Dynamic Evolution of Traveler ’ s Bounded-Rational Route Choice Behavior

Travelers’ route choice shows bounded-rationality because of different perceptions of route attributes. Based on the boundedrationality, the paper proposes the dynamic evolution rules and route choice model, and simulation method is applied to study the evolution process and results.Themodel includes three parameters reflecting the bounded-rationality of travelers. First, simulation results show that the bounded-rationality affects the evolution process.The switching threshold or the perception deviance is larger, convergence rate is faster, and shorter time is needed to reach equilibrium state. Also, fewer perfect rational travelers will lead to similar results. Second, the system can reach equilibrium and the final equilibrium volume of every route is almost unaffected by bounded-rationality. The equilibrium volume of every route is an approximately fixed value under all simulation scenes. At last, it is found that equilibrium volume of every route obeys normal distribution. That is, bounded-rationality affects the equilibrium convergence rate, but volume equilibrium results will not be influenced.


Introduction
Researchers have conducted many studies on route choice behavior, based on which various theories of traffic assignment have been derived.Expected utility theory has been widely used in route choice modeling.However, numerous studies show that even in the perfect experiment, travelers can only select the route according to incomplete information under bounded-rationality.That is, the traveler does not select the shortest route every time [1].Then theories based on bounded-rationality (Prospect Theory, Cumulative Prospect Theory, Regret Theory) and Game Theory have been developed gradually.
Because of the uncertainty of traffic environment, difference perception, and judgement deviance of traveler, it is unrealistic to find the optimal strategy for each traveler every time.The traveler must adjust his strategy according to the previous choice of other travelers.For traffic managers, it is of great significance to study the dynamic decisionmaking process of travelers' route choice and formulate traffic guidance measures to alleviate traffic congestion.
The traveler route choice behavior reflects traffic assignment results, and all traditional traffic assignments are based on traffic equilibrium theory.However, whether the traffic network achieves equilibrium requires further discussion.Some study indicated that the network traffic flow was difficult to reach balance [2,3].Then, from the view of Game Theory, relationship between bounded-rational route choice behavior and traffic assignment must be verified.
Based on the hypothesis of bounded-rationality and incomplete information, the paper proposes a learning mechanism and route choice model based on bounded-rationality by applying the evolutionary game.Three parameters representing bounded-rationality are added in the model to analyze the switching process of travelers' route choice under different rationality degree.

Literature Review
2.1.Bounded-Rational Route Choice.Perfect rationality was widely used in modeling traveler route choice behavior.
For perfect rationality, travelers are required to have perfect judgement and forecast ability, and they must have rational common cognition and trust other participants.To compensate for the hypothesis defect of economic man of perfect rationality, Simon proposed the concept of boundedrationality [4].He thinks that all decision-makers are not completely rational because of the uncertainty and perception deviance.Then, Kahneman and Tversky proposed the Prospect Theory (PT, 1979) and Cumulative Prospect Theory (CPT, 1992), which both assume that every individual was bounded-rational during decision-making process [5,6].Now, boundedly rational behavior has been widely analyzed for travel behavior.
Numerous experiments and investigation showed that the travel time and route choice behavior under uncertainty is consistent with the Prospect Theory.Di provides a survey on bounded-rational travel behavior modeling and estimation [7].Ye established a Bounded-Rational User Equilibrium (BRUE) model, studying the evolution processes under various bounded-rationality thresholds and different initial states [8].Zhang established the day-to-day route choice model under CPT and Bounded-Rational theory, respectively, [9,10].Evidence shows that bounded-rational behavior exists in route choice process.
Some researchers established traveler route choice model based on bounded-rationality.Sun established a route choice model considering the travelers' subjective preference for travel cost [11].Based on the hypothesis of boundedrationality, Li established the multiroute choice model and proposed the dynamic reference points and the evolution rules of heterogeneous travelers [12].Huang established a model considering both travel cost and comfort based on Prospect Theory [13].Considering the existence of selfish travelers in a network, Rambha proposed an average cost Markov decision process to describe the route switching behavior of day-to-day [14].The results show that the system equilibrium and toll pricing are influenced by the bounded-rationality.Zhao conducted a Bounded-Rational User Equilibrium (BRUE) model and numerical simulation was used to obtain the equilibrium point [15].
Perception deviance is the crucial origination of bounded-rationality.Luca and Zhou studied the risk perception under uncertainty and proved the existence of preference [16,17].Yang established a Stochastic User Equilibrium (SUE) model base on Cumulative Prospect Theory [18].Zhang [9], Jiang [19], Di [20], and Li [21] established the model of dynamic learning and updating regular based on the theory of bounded-rational, revealing the existence of perception deviance in microcosmic decision process.
Some researches focused on factors influencing traveler route choice behavior.Findings show that information release [22][23][24], socioeconomic attributes [25], accidents [26], driver's familiarity with road network [27], and the threshold of route effective difference [28] all have influence on route choice behavior.

Dynamic Evolution of Route
Choice.Dynamic evolution reflects the learning mechanism of route choice process and the most representative theory is Game Theory.Nash published two articles about noncooperative game in 1950 and 1951 and proposed the Nash Equilibrium.In 1973, Maynard and Price proposed the fundamental concept of evolutionary game [31,32].The hypothesis of perfect rationality for classical Game Theory is inconsistent with the real decision process.Evolutionary Game Theory has developed quickly during the skepticism and confusion of the rationality.Evolutionary Game Theory supposes the participants are bounded-rational and cannot make the best response to information changes.Duplicate dynamic function is usually applied in evolutionary game to reflect the strategies updating process.
Application of Game Theory focuses on two aspects, route choice modeling and optimal route solving [33].The utility function matrix considering some influence factors was proposed to establish the route choice model based on Game Theory; the game process and results were studied when utility function changes [34][35][36].These studies discussed the game results' difference through analyzing the Evolution Stable Strategy (ESS).Some researchers focused on solving the game question applying Nash Equilibrium [37,38].However, evolutionary game cannot reveal the learning and updating mechanisms of travelers' route switching behavior deeply at microlevel.

Summary.
As noted in the previous subsections, there is a huge body of literature on travelers bounded-rational route choice.Previous references paid more attention to the modeling process and route choice results.For route choice dynamic evolution, some papers focused on the equilibrium point solving methods.As a whole, less papers analyzed the bounded-rational microcosmic behavior evolution mechanism.How bounded-rationality affects the route switching behavior cannot be revealed clearly.
In the paper, route choice evolution rules and three parameters representing the bounded-rationality were proposed and discussed.Combining the evolution game theory, we proposed route switching rules and establish the route choice models.Different from the previous studies, the evolution rules consider the travelers' heterogeneity and bounded-rationality and can describe the individual route switching mechanism, not only the route choice results.The route choice model is based on the concept of switching threshold, which is different from Kahneman and Tversky [5,6].According to the individual heterogeneity, we assume that the bounded-rational parameters follow normal distribution.The results were studied deeply under different distribution parameters.Results show that the bounded-rational parameters affect the convergence rate and equilibrium volumes.The study reveals the relationship between network equilibrium and traveler's individual characteristic.
As a whole, the study can manifest travelers' route choice behavior at the microlevel.And it is the complement and improvement for current research on traffic behavior at macrolevel.

Dynamic Evolution Model Based on
Bounded-Rationality

Dynamic Evolution Rules. Evolutionary Game
Theory uses replicator dynamic model to manifest the participant's decision process.However, the replicator dynamic model cannot express the behavior adjustment process under different specific game rules.
Then, the authors propose a dynamic evolution rule assuming that all the travelers constantly adjust their routes and decide whether to switch to another route or keep the current route unchanged.All travelers are bounded-rational and do not always select the shortest route.Travelers sequentially select the routes, and the traveler who selected later can observe the behavior of other travelers who select earlier.The travelers can adjust their strategies at next round according to the behavior of other participants at last round.With continuous adjustment, the system may reach equilibrium state finally.
Assume the following: (1) Every round, travelers sequentially enter into the system and select their routes successively.
(2) Travelers can obtain the routes' history information (that is, the travel time after all travelers' selection at last round) and cannot obtain real-time information (the travel time before selection at current round).( 3) Travelers can obtain the selection results of other travelers who selected before them.(4) Travels will not switch the current route if the time difference between shortest route and current route do not exceed the switching threshold.There is an OD pair.The total number of travelers is q and number of travelers is constant.
In this paper, "traveler" refers to motor drivers, "round" means evolution times.
Assume that there are some routes between the origin and the destination.The travel time (h) for route k can be expressed as where  0, is the travel time (h) for free-flow.  is the volume (persons/h) for route k, and the total volume for the network is  = ∑  .  is the capacity (persons/h) of route k.  and  are parameters.Generally,  = 0.15,  = 4.
Because travelers subsequently arrive at the system, travelers should only consider the strategies of other travelers who selected before them.Number each traveler according to the arriving sequence; then, traveler  should consider the strategies of travelers 1, 2, 3, . . ., -1.
(1) At first round, original selection strategies: all travelers select the shortest routes according to historical information.
(2) Strategy-updating scheme: from round 2, traveler  determines his selection at next round according to the strategies of all (-1) travelers in last round.Because of the universal information technology, such as mobile navigation, broadcast, and variable message signs, travelers can observe other travelers' previous choice.The principle can be explained as follows: traveler  calculates the actual travel time E for each route according to the results of travelers before him at last round and predicts the travel time  for each route at next round.Assume traveler  selected route  at last round; if the predicted travel time for route  at next round has no significant difference with the actual travel time at last round, traveler  will select route  at next round.Otherwise, travelers i will choose the shortest route.In other words, the traveler will change his selection result only if he believes that the route's travel time will sharply increase.
For the first traveler, there are no other travelers on the system when he enters first; then the first traveler chooses the shortest route always.

Bounded-Rational Route Choice Model.
Assuming that travelers will not switch to the new route unless travel time saving goes beyond the switching threshold, a boundedly rational route choice model is proposed.
In round , if travel  selects route , the actual travel time for traveler  can be explained as follows.
The perception travel time at next round of route  for traveler  is as follows.
Traveler  can only predict the travel time at next round according to the selection of other travelers  ( ∈ ).Traveler  cannot obtain the selection results of all travelers, so at next round, the number of travelers who select route  can be described as the number of travelers who select route  in set  and him (traveler ).
Further, all travelers are bounded-rational and have limited perception ability.Better perception ability means the perception travel time is closer to actual predicted travel time.  is the perception deviance of traveler .Traveler with smaller   has accurate predicted travel time.  ∈ [0, 1].
In round  + 1, for traveler , the optimal route should satisfy the following condition.
In other words, the optimal route (that is, the shortest route for travelers ) is the route with shortest perception travel time.
Because of the bounded-rationality, the travelers will not necessarily select the shortest route every time.Introducing the concept of switching threshold, in round , if the difference between the actual travel time for route  at last round and the predicted time at next round is acceptable to traveler , then at next round, traveler  continues to select route ; otherwise, traveler i switches to the shortest route, which can be explained as follows.
In this formula, Δ  measures the rational degree of traveler .Smaller Δ  means lower switching threshold and traveler is more rational.When Δ  = 0, the traveler is completely rational and will select the shortest route every time.
Δ  is used as the standard to determine whether the traveler will switch his choice.Travelers with larger Δ  are more tolerant for travel time and are inclined to boundedrationality.Meanwhile, travelers with smaller Δ  are more rational and prone to change their selection frequently.
For traveler , the decision process is explained in Figure 1.

Basic Scene and Parameters of Numerical Simulations.
Considering travelers with different bounded-rationality degrees, study the dynamic evolution process of travelers' route choice.
Figure 2 shows the routes between Xi'an Jiaotong University and Chang'an University.Generally, travelers will make choice between the 5 routes shown in Figure 2. Routes 1 and 2 are based on the expressway-Nan' er Huan road.Routes 3, 4, and 5 are based on arterial roads, such as Youyi Road and Yanta Road.Attributes of the roads are shown in Table 1.
Nonpeak traveling is chosen as simulating scene.Assume  is constant.The proportion of completely rational (Δ  = 0) travelers is , the rationality degree of each traveler is Δ  , and the perception deviance is   .Parameters of basic scene are shown in Table 2.

Numerical Simulations Results.
For the basic simulation scenes, when the system reaches equilibrium, volume of each route is 169, 194, 255, 177, and 205, respectively.Figure 3 shows the evolution process.
After 929 rounds' evolution, the system reaches equilibrium state.All route's volumes will not change when the evolution times are increased continually.During the evolution process, volumes changing amplitudes of each route gradually decrease and volume is converged to the equilibrium point.
Then study the effects of one parameter when other parameters are constant, and analyze the equilibrium volume of each route and evolution rounds required to converge.
(1) Effect of the Rationality Degree.A Assume all travelers have the same rational degree and analyze the effects of Δ's increase.
Figure 4 shows the equilibrium volume of each route and evolution rounds required.We can conclude that smaller Δ leads to longer evolution time.In contrast, when Δ is large, the volume can fast converge.
A smaller Δ indicates that the travelers are more rational, and the evolutionary process is more similar to the user equilibrium under complete rationality.Then most travelers may change their routes frequently to choose the shortest routes.However, when Δ is large, routes switching frequency declines and most travelers keep their choice unchanged, so it takes less time to reach equilibrium, which implies that when the travelers are more rational, the network requires more time to reach equilibrium.
There is minor change for equilibrium volume of each route comparing to the basic scene.Equilibrium volume of each route almost keeps constant when Δ is increasing.
B Assume Δ  is subject to normal distribution.Δ  ∼ (0.2,  1 ).Then analyze the effects of  1 on volume evolution.Evolution result is shown in Figure 5.
Results are similar to those in Figure 4.With  1 increase, evolution rounds show decline tendency overall.However, there is some vibration in the declining process.For different deviance of rationality degree, equilibrium volume of each route can remain stable approximately with Figures 3 and 4.
C Assume Δ  is subject to normal distribution.Δ  ∼ ( 1 , 0.05).Δ  is different for every traveler and randomly obtained from the normal distribution.Evolution result is shown as Figure 6.
Results are the same as those in Figures 4 and 5.
Results of Figures 4∼6 show that travelers are more rational, the equilibrium process is closer to user equilibrium, and evolution rounds are longer.However, when the system reaches equilibrium, every route has the approximate same volume for different simulation scene with different rational degree.
(2) Effect of the Proportion of Completely Rational Travelers.A Assume other parameters are constant and analyze the effects of .Values of parameter are shown in Table 3.
Results of Table 3 show that, for a constant Δ, the system needs more time to reach equilibrium with the increase of parameter .Obviously, the comparatively large  indicates that there are more completely rational travelers, so in each Updating the route choice for next time, r i (t+1) For all i, r i (t + 1) = r i (t) evolution time, more travelers will switch to the shortest route.In other words, the travelers change routes more frequently, so the evolution time is longer.In contrast, when  is small, more travelers can accept the selection results at last round under the bounded-rationality principle, which indicates that more travelers keep the route selection stable, and the evolution time is shorter.For a constant , smaller Δ results in longer evolution time.The results are consistent with (1) A. No matter what values of Δ and  are, every route has similar equilibrium evolution volume for every simulation scene.
(3) Effects of Perception Deviance   .A Assume all travelers have the same perception deviance  and analyze the effects of  when  is increased.Evolution results are shown in Figure 7.
We can find that, as the perception deviance () increases, equilibrium rounds gradually decline.Smaller  means the travelers have accurate perception travel time and are more rational, which will lead to longer equilibrium rounds.
B Assume   is subject to normal distribution.  ∼ (0.2,  2 ).  is different for every traveler and randomly obtained from the normal distribution.Evolution process is shown in Figure 8.
C Assume   is subject to normal distribution.  ∼ ( 2 , 0.05).  is different for every traveler and randomly obtained from the normal distribution.Evolution process is shown in Figure 9.
Combining Figures 8 and 9, we can conclude that it needs less rounds to reach equilibrium for both larger  2 and  2 . 2 and  2 indicate the perception deviance.Larger perception deviance means traveler is bounded-rational; he cannot forecast the travel time accurately and may choose the near-optimal route.Further, if   obeys normal distribution, the evolution process is different from that when   is a constant.At the basic simulation scene, all routes can evenly reach equilibrium state.That is, volume of all routes evenly declines or rises to convergent point and has no obvious vibration.However, when   obeys normal distribution, volume of some routes fluctuates greatly in the equilibrium process.When  2 = 0.3, the irregular vibration lasts for a long time.We can deduce that, when distribution of perception deviance is discrete, the evolution process exhibits some vibration and uneven convergence characteristic.
(4) Volume Evolution of the Routes.We can find that, no matter what the value of each parameter is, every route  has similar equilibrium volume for all simulation scenes.Then, 29 scenes are randomly chosen to simulate and obtain the equilibrium volume.Volume for each route focuses on a certain range.From the distribution of volume, we can assume and prove the equilibrium volume for each route obeys normal distribution.Test results are shown as Figure 10.
Assume the equilibrium volume of each route is

Discussion
According to the simulation results, the bounded-rationality is objectively present in the route choice behavior of travelers.
The conclusion is consistent with Di [7,20] and Li [12].All results indicate that the bounded-rationality is applicable to travel behavior analysis and traveler's behavior is affected by individual rational degree.In this paper, the indicator Δ  practically represents the traveler's habit.Conclusions suggest the habit was the important factor that affected the traveler's selection.A larger Δ  indicates that the traveler is more accustomed to the previous selection.The conclusion is identical to Di's conclusion [28].
The selection results show that not all travelers select the shortest route, which depends on their habits and tolerance.Our findings are consistent with Zhu [1], which proves the effect of habit and other uncertain factors on the traveler's behavior.In practical travel, when travelers face uncertain factors, they should only select the near-optimal path because  they generally could not obtain the optimal scheme as a result of information limitation and variability.
The traffic volume can finally reach an equilibrium state.The conclusion answers the question asked by Lida [2] and He [3], who suggested whether the traffic volume can reach equilibrium.According to our conclusions, when all travelers have a uniform objective, volume equilibrium is the final state of the network, regardless of travelers' rationality degree.
In addition, authors find the rationality degree is closely related to volume evolution process.When travelers are more rational, volume convergence rate is slower and it needs more time to reach equilibrium.The results are similar with more rational travelers and less perception deviance.
From this discussion, most of our conclusions are consistent with existing studies, which illustrates that the proposed evolution rules and regulations are proper to travelers' decision-making.

Conclusions
In the examples, authors analyze the traveler's route switching under different rationality degree.The conclusions are shown as follows.
(1) More rational travelers require more time to achieve equilibrium.With the increase of complete rational travelers, more travelers select the shortest route.Then, the travelers may frequently change routes.Therefore, the travelers take more time to obtain the optimal strategies.
(2) Larger perception deviance and less rational behavior both lead to shorter equilibrium rounds, which means the travelers are more tolerant to the travel time, and fewer travelers will select the shortest route.Then the system can easily obtain the optimal strategy.
(3) The volume of each route fluctuates around the equilibrium point, and the vibration amplitude decreases.In the evolution process, all travelers adjust their decision according the behavior of the travelers who arrived earlier.Then, the travelers who arrive earlier can more quickly find the optimal routes.Therefore, the volume fluctuation of each route decreases and gradually stabilizes with increasing evolution time.Further, when   obeys normal distribution, convergence process illustrates some uneven fluctuation.
Therefore, the bounded-rationality of travelers can improve the evolution process.When most travelers select the near-optimal route instead of the shortest route, then the system can quickly achieve equilibrium.
For city managers, dynamic traffic information could be published to public.In terms of information content, fuzzy information is applicable so that travelers can not calculate travel time of each route but can only understand the traffic status.Then, the travelers can only select the near-optimal route according to the published information.In terms of information style, a combination of image and linguistic information should be published.Image information is used to inform the traffic condition, and linguistic information is designed to provide the suggested information.Additionally, suggestive induce information can be published to guide the selection of travelers.

Data Availability
Data of this work comprised output of simulation and some figures.Output.xlsxcontains the volumes of each route under different simulation scene.All the figures constitute the  evolution equilibrium process involving equilibrium volumes and rounds under different simulation scene (Supplementary Materials available here).

Figure 10 :
Figure 10: Normal distribution test for the equilibrium volume of each route.

Table 1 :
Attributes of the roads.

Table 2 :
Parameters of basic scene.