The Burden of Risk Aversion in Mean-Risk Selfish Routing

TL;DR

This paper analyzes how risk-averse agents in network routing can cause inefficiency, quantifies this via the price of risk aversion, and finds that network structure influences the impact of risk preferences.

Contribution

It introduces the concept of the price of risk aversion, providing bounds and properties for general and series-parallel networks, and employs combinatorial proofs involving alternating paths.

Findings

PRA depends linearly on risk tolerance and variability.

In series-parallel networks, PRA is independent of network size.

Wardrop equilibria maximize shortest-path objectives in SP networks.

Abstract

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 {\em 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 {\em 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 {\em 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 {\em maximize} the shortest-path objective among all feasible flows.