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Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

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Abstract

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms.

We increase the latency functions of the edges in the network, i.e., if ℓ e (x) is the latency function of an edge e, we replace it by \(\hat{\ell}_{e}(x)\) with \(\ell_{e}(x) \le \hat{\ell}_{e}(x)\) for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \(\hat{C}_{N} (r)\) denotes the cost of the worst Nash flow in the modified network for rate r and C opt (r) denotes the cost of the optimal flow in the original network for the same rate then

$$\mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}. $$

We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.

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Notes

  1. This assumes continuity and monotonicity of the latency functions. For non-continuous functions, see the discussion later in this section.

  2. Technically, we consider symmetric coordination mechanisms in this work, as defined in [8] i.e., the latency modifications affect the users in a symmetric fashion.

  3. One can interpret the difference \(\hat {\ell }_{e}-\ell_{e}\) as a flow-dependent toll imposed on the edge e.

  4. See [24, 26] for an excellent exposure of the relevant concepts, the relation among them, as well as for conditions that guarantee their existence.

  5. In [14], Dafermos weakened the original definition by [15] to make it closer to the concept of Nash Equilibrium.

  6. See Sect. 5 for a formal definition.

  7. See for example [4, 16] for examples where equilibria do not exist even for the simplest case of two parallel links and non-decreasing functions.

  8. See [4] for a definition of regular functions.

  9. The lower bound that we provide in Sect. 5 holds for all deterministic coordination mechanisms that use non-decreasing modified latencies, with respect to both notions of equilibrium described in the previous paragraph.

  10. It is important to observe that although the Nash flow is equal to the optimum flow, its cost with respect to the marginal cost function can be twice as large as its cost with respect to the original cost function. For Pigou’s network, the marginal costs are \(\hat {\ell }_{1}(x) = 2x\) and \(\hat {\ell }_{2}(x) = 1\). The cost of a Nash flow of rate r with r≤1/2 is 2r 2 with respect to marginal costs; the cost of the same flow with respect to the original cost functions is r 2.

  11. It is not hard to see that, similarly to the case where the demand is known, using global flow information (at least for the case of parallel links) can lead to mechanisms with ePoA=1. We would like to thank Nicolás Stier Moses for making us emphasizing that distinction.

  12. In a Nash flow all used links have the same latency. Thus, if j links are used at rate r and \(f_{i}^{N}\) is the flow on the i-th link, then \(a_{1} f_{1}^{N} + b_{1} = \cdots = a_{j} f_{j}^{N} + b_{j} \le b_{j+1}\) and \(r = f_{1}^{N} + \cdots + f_{j}^{N}\). The values for r j and \(f_{i}^{N}\) follow from this. Similarly, in an optimal flow all used links have the same marginal costs.

  13. In Pigou’s network we have ℓ 1(x)=x and ℓ 2(x)=1. Thus λ 2=∞. The modified cost functions are \(\hat {\ell }_{2}(x) = \ell_{2}(x)\) and \(\hat {\ell }_{1}(x) = x\) for x≤r 2/2=1/2 and \(\hat {\ell }_{1}(x) = \infty\) for x>1/2. The Nash flow with respect to the modified cost function is identical to the optimum flow in the original network and \(\hat {C}_{N}(f^{*}) = C(f^{*})\). Thus ePoA=1 for Pigou’s network.

  14. The optimal choice for x 1 and r ∗ is such that both terms are equal and as small as possible. We were unable to solve the resulting system explicitly. We will prove in the next section that the mechanism defined by these optimal choices of the parameters x 1 and r ∗ is optimal.

  15. It remains open whether similar arguments can be applied for showing the lower bound for non-monotone mechanisms with respect to User Equilibria.

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Acknowledgements

We would like to thank Elias Koutsoupias, Spyros Angelopoulos and Nicolás Stier Moses for many fruitful discussions.

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Authors and Affiliations

  1. University of Liverpool, Ashton Building, Ashton Street, Liverpool, L69 3BX, UK

    Giorgos Christodoulou

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Kurt Mehlhorn

  3. Technische Universität München, Munich, Germany

    Evangelia Pyrga

Authors
  1. Giorgos Christodoulou
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  2. Kurt Mehlhorn
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  3. Evangelia Pyrga
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Correspondence to Giorgos Christodoulou.

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Christodoulou, G., Mehlhorn, K. & Pyrga, E. Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms. Algorithmica 69, 619–640 (2014). https://doi.org/10.1007/s00453-013-9753-8

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