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. /sets-and-relations/countable-set
2. /analysis/sup-inf
3. σ-algebra
4. Set function
5. Fair division
6. Envy-freeness
7. PROP1
8. Epistemic fairness
9. EF1
10. Maximin share of a set function
11. Maximin share allocations
12. Additive set function
13. Optimization: Dual and Lagrangian
14. Dual of a linear program
15. Linear programming: strong duality (incomplete)
16. AnyPrice share