PROP implies EF for n=2

Dependencies:

1. Fair division
2. Proportional allocation
3. Envy-freeness
4. Subadditive and superadditive set functions

$\newcommand{\defeq}{:=}$ Let $([2], M, V, w)$ be a fair division instance (each agent $i$ has entitlement $w_i$). If agent 1 is PROP-satisfied by allocation $A$, then she is envy-free in $A$ if $v_1$ is superadditive.

Proof

\begin{align} & \frac{v_1(A_1)}{w_1} ≥ v_1([m]) ≥ v_1(A_1) + v_1(A_2) \\ &\implies v_1(A_2) ≤ v_1(A_1)\left(\frac{1}{w_1} - 1\right) = w_2 \frac{v_1(A_1)}{w_1}. \end{align}

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. Subadditive and superadditive set functions