PMMS Allocations Need Not Exist for 3 Agents with Additive Valuations

This note gives an instance demonstrating that the pairwise maximin share (PMMS) property cannot be satisfied by any allocation in certain fair division problems with indivisible goods and additive valuations. The instance requires only $n=3$ agents and $m=9$ goods, and is accompanied by a proof that no PMMS allocation exists. A separate instance shows (certified by exhaustive enumeration with a computer) that PMMS cannot be approximated within a ratio above $78\mathrm{/}79\approx 0.987$.