Robust network pricing and system optimization under combined long-term stochasticity and elasticity of travel demand

Abstract

Network pricing serves as an instrument for congestion management, however, agencies and planners often encounter problems of estimating appropriate toll prices. Tolls are commonly estimated for a single-point deterministic travel demand, which may lead to imperfect policy decisions due to inherent uncertainties in future travel demand. Previous research has addressed the issue of demand uncertainty in the pricing context, but the elastic nature of demand along with its uncertainty has not been explicitly considered. Similarly, interactions between elasticity and uncertainty of demand have not been characterized. This study addresses these gaps and proposes a framework to estimate nearest optimal first-best tolls under long-term stochasticity in elastic demand. We show first that the optimal tolls under the deterministic-elastic and stochastic-elastic demand cases coincide when cost and demand functions are linear, and the set of equilibrium paths is constant. These assumptions are restrictive, so three larger networks are considered numerically, and the subsequent pricing decisions are assessed. The results of the numerical experiments suggest that in many cases, optimal pricing decisions under the combined stochastic-elastic demand scenario resemble those when demand is known exactly. The applications in this study thus suggest that inclusion of demand elasticity offsets the need of considering future demand uncertainties for first-best congestion pricing frameworks.

Appendix

The results in this “Appendix” do not depend on the number of links, nodes, or OD pairs in a network, but only on linearity of the cost and demand functions, and conditions which intuitively express that the set of used paths does not change with the demand realization. The general result is followed by an illustrative example. The theorem below requires several restrictive assumptions, motivating the numerical analyses in the main text on large-scale networks which do not satisfy these conditions.

Let $\Xi$ denote the support of the random vector $\epsilon$, and T the set of feasible toll vectors. We will divide the space $\Xi \times T$ into subsets $\Pi_{i}$ according to the sets of minimum-cost paths at the user equilibrium solutions, depending on the demand realization and applied tolls. Formally, there exist a finite number of disjoint open sets $\Pi_{1} ,\Pi_{2} , \ldots \Pi_{P}$, all subsets of $\Xi \times T$, which satisfy the following properties:

• 1. For any $\left({\epsilon, \tau} \right) \in {\Pi}_{i}$, when the demand is given by $d + \epsilon$ and the toll vector is τ, the set of minimum-cost paths connecting all OD pairs at the user equilibrium solution is the same for all such pairs in the same set $\Pi_{i}$.

• 2. For each set $\Pi_{i} ,$ and for each $\left({\epsilon, \tau} \right) \in {\Pi}_{i}$, there exists at least one equilibrium path flow solution with strictly positive flow on all minimum-cost paths when demand is $d + \epsilon$ and tolls are τ.

• 3. The union of the closure of the sets $\Pi_{i}$ is the entire space $\Xi \times T$.

We now show that, among toll vectors $\hat{\tau }$ for which the set of used paths is the same regardless of the realization $\epsilon$, the tolls maximizing expected surplus with respect to $\epsilon$ also maximize the surplus when $\epsilon = 0$ deterministically when the cost and demand functions are linear and $\epsilon$ has zero mean. The first condition can be expressed as restricting the toll vectors under consideration to those $\tau$ such that $\Xi \times \left\{ \tau \right\} \subseteq {\text{cl}} \Pi_{i}$ for some set $\Pi_{i}$.

Theorem

Consider a network with any number of nodes, links, and OD pairs satisfying the following conditions:

• 1. The travel time on each link takes the form $t_{ij} \left( {x_{ij} } \right) = L_{ij} + M_{ij} x_{ij}$ where M _ij  > 0.

• 2. The inverse demand function for OD pair (r, s) is of the form $D_{rs}^{- 1} \left({d_{rs}} \right) = C_{rs} + \epsilon_{rs} - F_{rs} d_{rs}$ where F _rs  > 0.

• 3. The set of feasible tolls T has the property that $\Xi \times T \subseteq {\text{cl}} \Pi_{i}$ for some set $\Pi_{i}$.

• 4. For each OD pair (r, s), $E\left[{\epsilon_{rs}} \right] = 0$.

In such a network, a toll vector maximizes traveler surplus when $\epsilon = 0$ (the “deterministic case”) if and only if it maximizes expected traveler surplus when $\epsilon$ has any distribution satisfying conditions 1 and 2.

Proof

For any realization $\epsilon$, equilibrium link flows and OD demands exist and are unique, so functions $x(\epsilon)$ and $d(\epsilon)$ giving these values in terms of $\epsilon$ are well-defined, and the domain of these functions is the support of $\epsilon$. (Patriksson 1994). Furthermore, Condition 2 and the properties of $\Pi_{i}$ discussed above imply that the equilibrium solutions $x(\epsilon)$ are strictly complementary for all $\epsilon$ except possibly at the boundary (that is, there exists a corresponding path flow solution in which all minimum-cost paths have positive flow), so the functions $x(\epsilon)$ and $d(\epsilon)$ are differentiable on $\Pi_{i}$ (Proposition 2.2 and Theorem 2.3 of Lu 2008). Furthermore, these derivatives correspond to the solution of “linearized” elastic demand equilibrium problems (Patriksson 2004; Josefsson and Patriksson 2007; Lu 2008). Under Conditions 1, 2, and 3, these derivatives are identical for all realizations of $\epsilon$, the sensitivity analysis is exact, and these functions are linear in $\epsilon$. That is, we can express $x(\epsilon)$ and $d(\epsilon)$ in the forms

$$x_{ij} \left(\epsilon\right) = J_{ij} + \epsilon \cdot K$$

$$d_{rs} \left(\epsilon\right) = A_{rs} + \epsilon \cdot B$$

where K and B are constant vectors of the same dimension as $\epsilon$, and J _ij and A _rs are link- and OD-specific constants, respectively. Furthermore, of these constants, only the A _rs and J _ij directly depend on the specific toll vector τ; by linearity, the constants B and K depend only on the set of $\Pi_{i}$ which is independent of τ by Condition 3.

The total system travel time is thus a quadratic function of $\epsilon$:

$$\begin{aligned} TSTT & = \mathop \sum \limits_{{\left({i,j} \right)}} \left({J_{ij} + \epsilon \cdot K} \right)\left({L_{ij} + M_{ij} \left({J_{ij} +\epsilon \cdot K} \right)} \right) \\ & = \mathop \sum \limits_{{\left({i,j} \right)}} \left({\left[{J_{ij} L_{ij} + M_{ij} J_{ij}^{2}} \right] + \epsilon \cdot K\left[{2M_{ij} J_{ij} + L_{ij} } \right] + \left({\epsilon \cdot K} \right)^{2} \left[{M_{ij}} \right]} \right)^{{}} \\ \end{aligned}$$

which can be written as

$$TSTT = G + H \cdot \epsilon + \left({I \cdot \epsilon} \right)^{2}$$

for appropriate constants G, H, and I, and of these only G and H depend on the toll vector $\tau$ applied.

Therefore, the traveler surplus under any realization $\epsilon$ is:

$$\begin{aligned} TS\left(\epsilon\right) & = \mathop \sum \limits_{r,s} \mathop \int \limits_{0}^{{A_{rs} +\, \epsilon \cdot B}} \left({C_{rs} + \epsilon_{rs} - F_{rs} \omega} \right)d\omega - \left({G + H \cdot \epsilon + \left({I \cdot \epsilon} \right)^{2}} \right) \\ & = \left[{\mathop \sum \limits_{r,s} \left({A_{rs} C_{rs} - \frac{{F_{rs} A_{rs}^{2}}}{2}} \right) - G + A_{rs} \epsilon_{rs}} \right] + \left({\epsilon \cdot B} \right)\mathop \sum \limits_{r,s} \left({C_{rs} + \epsilon_{rs} - F_{rs} \left({A_{rs} + \epsilon \cdot \frac{B}{2}} \right)} \right) - H \cdot \epsilon - \left({I \cdot \epsilon} \right)^{2} \\ \end{aligned}$$

Applying Condition 4, the expected value of TS is thus

$$E\left[{TS}\right] = \left[{\mathop \sum \limits_{r,s} \left({A_{rs} C_{rs} - \frac{{F_{rs} A_{rs}^{2}}}{2}} \right) - G} \right] - \frac{1}{2}\mathop \sum \limits_{r,s} \mathop \sum \limits_{{r^{\prime},s^{\prime}}} \left({\left({F \cdot 1} \right)B_{rs} B_{{r^{\prime}s^{\prime}}} - 2I_{rs} I_{{r^{\prime}s^{\prime}}}} \right)E\left[{\epsilon_{rs} \epsilon_{r^{\prime}s^{\prime}}} \right]$$

Repeating this derivation for the deterministic case when $\epsilon = 0$ gives

$$TS\left( 0 \right) = \left[ {\mathop \sum \limits_{r,s} \left( {A_{rs} C_{rs} - \frac{{F_{rs} A_{rs}^{2} }}{2}} \right) - G} \right]$$

These expressions only differ in the term outside of the brackets in E[TS], but the only terms which depend on the toll vector are A _rs and G. Therefore, when maximizing TS(0) or E[TS] with respect to the link tolls, the objectives differ only by a constant term, and the tolls maximizing TS(0) also maximize E[TS]. □

Example

We use the Pigou–Knight–Downs network, which consists of two parallel links. Link 1 has a constant travel time,$t_{1} = 1,$ and link 2 has travel time equal to its flow, $t_{2} = x$. In the deterministic case, let the demand function be $d = D\left( \kappa \right) = 2 - \kappa$, where $\kappa = { \hbox{min} }(t_{1} , t_{2} )$ is the shortest-path travel time. If a toll of τ is applied to link 2, it is straightforward to see that the solution to this problem is $d = \kappa = 1$, and the total system travel time is $TSTT = \left( {1 - \tau } \right)^{2} + \left( {d - 1 + \tau } \right)$. Since this is an elastic demand problem, the appropriate metric is traveler surplus: $TS = \int_{0}^{d} {D^{ - 1} \left( \omega \right)d\omega - TSTT = d - \frac{1}{2}d^{2} + \tau - \tau^{2} }$. From this expression, the optimal toll is seen to be $\tau * = \frac{1}{2}$, resulting in $TS = \frac{3}{4}$.

We now introduce demand uncertainty by changing the demand function to $D\left( \kappa \right) = 2 + \varepsilon - \kappa$, where ɛ is a random term with zero mean and bounded support $\left[ { - \frac{1}{2},\frac{1}{2}} \right]$. For this distribution of ɛ, both paths will continue to be used at equilibrium, and the solution will be d = 1 + ɛ and $TSTT = \tau^{2} - \tau + \varepsilon + 1$. Therefore, after some simplification, the expected traveler surplus is $E\left( {TS} \right) = \frac{1}{2}E\left( {\varepsilon^{2} } \right) + \frac{1}{2} - \tau^{2} + \tau$, which results in the same optimal toll $\tau^{*} = \frac{1}{2}$, but a slightly different surplus value. In this example, demand uncertainty affects the expected traveler surplus, but not the optimal toll.