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Characterizing the Existence of Potential Functions in Weighted Congestion Games

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Abstract

Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let \(\mathcal{C}\) be an arbitrary set of locally bounded functions and let \(\mathcal{G}(\mathcal{C})\) be the set of weighted congestion games with cost functions in \(\mathcal{C}\). We show that every weighted congestion game \(G\in\mathcal{G}(\mathcal{C})\) admits an exact potential if and only if \(\mathcal{C}\) contains only affine functions. We also give a similar characterization for w-potentials with the difference that here \(\mathcal{C}\) consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facility-dependent demands and elastic demands, respectively.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Tobias Harks, Max Klimm & Rolf H. Möhring

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  1. Tobias Harks
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  2. Max Klimm
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  3. Rolf H. Möhring
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Corresponding author

Correspondence to Tobias Harks.

Additional information

An extended abstract of this paper appeared in the Proceedings of the Second International Symposium on Algorithmic Game Theory (SAGT), 2009.

The research of M. Klimm was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408).

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Harks, T., Klimm, M. & Möhring, R.H. Characterizing the Existence of Potential Functions in Weighted Congestion Games. Theory Comput Syst 49, 46–70 (2011). https://doi.org/10.1007/s00224-011-9315-x

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