Parameter Estimation in Optimal Tolling for Traffic Networks Under the Markovian Traffic Equilibrium

TL;DR

The paper addresses tolling in arc-based stochastic traffic networks where both arc latency parameters and the entropy controlling travelers' route choices are unknown. It proposes an online algorithm that alternates toll design with regularized least-squares estimation of latency parameters $\theta$ and a novel optimistic beta-estimation scheme to learn the entropy parameter $\beta$, guiding successive tolls toward reducing total latency. A key theoretical result is a sublinear regret bound of the form $R \le K g_o^2 \ln^2(g_o) |A| \sqrt{T} \ln(T g_o) \max\{ |I| \ln(|A|/|I|), B \}$, supported by proofs and validated on simulated networks. Practically, the method enables adaptive congestion pricing that converges toward socially efficient flows without requiring perfect knowledge of latency functions or traveler randomness, potentially improving traffic efficiency in real-time settings.

Abstract

Tolling, or congestion pricing, has emerged as an effective tool for preventing gridlock in traffic systems. However, tolls are currently mostly designed on route-based traffic assignment models (TAM), which may be unrealistic and computationally expensive. Existing approaches also impractically assume that the central tolling authority can access latency function parameters that characterize the time required to traverse each network arc (edge), as well as the entropy parameter $β$ that characterizes commuters' stochastic arc-selection decisions on the network. To address these issues, this work formulates an online learning algorithm that simultaneously refines estimates of linear arc latency functions and entropy parameters in an arc-based TAM, while implementing tolls on each arc to induce equilibrium flows that minimize overall congestion on the network. We prove that our algorithm incurs regret upper bounded by $O(\sqrt{T} \ln(T) |\arcsMod| \max\{|\nodesMod| \ln(|\arcsMod|/|\nodesMod|), B \})$, where $T$ denotes the total iteration count, $|\arcsMod|$ and $|\nodesMod|$ denote the total number of arcs and nodes in the network, respectively, and $B$ describes the number of arcs required to construct an estimate of $β$ (usually $\ll |I|$). Finally, we present numerical results on simulated traffic networks that validate our theoretical contributions.

Figures (2)

• Figure 1: (Left) A parallel 6-arc network; here, $i^\star = i_1$. (Right) A more general network with 6 arcs; here, $i^\star = i_2$, since there are two routes from $i_2$ to the destination $i_4$ which do not share an arc.
• Figure 2: (Left to right) The cumulative regret $R^t - L^\star t$, logarithm of stage-wise regret $\ln(L^t - L^\star)$, logarithm of $\theta$-estimation error $\ln(\Vert \theta^t - \theta^\star \Vert_2)$, and logarithm of stage-wise regret $\ln(|\beta^t - \beta^\star|)$ for the parallel-arc network in Figure \ref{['fig:Network Schematics']} (top) and the more general network in Figure \ref{['fig:Network Schematics']} (bottom), as a function of the iteration count $t$. Note the sub-linear growth of the cumulative regret with respect to the iteration count, and the rapid decay of the stage-wise regret, $\theta$-estimation error, and $\beta$-estimation error to 0.

Theorems & Definitions (41)

• Definition 1: Perturbed Socially Optimal Flow
• Proposition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof