Boundedly Rational User Equilibrium Research Articles

Influence network design via multi-level optimization considering boundedly rational user behaviours in social media networks

Social media networks have been playing an increasingly more important role for both socialization and information diffusion. Political campaign can gain more supporters by attracting more mass attention and influencing them directly, while commercial campaigns can increase their companies’ profits by expanding social media connection with new users. To build the optimal network structure to influence the whole, this paper studies mathematical models to simulate the users’ behaviours interacting with others in the information provider’s network. The behaviours of concerns include information re-posting and following/unfollowing other users. Linear threshold propagation model is used to determine the re-posting actions, Boundedly Rational User Equilibrium (BRUE) models are used to determine the following or unfollowing actions. Hence, the topology of the network changes and depends on the information provider’s plan to post various kinds of information. A three-level optimization model is proposed to maximize total number of connections, the goal of the top level. The second level simulates user behaviours under BRUE. The third level maximizes the each user’s utility defined in the second level. This paper solves this problem using exact algorithms for a small-scale synthetic network. For a large-scale problem, this paper uses heuristic algorithms based on large neighbourhood search. This paper also discusses possible reasons why the BRUE model may be a more accurate simulation of users’ actions compared to game theory. Comparisons from the BRUE model to game theoretical model show that the BRUE model performs significantly better than game theoretical model.

Contribution to the field of traffic assignment: A boundedly rational user equilibrium model with uncertain supply and demand

Since the existing models and algorithms cannot well deal with the traffic assignment problems under uncertain supply and demand conditions, this paper proposes a traffic assignment model based on cumulative prospect theory. Firstly, the cumulative prospect theory is extended on the basis of a large number of numerical analysis experiments and the road impedance function of road segments is improved. Then, by setting the reference point of fuzzy generalized cost and the reference point of staged dynamic risk degree of road segment, the comprehensive cumulative prospect value (CCPV) function is improved, a boundedly rational user equilibrium (BRUE) model under uncertain supply and demand conditions is constructed, and an isolation niche genetic simulated annealing algorithm (INGSAA) is designed to solve the model. Finally, taking the road network of the Sichuan-Tibet region in China as an example, the processes of boundedly rational equilibrium under the fixed and variable network structure are investigated, and the relevant parameters are analyzed. The research results verify that the BRUE model based on the extended cumulative prospect theory provides a good idea for solving the problem of traffic assignment under uncertain supply and demand conditions, which is of great theoretical significance and application value.

The Boundedly Rational User Equilibrium: A parametric analysis with application to the Network Design Problem

In this paper, we study a static traffic assignment that accounts for the boundedly rational route choice behavior of travelers. This assignment induces uncertainties to the ex-ante evaluation of a policy measure: the boundedly rational assignment is non-unique and the indifference band is an uncertain parameter. We consider two different ways to model the optimization problem that finds the best and worst-performing Boundedly Rational User Equilibrium with respect to the total travel time (Best/Worst-case BRUE). The first is the so-called branch approach, the second is a bilevel model. The latter approach is better suited to exploit techniques from parametric optimization and enables us, e.g., to prove the continuity of the optimal value function corresponding to the Best/Worst-case BRUE with respect to perturbations in the indifference band. We report on some numerical experiments. In addition, we extend our results to the Network Design Problem: we prove the existence of a second-best toll pricing scheme under bounded rationality.

Day-to-day departure time choice under bounded rationality in the bottleneck model

We extend the proportional swap system by Smith (1984a) and the network tatonnement process by Friesz et al. (1994) to formulate the day-to-day evolution of departure time choice under bounded rationality in the bottleneck model. It is shown that the stationary points of the doubly dynamical systems exist and are equivalent to the boundedly rational user equilibrium (BRUE) points. We propose a dynamic pricing policy implemented in the day-to-day evolution process. Despite that the stationary points correspond to a set of BRUE points and the day-to-day departure rate without pricing may not converge to the set of BRUE points, the pricing policy can drive the day-to-day departure rate to reach a new equilibrium state, which is system-optimal (SO). The convergence to the SO state is theoretically proved. Finally, by numerical examples, we demonstrate the properties of the doubly dynamical systems and the effectiveness of the pricing policy.

Travel time reliability with boundedly rational travelers

ABSTRACTThis paper presents a definition of boundedly rational confidence level (BRCL) which is the probability that a trip arrives within the shortest travel time budget plus the acceptable travel time difference (i.e. boundedly rational threshold). Using the method of data fitting, boundedly rational threshold is estimated. Then, a reliability-based boundedly rational user equilibrium (R-BRUE) model that explicitly considers both travel time reliability and travelers’ bounded rationality in the route choice decision process is proposed. This new model hypothesizes that for each origin–destination (O–D) pair no traveler can improve his/her BRCL by unilaterally changing routes. A route-based quasi-method of a successive average algorithm is developed to solve the R-BRUE model. Numerical examples illustrate the essential ideas of the proposed model and the applicability of the proposed solution algorithm.

Rational Behavior Adjustment Process with Boundedly Rational User Equilibrium

This paper extends the framework of “rational behavior adjustment process” (RBAP) to incorporating the “boundedly rational user equilibrium” (BRUE). The proportional-switch adjustment process (PSAP) and the network tatonnement process (NTP) are extended to the BRUE case, and their dynamical equations are shown to be Lipschitz continuous, which guarantees the global uniqueness of the classical solutions. A special group of the BRUE-RBAP is proposed, for which the path flows would increase if the paths are in an acceptable path set, and would decrease otherwise. Classical solutions to this special group of models may not exist. Stability of the BRUE-RBAP with classical solutions is proved with separable link travel cost functions. For nonseparable link travel cost functions, the stability of the BRUE-PSAP is proved. Numerical examples are presented to demonstrate the evolution processes of BRUE-PSAP and BRUE-NTP under various bounded rationality thresholds and different initial states. The applicability of BRUE-PSAP in larger networks with asymmetric link travel cost functions is also illustrated.

Day-to-day departure time choice under bounded rationality in the bottleneck model

Abstract We extend the proportional swap system by Smith (1984a) and the network tatonnement process by Friesz et al. (1994) to formulate the day-to-day evolution of departure time choice under bounded rationality in the bottleneck model. It is shown that the stationary points of the doubly dynamical systems exist and are equivalent to the boundedly rational user equilibrium (BRUE) points. We propose a dynamic pricing policy implemented in the day-to-day evolution process. Despite that the stationary points correspond to a set of BRUE points and the day-to-day departure rate without pricing may not converge to the set of BRUE points, the pricing policy can drive the day-to-day departure rate to reach a new equilibrium state, which is system-optimal (SO). The convergence to the SO state is theoretically proved. Finally, by numerical examples, we demonstrate the properties of the doubly dynamical systems and the effectiveness of the pricing policy.

The Restricted Stochastic User Equilibrium with Threshold model: Large-scale application and parameter testing

This paper presents the application and calibration of the recently proposed Restricted Stochastic User Equilibrium with Threshold model (RSUET) to a large-scale case-study. The RSUET model avoids the limitations of the well-known Stochastic User Equilibrium model (SUE) and the Deterministic User Equilibrium model (DUE), by combining the strengths of the Boundedly Rational User Equilibrium model and the Restricted Stochastic User Equilibrium model (RSUE). Thereby, the RSUET model reaches an equilibrated solution in which the flow is distributed according to Random Utility Theory among a consistently equilibrated set of paths which all are within a threshold relative to the cost on the cheapest path and which do not leave any attractive paths unused. Several variants of a generic RSUET solution algorithm are tested and calibrated on a large-scale case network with 18,708 arcs and about 20 million OD-pairs, and comparisons are performed with respect to a previously proposed RSUE model as well as an existing link-based mixed Multinomial Probit (MNP) SUE model. The results show that the RSUET has very attractive computation times for large-scale applications and demonstrate that the threshold addition to the RSUE model improves the behavioural realism, especially for high congestion cases. Also, fast and well-behaved convergence to equilibrated solutions among non-universal choice sets is observed across different congestion levels, choice model scale parameters, and algorithm step sizes. Clearly, the results highlight that the RSUET outperforms the MNP SUE in terms of convergence, calculation time and behavioural realism. The choice set composition is validated by using 16,618 observed route choices collected by GPS devices in the same network and observing their reproduction within the equilibrated choice sets generated by the RSUET model. Relevantly, the RSUET model is very successful in reproducing observed link.

A discrete dynamical system of formulating traffic assignment: Revisiting Smith’s model

Through relaxing the behavior assumption adopted in Smith’s model (Smith, 1984), we propose a discrete dynamical system to formulate the day-to-day evolution process of traffic flows from a non-equilibrium state to an equilibrium state. Depending on certain preconditions, the equilibrium state can be equivalent to a Wardrop user equilibrium (UE), Logit-based stochastic user equilibrium (SUE), or boundedly rational user equilibrium (BRUE). These equivalence properties indicate that, to make day-to-day flows evolve to equilibrium flows, it is not necessary for travelers to choose their routes based on actual travel costs of the previous day. Day-to-day flows can still evolve to equilibrium flows provided that travelers choose their routes based on estimated travel costs which satisfy these preconditions. We also show that, under a more general assumption than the monotonicity of route cost function, the trajectory of the dynamical system converges to a set of equilibrium flows by reasonably setting these parameters in the dynamical system. Finally, numerical examples are presented to demonstrate the application and properties of the dynamical system. The study is helpful for understanding various processes of forming traffic jam and designing an algorithm for calculating equilibrium flows.

Experiment of boundedly rational route choice behavior and the model under satisficing rule

In this paper, we study the boundedly rational route choice behavior under the Simon’s satisficing rule. A laboratory experiment was carried out to verify the participants’ boundedly rational route choice behavior. By introducing the concept of aspiration level which is specific to each person, we develop a novel model of the problem in a parallel-link network and investigate the properties of the boundedly rational user equilibrium (BRUE) state. Conditions for ensuring the existence and uniqueness of the BRUE solution are derived. A solution method is proposed to find the unique BRUE state. Extensions to general networks are conducted. Numerical examples are presented to demonstrate the theoretical analyses.

Boundedly Rational User Equilibrium with Restricted Unused Routes

A boundedly rational user equilibrium model with restricted unused routes (R-BRUE) considering the restrictions of both used route cost and unused route cost is proposed. The proposed model hypothesizes that for each OD pair no traveler can reduce his/her travel time by an indifference band by unilaterally changing route. Meanwhile, no route is unutilized if its travel time is lower than sum of indifference band and the shortest route cost. The largest and smallest used route sets are defined using mathematical expression. We also show that, with the increase of the indifference band, the largest and smallest used route sets will be augmented, and the critical values of indifference band to augment these two path sets are identified by solving the mathematical programs with equilibrium constraints. Based on the largest and smallest used route sets, the R-BRUE route set without paradoxical route is generated. The R-BRUE solution set can then be obtained by assigning all traffic demands to the corresponding generated route set. Various numerical examples are also provided to illustrate the essential ideas of the proposed model and structure of R-BRUE route flow solution set.

Braess paradox under the boundedly rational user equilibria

The Braess paradox and its variants have been studied under the perfectly rational behavior assumption. However, when the perfect rationality assumption is relaxed to bounded rationality, which assumes that travelers can take any route whose travel cost is within an ‘indifference band’ of the shortest path cost, it remains unclear under what conditions the Braess paradox occurs. This paper fills this gap by exploring relationships between the occurrence of the Braess paradox and the indifference band as well as the demand level in the setting of the boundedly rational user equilibrium (BRUE). The definition of the Braess paradox is extended based on planners’ risk-taking attitudes, i.e., risk-averse, risk-prone and risk-neutral, due to the non-uniqueness of BRUE. The paradox occurrence conditions under different risk-taking attitudes are investigated using the classical Braess network and compared with those under the user equilibrium. Then we generalize the paradox conditions to simple and ordinary grid networks with regular Bureau of Public Roads (BPR) link performance functions. The impact of the link cost congestion sensitivity along with the indfference band on the occurrence of the Braess paradox is also studied.

Toll sequence operation to realize target flow pattern under bounded rationality

Because boundedly rational user equilibrium (BRUE) always has a set of solutions instead of a unique one, from a static network equilibrium viewpoint, under BRUE there is no guarantee of attainability of any specific target flow by implementing tolls. In this study, from a disequilibrium flow evolution perspective, we design toll sequence operations (TS-operations) to guide the network flow to evolve towards the traditional Wardrop user equilibrium (UE) flow pattern. Under homogeneous bounded rationality (BR), iteratively implementing our TS-operations can make the network flow pattern converge to UE, which essentially solves the nonuniqueness problem of BRUE and re-establishes the effectiveness of link tolls in realizing any target link flow pattern. In particular we show that under homogenous BR the best-case untolled link-based BRUE can be realized as the untolled equilibrium. Under heterogeneous BR among different OD pairs, our TS-operations can make the flow converge to reduced BRUE and/or sub-network UE, which give smaller estimate intervals of the equilibrium flow pattern as compared to the original BRUE.

Bounded rationality and irreversible network change

A network change is said to be irreversible if the initial network equilibrium cannot be restored by revoking the change. The phenomenon of irreversible network change has been observed in reality. To model this phenomenon, we develop a day-to-day dynamic model whose fixed point is a boundedly rational user equilibrium (BRUE) flow. Our BRUE based approach to modeling irreversible network change has two advantages over other methods based on Wardrop user equilibrium (UE) or stochastic user equilibrium (SUE). First, the existence of multiple network equilibria is necessary for modeling irreversible network change. Unlike UE or SUE, the BRUE multiple equilibria do not rely on non-separable link cost functions, which makes our model applicable to real-world large-scale networks, where well-calibrated non-separable link cost functions are generally not available. Second, travelers’ boundedly rational behavior in route choice is explicitly considered in our model. The proposed model is applied to the Twin Cities network to model the flow evolution during the collapse and reopening of the I-35 W Bridge. The results show that our model can to a reasonable level reproduce the observed phenomenon of irreversible network change.

Robust congestion pricing under boundedly rational user equilibrium

This paper investigates congestion pricing strategies in static networks with boundedly rational route choice behavior. Under such behavior, users do not necessarily choose a shortest or cheapest route when doing so does not reduce their travel times by a significant amount. A general path-based definition and a more restrictive link-based representation of boundedly rational user equilibrium (BRUE) are presented. The set of BRUE flow distributions is generally non-convex and non-empty. The problems of finding best- and worst-case BRUE flow distributions are formulated and solved as mathematical programs with complementarity constraints. Because alternative tolled BRUE flow distributions exist, our congestion pricing models seek a toll vector or pattern that minimizes the system travel time of the worst-case tolled BRUE flow distribution. As formulated, the models are generalized semi-infinite min–max problems and we propose a heuristic algorithm based on penalization and a cutting-plane scheme to solve them. Numerical examples are presented to illustrate key concepts and results.

On Boundedly Rational User Equilibrium in Transportation Systems

A boundedly rational user equilibrium (BRUE) is achieved in a transportation system when all users are satisfied with their current travel choices. The theoretical and behavioral background for such a state is given in this paper. The properties of a BRUE in an idealized commuting system with a single bottleneck are investigated, and conditions for the existence of a BRUE are given. More general situations with multiple bottlenecks are also addressed. In general, BRUE flows are not unique, raising methodological and practical issues in flow prediction.