APS implies pessShare

Dependencies:

1. Fair division
2. AnyPrice share
3. Maximin share allocations
4. Bound on k extreme values

$\newcommand{\defeq}{:=}$ $\newcommand{\APS}{\mathrm{APS}}$ $\newcommand{\pessShare}{\mathrm{pessShare}}$ For any fair division instance of indivisible items and any agent $i$, we have $\APS_i ≥ \pessShare_i$. Hence, if $i$ is APS-satisfied, then she is also pessShare-satisfied.

Proof

Proposition 2 of https://doi.org/10.1287/moor.2021.0199. (Although this part of the paper only considers the goods case, the proof works for mixed manna too.)

Dependency for:

1. APS and MMS don't imply PROPm
2. Share vs envy for identical valuations (goods)
3. Share vs envy for identical valuations (chores)
4. APS = MMS for n=2

Info:

• Depth: 5
• Number of transitive dependencies: 12

Transitive dependencies:

1. /analysis/sup-inf
2. /sets-and-relations/countable-set
3. Bound on k extreme values
4. Optimization: Dual and Lagrangian
5. Dual of a linear program
6. Linear programming: strong duality (incomplete)
7. σ-algebra
8. Set function
9. Fair division
10. AnyPrice share
11. Maximin share of a set function
12. Maximin share allocations