PROP implies EF for superadditive identical valuations

Dependencies:

1. Fair division
2. Subadditive and superadditive set functions

In fair division with identical superadditive valuations, a PROP allocation is also an EF allocation. This is true even for unequal entitlements.

Proof

Let $A$ be a PROP allocation. Then for each agent $i$, we have $v(A_i) ≥ w_iv(M)$. Suppose $v(A_k) > w_kv(M)$ for some agent $k$. Sum these inequalities to get $\sum_{i=1}^n v(A_i) > v(M)$. This contradicts superadditivity of $v$, so $v(A_i) = w_iv(M)$ for each agent $i$. Hence, $v(A_i)/w_i = v(M)$ for all $i$, so $A$ is EF.

Dependency for: None

Info:

• Depth: 4
• Number of transitive dependencies: 5

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra
3. Set function
4. Fair division
5. Subadditive and superadditive set functions