Abstract
Every finite game can be represented as a weighted network congestion game on some undirected two-terminal network. The network topology may reflect certain properties of the game. This paper solves the topological equilibrium-existence problem of identifying all networks on which every weighted network congestion game has a pure-strategy equilibrium.
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Notes
The existence of representations of a similar kind was first indicated by Monderer (2007).
Renumbering effectively pairs each player \(i\) in \({\Gamma }\) with a specific player in \({\Gamma }^{{\prime }}\), namely, the one (re)assigned the same number \(i\). Therefore, it could alternatively be defined as a bijection between the two sets of players, that is, a one-to-one mapping from the player set in \({\Gamma }\) onto that in \({\Gamma }^{{\prime }}\).
Embedding in the wide sense, which was introduced in Milchtaich (2005), is more inclusive than the narrower notion of embedding used in Milchtaich (2006a). The difference is that, in the latter, the only kind of terminal subdivision (operation (c) above) allowed is terminal extension, in which all the edges originally incident with the terminal vertex become incident with the new vertex \(v\) instead. Whereas embedding in the wide sense roughly corresponds to the notion of a minor of a graph, embedding in the narrower sense corresponds to a topological minor (see Diestel 2005).
The assumption of a single origin–destination pair may be viewed as a normalization. Any weighted network congestion game on a multi-commodity network, which has multiple origin–destination pairs, may also be viewed as a game with a single such pair. In that game, each of the two terminal vertices is incident with a single allowable edge (see below) for each player, which joins it with the player’s corresponding terminal vertex in the original game.
The definition of cost function allows for negative costs, which may be interpreted as (net) gains from using the edge. However, negative costs do not play any role in Sect. 3, where all the results would hold also with the more restrictive definition that only allows nonnegative cost functions, \(c_e :\left( {0,\infty } \right) \rightarrow [0,\infty )\).
The cardinality assumption is used in the proof of Lemma 5. Whether or not it can be dispensed with I do not know. In one, important sense, the assumption is not overly restrictive. The proof of the representation theorem (Theorem 1 below) only uses network games that satisfy the assumption, which means that the theorem’s results hold with as well as without it (Milchtaich 2014). Note that the cardinality assumption trivially holds if all players have the same number of strategies, or if the allowable users of each edge are those whose weight does not exceed a certain threshold.
This condition can be expressed as the requirement that \({\Gamma }^{{\prime }} = {\Gamma }^{{\prime }{\prime }}\).
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Acknowledgments
I thank a referee and an Associate Editor for their comments. This research was supported by the Israel Science Foundation (Grants No. 1082/06 and 1167/12).
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Milchtaich, I. Network topology and equilibrium existence in weighted network congestion games. Int J Game Theory 44, 515–541 (2015). https://doi.org/10.1007/s00182-014-0443-9
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DOI: https://doi.org/10.1007/s00182-014-0443-9