PROPx doesn't exist even if APS does

Dependencies:

1. Fair division
2. PROPx
3. AnyPrice share
4. Restricted agents, pairwise fairness, and groupwise fairness

$\newcommand{\defeq}{:=}$ Consider a fair division instance with 3 agents with equal entitlements and identical additive valuations. There are 2 goods of values 5 and 1. Then no PROPx allocation exists.

Let $A$ be the allocation where the first agent gets the good of value 5, and the second agent gets the good of value 1. Then $A$ is a groupwise APS allocation.

Proof

In every allocation, some agent doesn't get any good. That agent is not PROPx-satisfied.

Set the price of the goods to $1.1$ and $0.9$. This shows that $A$ is groupwise APS.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 12

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra
3. Set function
4. Fair division
5. Proportional allocation
6. Restricted agents, pairwise fairness, and groupwise fairness
7. Submodular function
8. PROPx
9. Optimization: Dual and Lagrangian
10. Dual of a linear program
11. Linear programming: strong duality (incomplete)
12. AnyPrice share