THE GEOMETRY OF EFFICIENT FAIR DIVISION

By CHARLOTTE JONES

CONTENTS 1 NOTATION AND PRELIMINARIES 2 GEOMETRIC OBJECT #1A: THE INDIVIDUAL PIECES SET (IPS) FOR TWO PLAYERS 3 WHAT THE IPS TELLS US ABOUT FAIRNESS AND EF?CIENCY IN THE TWO-PLAYER CONTEXT 3A FAIRNESS 3B EF?CIENCY 3C FAIRNESS AND EF?CIENCY TOGETHER: PART 1A 3D THE SITUATION WITHOUT ABSOLUTE CONTINUITY 4 THE INDIVIDUAL PIECES SET (IPS) AND THE FULL INDIVIDUAL PIECES SET (FIPS) FOR THE GENERAL N-PLAYER CONTEXT 4A GEOMETRIC OBJECT #1B: THE IPS FOR N PLAYERS 4B WHY THE IPS DOES NOT SUF?CE 4C GEOMETRIC OBJECT #1C: THE FIPS 4D A THEOREM ON THE POSSIBILITIES FOR THE FIPS 5 WHAT THE IPS AND THE FIPS TELL US ABOUT FAIRNESS AND EF?CIENCY IN THE GENERAL N-PLAYER CONTEXT 5A FAIRNESS 5B EF?CIENCY 5C FAIRNESS AND EF?CIENCY TOGETHER: PART 1B 5D THE SITUATION WITHOUT ABSOLUTE CONTINUITY 5E EXAMPLES AND OPEN QUESTIONS 6 CHARACTERIZING PARETO OPTIMALITY: INTRODUCTION AND PRELIMINARY IDEAS 7 CHARACTERIZING PARETO OPTIMALITY I: THE IPS AND OPTIMIZATION OF CONVEX COMBINATIONS OF MEASURES INTRODUCTION CONTENTS 7A INTRODUCTION: THE TWO-PLAYER CONTEXT 7B THE CHARACTERIZATION 7C THE SITUATION WITHOUT ABSOLUTE CONTINUITY 8 CHARACTERIZING PARETO OPTIMALITY II: PARTITION RATIOS 8A INTRODUCTION: THE TWO-PLAYER CONTEXT 8B THE CHARACTERIZATION 8C THE SITUATION WITHOUT ABSOLUTE CONTINUITY 9 GEOMETRIC OBJECT #2: THE RADON–NIKODYM SET (RNS) 9A THE RNS 9B THE SITUATION WITHOUT ABSOLUTE CONTINUITY 10 CHARACTERIZING PARETO OPTIMALITY III: THE RNS, WELLER’S CONSTRUCTION, AND W-ASSOCIATION 10A INTRODUCTION: THE TWO-PLAYER CONTEXT 10B THE CHARACTERIZATION 10C THE SITUATION WITHOUT ABSOLUTE CONTINUITY 11 THE SHAPE OF THE IPS 11A THE TWO-PLAYER CONTEXT 11B THE CASE OF THREE OR MORE PLAYERS 12 THE RELATIONSHIP BETWEEN THE IPS AND THE RNS 12A INTRODUCTION 12B RELATING THE IPS AND THE RNS IN THE TWO-PLAYER CONTEXT 12C RELATING THE IPS AND THE RNS IN THE GENERAL N-PLAYER CONTEXT 12D THE SITUATION WITHOUT ABSOLUTE CONTINUITY 12E FAIRNESS AND EF?CIENCY TOGETHER: PART 13 OTHER ISSUES INVOLVING WELLER’S CONSTRUCTION, PARTITION RATIOS, AND PARETO OPTIMALITY 13A THE RELATIONSHIP BETWEEN PARTITION RATIOS AND W-ASSOCIATION 13B TRADES AND EF?CIENCY 13C CLASSIFYING THE FAILURE OF PARETO OPTIMALITY 13D CONVEXITY 13E THE SITUATION WITHOUT ABSOLUTE CONTINUITY 14 STRONG PARETO OPTIMALITY 14A INTRODUCTION 14B THE CHARACTERIZATION 14C EXISTENCE QUESTIONS IN THE TWO-PLAYER CONTEXT 14D EXISTENCE QUESTIONS IN THE GENERAL N-PLAYER CONTEXT 14E THE SITUATION WITHOUT ABSOLUTE CONTINUITY 14F FAIRNESS AND EF?CIENCY TOGETHER: PART CONTENTS 15 CHARACTERIZING PARETO OPTIMALITY USING HYPERREAL NUMBERS 15A INTRODUCTION 15B A TWO-PLAYER EXAMPLE 15C THREE-PLAYER EXAMPLES 15D THE CHARACTERIZATION 16 GEOMETRIC OBJECT #1D: THE MULTICAKE INDIVIDUAL PIECES SET (MIPS) SYMMETRY RESTORED 16A THE MIPS FOR THREE PLAYERS 16B THE MIPS FOR THE GENERAL N-PLAYER CONTEXT