Asymptotically tight bounds for inefficiency in risk-averse selfish routing

Abstract

We consider a nonatomic selfish routing model with independent stochastic travel times, represented by mean and variance latency functions for each edge that depend on their flows. In an effort to decouple the effect of risk-averse player preferences from selfish behavior on the degradation of system performance, Nikolova and Stier- Moses [16] defined the concept of the price of risk aversion as the worst-case ratio of the cost of an equilibrium with risk-averse players and that of an equilibrium with risk-neutral users. For risk-averse users who seek to minimize the mean plus variance of travel time on a path, they proved an upper bound on the price of risk aversion, which is independent of the latency functions, and grows linearly with the size of the graph and players' risk-aversion. In this follow-up paper, we provide a matching lower bound for graphs with number of vertices equal to powers of two, via the construction of a graph family inductively generated from the Braess graph. We also provide conceptually different bounds, which we call functional, that depend on the class of mean latency functions and provide characterizations that are independent of the network topology (first derived, in a more complicated way, by Meir and Parkes [10] in a different context with different techniques). We also supplement the upper bound with a new asymptotically-tight lower bound. Our third contribution is a tight bound on the price of risk aversion for a family of graphs that generalize series-parallel graphs which applies to users minimizing the mean plus standard deviation of a path, a much more complex model of risk-aversion due to the cost of a path being non-additive over edge costs. This is a refinement of previous results in [16] that characterized the price of risk-aversion for series-parallel graphs and for the Braess graph.

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