Share vs envy for identical valuations (goods)

Dependencies:

1. Fair division
2. AnyPrice share
3. Maximin share allocations
4. PROPx
5. EF1
6. APS implies pessShare
7. PROP implies APS

$\newcommand{\MMS}{\mathrm{MMS}}$ $\newcommand{\APS}{\mathrm{APS}}$ Consider a fair division instance having $n ≥ 3$ agents having equal entitlements and identical additive valuations, and $m = 2n-1$ goods, each of value 1. Let $A$ be an allocation where agent $n$ gets $n$ goods, and all other agents get 1 good each. Then this allocation is APS, MMS, and PROPx, but not EF1.

Proof

$1 = \MMS_i ≤ \APS_i ≤ v([m])/n = 2 - 1/n$. Hence, $APS_i = 1$, since the APS is the value of some bundle. Hence, $A$ is APS, MMS, and PROPx.

$A$ is not EF1 because the first $n-1$ agents EF1-envy agent $n$.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 20

Transitive dependencies:

1. /analysis/sup-inf
2. /sets-and-relations/countable-set
3. Bound on k extreme values
4. Optimization: Dual and Lagrangian
5. Dual of a linear program
6. Linear programming: strong duality (incomplete)
7. σ-algebra
8. Set function
9. Fair division
10. AnyPrice share
11. Proportional allocation
12. Envy-freeness
13. EF1
14. Additive set function
15. PROP implies APS
16. Maximin share of a set function
17. Maximin share allocations
18. APS implies pessShare
19. Submodular function
20. PROPx