The drivers’ lane selection model based on mixed fuzzy many-person multi-objective non-cooperative game

2.1 Vehicle aggregation situation

The traffic situation refers to all states and situations of traffic entity deployment and behavior in a driver’s interest-sensitive area (the interest-sensitive area is the area that has the greatest impact on vehicle safety; increased driver distraction increases this area) in state transport entity to deploy conduct constituted, and it contains all the information of traffic entity [15–17]. The traffic situation includes the characteristics of all traffic entities and vehicle group relationships needed by drivers to obtain complete scene interpretation information and accomplish their driving tasks. The characteristics of a traffic entity include vehicle type, velocity, acceleration, and driver type. The vehicle group relationship describes the changing positional relationship that self-organizes and evolves through aggregating behavior. The scene interpretation information is a description of the natural, artificial, and social environments inhabited by a car. In this article, we assume a vehicle aggregation situation (encompassing the characteristics of the traffic entities and the vehicle group relationship) in which drivers can perceive their interest-sensitive areas.

2.1.1 Definition of vehicle aggregation situation

To a better analyze drivers’ game-playing behavior under vehicle aggregation situations; the research scenario is defined in a three-lane scenario as an example. Figure 1 show that, the most typical case is when the target car located in n₁. With target vehicle n₁, the interest-sensitive area is divided into the left-front, left-rear, front, rear, right-front, and right-rear sub-regions according to the position of target vehicle’s front bumper, and other conditions can be calculated in the same way.

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Fig.1

Vehicle aggregation situation when target vehicle is located in the middle line.

2.2 Analysis of lane selection game behavior

Drivers have two options—remain in lane or change lane—when driving on multi-lane roads. Considering the impact of the surrounding vehicle aggregation situation on his or her interest and security, a driver determines the vehicle’s optimal running lane based on the current state and other vehicles’ actions. The driver’s lane selection process is an interest (including safety, efficiency, time and space, comfort, etc.) pursuing process.

In a three-lane scenario in which a target vehicle, left-front vehicle, left-rear vehicle, front vehicle, rear vehicle, right-front vehicle, and right-rear vehicle are players (see Fig. 1), there are three pure strategies for the middle lane vehicle: No Change (NC), Change Left (CL), and Change Right (CR). There are two pure strategies for a left lane vehicle: No Change (NC), and Change Right (CR). There are two pure strategies for a right lane vehicle: No Change (NC), and Change Left (CL). Thus, vehicles involved in the game will have a pure strategy that is an element of Λ _i  = {NC, CL, CR}. In the process of selecting a driving strategy, each driver must not only choose their own policy decision variable space (keep lane or change lane) but also consider other drivers’ strategy selections; the strategy selections for the driver are therefore interdependent and mutuallyrestrained.

A driver’s benefits are related not only to the vehicle aggregation situation before and after the driver perform actions but also to the conflicts among vehicles. The driving revenue function of the driver can reflect traffic safety, travel time, and driving conflict benefits.

1) Calculation of traffic safety benefit

The traffic safety benefit (ΔP _i ) refers to the change in a target vehicle’s experienced integrated force P _i (the weighted sum of forces from all of the vehicle’s sub-regions) before and after the driver performs an action. The vehicle aggregation situation of the target vehicle (the set of forces from sub-region vehicles applied to the target vehicle) when the target vehicle changes lanes can be obtained using the method in Section 2.1.2. The contributions of each sub-region force to the overall target vehicle integrated force differ; for example, when the target vehicle is in the left lane, the force contribution to the integrated force of the separated lane right-front vehicle is smaller than that of the adjacent lane right-front vehicle. For a target vehicle in the middle lane, the integrated force is P i = ξ i q F i q + ξ i h F i h + ξ i zq F i zq + ξ i zh F i zh + ξ i yq F i yq + ξ i yh F i yh . For a target vehicle in the left lane, the integrated force is P i = ξ i ql F i ql + ξ i hl F i hl + ξ i lyq F i lyq + ξ i lyh F i lyh + ξ i gyq F i gyq + ξ i gyh F i gyh . A target vehicle in the right lane experiences an integrated force of P i = ξ i qr F i qr + ξ i hr F i hr + ξ i lzq F i lzq + ξ i lzh F i lzh + ξ i gzq F i gzq + ξ i gzh F i gzh . Here, ξ i * represents the contributory force of a vehicle located in sub-region * to the integrated force on target vehicle n _i , with i = 1, 2, 3, 4, 5, 6, 7. Thus, ξ 1 q represents the contribution to the integrated force on n₁ by the vehicle located in the front sub-region. According to reference [10, 14], based on responses to questionnaires, an analytic hierarchy process (AHP) was used to obtain contribution rates associated with different driver types for the sub-regional contributions to target vehicle integrated force (Table 1). Denoting the respective study subjects as n₁, n₂, n₃, n₄, n₅, n₆, and n₇ using the above methods, and the game-incurred integrated forces P _i for seven vehicles can be obtained. When each driver carries out their operations, each vehicle experiences an integrated force P i ′ (i = 1, 2, 3, 4, 5, 6, 7), which is also obtained using the method described above. By calculating Δ P i = P i ′ - P i , the traffic safety benefits accruing to each vehicle before and after the driver performs actions can be obtained.

2) Calculation of travel time benefit

The travel time benefit (D _i ) can be enhanced by improving the front road traffic situation before and after each driver performs actions. This benefit can be represented by the difference in the values of cumulative force with two vehicles in front on a target vehicle before and after each driver performs actions. For the target vehicle n₁, the cumulative force with two vehicle in front of before each driver performs actions is F 1 q + F 4 q and the cumulative force after each driver performs actions is F 1 ″ q ′ + F 4 ″ q ′ ( F 1 ″ q ′ is the force from vehicle n_1″ and F 4 ″ q ′ is the force from vehicle n_4″ on n_1″). By calculating F 1 ″ q ′ + F 4 ″ q ′ - F 1 q + F 4 q , the travel time benefit can be obtained; the travel time benefits of other vehicles in the game can similarly be obtained.

3) Calculation of driving conflict benefit

The driving conflict benefit (C _i ) is the conflict strength among vehicles during the drivers’ implementation of their strategy processes. Drivers have two strategies when they determine a travel lane: maintain or change lane (Fig. 2). When a driver takes a given strategy, vehicle conflicts will be increased or decreased. The more serious the conflict between vehicles, the more driving conflict benefit is lost; the more obvious the benefits of reducing conflict between vehicles, the more driving conflict benefit will be obtained.

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Fig.2

Analysis of lane selection game behavior for three-lane game.

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Fig.3

Schematic of vehicle conflict types.

By calculating the relative velocity, relative distance and the combined forces of the target vehicle and the surrounding vehicles, the severity of each type of conflict is judged. In Fig. 2, if vehicles n₃ and n₅, which are in different lanes, choose to change lanes (seize change lanes, as shown in Fig. 3(1)), the conflict between the two vehicles will be very serious and the driving conflict benefit will be reduced. If vehicle n₃ changes lane and vehicle n₅ maintains its lane (pressing change lane, as shown in Fig. 3(2)), the conflict between the two vehicles will be serious and the driving conflict benefit will be even lower; if the two vehicles choose to maintain their lanes (no conflict, as shown in Fig. 3(3), (6)), there will be no conflict between the two vehicles and the driving conflict benefit will be high. By contrast, if vehicle n₃ changes lane and vehicle n₁ maintains lane (keep escaping conflict, as shown in Fig. 3(4)), they can avoid conflict and the driving conflict benefit will improve; if two vehicles both choose to change lanes (change lane to avoid conflict, as shown in Fig. 3(5)), the driving conflict benefit will be even higher. If both n₁ and n₅, in the same lane, choose to change lanes (change lane conflict, as shown in Fig. 3(7)), the conflict between the two will be serious and driving conflict benefit will be low. If n₁ changes lane and n₅ maintains lane (change lane to avoid conflict, as shown in Fig. 3(8)), the two vehicles will avoid conflict and the driving conflict benefit will be high; if the two vehicles choose to maintain (keep conflict, as shown in Fig. 3(9)), the conflict between the two vehicles will be more serious, and driving conflict benefit will be moderate. The conflict benefit is assumed to fall in the range [0, 1], with increasing value equating to improve benefit. Different types of conflict that show the correspondence between semantic driving conflict benefit and triangular fuzzy numbers are listed in Table 2.

In the vehicle game, each vehicle has some conflict with others. Depending on the conflict types that may arise between two vehicles, a combination of various conflicts may arise in the vehicle game process, the respective driving conflict benefits of which can be calculated.

4) Driving benefit calculations

To facilitate a unified balance, the traffic safety benefit ΔP _i , travel time benefit D _i , and driving conflict benefit C _i were converted to driving benefit W _i as follows: W i = ω i 1 Δ P i + ω i 2 D i + ω i 3 C i(1)

In Equation (1), ω i 1 , ω i 2 , ω i 3 show the respective fuzzy objective weights of traffic safety benefit, travel time benefit, and driving conflict benefit for player n _i . The fuzzy objective weights differ by driver type. For example, aggressive drivers have more emphasis on traveling benefit while conservative drivers pay more attention to traffic safety benefit. The assorted fuzzy weights are shown by driver type in Table 3.

A driver’s lane selection process is a multi-objective game process that takes into consideration many factors. The driving benefits for the multiple vehicles participating in a game need to be considered from the point of view of the three types of benefit listed above. The vehicle aggregation situation and strategy information elements are both associated with a certain degree of ambiguity and subjectivity that are difficult to describe by precise mathematical modeling. In addition, differences exist in terms of drivers’ propensities [15], leading to an interests and preferences structure with obvious conflicts. To address these issues, a mixed fuzzy many-person, multi-objective non-cooperative game based on drivers’ preference information is used to describe the behavior of the drivers’ lane selection game.

3.1 Model description

This article focuses on a mixed fuzzy many-person, multi-objective, non-cooperative game based on drivers’ preference information encoded as fuzzy objective weight vectors and fuzzy objective payoffs vectors [19]. The game can be represented by the mathematical model Γ ˜ M N i = [ I ; { X i } , { E i ′ i ( x ) } ( n i ′ ∈ I ) ] ( n i ∈ I ) .

Usually I represents a set of players; if there are N players, I = {n₁, n₂, ⋯ , n _i , ⋯ , n _N }. Each player(players are drivers of interest-sensitive area) n _i  (n _i  ∈ I) has its own set of strategies S _i , S _i  = {s _{j i} |j _i  ∈ Λ _i } (Λ _i is the index set of pure strategies). In each game, the set of strategies comprising all players’ strategies is called a situation; if s _{j i} is the strategy of player i, then the set of strategies s = (s _{j 1} , s _{j 2} , ⋯ , s _{j N} ) for N players is the situation. The overall set of situation S is represented by the Cartesian product of each player’s strategy set, i.e., S = s _{j 1}  × s _{j 2}  × ⋯ × s _{j N} .

X _i  = {x _i  = (x _{j i} |j _i  ∈ Λ _i )  ^T |∑_{j i ∈Λ i} x _{j i}  = 1, x _{j i}  ≥ 0} is a mixed strategy set of player n _i  (n _i  ∈ I), where x _{j i} is the probability that the player selects the pure strategy s _{j i} from the strategy set S _i  = {s _{j i} |j _i  ∈ Λ _i } and x _i  = (x _{j i} |j _i  ∈ Λ _i )  ^T is the mixed strategy of the player. Note that X = X₁ × X₂ × ⋯ × X _i  × ⋯ × X _N ; when the mixed strategy x _i  ∈ X _i is selected by player n _i  (n _i  ∈ I), the mixed strategy situation is x = (x₁, x₂, ⋯ , x _N ) ∈ X.

As the driver does not know the objective weights of other players, she will assume that her own objective weight vector reflects the others’ to analyze the problem. In this case, each player n _i  (n _i  ∈ I) uses synthetic weights for players n_i′ (n_i′ ∈ I) ‘s fuzzy payoffs based on her objective weight vector. Note the weighted game of player n _i  (n _i  ∈ I) is Γ ˜ M N i = [ I ; { X i } , { E i ′ i ( x ) } ( n i ′ ∈ I ) ] ( n i ∈ I )(2)

Where

E i ′ i ( x ) = ∑ m = 1 M ω i m E i ′ m ( x ) = ∑ j k ∈ Λ k n k ∈ I ∑ m = 1 M ω i m u i ′ m ( s j 1 , s j 2 , ⋯ , s j N ) ∏ n k ∈ I x j k ≜ ∑ j k ∈ Λ k n k ∈ I u i ′ i ( s j 1 , s j 2 , ⋯ , s j N ) ∏ n k ∈ I x j k ( n i ′ ∈ I )(3)

are the fuzzy weighted payoffs of player n_i′ (n_i′ ∈ I) based on the representation of player n _i  (n _i  ∈ I) in strategy situation x ∈ X. E i ′ m ( x ) = ∑ j k ∈ Λ k n k ∈ I u i ′ m ( s j 1 , s j 2 , ⋯ , s j N ) ∏ n k ∈ I x j k representsthe fuzzy payoffs of player n_i′ (n_i′ ∈ I) for objective m (m = 1, 2, ⋯ , M) in strategy situationx ∈ X, where u i ′ m ( s j 1 , s j 2 , ⋯ , s j N ) are the fuzzy payoffs of player n_i′ (n_i′ ∈ I) for objective m (m = 1, 2, ⋯ , M) in pure strategy (s _{j 1} , s _{j 2} , ⋯ , s _{j N} ). Also note that ω i = ( ω i 1 , ω i 2 , ⋯ , ω i m , ⋯ , ω i M ) T ( n i ∈ I )  is the fuzzy objective weight vector of player n _i  (n _i  ∈ I), where ω i m (assuming it is already standardized) represents fuzzy weight for objective m (m = 1, 2, ⋯ , M).

The player n _i  (n _i  ∈ I) can select a fuzzy ranking function f to clear game Γ ˜ MN i = [ I ; { X i } , { E i ′ i ( x ) } ( n i ′ ∈ I ) ] 1 pt 1 pt 1 pt 1 pt 1 pt ( n i ∈ I ) into a weighted countermeasure Γ MN i = [ I ; { X i } , { E i ′ i ( x ) } ( n i ′ ∈ I ) ] ( n i ∈ I ) ,where E i ′ i ( x ) = f ( E i ′ i ( x ) ) ( n i ′ ∈ I ) .

The steps of the game solution are solved [1] as follows:

Taking x i ′ = ( x j i ′ | j i ∈ Λ i ) T as a constant, calculate the minimum of function E i ′ i ( x | | x i ′ ) (where, x | | x i ′ = ( x 1 , … , x i - 1 , x i ′ , x i + 1 , … , x N ) ) on the compact sets X₁ × X₂ × ⋯ × X_i-1 × X_i+1 ⋯ × X _N , and denote it as g i ( x i ′ ) ( n i ∈ I ) .

Using the above steps, player n _i  (n _i  ∈ I) can get her desired solution, which is denoted as x i = ( x 1 i , x 2 i , ⋯ , x i i , ⋯ , x N i ) ( n i ∈ I ) , and an expected fuzzy weighted safety value of v i = m in ˜ x ∈ X E i i ( x | | x i i ) ( n i ∈ I )(4)

However, each player selects a strategy and synthetic weighted game based on their own objective weight vector; therefore, the actual strategy situation should be x ̇ = ( x 1 1 , x 2 2 , ⋯ , x i i , ⋯ , x N N ) and the fuzzy weighted payoff of the players should in reality be

v ˙ ˜ i = E ¯ i ( x ˙ ) = ∑ m = 1 M ω i m E i m ( x ˙ ) ( n i ∈ I )(5)

4.1 Identification of the driver’s propensity

Using the method in reference [14, 15], drivers’ real-time propensities were distinguished for online identification.

4.2 Data processing

Each record in NGSIM contains only a single vehicle data with one-time status information, so the information on surrounding vehicles and time-evolved status information must be obtained through a related query that follows the vehicle number and time information, after which the accumulated records must be combined. Vehicle type, relative distance, and relative speed between the target and surrounding vehicles can be obtained from video and document data, from which micro data to calculate the vehicle aggregation situation can also be obtained. The NGSIM preprocessing data are shown inTable 4.

Based on the reference values, parameters calibration and model verification of a lane selection model that correctly reflects the traffic flow characteristics can be completed through scientific analysis and processing of NGSIM data. Some calibration parameters are shown in Table 5.

5.1 Verification of lane selection model based on vehicle road test

Typical road sections of Zibo city were selected and road, traffic, and environmental condition data were collected to carry out a vehicle road test (test equipment is shown in Fig. 4). Different types of drivers and vehicle types were selected for experiments over different time periods and speeds. Using video monitoring, all experimental data were stored. Based on video monitoring, the drivers’ real-time lane changing behavior was predicted by the model and compared with the actual recorded behavior after the experiment. In this way, the validity and reliability of the model could be determined and the model parameters could be calibrated to develop a finalized drivers’ lane selection model. The results of vehicle road test as shown in Table 6.

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Fig.4

Dynamic human-vehicle-environment information acquisition systems.

We designed a variety of typical traffic scenes in a driving simulator using different types of driver to perform a driving simulation test in particular environments. The simulation experiments were carried out using a simulator made by the Japanese FORUM8 enterprise and employing the driving simulator software UC-Win road (shown in Fig. 5). Simulated three-lane driving scenarios were built up through experiments with urban roads simulated according to actual vehicle road tests in identical environments (in terms of road facilities, signals, traffic, etc.). Drivers were trained to use the simulator before the experiments were conducted and interference with them was avoided during the experimental process. Experimental data from video surveillance were stored. The resulting interactive parallel driving simulated experiments data were input to the lane selection model to obtain optimal game results for real-time computing. The simulation results are compared with actual lane selection results in Table 7.

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Fig.5

Interactive parallel driving simulated experiments.

According to above table, the prediction model established in this paper produces results that are highly compatible with recognition results.

Using the above road experiments, car-following and lane change decision models were built based on optimal control and fuzzy multi-objective decision theory, respectively. The drivers’ experimental data obtained in the experiments were input to micro-simulation models that used (simulation 1) and did not use (simulation 2) the lane selection game, and the macro- (such as flow rate, lane changing frequency, lane changing time, etc.) and micro-laws (such as speed, acceleration, displacement, etc.) of traffic flow in each simulation were compared with road test cases to understand the functioning of the lane selection model.

The lane change frequency, lane changing time, and lane utilization ratio were obtained for verification, the results of which are shown in Figs. 6 to 8. Figure 6 shows the lane changing frequency at various vehicle densities for various driver types. As mentioned, simulation 1 is the case that uses the driver’s lane selection game for real-time simulation, while simulation 2 does not.

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Fig.6

Lane changing frequency simulation results for conservative, common, and radical driver types.

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Fig.7

Lane-changing time comparison of simulation and actual measured values.

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Fig.8

Lane utilization ratio comparison of simulation and actual measured values in left, middle, and right lanes.

The simulation results show that a driver’s (simulated) lane changing frequency can be enhanced by a certain degree using the dynamics of the driver lane selection game (simulation 1), and that, at a given density of vehicles, an aggressive driver’s lane changing frequency is higher than that of a conventional, which in turn is higher than that of a conservative driver.

The model was then used to conduct comparisons between simulated and actual measured lane-changing times and lane utilization ratios. Figure 7 shows the lane-changing times for different traffic flows under a three-lane scenario. Figure 8 shows the lane utilization ratio of each lane in a three-lane scenario. Again in this case, simulation 1 used the lane selection game for real-time simulation while simulation 2 did not.

The average speed, density, and delay were obtained as evaluation indicators for macro verification against experimentally measured data from Zibo city’s Beijing Road; a comparative analysis of simulation and measured values shows that the error is within an acceptable range (Table 8).