  One of the most important topics in discrete fair division is whether an EFX
allocation exists for any instance. Although the existence of EFX allocations
is a standing open problem for both goods and chores, the understanding of the
existence of EFX allocations for chores is less established compared to goods.
We study the existence of EFX allocation for chores under the assumption that
all agent's cost functions are additive. Specifically, we show the existence of
EFX allocations for the following three cases: (i) the number of chores is at
most twice the number of agents, (ii) the cost functions of all agents except
for one are identical ordering, and (iii) the number of agents is three and
each agent has a personalized bi-valued cost function. Furthermore, we provide
a polynomial time algorithm to find an EFX allocation for each case.

Computer Science and Game Theory