The Burden of Risk Aversion in Mean-Risk Selfish Routing

Considering congestion games with uncertain delays, we compute the inefficiency introduced in network routing by risk-averse agents. At equilibrium, agents may select paths that do not minimize the expected latency so as to obtain lower variability. A social planner, who is likely to be more risk neutral than agents because it operates at a longer time-scale, quantifies social cost with the total expected delay along routes. From that perspective, agents may make suboptimal decisions that degrade long-term quality. We define the price of risk aversion (PRA) as the worst-case ratio of the social cost at a risk-averse Wardrop equilibrium to that where agents are risk-neutral. For networks with general delay functions and a single source-sink pair, we show that the PRA depends linearly on the agents' risk tolerance and on the degree of variability present in the network. In contrast to the price of anarchy, in general the PRA increases when the network gets larger but it does not depend on the shape of the delay functions. To get this result we rely on a combinatorial proof that employs alternating paths that are reminiscent of those used in max-flow algorithms. For series-parallel (SP) graphs, the PRA becomes independent of the network topology and its size. As a result of independent interest, we prove that for SP networks with deterministic delays, Wardrop equilibria maximize the shortest-path objective among all feasible flows.