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

Toll caps in privatized road networks

“…First, potentially each edge in the network is tolled; an issue that is considered by Hoefer et al [27] and Harks et al [24]. Second, the imposed tolls can be arbitrary large; an issue that is considered by Bonifaci et al [6], Fotakis et al [21], Jelinek et al [28] and Kleer and Schäfer [32]. Cole et al [12] show that optimal tolls also exist when users are heterogeneous with respect to the tradeoff between time and money.…”

Path Deviations Outperform Approximate Stability in Heterogeneous Congestion Games

“…To this aim, we study the (relative) worst-case deterioration in social cost of a β-deviated Nash flow with respect to an original (unaltered) Nash flow; we use the term β-deviation ratio to refer to this ratio. This ratio has recently been studied in the context of risk aversion [8,12] and in the more general context of bounded path deviations [6]. Similarly, for approximate Nash flows we are interested in bounding the ǫ-stability ratio, i.e., the worst-case deterioration in social cost of an ǫ-approximate Nash flow with respect to an original Nash flow.…”

“…Note that these notions differ from the classical price of anarchy notion [7], which refers to the worst-case deterioration in social cost of a β-deviated (respectively, εapproximate) Nash flow with respect to an optimal flow. While the price of anarchy typically depends on the class of latency functions (see, e.g., [1,2,6,12] for results in this context), the deviation ratio is independent of the latency functions but depends on the topology of the network (see [6,12]).…”

“…Christodoulou et al [2] study the inefficiency of approximate equilibria in terms of the price of anarchy and price of stability (for homogeneous populations). Generalized Braess graphs were introduced by Roughgarden [13] and are used in many other lower bound constructions (see, e.g., [3,6,13]). Chen et al [1] study an altruistic extension of the Wardrop model and, in particular, also consider heterogeneous altruistic populations.…”

“…In particular, Fujishige et al [5] show that matroid congestion games are immune against the Braess paradox (and their analysis is tight in a certain sense). We refer the reader to [6] for additional references and relations of other models to the bounded path deviation model considered here.…”