PROP but not MEFS

Dependencies:

1. Fair division
2. Minimum fair share
3. Envy-freeness
4. Proportional allocation

$\newcommand{\defeq}{:=}$ Consider a fair division instance with 3 goods and 3 agents having equal entitlements and additive valuations. Valuations are given by this table:

$j$	1	2	3
$v$1( $j$ )	10	20	30
$v$2( $j$ )	20	10	30
$v$3( $j$ )	10	20	30

Then the allocation $(\{2\}, \{1\}, \{3\})$ is PROP, but no allocation is MEFS (every agent's minimum EF share is 30).

Consider a fair division instance with 3 chores and 3 agents having equal entitlements and additive valuations. Disutilities are given by this table:

$j$	1	2	3
$d$1( $j$ )	30	20	10
$d$2( $j$ )	20	30	10
$d$3( $j$ )	30	20	10

Then the allocation $(\{2\}, \{1\}, \{3\})$ is PROP, but no allocation is MEFS (every agent's minimum EF share is $-10$).

Dependency for: None

Info:

• Depth: 5
• Number of transitive dependencies: 7

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