Abstract
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.
Highlights
The theory of non-cooperative games studies situations that involve rational and selfish agents who are motivated by optimizing their own utilities rather than reaching some social optimum
It is well known that mixed or correlated strategies have no meaningful physical interpretation for many strategic games arising in practice; see the discussion by Osborne and Rubinstein (1994, § 3.2) about critics on mixed Nash equilibria
We prove that continuity of φ in the definition of the pairwise lexicographical improvement property (LIP)
Summary
The theory of non-cooperative games studies situations that involve rational and selfish agents who are motivated by optimizing their own utilities rather than reaching some social optimum. One defines a real-valued function P on the set of strategy profiles of the game and shows that every improving move of a single player strictly reduces the value of P. Referring to Banner and Orda (2007), Cole et al (2006), Keshav (1997) and Qiu et al (2006), the throughput of a stream of packets in a communication network is usually determined by the available bandwidth or the capacity of the weakest links This aspect is captured more realistically by bottleneck congestion games in which the individual cost of a player is the maximum (instead of the sum) of the delays in her strategy. These games constitute a more realistic model for network routing than classical congestion games, they have not received similar attention in the literature
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