Equitability (EQ) in fair division requires that items be allocated such that all agents value the bundle they receive equally. With indivisible items, an equitable allocation may not exist, and hence we instead consider a meaningful analog, EQx, that requires equitability up to any item. EQx allocations exist for monotone, additive valuations. However, if (1) the agents' valuations are not additive or (2) the set of indivisible items includes both goods and chores (positively and negatively valued items), then before the current work it was not known whether EQx allocations exist or not.

We study both the existence and efficient computation of EQx allocations. (1) For monotone valuations (not necessarily additive), we show that EQx allocations always exist. Also, for the large class of weakly well-layered valuations, EQx allocations can be found in polynomial time. Further, we prove that approximately EQx allocations can be computed efficiently under general monotone valuations. (2) For non-monotone valuations, we show that an EQx allocation may not exist, even for two agents with additive valuations. Under some special cases, however, we show the existence and efficient computability of EQx allocations. This includes the case of two agents with additive valuations where each item is either a good or a chore, and there are no mixed items.

The focus of this talk will be on the results obtained under monotone valuations. This is joint work with Siddharth Barman(IISC, Bangalore), Umang Bhaskar(TIFR, Mumbai) and Soumyajit Pyne(TIFR, Mumbai).