MMS implies EFX for two agents (goods)

Dependencies:

1. Fair division
2. /fair-division/notions/ef1x-goods
3. Maximin share allocations

In fair division of indivisible goods with two agents, if an agent is MMS-satisfied and has positive marginal value for every good, then she is also EFX-satisfied.

Proof

Let $(P_1, P_2)$ be the agent 1's MMS partition, where $v_1(P_1) ≤ v_1(P_2)$. Assume that in allocation $(A_1, A_2)$, agent 1 is MMS-satisfied but not EFX-satisfied. Then $v_1(P_1) ≤ v_1(A_1) < v_1(A_2 \setminus \{g\})$ for some $g \in A_2$.

Let $B = (A_1 \cup \{g\}, A_2 \setminus \{g\})$. Since agent 1 has positive marginal value for every good, we get $v_1(B_1) > v_1(A_1) ≥ v_1(P_1)$ and $v_1(B_2) = v_1(A_2 \setminus \{g\}) > v_1(P_1)$. Hence, in $B$, both bundles have value more than $v_1(P_1)$. This contradicts the fact that $P$ is an MMS partition for agent 1. Hence, it's not possible for agent 1 to be MMS-satisfied but not EFX-satisfied.

Dependency for: None

Info:

• Depth: 5
• Number of transitive dependencies: 8

Transitive dependencies:

1. /sets-and-relations/countable-set
2. /fair-division/notions/ef1x-goods
3. /analysis/sup-inf
4. σ-algebra
5. Set function
6. Fair division
7. Maximin share of a set function
8. Maximin share allocations