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