  The quest for optimal/stable paths in graphs has gained attention in a few
practical or theoretical areas. To take part in this quest this chapter adopts
an equilibrium-oriented approach that is abstract and general: it works with
(quasi-arbitrary) arc-labelled digraphs, and it assumes very little about the
structure of the sought paths and the definition of equilibrium, \textit{i.e.}
optimality/stability. In this setting, this chapter presents a sufficient
condition for equilibrium existence for every graph; it also presents a
necessary condition for equilibrium existence for every graph. The necessary
condition does not imply the sufficient condition a priori. However, the
chapter pinpoints their logical difference and thus identifies what work
remains to be done. Moreover, the necessary and the sufficient conditions
coincide when the definition of optimality relates to a total order, which
provides a full-equivalence property. These results are applied to network
routing.
