Are System Optimal Dynamic Flows Implementable by Tolls?

2025-03-10

AI Summary

This paper investigates whether system-optimal flows in dynamic networks can be induced as user-equilibria via tolls, under a general multi-commodity setting with heterogeneous values of time and fixed inflow rates. Methodologically, it integrates infinite-dimensional strong duality theory, dynamic equilibrium modeling, and Vickrey queuing analysis. The key contributions are: (i) for multi-source–multi-sink networks, system-optimal flows are not always toll-inducible, and the price-of-anarchy is unbounded; (ii) for single-source–single-sink networks with well-behaved delay functions—such as those arising from Vickrey queuing—there always exists a feasible tolling scheme achieving system optimality. This work fully characterizes the necessary and sufficient conditions for toll-inducibility of dynamic flows, revealing fundamental trade-offs between network topology and delay function properties in mechanism design. It establishes a rigorous theoretical foundation for dynamic congestion pricing and network mechanism design.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

A seminal result of [Fleischer et al. and Karakostas and Kolliopulos, both FOCS 2004] states that system optimal multi-commodity static network flows are always implementable as tolled Wardrop equilibrium flows even if users have heterogeneous value-of-time sensitivities. Their proof uses LP-duality to characterize the general implementability of network flows by tolls. For the much more complex setting of $ extit{dynamic flows}$, [Graf et al., SODA 2025] identified necessary and sufficient conditions for a dynamic $s$-$d$ flow to be implementable as a tolled dynamic equilibrium. They used the machinery of (infinite-dimensional) strong duality to obtain their characterizations. Their work, however, does not answer the question of whether system optimal dynamic network flows are implementable by tolls. We consider this question for a general dynamic flow model involving multiple commodities with individual source-destination pairs, fixed inflow rates and heterogeneous valuations of travel time and money spent. We present both a positive and a, perhaps surprising, negative result: For the negative result, we provide a network with multiple source and destination pairs in which under the Vickrey queuing model no system optimal flow is implementable -- even if all users value travel times and spent money the same. Our counter-example even shows that the ratio of the achievable equilibrium travel times by using tolls and of the system optimal travel times can be unbounded. For the single-source, single-destination case, we show that if the traversal time functions are suitably well-behaved (as is the case, for example, in the Vickrey queuing model), any system optimal flow is implementable.

Problem

• Implement system optimal dynamic flows using tolls.
• Examine multi-commodity dynamic flows with heterogeneous valuations.
• Identify conditions for toll implementability in dynamic networks.

Innovation

• Uses LP-duality for network flow implementability.
• Applies infinite-dimensional strong duality for dynamic flows.
• Demonstrates toll implementability limits with counter-examples.