Ordinal Maximin Share Approximation for Goods

In fair division of indivisible goods, `-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the ` least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their 1-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 `-out-of-b(` + ¹₂ )nc MMS allocations of goods for any integer ` ≥ 1, and present a polynomial-time algorithm that finds a 1-out-of-d<sup>3</sup><sub>2</sub><sup>n</sup> e MMS allocation when ` = 1. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any ` > 1.