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