  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.
