Strong equilibria in games with the lexicographical improvement property

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 abovementioned 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. An extended abstract of this paper appeared in the Proceedings of the 3rd Workshop on Internet and Networks Economics. Research supported by the Federal Ministry of Education and Research (BMBF grant 03MOPAI1) and by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408). T. Harks School of Business and Economics, Maastricht University, Tongersestraat 53, 6211 LM Maastricht, The Netherlands e-mail: [email protected] M. Klimm (B) · R. H. Möhring Institut fürMathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail: [email protected] R. H. Möhring e-mail: [email protected]