We study a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the lexicographical improvement property (LIP) and show that, in finite games, it is equivalent to the existence of a generalized strong potential function. We use this characterization to derive existence, efficiency and fairness properties of strong equilibria (SE). As our main result, we show that an important class of games that we call bottleneck congestion games has the LIP and thus the above mentioned properties. For infinite games, the LIP does neither imply the existence of a generalized strong potential nor the existence of SE. We therefore introduce the slightly more general concept of the pairwise LIP and prove that whenever the pairwise LIP is satisfied for a continuous function, then there exists a SE. As a consequence, we show that splittable bottleneck congestion games with continuous facility cost functions possess a SE.
