Fair allocation of indivisible goods with minimum inequality or minimum envy criteria

For the classic problem of fair allocation of indivisible goods, we introduce the notion of minimum social inequality allocations and discuss its connection to other fair allocation rules such as minimum envy. We show that a fair allocation problem can always be cast as the problem of finding an optimal rearrangement of multiple matrices. Based on this reformulation, we provide two novel algorithms to find allocations with a minimum level of social inequality or with a minimum level of envy. Numerical illustrations show the efficiency of the newly developed algorithms, especially in problems with a large number of objects/agents.