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Collusion in Atomic Splittable Routing Games

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Abstract

We study how collusion affects the social cost in atomic splittable routing games. Suppose that players form coalitions and each coalition behaves as if it were a single player controlling all the flows of its participants. We investigate the following question: under what conditions would the social cost of the post-collusion equilibrium be bounded by the social cost of the pre-collusion equilibrium?

We show that if (i) the network is “well-designed” (satisfying a natural condition), and (ii) the delay functions are affine, then collusion is always beneficial for the social cost in the equilibrium flows. On the other hand, if either of the above conditions is unsatisfied, collusion can worsen the social cost.

Our main technique is a novel flow-augmenting algorithm to build equilibrium flows. Our positive result for collusion is obtained by applying this algorithm simultaneously to two different flow value profiles of players and observing the difference in the derivatives of their social costs. Moreover, for a non-trivial subclass of selfish routing games, this algorithm finds the exact equilibrium flows in polynomial time.

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Notes

  1. More precisely, the examples in [12, 16] show that the social cost in the post-collusion equilibrium can be higher than in the pre-collusion Wardrop equilibrium. See the discussion in the related work for details.

  2. Even though the result for affine delays as stated in [9] is only for series-parallel networks, the technique of [9] indeed also works for well-designed networks. This is because a well-designed network always satisfies the nesting property (see Sect. 3.3 for definition). A minor contribution of this paper is to identify a larger class of networks for this result of [9] to hold.

  3. The proof proceeds as follows. Let σ′ be the profile of k players, each with the same amount of flow. Let σ be the profile of the post-collusion equilibrium flow. Observe that σ dominates σ′. It is established in [16] that the social cost of the equilibrium flow for σ′ has social cost no larger than the pre-collusion Wardrop equilibrium. Now applying Lemma 5 to compare social costs of the equilibria flows for σ′ and for σ would give the proof.

  4. A Wardrop equilibrium f has the following characterization [40, 48]: if p and q are directed paths between the same pair of vertices and f uses p, ∑ e∈p l e (f e )≤∑ e∈q l e (f e ).

  5. This follows from the fact that in an equilibrium flow of symmetric players, the flows of all players are identical [16].

  6. To simplify our presentation, we slightly abuse notation by writing a value that goes to infinity as a fixed value t x .

  7. Since we increase the flow by an infinitesimal amount ϵ, this algorithm, along with the main algorithm in the next section, do not run in polynomial time. But they can be easily modified to run in polynomial time. We choose to present in this manner since we consider it as simpler and it makes the analysis in Sect. 3.4 go smoother.

  8. To be precise, C(σ,t) is not differentiable in the breakpoints t i (σ). Hence the domain of \(\frac{d C(\sigma,t)}{d t}\) is [0,1]∖{t i (σ)}∀i . For our purpose, it suffices to consider the open intervals between these breakpoints, because the functions C(σ,t) and C(σ′,t) are both continuous.

  9. Recall our earlier remark at the end of Sect. 3.1: the growing speed of the social cost using a larger set of paths \(\mathcal {P}_{h(\sigma, t^{*})}\), which is \(Y_{h(\sigma,t^{*})} t^{*} + Z_{h(\sigma,t^{*})}\), is less than the growing speed of the social cost using a smaller set of paths \(\mathcal {P}_{h(\sigma', t^{*})}\), which is \(Y_{h(\sigma',t^{*})} t^{*} + Z_{h(\sigma',t^{*})}\), under a certain condition, which is \(t^{*} > t_{h(\sigma,t^{*})-1}\) and it is proved in Lemma 22.

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Acknowledgements

I thank Lisa Fleischer and Kurt Mehlhorn for their comments on earlier drafts and Khaled Elbassioni and Michael Sagraloff for discussions on several technical issues. I also thank the two anonymous reviewers for their comments that help improve the presentation of this paper. Research supported by an Alexander von Humboldt fellowship.

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  1. Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Chien-Chung Huang

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Appendix: Proof of Proposition 2

Appendix: Proof of Proposition 2

In the following proof, we implicitly assume that all edges are directed from the origin to the destination and the graph is acyclic in the directed edges. This does not hurt generality, since any edge not on an o-d path is not used in an optimal flow, and no optimal flow is sent along a directed cycle.

Proof of Proposition 2

Let O(t) and O(t′) be optimal flows of values t and t′ and t>t′. Suppose that the proposition does not hold. Then in the pseudo-flow z=O(t)−O(t′), there exists some edge e ∗ on which \(z_{e^{*}}<0\). Let G′=(V′,E′) be the largest subgraph of G containing e such that G′ is also series-parallel and ∑ e∈E′ z e <0. Then there exists another subgraph G″=(V″,E″) such that ∑ e∈E″ z e >0 and G″ and G′ share the same two terminals o′ and d′.

Claim

There exists an o′-d′ path p′ in G′ so that z e <0, ∀e∈p′ and another o′-d′ path p″ in G″ so that z e >0, ∀e∈p″.

If the claim holds, then ∑ e∈p′ O e (t)<∑ e∈p′ O e (t′) and ∑ e∈p″ O e (t)>∑ e∈p″ O e (t′). Recall that optimal flow is an equilibrium flow of one player, so we can apply Lemma 8 and the preceding two inequalities to derive

a contradiction.  □

Proof of the Claim

We only prove the first part. Suppose that G′ is composition of \(G'_{1}=(V'_{1},E'_{1})\) and \(G'_{2}=(V'_{2},E'_{2}) \). If G′ is resulted from the series-composition of \(G'_{1}\) and \(G'_{2}\), then \(\sum_{e\in E'_{1}} z_{e} = \sum_{e\in E'_{2}} z_{e}<0\). If G′ is resulted from the parallel-composition of \(G'_{1}\) and \(G'_{2}\), then either \(\sum_{e\in E'_{1}} z_{e} <0\) or \(\sum_{e\in E'_{2}} z_{e}<0\). In both cases, we can apply induction to find out a path p′ in \(G'_{1}\) and/or \(G'_{2}\) so that z e <0 if e∈p′. □

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Huang, CC. Collusion in Atomic Splittable Routing Games. Theory Comput Syst 52, 763–801 (2013). https://doi.org/10.1007/s00224-012-9421-4

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