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On Stackelberg Strategies in Affine Congestion Games

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Abstract

We investigate the efficiency of some Stackelberg strategies in congestion games with affine latency functions. A Stackelberg strategy is an algorithm that chooses a subset of players and assigns them a prescribed strategy with the purpose of mitigating the detrimental effect that the selfish behavior of the remaining uncoordinated players may cause to the overall performance of the system. The efficiency of a Stackelberg strategy is measured in terms of the price of anarchy of the pure Nash equilibria they induce. Three Stackelberg strategies, namely Largest Latency First, Cover and Scale, were already considered in the literature and non-tight upper and lower bounds on their price of anarchy were given. We reconsider these strategies and provide the exact bound on the price of anarchy of both Largest Latency First and Cover and a better upper bound on the price of anarchy of Scale.

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Correspondence to Vittorio Bilò.

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This work was partially supported by the PRIN 2010–2011 research project ARS TechnoMedia: “Algorithmics for Social Technological Networks” funded by the Italian Ministry of University. An extended abstract of this paper appeared in the Proceedings of the 11th International Conference on Web and Internet Economics (WINE 2015) [6].

Appendix

Appendix

Proof of Lemma 1

(sketch). Given an affine congestion game CG, consider the linear congestion game CG whose set of resources R is defined as

$$R^{\prime}:=\{f_{0}(r),f_{1}(r),f_{2}(r),\ldots, f_{n}(r): r\in R\},$$

where f is a function mapping resource rR into n + 1 resources of R such that \(a_{f_{0}(r)}=a_{r}\) and \(a_{f_{i}(r)}=b_{r}\) for any iN, where arx + br (resp. \(\ell _{r^{\prime }}(x)=a_{r^{\prime }}x\)) is the latency function of resource rR (resp. rR). For each iN, let gi be the function mapping each strategy s ∈Σi of player i in game CG into strategy \(g_{i}(s):=\{f_{0}(r), f_{i}(r):r\in s\}\in {\Sigma }^{\prime }_{i}\) of player i in game CG. Given a strategy profile σ of CG, let g(σ) := (g1(σ1),g2(σ2),…,gn(σn)). It is easy to show that g biunivocally maps any strategy profile σ of CG into a strategy profile g(σ) of CG such that ci(σ) = ci(g(σ)) for every player iN. Thus CG and CG are isomorphic and have the same price of anarchy.

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Bilò, V., Vinci, C. On Stackelberg Strategies in Affine Congestion Games. Theory Comput Syst 63, 1228–1249 (2019). https://doi.org/10.1007/s00224-018-9902-1

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