Improved maximin fair allocation of indivisible items to three agents

We consider the problem of approximate maximin share (MMS) allocation of indivisible items among three agents with additive valuation functions. For goods, we show that an frac1112 - MMS allocation always exists, improving over the previously known bound of frac89 . Moreover, in our allocation, we can prespecify an agent that is to receive her full proportional share (PS); we also present examples showing that for such allocations the ratio of frac1112 is best possible. For chores, we show that a frac1918-MMS allocation always exists. Also in this case, we can prespecify an agent that is to receive no more than her PS, and we present examples showing that for such allocations the ratio of frac1918 is best possible.