The Impact of Worst-Case Deviations in Non-Atomic Network Routing Games

Abstract

We introduce a unifying model to study the impact of worst-case latency deviations in non-atomic selfish routing games. In our model, latencies are subject to (bounded) deviations which are taken into account by the players. The quality deterioration caused by such deviations is assessed by the Deviation Ratio, i.e., the worst case ratio of the cost of a Nash flow with respect to deviated latencies and the cost of a Nash flow with respect to the unaltered latencies. This notion is inspired by the Price of Risk Aversion recently studied by Nikolova and Stier-Moses (Nikolova and Stier-Moses 2015). Here we generalize their model and results. In particular, we derive tight bounds on the Deviation Ratio for multi-commodity instances with a common source and arbitrary non-negative and non-decreasing latency functions. These bounds exhibit a linear dependency on the size of the network (besides other parameters). In contrast, we show that for general multi-commodity networks an exponential dependency is inevitable. We also improve recent smoothness results to bound the Price of Risk Aversion.

Appendices

Appendix A: Proof of Lemma 1

Lemma 1

Let− 1 < α ≤ 0 ≤ βbe fixed. Thenf is inducible with an (α, β)-deviationif and only if it is inducible with a$(0,\frac {\beta - \alpha }{1 + \alpha })$-deviation.

Proof

Let f be δ-inducible for some αl ≤ δ ≤ βl, and for a ∈ A, write δ_a(f_a) = d_al_a(f_a). Without loss of generality we may assume that δ_a(x) = d_al_a(x)(since by definition d_al_a(x) also induces f). From the equilibrium conditions (2), we know that

$$\forall i \in [k], \forall P \in \mathcal{P}_{i}, f_{P} > 0: \quad \sum\limits_{a \in P} l_{a}(f_{a}) + \delta_{a}(f_{a}) \leq \sum\limits_{a \in P^{\prime}} l_{a}(f_{a}) + \delta_{a}(f_{a}) \ \ \forall P^{\prime} \in \mathcal{P}_{i}. $$

This is equivalent to $\forall i \in [k], \forall P \in \mathcal {P}_{i}, f_{P} > 0:$

$$\sum\limits_{a \in P} \left(1 + \frac{d_{a} - \alpha}{1 + \alpha}\right)l_{a}(f_{a}) \leq \sum\limits_{a \in P^{\prime}} \left(1 + \frac{d_{a} - \alpha}{1 + \alpha}\right)l_{a}(f_{a}) \ \ \forall P^{\prime} \in \mathcal{P}_{i} $$

which can be seen bywriting

$$l_{a}(f_{a}) + \delta_{a}(f_{a}) = (1 + d_{a})l_{a}(f_{a}) = (1 + \alpha + d_{a} - \alpha)l_{a}(f_{a}),$$

and then dividingthe inequality by 1 + α.We then see that δ^′,defined by $\delta _{a}^{\prime }(x) = \frac {d_{a} - \alpha }{1 + \alpha } l_{a}(x)$ for all a ∈ A and x ≥ 0, also induces f since

$$\alpha l_{a}(x) \leq d_{a}l_{a}(x) \leq \beta l_{a}(x) \ \ \ \Leftrightarrow \ \ \ 0 \leq \frac{d_{a} - \alpha}{1 + \alpha} l_{a}(x) \leq \frac{\beta - \alpha}{1 + \alpha} l_{a}(x). $$

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Appendix B: Computing Optimal Deviations

The bounded deviation model introduced in Section 2.2 naturally gives rise to the following two optimization problems:¹¹

• 1. Best deviation problem: compute a deviation δ^∗∈ Δ(𝜃) such that

  $$\delta^{*} = \arg\inf_{\delta \in {\Delta}(\theta)} \inf \{C(f^{\delta}) \; | \; f^{\delta}\text{ is }\delta\text{-inducible}\}. $$

• 2. Worst deviation problem: compute a deviation δ^∗∈ Δ(𝜃) such that

  $$\delta^{*} = \arg\sup_{\delta \in {\Delta}(\theta)} \sup \{C(f^{\delta}) \; | \; f^{\delta}\text{ is }\delta\text{-inducible}\}. $$

Recall that the social cost function C only takes into account the latencies but not the deviations. A somewhat subtle point here is that for a fixed deviation δ ∈ Δ(𝜃), the social cost of a δ-inducible flow might not be unique. In particular, in the best deviation problem we seek a feasible deviation δ such that the social cost of the best Nash flow that is δ-inducible is minimized (similar as in [2]). In contrast, in the worst deviation problem we want to determine a feasible deviation δ such that the social cost of the worst Nash flow that is δ-inducible is maximized.

Below we elaborate on relations between the best deviation problem and various network toll problems. As a side result, we also show that the worst deviation problem is NP-hard, even for single-commodity instances with linear latencies (Theorem 12).

2.1 B.1 Relations to Network Toll Problems

The best deviation problem is a direct generalization of the restricted network toll problem introduced by Bonifaci et al. [2]. We obtain this model for 𝜃^min = 0. The deviations are interpreted as non-negative tolls on the arcs. The objective minimized in [2] is measured against the social optimum, i.e., the authors are interested in the ratio C(f^δ)/C(f^∗), where f^∗ is an optimal flow for the instance $\mathcal {I}$. Also, our definition of (0, β)-deviations is equivalent to the definition of β-restricted tolls in [2].

The work by Fotakis et al. [8] can technically be seen as a variant of the restricted network toll problem in which the tolls are interpreted as risk-averse behavior of players. Here, we have $\theta ^{\min }_{a} = 0$ and $\theta ^{\max }_{a} = \gamma l_{a}$ for all a ∈ A. The authors consider deviations of the form δ_a(x) = γ_al_a(x) for 0 ≤ γ_a ≤ γ for all a ∈ A. In particular, deviations of this form induce an approximate Nash flow as studied by Christodoulou et al. [4]. For example, if all latency functions in the network are polynomials of degree at most d, then we obtain a γd-approximate Nash flow.

Hoefer et al. [9] consider the taxing subnetwork problem, which is a special case of the restricted network toll problem. Here only a designated subset of the arcs can be tolled, which is equivalent to $\theta ^{\min }_{a} = 0$ and $\theta ^{\max }_{a} \in \{0,\infty \}$ for all a ∈ A. They show that best deviation problem is NP-complete, even for two commodities. To the best of our knowledge, the single-commodity case is still an open problem. On the positive side, Hoefer et al. [9] and Bonifaci et al. [2] give polynomial time algorithms for parallel-arc networks, solving the best deviation problem for their respective definitions of the threshold functions.

Beckmann et al. [1] proved that the social optimum can be induced as a Nash flow using marginal tolls, that is, by setting δ_a(x) = x ⋅ la′(x), where $l^{\prime }_{a}(x)$ is the derivative of l_a(x) (assuming the existence of $l_{a}^{\prime }$). In particular, if these tolls are feasible, i.e., δ ∈ Δ(𝜃), then δ is an optimal solution for the best deviation problem.

There are several models that study deviations in the form of scaled marginal tolls, i.e., deviations defined by δ_a(x) = ρxla′(x) for some $\rho \in \mathbb {R}$. We elaborate on two such models in more detail:

In the standard non-atomic routing model it is assumed that players are completely selfish in the sense that they want to minimize their own latencies. However, more recently researchers also considered settings where players are (partially) altruistic. Chen et al. [3], for example, model such altruistic behavior by including scaled marginal tolls in the objective of the players. In particular, they study scaled marginal tolls with − 1 ≤ ρ ≤ 1.

Meir and Parkes [13] also study deviations in the form of scaled marginal tolls, which are interpreted as behavioral biases towards the marginal tolls. A conceptual difference here is that the parameter ρ is chosen by the players, instead of the system designer (as, for example, in the restricted network toll model). Here, the authors are also interested in the case ρ ≥ 1 (which is less relevant in the other models). The authors also study this model in [14], where these deviations are interpreted as distance-based strict uncertainty.

2.2 B.2 Hardness of the Worst Deviation Problem

As a side-result, we prove that the problem of determining worst-case deviations is NP-hard.

Theorem 12

It isNP-hard to compute deviationsδ ∈ Δ(𝜃) such thatC(f^δ) is maximized, even for single-commodity networks with linear latencies.

Proof

We give a reduction from the Directed Hamiltonians, t-Path problem: We are given a directed graph G = (V, A),and fixed s, t ∈ V,and the goal is to decide whether or not there exists a simple directeds, t-pathin G that visits every node exactly once. Let$\mathcal {J}$ be an instance of Directed Hamiltonians, t-Path problem.

Now, define an instance $\mathcal {I}$of the bounded deviation model on the graph G by takingl_a(x) = x for alla ∈ A,$\theta ^{\min }_{a} = 0$for alla ∈ A, and$\theta ^{\max }_{a} = n - 1$for alla ∈ A. Furthermore,take r = 1. We claim that G has a Hamiltonian path from s to t if and only if there is a deviationδ ∈ Δ(𝜃)suchthat C(f^δ) ≥ n − 1.First, let G have a Hamiltonian path P from s to t, and defineδ byδ_a = 0ifa ∈ P, andδ_a = n − 1otherwise. Wethen have that f^δis given by $f_{a}^{\delta } = 1$if a ∈ P and$f^{\delta }_{a} = 0$otherwise, since the perceived latency along P is then equal tol_P(f^δ) = n − 1, and any other pathP^′uses at least one differentarc a^′∉P, which gives usthat

$$l_{P^{\prime}}(f^{\delta}) + \delta_{P^{\prime}}(f^{\delta}) \geq l_{a^{\prime}}(f^{\delta}) + \delta_{a^{\prime}}(f^{\delta}) \geq n-1 = l_{P}(f^{\delta}) + \delta_{P}(f^{\delta}). $$

Notethat f^δis the unique Nash flow in this case (since all the perceived latenciesl_a + δ_aarestrictly increasing).

Conversely, suppose there is a δ ∈ Δ(𝜃)such that C(f^δ) ≥ n − 1. For any feasible flow g we have that l_P(g) ≤ n − 1, withstrict inequality if $f^{\delta }_{P} < 1$(since thenthere will be at least one arc a ∈ P with $f^{\delta }_{a} < 1$). This meansthat

$$C(g) = \sum\limits_{P \in \mathcal{P}} g_{P}l_{P}(g) \leq \sum\limits_{P \in \mathcal{P}} g_{P}(n-1) = n - 1,$$

using thatr = 1. Again, we havestrict inequality if 0 < g_P < 1for some path P, i.e., if not all players use the same path. This means that forf^δthere is at mostone path P^∗with$f^{\delta }_{P^{*}} > 0$, which then impliesthat $f^{\delta }_{P^{*}} = 1$. Furthermore,we can conclude that $|A(P^{*})| = l_{P^{*}}(f^{\delta }) = C(f^{\delta }) = n - 1$,which implies that P^∗is a Hamiltonian path from s to t, since it is a simple path by assumption.□

Appendix C: Necessity of Common Source Assumption in Theorem 2

The example below shows that Theorem 2 does not hold if the assumption that all commodities share a common source is dropped.

Example 2

Consider the graph G = (V, A)in Fig. 6 and suppose that r₁ = r₂ = 1.Then the flow f that routes one unit of flow over both paths(s₁, v₁, 1, 2, t₁)and (s₂, v₂, 3, 4, t₂)is feasible and inducible (take δ = 0).However, looking at the graph $\hat {G}(f)$,we obtain a negative cost cycle (1, 4, 3, 2, 1)(by using the reversed arcs of (1,2) and (3,4)).

Fig. 6

All the values of $l_{a}, \ \theta ^{\min }_{a}$ and $\theta ^{\max }_{a}$ that are not explicitly stated are zero