A traffic congestion analysis by user equilibrium and system optimum with incomplete information

Abstract

Nowadays, the rapid development of intelligent navigation systems has profound impacts on the routing of traffic users. With the assistance of these intelligent navigation systems, traffic users can obtain more accurate information about a traffic network such as traffic capacities, feasible paths, congestion status, etc. In this paper, we focus on a game-theory-based traffic congestion analysis model which considers incomplete traffic information (e.g., variabilities of path information) generated by intelligent navigation systems. The variabilities of path information are treated as incomplete information associated with different subsets of arcs. We adopt the notions of user equilibrium with incomplete information (UEII) and system optimum with incomplete information (SOII) in this study. Based on these two new notions, we extend two classical theorems and combine them into a new model to analyze the relationship between UEII and SOII. Finally, numerical cases are given to illustrate the implication of UEII and SOII in practical implementations.

Appendices

Appendix A: Illustration of Braess paradox

Figure 2 is drawn to illustrate a small-size BP example, in which the label on each arc is a travel time–cost function associated with the traffic amount. Suppose that the traffic amount consists of 1000 users who will travel from the source node $S$ to the terminal node $T$. All users are acting selfishly to find the minimum-cost path and they are indifferent to the total time-cost.

Fig. 2

Illustration of Braess paradox by a numerical example. a The original network. b The updated network by adding a new arc

In Fig. 2a, there are two paths, $S \to W \to T$ and $S \to V \to T$, which can be used to direct users from source node $S$ to terminal node $T$. Because of the symmetry of the traffic network structure, half of the users choose the path $S \to W \to T$ and another half select the path $S \to V \to T$. In this status, each user can not choose another path to reduce its time cost. The equilibrium status is user equilibrium (UE), and each user spends 70 time units form $S$ to $T$. Because there is no traffic assignment that can minimize the total time cost of all users, the system optimum (SO) is the same as the UE, as demonstrated in Fig. 2a.

In Fig. 2b, a zero-cost arc is constructed to connect node $W$ and node $V$ and the new arc will affect all users’ route selection. When arc $W \to V$ is added, there are three paths that can be used to direct users from $S$ to $T$: $S \to W \to T$, $S \to W \to V \to T$, $S \to V \to T$. We call them $P_{1}$,$P_{2}$,$P_{3}$ respectively. Suppose that there are users that choose path $P_{i}$, $i = 1,2,3$. The number of users on each arc is calculated in Table 5. Based on the number of users on each arc, the time cost on each path is summarized in Table 6.

Table 5 A summary of the number of users on each arc in Fig. 2b

Table 6 A summary of the time cost of each path in Fig. 2b

As shown in Table 6, because $d_{1} + d_{2} \le 1000$ and $d_{2} + d_{3} \le 1000$, the time cost of path $P_{2}$ always be less than those of paths $P_{1}$ and $P_{3}$. Therefore, all users choose path $P_{2}$ and each user spends 80 time units from $S$ to $T$. The newly added arc and more paths may lead to more time costs in UE. This situation is regarded as the well-known Braess Paradox (BP). For clarity, the values of UEs (User equilibriums) and SOs (System optimums) of this example are summarized in Table 7.

Table 7 A summary of UEs and SOs in Fig. 2

Appendix B: Proof of Theorem 2

Based on an assumption that each time-cost function $c^{a} \left( x \right), a \in A$ is differentiable, the NMP model satisfies the conditions of the Karush–Kuhn–Tucker Theorem (Bertsekas 2016). Then, for all $i \in K,p \in P_{i}$, there are parameters $\mu_{i}^{p} \ge 0$ and $\lambda_{i}$ such that:

$$\frac{\partial }{{\partial f_{i}^{p} }}\left( {\mathop \sum \limits_{a \in A} t^{a} \left( {f^{a} } \right) - \mathop \sum \limits_{i = 1}^{K} \lambda_{i} \left( {\mathop \sum \limits_{{p \in P_{i} }} f_{i}^{p} - d_{i} } \right) - \mathop \sum \limits_{i = 1}^{K} \mathop \sum \limits_{{p \in P_{i} }} \mu_{i}^{p} f_{i}^{p} } \right) = 0$$

where $f^{a} = \sum\nolimits_{j = 1}^{K} {\sum\nolimits_{{p \in P_{j} :a \in p}} {f_{j}^{p} } }$ and $\mu_{i}^{p} = \left\{ {\begin{array}{*{20}c} { = 0} & {{\text{if}}\;f_{i}^{p} > 0} \\ { \ge 0} & {{\text{if}}\;f_{i}^{p} = 0} \\ \end{array} } \right..$

Note that $\sum\nolimits_{a \in A} {\frac{{\partial f^{a} }}{{\partial f_{i}^{p} }}\left( {t^{a} \left( {f^{a} } \right)} \right)^{'} = \left( {t^{p} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'} }$. Then, we have the following two cases: if $f_{i}^{p} > 0$, $\left( {t^{p} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'} = \lambda_{i}$ and if $f_{i}^{p} = 0$, $\left( {t^{p} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'} - \lambda_{i} = \mu_{i}^{p} \ge 0$, that is,

$$\left( {t^{p} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'} = \left\{ {\begin{array}{*{20}c} { = \lambda_{i} } & {{\text{if}}\;f_{i}^{p} > 0} \\ { \ge \lambda_{i} } & {{\text{if}}\;f_{i}^{p} = 0} \\ \end{array} .} \right.$$

Therefore, for each $i \in K$ and each pair of paths $p_{1} ,p_{2} \in P_{i}$ with $f_{i}^{{p_{1} }} > 0$, we have $\left( {t^{{p_{1} }} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'} \le \left( {t^{{p_{2} }} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'}$.

Suppose that $\left( {t^{p} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'}$ is $\lambda_{i}$ for each type $i$; and $\left( {t^{{p_{1} }} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'} \le \left( {t^{{p_{2} }} \left( {{\mathbb{F}}_{K} } \right)} \right)^{'}$ for each $i \in K$ and each pair of paths $p_{1} ,p_{2} \in P_{i}$ with $f_{i}^{{p_{1} }} > 0$, we have

$$\mu_{i}^{p} = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if}}\quad f_{i}^{p} > 0} \\ {(t^{p} ({\mathbb{F}}_{K} ))^{'} } - \lambda_{i} & {{\text{if}}\quad f_{i}^{p} > 0} \\ \end{array} .} \right.$$

and

$$\mu_{i}^{p} = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if}}\quad f_{i}^{p} > 0} \\ {(t^{p} ({\mathbb{F}}_{K} ))^{'} } & {{\text{if}}\quad f_{i}^{p} > 0} \\ \end{array} .} \right.$$

In addition, we have $\mu_{i}^{p} \ge 0$ and

$$\frac{\partial }{{\partial f_{i}^{p} }}\left( {\mathop \sum \limits_{a \in A} t^{a} \left( {f^{a} } \right) - \mathop \sum \limits_{i = 1}^{K} \lambda_{i} \left( {\mathop \sum \limits_{{p \in P_{i} }} f_{i}^{p} - d_{i} } \right) - \mathop \sum \limits_{i = 1}^{K} \mathop \sum \limits_{{p \in P_{i} }} \mu_{i}^{p} f_{i}^{p} } \right) = 0.$$

Because of the convexity of the objective function and the affine property of constraints, the so-called Karush–Kuhn–Tucker conditions are satisfied, implying that the feasible flow pattern ${\mathbb{F}}_{K}$ is at system optimum with incomplete information.