Nobody Left Behind : Fair Allocation of Indivisible Goods

The Max-Min Fairness problem is as follows: Given m indivisible goods and k players, each with a specified valuation function on the subsets of the goods, how should the goods be split between the players so as to maximize the minimum valuation. Viewing the problem from a game theoretic perspective, we show that for two players and additive valuations the expected minimum of the (randomized) cut-and-choose mechanism is a 1/2-approximation of the optimum. To complement this result we show that no truthful mechanism can compute the exact optimum. We also consider the algorithmic perspective when the (true) additive valuation functions are part of the input. We present a simple 1/(m− k + 1) approximation algorithm which allocates to every player at least 1/k fraction of the value of all but the k−1 heaviest items. We also give an algorithm with additive error against the fractional optimum bounded by the value of the largest item. The two approximation algorithms are incomparable in the sense that there exist instances when one outperforms the other. We conclude with a 1/2 + ε factor NP-hardness of approximation result.