  A systematic study is carried out regarding universally optimal designs under
the interference model, previously investigated by Kunert and Martin [Ann.
Statist. 28 (2000) 1728-1742] and Kunert and Mersmann [J. Statist. Plann.
Inference 141 (2011) 1623-1632]. Parallel results are also provided for the
undirectional interference model, where the left and right neighbor effects are
equal. It is further shown that the efficiency of any design under the latter
model is at least its efficiency under the former model. Designs universally
optimal for both models are also identified. Most importantly, this paper
provides Kushner'ss type linear equations system as a necessary and sufficient
condition for a design to be universally optimal. This result is novel for
models with at least two sets of treatment-related nuisance parameters, which
are left and right neighbor effects here. It sheds light on other models in
deriving asymmetric optimal or efficient designs.
