WMMS implies EFX for two agents

Dependencies:

1. Fair division
2. Maximin share allocations
3. EFX
4. WMMS-sat and envious of everyone implies EFX-sat

$\newcommand{\defeq}{:=}$ $\newcommand{\WMMS}{\mathrm{WMMS}}$ Consider a fair division instance $([2], [m], V, w)$ of indivisible items (where each agent $i$ has entitlement $w_i$). Suppose an allocation $A$ is WMMS-fair to agent 1. Then $A$ is also EFX-fair to agent 1 if one of these conditions hold:

1. The items are goods to agent 1.
2. $v_1$ is additive and $w_1 ≤ w_2$.

Proof

If agent 1 doesn't envy agent 2, she is EFX-satisfied. Otherwise, she is EFX-satisfied because of mms-and-all-envy.html.

Dependency for:

1. MMS implies MXS

Info:

• Depth: 7
• Number of transitive dependencies: 11

Transitive dependencies:

1. /sets-and-relations/countable-set
2. /analysis/sup-inf
3. σ-algebra
4. Set function
5. Fair division
6. Envy-freeness
7. Maximin share of a set function
8. Maximin share allocations
9. Submodular function
10. EFX
11. WMMS-sat and envious of everyone implies EFX-sat