This paper proposes a route choice analytic method that embeds cumulative prospect theory in evolutionary game theory to analyze how the drivers adjust their route choice behaviors under the influence of the traffic information. A simulated network with two alternative routes and one variable message sign is built to illustrate the analytic method. We assume that the drivers in the transportation system are bounded rational, and the traffic information they receive is incomplete. An evolutionary game model is constructed to describe the evolutionary process of the drivers’ route choice decisionmaking behaviors. Here we conclude that the traffic information plays an important role in the route choice behavior. The driver’s route decisionmaking process develops towards different evolutionary stable states in accordance with different transportation situations. The analysis results also demonstrate that employing cumulative prospect theory and evolutionary game theory to study the driver’s route choice behavior is effective. This analytic method provides an academic support and suggestion for the traffic guidance system, and may optimize the travel efficiency to a certain extent.
In recent years, with the rapid development of information technology, traffic information system has had a great effect on travel decisionmaking behavior. Drivers may respond to the information through adjusting the travel mode, destination, departure time, and speed, but most commonly by altering routes [
Researches related to route choice have been conducted in many perspectives. Chen and Jovanis [
The above researchers studied the route choice behavior in the perspective of expected utility theory (EUT) [
Looking at the issue from another point, route choice is a dynamic selection process because of the realtime traffic information and the updated road condition. Little work has been done from the point of dynamic selection process to discuss how drivers make route choice decisions considering traffic information. Evolutionary game theory is the theory that discusses system’s dynamic evolution process under bounded rational conditions.
The purpose of this paper is to describe how drivers adjust their route choice behaviors under the influence of traffic information from a bounded rational and dynamic selection process perspective. The remainder of the paper is organized as follows. Section
Cumulative prospect theory (CPT) is a method for descripting decisions under risk and crisis which was introduced by Tversky and Kahneman in 1992. CPT distinguishes the choice process into two phases: framing and valuation. In the phase of framing, the decision maker constructs a representation of the acts, contingencies, and outcomes that are relevant to the decision. In the phase of valuation, the decision maker assesses the representation value of each prospect and chooses the largest one accordingly [
The main opinion of CPT is that people tend to think of possible outcomes relative to a certain reference point rather than to the final status, a phenomenon which is called framing effect. Moreover, they have different risk attitudes towards gains (i.e., outcomes above the reference point) and losses (i.e., outcomes below the reference point) and care generally more about the potential losses than the potential gains. Finally, people usually overweigh the extreme, but unlikely, events, however, underweigh the “average” events.
CPT incorporates these opinions in a modification of the expected utility theory by replacing the final wealth with the payoffs relative to the reference point, replacing the utility function with the value function that depends on the relative payoffs, and replacing the cumulative probabilities with the weighting cumulative probabilities. The subjective utility of a risky outcome is described by a probability measure
Shape of value function.
Shape of weighting function.
The value function proposed by Tversky and Kahneman [
It is apparent from Figure
Based on the research of Tversky and Kahneman [
Based on CPT, the representation value of a prospect
Evolutionary game theory (EGT) is a theory that combines game theory with dynamic evolution process analysis. EGT is useful in this context by defining a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. EGT originated in 1973 with Smith and Price’s formulization of the way in which such contests can be analyzed as “strategies” and the mathematical criteria that can be used to predict the resulting prevalence of such competing strategies [
Evolutionary stable strategy (ESS) was defined and introduced by Smith and Price in a 1973 Nature paper [
ESS presumes that individuals have no control over their strategies and need not be aware of the game. To be an ESS, a strategy must be resistant to alternatives. Every ESS corresponds to a Nash equilibrium solution, but not all Nash equilibrium solutions are ESSes.
The mathematical definition of ESS can be expressed as follows. For a very small positive
There are two properties for a strategy
The limitation of ESS is that it is a static equilibrium without considering the dynamic evolutionary process. The stability equilibrium of evolution should be associated with the specific evolutionary process.
The common methodology to study the evolutionary process is through the selection dynamics. It shows the growth rate of the proportion of people using a certain strategy. The basic expression of the selection dynamics is presented as
Among all kinds of game dynamic schemes, replicator dynamics (RD) by Taylor and Jonker [
Cumulative prospect theory and evolutionary game theory deal with bounded rationality from two different perspectives: the former tries to handle individual irrationality from the perspective of psychological perception, while the latter focuses on the limited rationality in selection and decision [
In this section, we will take a tworoute network, for example, to illustrate the route choice modeling process. The network consisted of route
An example of tworoute network.
Flow chart of route choice modeling process.
For a specific road network, drivers determine the perceptive time of each alternative route based on their previous travel experiences. The travel time distribution of each alternative route is assumed to be identical and independent of each other. According to the centrallimit theorem, the distribution of the perceived travel time of the alternative route approximately obeys the normal distribution. The distribution of the perceptive travel time is written as follows:
Because the free flow time can reflect the physical properties of the route in a certain extent, the reference point in this research is defined as the average value of the free flow time of all alternative routes. The reference point is represented as follows:
Based on CPT, we assign each route
During a travel activity, the variable message signs are used to provide travel related information in real time. Each type of drivers has two route choice strategies:
According to the difference of individual preference, we assume that participant
The payoff of each participant under different decision conditions is represented as follows:
When the two participants choose different routes, the participant who chooses route
Payoff matrix under different decision conditions.

 

Route 
Route 

Route 


Route 


Assume that the probabilities of choosing route
In conclusion, the route choice game model which embeds CPT can be expressed as follows:
Section
The utility of strategy
The average utility of strategies
In evolutionary game theory, the dynamic change rate of strategy proportion is the core of the bounded rational game analysis. The change rate depends on the player’s learning ability and learning rate. This process can be represented by the replicator dynamics. The replicator dynamics of strategy
To participant
The average utility of strategies
The replicator dynamics of strategy
A fixed point of the replicator dynamics is a population that satisfies
The Jacobin matrix is
The determinant of the Jacobin matrix is
The trace of the Jacobin matrix is
Local stability analysis.
Equilibrium point  det 
Sign of det 
tra 
Sign of tra 
Local stability 



+ 

−  ESS 


− 

−  Instability 


− 

−  Instability 


+ 

+  Instability 
From Table
From the evolutionary equilibrium analysis (Table
Local stability analysis.
Equilibrium point  det 
Sign of det 
tra 
Sign of tra 
Local stability 



+ 

−  ESS 


− 

−  Instability 


− 

Instability  


+ 

+  Instability 
Local stability analysis.
Equilibrium point  det 
Sign of det 
tra 
Sign of tra 
Local stability 



− 

Instability  


+ 

−  ESS 


− 

Instability  


+ 

+  Instability 
Table
Local stability analysis.
Equilibrium point  det 
Sign of det 
tra 
Sign of tra 
Local stability 



+ 

+  Instability 


+ 

−  ESS 


+ 

−  ESS 


+ 

+  Instability 
From Table
The above uncertainty can be solved by the stability theorem. The stability theorem of differential equations to distinguish the different stable states can be expressed in mathematics:
To participant
Take
When
When
When
The group replicated dynamic phase of
Replicated dynamic phase of participant
To participant
Take
Replicated dynamic phase of participant
When
When
When
The stability of groups
Group replicated dynamic phase of
When the initial state of
This paper has embedded cumulative prospect theory into evolutionary game theory in order to integrate the individual perception and decision schemes with the group learning and evolutions. This paper discussed the drivers’ route choice behaviors and the corresponding stable state of the dynamic traffic system according to the different information shown on VMS.
When there is no VMS in the transportation system (
However, there are some limitations in this research. First, the modeling method presented here is effective but needs to be validated in the empirical work. Another issue is that our findings are valid only for the assumption that the distribution of driver’s characteristic is identical in the same participant.
We suggest that the survey data should be collected in order to calibrate the parameters of the proposed model and to investigate the capability of the model to explain the field observations. In the future, an investigation on the effect of the drivers with different characteristic distributions should be carried out.
The results of this study may be useful to learn the driver’s route choice behavior and to alleviate the urban traffic congestion. The potential applications of the proposed method involve the modeling and describing the group choice evolution process from the perspective of the individual risk attitude as well as the decisionmaking schemes. It is suitable for capturing the adaptation course of the group choice.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The key National Natural Science Foundation of China (no. 51338003) and the financial support from the National Key Basic Research Program of China (no. 2012CB725402) are gratefully acknowledged.