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We study the efficient computation of Nash and strong equilibria in weighted bottleneck games. In such a game different players interact on a set of resources in the way that every player chooses a subset of the resources as her strategy. The cost of a single resource depends on the total weight of players choosing it and the personal cost every player tries to minimize is the cost of the most expensive resource in her strategy, the bottleneck value. To derive efficient algorithms for finding Nash equilibria in these games, we generalize a tranformation of a bottleneck game into a special congestion game introduced by Caragiannis et al. [1]. While investigating the transformation we introduce so-called lexicographic games, in which the aim of a player is not only to minimize her bottleneck value but to lexicographically minimize the ordered vector of costs of all resources in her strategy. For the special case of network bottleneck games, i.e., the set of resources are the edges of a graph and the strategies are paths, we analyse different Greedy type methods and their limitations for extension-parallel and series-parallel graphs.

We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it depends on the graph topology and the number of users. In our context best and worst equilibria are those with minimum respectively maximum total latency. We establish that both problems can be solved by a Greedy algorithm with a suitable tie breaking rule on parallel links. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. Additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.

Selfish Bin Coloring
(2009)

We introduce a new game, the so-called bin coloring game, in which selfish players control colored items and each player aims at packing its item into a bin with as few different colors as possible. We establish the existence of Nash and strong as well as weakly and strictly Pareto optimal equilibria in these games in the cases of capacitated and uncapacitated bins. For both kinds of games we determine the prices of anarchy and stability concerning those four equilibrium concepts. Furthermore, we show that extreme Nash equilibria, those with minimal or maximal number of colors in a bin, can be found in time polynomial in the number of items for the uncapcitated case.