Networks

Much recent interest has focused on applying game-theoretic analyses to problems of congestion defined on networks. For example, the arcs can be considered as resources, and users (Followers) wish to select optimal paths with respect to prices externally set (by a Leader).

This virtual issue contains papers that deal with such bilevel/Stackelberg games on networks, as well as those investigating cooperative network games and their associated cost allocation problems. Several papers deal with complexity and approximation results, in particular for welfare maximization in network congestion games.

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An approximation algorithm for Stackelberg network pricing

Sébastien Roch, Gilles Savard, Patrice Marcotte
From the abstract:  We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of , where mT denotes the number of toll arcs. Finally, we show that the approximation is tight with respect to a natural relaxation by constructing a family of instances for which the relaxation gap is reached.

Stackelberg network pricing is hard to approximate

Gwenaë Joret
From the abstract:  In the Stackelberg network pricing problem, one has to assign tariffs to a certain subset of the arcs of a given transportation network. The aim is to maximize the amount paid by the user of the network, knowing that the user will take a shortest st-path once the tariffs are fixed. (Roch et al., Networks, 46 (2005), 57–67) proved that this problem is NP-hard, and gave an O(log m)-approximation algorithm, where m denote the number of arcs to be priced. In this note, we show that the problem is also APX-hard.

Public congestion network situations and related games (2009)

John Kleppe, Marieke Quant, Hans Reijnierse
From the abstract:  This article analyses congestion in network situations from a cooperative game theoretic perspective. In network situations, players have to connect themselves to a source. As we consider publicly available networks, any group of players is allowed to use the entire network to establish their connections. We deal with the problem of finding an optimal network and discuss the associated cost allocation problem. For the latter, we introduce two different transferable utility cost games. For concave cost functions, we use the direct cost game, in which coalition costs are based on what a coalition can do in the absence of other players...

A note on Steiner tree games

Darko Skorin-Kapov, Jadranka Skorin-Kapov
From the abstract:  We investigate the cost allocation strategy associated with the problem of providing some service of common interest from some source to a number of network users, via the minimum cost directed Steiner tree (ST) network that spans the source and all the receivers. The cost of such ST is distributed among its receivers who may be individuals or organizations with possibly conflicting interests. The objective of this article is to develop a reasonably fair and efficient cost allocation scheme associated with the above cost allocation problem...

The complexity of welfare maximization in congestion game

Carol A. Meyers, Andreas S. Schulz
From the abstract:  We investigate issues of complexity related to welfare maximization in congestion games. In particular, we provide a full classification of complexity results for the problem of finding a minimum cost solution to a congestion game, under the model of Rosenthal. We consider both network and general congestion games, and we examine several variants of the problem concerning the structure of the game and the properties of its associated cost functions...

Computing approximate Nash equilibria in network congestion games

Andreas Emil Feldmann, Heiko Röglin, Berthold Vöcking
From the abstract:  We consider the problem of computing ε -approximate Nash equilibria in network congestion games. The general problem is known to be PLS-complete for every ε > 0, but the reductions are based on artificial and steep delay functions with the property that already two players using the same resource cause a delay that is significantly larger than the delay for a single player. We consider network congestion games with delay functions such as polynomials, exponential functions, and functions from queuing theory...