Ordinal Maximin Share Approximation for Goods

In fair division of indivisible goods, $\ell$-out-of-$d$ maximin share (MMS) is the value that an agent can guarantee by partitioning goods into $d$ bundles and choosing $\ell$ least preferred bundles. Most existing works aim to all agents a constant fraction their 1-out-of-$n$ MMS. But this sensitive small perturbation in agents' cardinal valuations. We consider more robust approximation notion, which depends only on \emph{ordinal} rankings prove existence $\ell$-out-of-$\lfloor(\ell+\frac{1}{2})n\rfloor$ MMS allocations for any integer $\ell\geq 1$, present polynomial-time algorithm finds $1$-out-of-$\lceil\frac{3n}{2}\rceil$ allocation when $\ell = 1$. further develop provides weaker ordinal >