In The Geometry of Efficient Fair Division, Julius B. Barbanel, addresses the question "Can a cake be cut so that every diner is satisfied?" The "cake" is a measure space, and each diner judges the division of the cake using their own measure. Barbanel focuses of the existence of divisions with certain properties. In particular, he considers fairness properties (Does each diner believe she received a piece at least as big as the others?) and efficiency properties (Is there a way to divide the cake so that at least one diner would be happier?) in the two-diner and the more complicated n-diner settings. The author does not present algorithms for constructing divisions.
The monograph is a clearly-written, matter-of-fact presentation of definitions, theorems, and proofs. There are no exercises, though there are some statements of unsolved problems. It is aimed at researchers and graduate students in the area, though an economist or other researcher needing these results should find it accessible as long as they have a background in basic real analysis.
Stephen Ahearn teaches at Macalester College.
0. Preface; 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 efficiency in the two-player context; 4. The general case of n players; 5. What the IPS and the FIPS tell us about fairness and efficiency in the n-player context; 6. Characterizing Pareto optimality: introduction and preliminary ideas; 7. Characterizing Pareto optimality I: the IPS and optimization of convex combinations of measures; 8. Characterizing Pareto optimality II: partition ratios; 9. Geometric object #2: The Radon-Nikodym set (RNS); 10. Characterizing Pareto optimality III: the RNS, Weller's construction, and w-association; 11. The shape of the IPS; 12. The relationship between the IPS and the RNS; 13. Other issues involving Weller's construction, partition ratios, and Pareto optimality; 14. Strong Pareto optimality; 15. Characterizing Pareto optimality using hyperreal numbers; 16. The multi-cake individual pieces set (MIPS): symmetry restored.