MMS+APS doesn't imply PROP1 for chores

Dependencies:

1. Fair division
2. Maximin share allocations
3. AnyPrice share
4. EF1
5. Epistemic fairness
6. PROP1
7. Additive set function

$\newcommand{\defeq}{:=}$ Consider a fair division instance with 3 agents with identical additive disutilities and equal entitlements, and one chore of disutility 18 (large chore) and 5 chores of disutility 3 each (small chores). Let $X$ be an allocation where agent 1 gets all the small chores and agent 2 gets the large chore. Then $X$ is an MMS and APS allocation, but it is not EEF1-fair or PROP1-fair to agent 1.

Proof

In every allocation, some agent gets the large chore, so the MMS is $-18$. The APS is also $-18$: set the payment (i.e., negative of price) of the large chore to 1 and the payment of small chores to 0.

The proportional share is $-11$, and agent 1's disutility in $X$ after removing any chore is $12$, so $X$ is not PROP1-fair to agent 1.

$X$ is not EEF1-fair to agent 1 because even after redistributing chores among the remaining agents, someone will always have no chores.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 16

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. PROP1
10. Epistemic fairness
11. AnyPrice share
12. Envy-freeness
13. EF1
14. Additive set function
15. Maximin share of a set function
16. Maximin share allocations