GMMS doesn't imply APS

Dependencies:

1. Fair division
2. Additive set function
3. AnyPrice share
4. Maximin share allocations
5. Restricted agents, pairwise fairness, and groupwise fairness
6. Leximin implies GMMS for idval
7. APS can be > MMS

$\newcommand{\defeq}{:=}$ Consider a fair division instance with 15 goods and 3 agents having equal entitlements and identical additive valuations. The values of the goods are 65, 31, 31, 31, 23, 23, 23, 17, 11, 7, 7, 7, 5, 5, 5. Then a GMMS allocation exists but an APS alloction doesn't exist.

Even for the instance of chores obtained by multiplying every good's value by $-1$, a GMMS allocation exists but an APS alloction doesn't exist.

Proof

For these instances, the leximin allocation is GMMS. However, APS $>$ MMS for both these instances, and the minimum value across all bundles is at most the MMS, so no APS allocation exists.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 17

Transitive dependencies:

1. /sets-and-relations/countable-set
2. /analysis/sup-inf
3. σ-algebra
4. Set function
5. Fair division
6. Proportional allocation
7. Restricted agents, pairwise fairness, and groupwise fairness
8. Maximin share of a set function
9. Maximin share allocations
10. Additive set function
11. Leximin partition of a set function
12. Leximin implies GMMS for idval
13. Optimization: Dual and Lagrangian
14. Dual of a linear program
15. Linear programming: strong duality (incomplete)
16. AnyPrice share
17. APS can be > MMS