Journal of Advanced Transportation

Volume 2018, Article ID 9876598, 20 pages

https://doi.org/10.1155/2018/9876598

## Effects of Users’ Bounded Rationality on a Traffic Network Performance: A Simulation Study

^{1}Univ. Lyon, ENTPE, IFSTTAR, LICIT, F-69518, Lyon, France^{2}School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

Correspondence should be addressed to Ludovic Leclercq; rf.eptne@qcrelcel.civodul

Received 16 March 2018; Revised 27 June 2018; Accepted 12 August 2018; Published 10 September 2018

Academic Editor: Zhi-Chun Li

Copyright © 2018 S. F. A. Batista et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we revisit the principle of bounded rationality applied to dynamic traffic assignment to evaluate its influences on network performance. We investigate the influence of different types of bounded rational user behavior on (i) route flows at equilibrium and (ii) network performance in terms of its internal, inflow, and outflow capacities. We consider the implementation of a bounded rational framework based on Monte Carlo simulation. A Lighthill-Whitham-Richards (LWR) mesoscopic traffic simulator is considered to calculate time-dependent route costs that account for congestion, spillback, and shock-wave effects. Network equilibrium is calculated using the Method of Successive Averages. As a benchmark, the results are compared against both Deterministic and Stochastic User Equilibrium. To model different types of bounded rational user behavior we consider two definitions of user search order (indifferent and strict preferences) and two settings of the indifference band. We also test the framework on a toy Braess network to gain insight into changes in the route flows at equilibrium for both search orders and increasing values of aspiration levels.

#### 1. Introduction

The first notions of traffic assignment were introduced by Wardrop [1]. According to the first Wardrop principle, users aim to minimize their personal route travel times. This leads to a network equilibrium called the Deterministic User Equilibrium (DUE) and it is that most commonly used in dynamic traffic assignment (DTA) problems. Under DUE conditions, no user can decrease his/her own travel time by unilaterally switching routes. However, the first Wardrop principle assumes that users are perfectly rational and perceive all routes and network traffic states perfectly although information on route travel times (i.e., traffic states) is not necessarily perfect. To overcome this problem, Daganzo and Sheffi [2] and Daganzo [3] introduced the Stochastic User Equilibrium (SUE), to take into account the uncertainty of route travel times. The Multinomial Logit and C-Logit are the Random Utility models (RUM) most commonly used in DTA problems. Nonetheless, both these models present several limitations when dealing with correlations between routes. In this study we focus in particular on the Probit model solved using Monte Carlo simulations [4].

Revealed [5] and stated [6] preference surveys show that users tend to choose suboptimal routes instead of optimal ones [7]. We emphasize that a suboptimal route is understood as a route with a longer travel time than the minimum one for the origin-destination (od) pair. In the literature on static traffic assignment, there are other alternative model frameworks that take into account different types of user behavior. One example is the Prospect Theory [8, 9] which considers the users risk-seeking and risk-aversion behavior. It was adapted to the context of route choice by Avineri [10]. In the Prospect Theory, users evaluate the different routes in terms of time prospect and choose the route with the maximum prospect. Users are risk-averse when confronted with prospects of gains and risk-seekers when confronted with prospects of losses and are more sensitive to losses than gains (*loss effect*). Another example is the Regret Theory [11, 12]. The users aim to minimize their regret with respect to the nonselected routes [13, 14]. If the users choose the route with the minimum travel time, they will feel joy or feel regret otherwise. Another example is the notion of bounded rationality introduced by the seminal works of Simon [15–18]. He stated that users choices are driven by aspiration levels (), which represent a set of goal or target variables that should be achieved or exceeded for the users satisfaction. In his original idea, the user searches until a satisfactory alternative is found. This term used to describe this process was coined by Simon as* satisficing*, which stands for the combination of satisfy and suffice. In this study, we focus on the application of the notion of bounded rationality in a dynamic context, by considering distributions of route travel times and a traffic simulator. The goal of this study is to investigate the influence of bounded rational user behavior on individual route flows and network performance. This type of study is very important for decision-making in transportation planning.

Mahmassani and Chang [22] discussed the first notion of bounded rationality applied to traffic assignment, but no mathematical formulation was given. To define users , Mahmassani and Chang [22] introduced the concept of* indifference band* (IB), where a route is* satisficing* if the difference between its travel cost and that of the best available route is lower than a given threshold (or IB). The implementation of bounded rationality in traffic assignment is challenging as (i) the calibration of the is context dependent [23] and (ii) the BR-UE solutions are not unique [24–26]. Thus, to analyze the BR-UE solutions, some authors have focused on the analysis of the best and worst BR-UE flows of the network [24, 25, 27]. Moreover, the can change from user to user. A thorough review of bounded rationality in traffic assignment was provided in Di and Liu [28]. There are two main ingredients that dictate bounded rational network equilibrium: (i) the definition of the that dictates whether a route is* satisficing* or not and (ii) the users search order that defines how users are guided in their choice of a* satisficing* route.

For a route to be considered as* satisficing*, its route utility must satisfywhere is the perceived route utility; is the aspiration level we consider in this paper, to be defined at the od level; is the route choice set for the od pair; and is the set of all od pairs of the network.

The can be calibrated exogenously by route choice surveys or calibrated endogenously by explicit formulations. The most commonly used definition is based on the concept of indifference band [22, 29, 30]:where is a vector containing all deterministic route utilities , and contains routes. is the tolerance or IB at the od level.

Ge and Zhou [20] propose a variable definition of the IB ():where and are dummy variables that equal 1 if routes and belong to , respectively.

Ge et al. [31] analyzed the BR-DUE equilibrium, considering exogenously fixed and fixed and endogenously variable . In their model framework, the authors showed that the DUE is a special case of the BR-DUE and discussed the existence conditions of the BR-DUE. However, the uncertainty on the travel times was not considered.

Di and Liu [28] highlighted that a bounded rational behavior can be due to the users habits and inertia or their cognitive costs or individual preferences. In this paper, we focus our attention on the users preferences as a bounded rational behavior to define the search order for the* satisficing* alternatives. Zhao and Huang [19] defined a search order based on a strict preference order for all users sharing the same od pair. This strong assumption allowed obtaining unique BR-UE solutions. To the authors knowledge, the framework of Zhao and Huang [19] has never been tested in a dynamic context, i.e., considering a traffic simulator and time-dependent path costs. In addition, its dynamic implementation using a traffic simulator is highly challenging because it requires solving suboptimization problems to calibrate the of the sub-most preferred routes. Thus, a framework capable of solving the global optimization problem is required and discussed further on in this paper. On the other hand, users may also have an indifferent preference for any of the* satisficing* routes (i.e., that satisfy (1)). This is adopted from the notions discussed in [32]. In this case, we consider the fact that all users sharing the same od pair have a similar indifference preference. The choice is modeled by uniform random sampling of any of the* satisficing* routes. Users are then assigned to the* satisficing* route sampled.

Szeto and Lo [30] (the BR-UE [22] and Tolerance-Based Dynamic User Optimum Principle [30] have been used interchangeably in the traffic assignment literature; for the sake of simplicity, we refer to both as BR-UE) discussed an analytical BR-UE dynamic traffic assignment model. The authors proposed a route swapping algorithm, but no clear definition of the users search order was discussed. Instead, the authors targeted certain users on the most congested routes and switched them to less congested ones for each od pair. Moreover, the BR-UE solutions were not unique. Han et al. [33] discussed a dynamic simultaneous departure time and route choice bounded rational framework. However, neither of these frameworks included travel time distributions. In this paper, we revisit the notions of bounded rationality by considering the distribution of travel times rather than deterministic values.

The literature includes a large number of applications of a bounded rational framework to static [25] and dynamic traffic assignment [30, 33], transportation planning [34], traffic policy-making [35], congestion pricing [24], and traffic safety [36]. However, to the authors knowledge, there is no study in the literature that investigates the influence of users preferences (indifferent and strict) for a bounded rational behavior on individual route flows and network performance in terms of the internal level of congestion and inflow and outflow capacities. The goal of this paper is to fill this gap. We consider time-dependent path costs that account for congestion, shock-waves, and spillback effects calculated using a mesoscopic Lighthill-Whitham-Richards (LWR) model [21]. A spillback effect is the reduction of a link capacity that spreads over other connected links in the network. To model bounded rationality behavior, we relax the definition of the search order of the DUE and SUE frameworks [4]. In both the DUE and SUE cases, users are assigned to the routes with the minimum travel times based on an all-or-nothing procedure. The search order is relaxed to account for the users indifferent and strict preferences. In the case of the indifferent preference search order, users present indifference behavior when choosing any of the* satisficing* routes, whereas in the case of the strict preference search order [19], users are assigned to the most preferred route if this route is perceived as* satisficing* (see (1)), or to the first sub-most preferred route that satisfies (1). We make use of Monte Carlo simulations [4] to account for travel times distribution and consider the classical Method of Successive Averages to calculate the network equilibrium. First, we test the bounded rationality methodology in a toy Braess network and consider a simple linear static and flow dependent utility function. We then consider the two settings of the search order previously mentioned and the defined exogenously. These initial tests allow acquiring insight into how the route flows at equilibrium change according to the two definitions of the search order and increasing values of . Second, for the dynamic implementation, we also consider the two settings of the users search order (i.e., indifferent and strict preferences) and the concept of the IB ((2) and (3)) to define the . The dynamic tests are performed on a Manhattan network. We investigate the influence of the definition of the search order on the individual route flows and analyze the network performance in terms of the internal, inflow, and outflow capacities, given the two search orders and different values of the . The results are compared against both DUE and SUE as benchmarks.

This paper is organized as follows. In Section 2, we discuss the bounded rational model framework considered in this paper. In Section 3, we discuss a simple static test scenario on the Braess network, considering both the indifferent and strict preferences search order. In Section 4, we discuss the influence of the bounded rationality behavior on the network performance also considering the two search orders. In Section 5, we outline the conclusions of this paper.

#### 2. Bounded Rational Framework

The analysis of the effect of users’ behavior on network performance in terms of its internal inflow and outflow capacities is very important for policy-makers, in particular when determining policies aimed at increasing network performance. In this paper, we focus on two types of bounded rational user behavior.

We start by introducing the general formulation of the route utilities. The perceived route utility, , iswhere is the deterministic route utility and is the uncertainty or error term as often referred to in the literature.

The DUE assumes that users are utility minimizers and the error terms are set to 0. Users are assigned based on an all-or-nothing procedure to the route with the minimum travel time. In the case of the SUE, users are also utility minimizers, but they perceive travel times with uncertainties, meaning that the error terms are not 0. Theoretically, the Probit model [2] is the most attractive model for solving the SUE. However, it requires the computation of a covariance matrix and integrating the multivariate normal distribution. The complexity of the computation increases with the number of routes per od pair. An alternative to this is to use Monte Carlo to consider the distributions of route travel times [4]. We consider that the error terms are defined at the link level (i.e., ) instead at the route level. This allows capturing existing correlations between different routes sharing the same links. Sheffi [4] proposed to consider link travel times that are normal distributed and to truncate the negative travel times. This skews the distributions and a positive defined travel times distribution is to be preferred. In this paper, we consider that the terms are gamma distributed following Nielsen [37]. The principle of the Monte Carlo simulations is to discretize the error terms into samples or draws and locally solve DUE problems. For each error draw, the deterministic utility for route is defined as . In short, DUE is solved one for all considering the utility for each route. The SUE corresponds to the average of Monte Carlo trials, where the utilities are adjusted for each trial considering the link error terms values. It is sufficient to explain how we extend DUE to account for bounded rational (BR-DUE), as the extension of SUE is exactly similar; i.e., we have to solve a BR-DUE problem for each Monte Carlo trial of link error terms and then average the results to get the BR-SUE solutions. Mahmassani and Chang [22] introduced the first notions of bounded rationality applied to route choice. Lou et al. [24] and Di et al. [25] formulated the BR-DUE mathematically, but without defining preference rules among* satisficing* routes. Under BR-DUE, all users are satisfied with their choices and no longer consider switching routes. It should be noted that the DUE is an extreme case of the BR-DUE (i.e., when ). The idea is that users are assigned to* satisficing* routes instead of optimal routes (i.e., routes with the minimum perceived travel times). A* satisficing* route should satisfy (1); i.e., . Let us define as the set of* satisficing* routes. The users are then assigned to one route depending on the preference rule. In this paper, we consider that the search order is defined according to the user’s preferences (indifferent and strict preferences, see below). The previous description of BR-SUE was given based on Monte Carlo trials as it helps to make the connection with BR-DUE. We can also provide a formal definition of the BR-SUE independently of the solution method. Let be the probability that route is chosen for one od pair:

The second part of the equation is the conditional probability of route being chosen when is perceived as* satisficing*. Its value depends on the preference rule. We consider two behavioral rules to represent the user’s search order. We consider indifferent preferences, where users are uniformly assigned to* satisficing* routes. In this case, , where is the number of* satisficing* routes listed in . This number is at least equal to 1, as route is* satisficing*. A close form for this probability cannot be provided as the number of other* satisficing* routes than is a random variable whose distribution is intractable in the general case. Interestingly, Monte Carlo trials for link error terms make the calculation of straightforward, for each Monte Carlo iteration step since all* satisficing* paths are then known. We also consider a strict preference order [19], where users are selecting a* satisficing* route according to a predefined list of preferences . The users are always choosing the* satisficing* route of highest rank in this list . Again, the probability does not have a closed form, but Monte Carlo trials permit determining the unique assignment at each iteration.

The idea of the strict preference order was introduced by Zhao and Huang [19], to deal with the nonuniqueness of the equilibrium solution. However, we highlight two main differences between our methodology and that discussed in Zhao and Huang [19]. First, we consider the fact that routes are* satisficing* if and only if their perceived utility satisfies for the BR-DUE or for the BR-SUE. Zhao and Huang [19] consider that routes are* satisficing* according to the strict preference order; i.e., the users are first assigned to the most preferred route and then consecutively to the sub-preferred routes, until all the users are assigned. Second, we consider that is defined at the od level (i.e., ), while Zhao and Huang [19] consider its definition at the route level. We also assume that all users sharing the same od pair have the same . We consider that it is more realistic from the user’s perspective to set a global instead of establishing for the sub-preferred routes based on the most preferred ones.

In this paper, we consider the two definitions of as defined in (2) and (3).

To reach a solution for the BR-SUE and simulate the probability term (route k is satisficing) (see (5)), we consider Monte Carlo simulations as discussed in Sheffi [4] and the classical Method of Successive Averages (MSA). The MSA solves a fixed point problem and is commonly used in traffic assignment to solve both the DUE and SUE [4]. The Monte Carlo simulations consist in discretizing the distributions of the link travel times into samples or draws and solving BR-DUE problems locally. For each discretization, we identify the* satisficing* routes and assign the users based on an all-or-nothing assignment following the search order established. If the search order is considered to be the indifferent preferences, all the users are assigned randomly to any of the* satisficing* routes. On the other hand, if the search order follows a strict user preference order, all users are assigned to the first* satisficing* route found on this strict sequence of preferences. The new temporary route flows, , correspond to averaging all the local BR-DUE solutions. This corresponds to the temporary route flows , which will be used to update the new route flows at iteration , aswhere represent the route flows at iteration of the MSA and is the descent step. This process is repeated at every descent step of the MSA algorithm.

The sequence of descent steps guarantees the convergence of the MSA. For the theoretical convergence of the algorithm, the following two conditions must be satisfied [4]:

One definition of that satisfies both of the previous conditions is . We consider this definition of in this paper. Other definitions of the descent step size are discussed in the literature [38–41].

A commonly used convergence or stopping criterion is based on the comparison between the current and the previous descent step of the MSA that must be lower to a predefined threshold. Instead we consider the number of violations and the relative Gap [42]. represents the number of cases where is higher than a predefined path convergence threshold . Note that represents an upper bound. The convergence of the algorithm is achieved if . The relative Gap for the DUE is as follows [42]:where is the total demand for the od pair; and is the average travel time of route ; and is the minimum route travel time for the od pair.

The Gap function (see (9)) represents the difference between the travel costs and the equilibrium travel costs. Thus, under perfect DUE conditions, . This means that all users choose the routes with the minimum travel times. Under SUE conditions, the Gap is higher than 0, however small. In this case not all users choose the routes with the minimum travel times. In the case of bounded rationality, the Gap value increases as increases. The Gap function is also a measure of how close users are to the equilibrium route travel times (or ). The definition of the Gap as defined in (9) is valid for DUE and SUE and informs on how far we are from the DUE. For both the BR-DUE and BR-SUE convergence, we modify the Gap function as follows:

Thus, under BR-DUE or BR-SUE conditions, the Gap is about 0 if and the equilibrium condition is fulfilled. Note that, throughout the paper, we use the definition of the Gap as in (9) as an indicator that measures how far the bounded rational equilibria are from the DUE and (10) as the equilibrium convergence criterion for the MSA.

We present the solution algorithm of this framework in Algorithm 1. Note that the difference between Algorithm 1 and that proposed by Sheffi [4] is that we assign the users to* satisficing* routes instead of routes with the minimal travel times. They are assigned to these* satisficing* routes according to one of the search orders discussed previously (i.e., indifferent or strict preferences) at every descent step of the MSA. The first step before entering the MSA loop consists in calculating the route choice set , for each od pair. It defines the set of routes for the users choices. We then perform an initial loading on these routes and consider the number of violations, the Gap (see (10)), and the maximum number of iterations for the MSA convergence criteria. The corresponding , , and are set. It is also necessary to define the input scale () and shape () parameters of the link travel time gamma distributions for the first Monte Carlo simulations. We then enter in the MSA loop and the is first updated based on the average route travel times (see (2) or (3)). The next step consists in performing the link error sampling considering the and parameters. This is done through Monte Carlo simulations. The algorithm then loops over all the error samples and locally solves the BR-DUE problems. For each sample, the route utilities are computed to identify the* satisficing* routes based on . This defines the* satisficing* set of routes . Users are assigned based on the predefined search order (indifferent or strict preferences) based on an all-or-nothing procedure to one route in . It should be noted that in the case of solving the BR-SUE and taking the indifferent preferences into account, it is necessary to repeat the all-or-nothing assignment on the* satisficing* routes times. The users choices for the local BR-DUE correspond to averaging the previous choices. By applying the law of large numbers, when is large, we converge to the same average values. The new temporary route flows correspond to the average of all the local BR-DUE choices. The new route flows are updated according to (6) and the network loading is updated. To determine time-dependent link costs that consider congestion, shock-waves, and spillback effects, we run an LWR mesoscopic traffic simulator [21]. The link travel time distributions are obtained based on the simulated vehicle travel times. To update and , we fit a gamma distribution to each link travel time distribution. The updated values of and will be used to perform the error samplings in the next MSA descent step. The (see (10)) and number of violations are updated based on the new average route travel times through the individual vehicles travel times. This process is repeated until convergence is achieved. Note that Algorithm 1 also allows solving the BR-DUE by setting and .