MEFS+PROP doesn't imply EEF1 for chores

Dependencies:

1. Fair division
2. Proportional allocation
3. Envy-freeness
4. EF1
5. Epistemic fairness
6. Minimum fair share

$\newcommand{\defeq}{:=}$ $\newcommand{\PROP}{\mathrm{PROP}}$ $\newcommand{\MEFS}{\mathrm{MEFS}}$ Consider the problem of fairly dividing 12 additive bivalued chores among 3 agents. Agent 1 has disutility 7 for each of the first 3 chores, and disutility 1 for each remaining chore. Agents 2 and 3 have disuitlity 1 for each chore. Then $A \defeq ([12] \setminus [3], [2], \{3\})$ is a MEFS+PROP allocation where agent 1 is not epistemic-EF1-satisfied.

Proof

$\PROP_1 = -10$ and $\PROP_2 = \PROP_3 = -4$. $\MEFS_1 ≤ -10$ because of the allocation $(\{1, 4, 5, 6\}, \{2, 7, 8, 9\}, \{3, 10, 11, 12\})$. $\MEFS_i ≤ -4$ for $i \in \{2, 3\}$ because of the allocation $([4], [8] \setminus [4], [12] \setminus [8])$. Agent 1 has disutility $9$ in $A$, so $A$ is MEFS-fair and PROP-fair to agent 1. Agents 2 and 3 have disutility at most 2 in $A$, so $A$ is MEFS-fair and PROP-fair to them.

Agent 1 is not epistemic-EF1-satisfied by $A$, since in any epistemic-EF1-certificate $B$, some agent $j \in \{2, 3\}$ receives at most one chore of value 7, and agent 1 would EF1-envy $j$.

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. Proportional allocation
6. Envy-freeness
7. Epistemic fairness
8. Minimum fair share
9. EF1