Are System Optimal Dynamic Flows Implementable by Tolls?

TL;DR

The paper studies whether system-optimal dynamic flows in a general multi-commodity network can be implemented as toll-based dynamic Wardrop equilibria, extending classical static results to dynamic settings. It proves a negative result: in multi-source, multi-destination networks, there exist system-optimal flows that are not toll-implementable under the Vickrey model, and the toll-based price of stability can be unbounded. Conversely, for the single-source, single-destination case, under natural monotonicity conditions on travel times (as in Vickrey queues or linear edge delays), every system-optimal dynamic flow is toll-implementable. The results rely on a rigorous dynamic flow framework with walk inflows, aggregated edge flows, and infinite-dimensional duality, and they distinguish a sharp boundary between multi-pair and single-pair networks. The work clarifies the limits of toll-based interventions for dynamic traffic and points to further research on computation and broader classes of travel-time dynamics.

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 $\textit{dynamic flows}$, [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.

Figures (7)

• Figure 1: A 5-commodity network using the Vickrey model where the system optimum is not implementable (see \ref{['SystemOptimaArxivVersionVonECVersion:thm: CounterSysopImpl']}). All free flow travel times not explicitly given (i.e. the ones on edges $e_1$, $e_2$ and $e_7$) are $1$. Capacities are as given on the edges.
• Figure 2: Another example for an implementable flow in the network considered in \ref{['SystemOptimaArxivVersionVonECVersion:thm: CounterSysopImpl']} (with $\varepsilon=1$). This is an example for a flow where \ref{['claim: ImplemenatbleImpliesHighTT:For6']} holds, i.e. a significant amount of XXX blue green red yellow pink and XXX blue green red yellow pink flow travels around the cycle creating a queue on edge $e_4$, which then delays all future particle of the XXX blue green red yellow pink flow.
• Figure 3: An example for an implementable flow in the network considered in \ref{['SystemOptimaArxivVersionVonECVersion:thm: CounterSysopImpl']} (with $\varepsilon=1$). This is an example for a flow where both \ref{['claim: ImplemenatbleImpliesHighTT:For4', 'claim: ImplemenatbleImpliesHighTT:For6']} hold. \ref{['SystemOptimaArxivVersionVonECVersion:claim: ImplemenatbleImpliesHighTT']} holds as large amounts of both XXX blue green red yellow pink and XXX blue green red yellow pink flow take the direct path towards their destination. The XXX blue green red yellow pink flow then creates a queue on edge $e_2$, delaying the later arriving XXX blue green red yellow pink flow which then, in turn, creates a queue on edge $e_3$ and delays all future particles of the XXX blue green red yellow pink flow. \ref{['claim: ImplemenatbleImpliesHighTT:For6']} holds as well since still a significant part of the flow uses the cycle $(e_6,e_5,e_4)$ and, therefore, creates a queue on edge $e_4$ delaying all future particles of the XXX blue green red yellow pink flow.
• Figure 4: The flow used in \ref{['SystemOptimaArxivVersionVonECVersion:claim: PropsSysop']} to show that there are flows with strictly smaller total travel time than any implementable flow (depicted here for $\varepsilon=1$).
• Figure 5: The 2-commodity network considered in \ref{['SystemOptimaArxivVersionVonECVersion:exa: CE_SystemOptimumWithoutMonotonicity']}. All travel times are constant (flow and time-independent) except on $e_2$ which has a travel time of $D_{e_2}(g_{e_2},t) \coloneqq \max\Set{2-\frac{1}{2}\int_0^{t-1}\min\set{g_{e_2}(t'),2}\;\mathrm{d}\sigma(t'),1}$.

Theorems & Definitions (46)

• Definition 2.3
• Definition 2.5
• Definition 2.6: Implementability
• Definition 2.7: System Optimum
• Theorem 3.1
• proof
• Claim 1
• proof
• Theorem 3.2
• proof