MXS doesn't imply PROPx or EFX

Dependencies:

1. Fair division
2. PROPx
3. EFX
4. Minimum fair share

$\newcommand{\defeq}{:=}$ Let $t \in \{-1, 1\}$. Consider a fair division instance with 2 agents having equal entitlements and identical additive valuations. Let there be 2 items of value $4t$ and 5 items of value $t$. Then the allocation $A = (\{4t, t\}, \{4t, t, t, t, t\})$ is not PROPx or EFX, but it is MXS because the agents have $(\{t, t, t, t, t\}, \{4t, 4t\})$ and $(\{4t, 4t\}, \{t, t, t, t, t\})$ as their certificates for $A$.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 10

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra
3. Set function
4. Fair division
5. Proportional allocation
6. Envy-freeness
7. Minimum fair share
8. Submodular function
9. PROPx
10. EFX