Information Design in Nonatomic Routing Games With Partial Participation: Computation and Properties

We consider a routing game among nonatomic agents where link latency functions are conditional on an uncertain state of the network. The agents have the same prior belief about the state, but only a fixed fraction receive private route recommendations or a common message, which are generated by a known randomization, referred to as
private
or
public signaling policy
, respectively. The remaining agents choose route according to Bayes Nash flow with respect to the prior. In this article, we develop a computational approach to solve the optimal information design problem, i.e., to minimize expected social latency over all public or
obedient
private signaling policies. For a fixed flow induced by nonparticipating agents, the design of an optimal private signaling policy has been shown to be a generalized problem of moments for polynomial link latency functions, and to admit an atomic solution with a provable upper bound on the number of atoms. This implies that, for polynomial link latency functions, information design can be equivalently cast as a polynomial optimization problem. This, in turn, can be arbitrarily lower bounded by a known hierarchy of semidefinite relaxations. The first level of this hierarchy is shown to be exact for the basic two link case with affine latency functions. We also identify a class of private signaling policies over which the optimal social cost is nonincreasing with an increasing fraction of participating agents for parallel networks. This is in contrast to existing results where the cost of participating agents under a
fixed
signaling policy may increase with their increasing fraction.