Ordinal Maximin Share Approximation for Goods

In fair division of indivisible goods, l-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the l least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their one-out-of-n MMS. But this guarantee is sensitive to small perturbation in agents' cardinal valuations. We consider a more robust approximation notion, which depends only on the agents' ordinal rankings of bundles. We prove the existence of l-out-of-(l+1/2)n MMS allocations of goods for any integer l >= 1, and develop a polynomial-time algorithm that achieves this guarantee when l = 1.