  The study of equilibrium concepts in congestion games and two-sided markets
with ties has been a primary topic in game theory, economics, and computer
science. Ackermann, Goldberg, Mirrokni, R\"oglin, V\"ocking (2008) gave a
common generalization of these two models, in which a player more prioritized
by a resource produces an infinite delay on less prioritized players. While
presenting several theorems on pure Nash equilibria in this model, Ackermann et
al.\ posed an open problem of how to design a model in which more prioritized
players produce a large but finite delay on less prioritized players. In this
paper, we present a positive solution to this open problem by combining the
model of Ackermann et al.\ with a generalized model of congestion games due to
Bil\`o and Vinci (2023). In the model of Bil\`o and Vinci, the more prioritized
players produce a finite delay on the less prioritized players, while the delay
functions are of a specific kind of affine function, and all resources have the
same priorities. By unifying these two models, we achieve a model in which the
delay functions may be finite and non-affine, and the priorities of the
resources may be distinct. We prove some positive results on the existence and
computability of pure Nash equilibria in our model, which extend those for the
previous models and support the validity of our model.
