows function give given Hence holds idea illustrated implies infinitely infinitely many mutually inner interior involves least Lemma line segment mi(A minimal non-s-equivalent notion outer boundary outer Pareto boundary p-class p-equivalent Pareto bigger Pareto maximal partition Pareto maximal points partition ratios piece of cake Player players named positive measure possible present proof proof of Theorem proportional and Pareto recall region result satisfy simplex situation statement strongly Pareto maximal strongly proportional subset Suppose tells Theorem transfer w-associated zero