Public Signals in Network Congestion Games

Abstract

We study a novel approach to information design in the standard traffic model of network congestion games. It captures the natural condition that the demand is unknown to the users of the network. A principal (e.g., a mobility service) commits to a signaling strategy, observes the realized demand and sends a (public) signal to agents (i.e., users of the network). Based on the induced belief about the demand, the users then form an equilibrium. We consider the algorithmic goal of the principal: Compute a signaling scheme that minimizes the expected total cost of the induced equilibrium. We concentrate on single-commodity networks and affine cost functions, for which we obtain the following results. First, we devise a fully polynomial-time approximation scheme (FPTAS) for the case that the demand can only take two values. It relies on several structural properties of the cost of the induced equilibrium as a function of the updated belief about the distribution of demands. We show that this function is piecewise linear for any number of demands, and monotonic for two demands. Second, we give a complete characterization of the graph structures for which it is optimal to fully reveal the information about the realized demand. This signaling scheme turns out to be optimal for all cost functions and probability distributions over demands if and only if the graph is series-parallel. Third, we propose an algorithm that computes the optimal signaling scheme for any number of demands whose time complexity is polynomial in the number of supports that occur in a Wardrop equilibrium for some demand. Finally, we conduct a computational study that tests this algorithm on real-world instances.

Information Design for Congestion Games with Unknown Demand

“…The proof of Lemma 1 has striking similarities to the proof of the same result for the model with (known demand and) affine costs and uncertain offsets in (Griesbach et al 2022, Lemma 1). We have not been able to derive a direct reduction between the two scenarios and discuss why it seems non-obvious to establish.…”

“…The reinterpretation per se does not appear to be very useful -games with uncertain affine costs are not very well-understood and in general do not admit, e.g., the linearity properties of Lemma 1 (in contrast to the case when only offsets are uncertain, c.f. (Griesbach et al 2022)).…”

“…Bhaskar et al (2016) considered games with affine costs where the offset depends on the state and showed that the problem of computing an optimal signaling scheme cannot be approximated by a factor of (4 3 − ε) for ε > 0, unless P = NP. For the same setting, Griesbach et al (2022) proved that revealing the realized state is always an optimal signaling scheme if and only if the underlying network is a seriesparallel graph. They also provided LP-based techniques to compute the optimal signaling schemes.…”

Enhancing the Efficiency of Altruism and Taxes in Affine Congestion Games through Signalling

“…Signalling (Kamenica and Gentzkow 2011) has been adopted in a variety of stochastic scenarios, as a mean for a centralized authority to persuade selfish agents to pursue a desired behaviour. Recently, it has also been used in congestion games with incomplete information to induce selfish agents to play optimally (Bhaskar et al 2016;Castiglioni et al 2021;Griesbach et al 2022). In such a setting, the resource latency functions are assumed to depend on possible outcomes of an uncertain state of nature, which follows a publicly known probability distribution.…”

“…Bhaskar et al (2016) show that the optimal signalling scheme cannot be approximated in polynomial time up to a factor 4/3 − , for any > 0, unless P = NP. As this negative result requires a huge degree of freedom in the network structure and a very large number of states of nature, Griesbach et al (2022) provide a set of conditions for which an optimal scheme can be devised. Vasserman, Feldman, and Hassidim (2015) define the notion of mediation ratio, as the ratio between the expected social cost of the best mediated equilibrium and the social optimum.…”