M1S doesn't imply PROP1

Dependencies:

1. Fair division
2. PROP1
3. EF1
4. Minimum fair share
5. Additive set function

$\newcommand{\defeq}{:=}$ Let $t \in \{-1, 1\}$. Consider a fair division instance with 2 agents having equal entitlements and identical additive valuations. There are 9 items, and $v(9) = 4t$ and $v(j) = t$ for $j \in [8]$. Let $A$ be an allocation where $A_1 = \{9\}$. Then for all $t \in \{-1, 1\}$, $A$ is M1S but not PROP1.

Proof

$v([m])/2 = 6t$. Let $B = ([4], [9] \setminus [4])$.

For $t = 1$ (goods), $B$ is agent 1's M1S-certificate for $A$, but agent 1 is not PROP1-satisfied by $A$.

For $t = -1$ (chores), $B$ is agent 2's M1S-certificate for $A$, but agent 2 is not PROP1-satisfied by $A$.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 9

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra
3. Set function
4. Fair division
5. Envy-freeness
6. PROP1
7. Minimum fair share
8. EF1
9. Additive set function