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Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials
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Contents lists available at ScienceDirect
International Journal of Transportation Science and Technology journal homepage: www.elsevier.com/locate/ijtst 5 6
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Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials
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Emmanouil N. Barmpounakis ⇑, Eleni I. Vlahogianni, John C. Golias
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National Technical University of Athens, 5 Iroon Polytechniou Str, Zografou Campus, 157 73 Athens, Greece
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a r t i c l e
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Article history: Received 2 September 2016 Received in revised form 26 November 2016 Accepted 28 November 2016 Available online xxxx Keywords: Powered two-wheelers Overtaking Urban arterials Game theory Structural equation modeling
a b s t r a c t A difference between Powered Two Wheelers (PTW) drivers’ behavior and other drivers’ behavior in urban arterials is the frequency of overtaking. The present paper focuses on PTW overtaking and models the specific behavior using concepts of Game Theory. Both the PTW driver and the lead vehicle’s driver are assumed rational decision-makers that develop strategies, trying to maximize their payoffs. These strategies may be cooperative or not with respect to the distances and safety gaps and other behavioral aspects. The payoff function is formulated based on a novel latent statistically determined driving indicator, which quantifies both the driving risk and comfort. The proposed model is evaluated using trajectory data from video recordings on an urban arterial. Results show that both drivers have maximized gains by following a cooperative strategy. Findings also reveal that the successful overtaking rate is higher, when the PTW driver is non-cooperative, whereas lower overtaking rates occur, when the driver of the lead vehicle is non-cooperative. Finally, the concepts of Dominant Strategies, bounded rationality and the construction of the optimum payoff function are further discussed. Ó 2016 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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One of the players that got vigorously into the game of city commuting is Power-Two Wheelers (PTW). PTW can take advantage of their smaller width and horse power to weight ratio to navigate through city traffic more efficiently through overtaking, filtering, or lane sharing (Barmpounakis et al., 2016a; Correa et al., 2016; Wong and Lee, 2015). Moreover, with the increasing popularity of PTW in European roads, their effect on multimodal urban environments has been magnified (ACEM, 2015). The interactions with the rest of the vehicles on urban road networks may be significant to traffic operations and safety, and may lead to increased delays at intersections, reduced level of service at arterials and increased accident risk for both PTW drivers and other road users (Barmpounakis et al., 2016b). Therefore, the manner PTW interact with the rest of the traffic could underlie a key issue of traffic operations management. Most studies model PTW macroscopic and microscopic traffic characteristics during normal conditions or attempt to address safety issues using macroscopic or individual based Intelligent Transportation Systems (ITS) applications (Barmpounakis et al., 2016b; Vlahogianni et al., 2012). Literature has emphasized on the need to address special cases of traffic flow conditions with emphasis on overtaking phenomena that, until recently, have been rarely addressed in the context of
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Peer review under responsibility of Tongji University and Tongji University Press. ⇑ Corresponding author. Fax: +30 210 772 1454. E-mail addresses: [email protected] (E.N. Barmpounakis), [email protected] (E.I. Vlahogianni), [email protected] (J.C. Golias). http://dx.doi.org/10.1016/j.ijtst.2016.11.001 2046-0430/Ó 2016 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: Barmpounakis, E.N., et al. Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials. International Journal of Transportation Science and Technology (2017), http://dx.doi.org/ 10.1016/j.ijtst.2016.11.001
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urban road networks, as well as PTW circulation (Lee et al., 2012; Vlahogianni et al., 2013, 2012). Most of these studies suggest that there is a strong interaction between PTW and surrounding vehicles, affecting how they move, which is still an ongoing research topic, characterized by complex traffic phenomena and trajectories (Barmpounakis et al., 2016a; Mallikarjuna and Kuzhiyamkunnath, 2014; Minh et al., 2005; Nguyen and Hanaoka, 2013; Theofilatos and Yannis, 2015; Wong and Lee, 2015). The principles of Game Theory can be implemented in overtaking conditions, since both PTW and the rest of the road users (drivers, pedestrians etc.) are continuously challenged to make decisions in order to move fast and safely on urban arterials. Given that drivers of different means of transportation have specific characteristics and different way of moving than others, the study of how they make decisions of conflict and cooperation should be studied under the prism of driving behavior. Thus, a PTW driver and the driver of the vehicle being overtaken are considered two different rational decision makers (players), both developing strategies trying to get the best outcome for their decisions. Each one has his own set of actions depending on his driver profile. This game theoretic concept is distinctive to drivers following specific driving patterns while overtaking. It is assumed that each driver has his own strategy when commuting, affecting the surrounding vehicles and traffic macroscopically, for example when it comes to special maneuvers, this strategic thinking can have a considerable effect on traffic flow and traffic volume, by changing the placement of the vehicles, density or distance and time headway between them. Therefore, the study and modeling of drivers’ strategies can be the square one for designing systems that would not only optimize traffic flow but also enhance safety in an automated or semi-automated vehicles environment. Especially for the future roadway conditions and Vehicle to Vehicle (V2V) environments, understanding and controlling the strategies of the drivers is of crucial importance for designing efficient communication protocols. The aim of this paper is to present a game theoretic approach to describe PTW overtaking phenomena and interactions with the rest of the traffic to set the framework of a Cooperative ITS design. First, a concise theoretical analysis on Game Theory and its solution concepts are presented in the context of traffic overtaking conditions. Next, a definition of cooperativeness among drivers is provided and the specifications of a novel payoff function are extracted using multivariate structural equation statistical modeling. Following, the study area and the driving experiment are described, and the payoffs for each player are estimated based on real data extracted from video recordings. The Game Theoretic overtaking model is, then, formed and evaluated. Finally, conclusions are summarized and future research directions are discussed.
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2. Game theory: a short introduction
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Game Theory includes mathematical tools and models for describing conflict and cooperation conditions between intelligent rational decision makers that will affect their welfare (Myerson, 2013). In transportation and traffic engineering, game theory applications refer mostly to logistics, urban networks and public transportation, mostly connected with economics (Hollander and Prashker, 2006; Zhang et al., 2010). Concepts of Game theory have been previously applied to traffic analysis and modeling with the aim to explain driving patterns and behavioral characteristics, for example the merging or lane changing maneuvers (Kita, 1999; Kita et al., 2002; Liu et al., 2007; Talebpour et al., 2015), the decrease in safety due to the selfish actions of drivers (Rass et al., 2008), or the formulation of control strategies for automated vehicles, while obeying safety and comfort requirements (Wang et al., 2015). In most of the above studies, the use of strategic decision making of explaining drivers behavior has acknowledged. Nevertheless, game theory applicability to driving behavior has been rarely supported by real world data. Towards this direction, in a recent study, a strategic game was formed to explain the PTW overtaking procedure in urban arterials, in which each player considers his plan of actions at the beginning of the game without evaluating the other player’s action (Barmpounakis et al., 2015). This study extends previous research and proposes an extensive-form game to model PTW overtaking behavior, which is an explicit description of sequential structure of the decision problems encountered by the players in a strategic situation, where a player can consider his plan of actions at any point of time at which he has to make a decision (Osborne and Rubinstein, 1994). As overtaking phenomena are common in everyday commuting, it is assumed that both players know the strategy that the other player has chosen. Therefore, the game formed is an extensive game with perfect information. A finite extensive game consists of a finite set of i = 1, 2, . . ., I players and a set X of decision points (nodes), which forms a tree, with Z X being the terminal nodes. Moreover, a set of functions that describe for each x R Z, the player i(x) who moves at x, the set of A(x) of possible actions at x and the successor noden(x, a) resulting from action a. Finally, there is a payoff function ui:Z ? assigning payoffs to players as a function of the terminal node reached and an information partition, that for each x, let h(x) denote the set of nodes that are possible given what player i(x) knows. Thus, if x0 e h(x), then i(x0 ) = i(x), A(x0 ) = A(x) and h(x0 ) = h(x) (Levin, 2002). The situation described above is could be represented by a ‘‘Stackelberg game” where one player is a ‘‘leader” and chooses an action from a set A1 and the other a ‘‘follower”, who informed of the leader’s choice chooses his action from an action set A2. A more realistic representation of the game would be to insert ‘‘nature” as a third player, who decides at the initial point of the game who plays first. Therefore, in our case the game will start with ‘‘nature” (N) deciding randomly who plays first. Next, depending the choice of ‘‘nature”, Player 1 (P1) or Player 2 (P2) will choose his/her action following the choice of the second player to end the game (Fig. 1). Although Fig. 1 describes a rather abstract representation of the game to describe the
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Please cite this article in press as: Barmpounakis, E.N., et al. Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials. International Journal of Transportation Science and Technology (2017), http://dx.doi.org/ 10.1016/j.ijtst.2016.11.001
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Fig. 1. The game theoretic representation of PTW overtaking phenomena.
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sequential decisions of the players, in the following paragraphs it will be enriched with the players’ strategies and payoffs for its full form. Every game has a number of solution concepts, depending on the application and desired outcome. One of the most widely used one is the Nash Equilibrium solution, in which no player has anything to gain by deviating from his strategy after considering the opponent’s choice, assuming that his opponent will also not deviate. However, the Nash equilibrium treats the strategies as choices that are made once and for all, therefore in an extensive-form game it ignores the sequential structure of the game; thus, detecting the Dominant Strategy for each player is also very important in a game. A strategy is dominant when the payoff is maximized regardless the actions of the other players. This means that, using backwards induction, every player chooses the action that gives the best outcome, while only keeping actions that are available to a player from reachable nodes.
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3. Modeling interactions between vehicles during overtaking
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3.1. Cooperativeness among drivers revisited
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How one can define the cooperativeness of a driver during overtaking? Cooperativeness is a latent concept, which cannot be measured, but needs to be estimated using observed variables (Talebpour et al., 2015). Cooperation between vehicles approaching an intersection or a stopped vehicle has been previously considered to occur when no collision is achieved (Li and Wang, 2006; Sengupta et al., 2007; van Arem et al., 2006; Yang et al., 2004). In another study, cooperativeness has been treated as being univariately defined based on speeds or spatial factors (Barmpounakis et al., 2015). In the framework of lane changing maneuvers, a previous research attempt defined cooperative driving when no critical situation arises or when the efficiency of the maneuver can be maximized with respect to fuel or time consumption (Heesen et al., 2012). Since overtaking is one of the most complex maneuvers, in this study we focus on safety and risk free driving conditions. Let Player 1 (P1) be the subject PTW and Player 2 (P2) the lead vehicle. Moreover, consider that cooperativeness of a driver during overtaking can be defined by the following measures: i. the spacing between the two players, ii. the available space for P1 to pass P2, iii. the lane P1 is using in order to overtake, and iv. the lead vehicle’s acceleration (Fig. 2). Specifically, the cooperativeness of the PTW (P1) can be described by the following rules: If the PTW driver keeps a low distance from P2 (tailgating), then his behavior is non-cooperative due to safety reasons. Based on recent statistical findings, this minimum safety gap is set to 4.95 m (Nguyen and Hanaoka, 2013). Moreover, the PTW driver has a cooperative behavior
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Fig. 2. The basic measures for defining cooperativeness.
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in relation to the lead vehicle (P2), when driving in the adjacent lane at the beginning of the overtake, since no lateral or other risky maneuvers are required and less conspicuity issues may occur. Concerning the cooperativeness of the lead vehicle’s driver, to our knowledge, there is no previous research on the safe gap for the PTW to complete an overtake, therefore, we extend the use of the safety gap of 4.95 m. Consider the case of a two lane arterial stream where P2 is the lead vehicle that is been overtaken by the PTW and V3 a vehicle in the adjacent lane in front of the PTW. Let the minimum safety gap of PTW and the lead vehicle (P2) equal to 4.95 m. Moreover, let the same minimum gap apply between the PTW and V3 at the time which the PTW overtakes. Then, the opening during overtake between the lead vehicle and V3 should be at least two times the safety gap, which equals to 9.90 m. If the average length of motorcycles and scooters (2.1 m) is also added, then, the opening required for a safe and comfortable pass is increased from 9.90 m to at least 12 m. Therefore, if the opening space is less than 12 m P2 is considered to be non-cooperative. In addition, P2 is considered non-cooperative, when he accelerates during a PTW overtake attempt.
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3.2. Comfort at overtake as a game’s payoff
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In order to quantify the gain of a driver during overtaking, the critical factor, which may significantly affect overtaking, should be identified and modeled. Although recent approaches focus on the effect of speeds during overtaking (Barmpounakis et al., 2015), literature has been systematically underlined the effect of other supplementary factors, such as Time To Collision (TTC), perceived safety of the individual driver-vehicle system, joy of driving, social welfare that should be taken into consideration (Altendorf and Flemisch, 2014; Arbis et al., 2016; Barmpounakis et al., 2015). The above show that the proposed payoff function should have a certain multivariate nature to account for the entire spatio-temporal aspects of interactions between PTW and other vehicles during overtaking. To account for the multivariate aspects of overtaking, a novel measure termed as Comfort at Overtake (CaO) is defined, which takes into consideration both safety measures (TTC and crash avoidance with surrounding vehicles) and the desire of PTW to move faster than other vehicles. The CaO is not observable (latent variable), but needs to be estimated in a certain consistent manner taking into consideration real world collected vehicle trajectories and is quantified using advanced statistical modeling. In order to quantify the unobserved variable CaO for both Player 1 and 2 at an overtaking attempt, a Multiple-InputsMultiple-Outputs (MIMIC) structural equation model is constructed. The MIMIC model is a special case of latent variable statistical modeling, a thorough technique for testing hypotheses for the relations between observed and unobserved (latent) variables. The model consists of two components: a measurement model which defines the relations between a latent variable and its indicators and a structural model which specifies the casual relationships among latent variables and explains the casual effects. MIMIC models consider the latent variable g to be scalar and relates the vector of indicators y and the
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observed exogenous variables x that cause y by the following system of equations
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?? ¼ Cx þ ??
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y ¼ K?? þ ??
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where C and K are matrices of unknown parameters to be estimated and e and f; are the error terms (Washington et al., 2010). These type of models has been previously implemented in (Barmpounakis et al., 2016a; Chung et al., 2012; Hassan and Abdel-Aty, 2011; Karlaftis et al., 2001). The goodness of fit of the models developed are assessed by implementing likelihood ratio tests for comparing the proposed model to the saturated one (the model that fits the covariances perfectly) and baseline models (model that includes the means and variances of all observed variables plus the covariances of all observed exogenous variables). In addition, other indicators that were examined were the root mean squared error of approximation (RMSEA) along with the probability of RMSEA being below 0.05, the standardized root mean squared residual SRMR and the coefficient of determination of the various models.
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4. Implementation and findings
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4.1. The study area
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In order to avoid complex overtaking phenomena, the proposed approach is implemented on data from a divided arterial with two lanes per directions (Fig. 3). Except from offering the ability to observe and monitor the upstream traffic, the spot is ideal for not affecting drivers’ behavior. More information on the database used for this research can be found in a previous study where the same database was used (Barmpounakis et al., 2016a). Data collected through trajectory tracking, consists of 850 cases of PTW overtaking. Table 1 and Fig. 4 depict the variables that are measured in each case and will be used in the formation of the game. All speeds and headways are measured at the start of the overtaking maneuver.
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Fig. 3. Camcorder setup on Calatrava’s pedestrian bridge in Athens (Greece).
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4.2. Quantifying comfort at overtake
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Two distinct models are developed to estimate the latent CaO parameter for players 1 and 2, the PTW and the lead vehicle respectively. For the case of PTW (P1), three variables are defined as indicators in the MIMIC model; i. The speed of the PTW (Vm) expresses the desire of the PTW driver to move faster, ii. the variable DistanceXY expresses the desire to avoid a crash with the preceding vehicle, and iii. Back_dis the desire to avoid a crash with the vehicle that is coming from the back. For the lead vehicle (P2), the indicators of comfort at overtake are: i. speed difference between the two vehicles (Diff) expresses the desire of Player 2 not to be overtaken with excessive speeds, ii. the distance d1 and iii. the Opening which expresses the desire of not having many vehicles around him. Table 2 shows the goodness of fit for the best-fitted models (Washington et al., 2010). The CaO for P1 is calculated using the model that is described in Table 3 and illustrated in Fig. 5. All variables depicted are significant in 5% significance level, while passenger is significant in 10% significance level. The coefficients of the variables show that the speed of the PTW (Vm) and the distance from the vehicle in the back (Back_dis) are critical parameters for defining CaO. As far as the predictors of the models are concerned, the placement of the vehicles concerning the lane each one is present (PTW_lane and Veh_lane) and whether the PTW driver is wearing a helmet or not are also significant parameters. It should be noted that the opening space for the PTW (Opening) is also of importance when calculating CaO.
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Table 1 Description of variables measured. Variable
Description
Vm V1 Diff d2 d3 Opening s_x s_y Distance d1 Back_dis Passenger Platoon Helmet PTW_lane Veh_lane Over_RL Lane_same Over
Subject PTW’s [Player 1] speed (km/h) Vehicle 1 [Player 2] speed (km/h) Speed difference between Player 1 and Player 2, diff = Vm V1 (km/h) Oblique distance between Player 2 and front vehicle (m) Lateral distance between Player 2 and front vehicle (m) Opening space in front of Player 2, Opening = (d22 + d23)0.5 (m). Lateral distance between Player 1 and Player 2 (m) Oblique distance between Player 1 and Player 2 (m) Distance between Player 1 and Player 2, (s_x2 + s_y2)0.5 (m) Distance between Vehicle 2 and 3 (m) Distance between Player 1 and Vehicle 2 (m) Is there a passenger on the PTW? (0/1) Existence of platoon, subject PTW is the leader (0/1) Does the PTW driver use a helmet? (0/1) Left or right lane in which PTW is moving (0/1) Right or left lane in which vehicle that is being overtaken is moving (0/1) Player 1 tried to overtake Player 2 from Right or Left? (0/1) Are Player 1 and Player 2 in the same lane? (0/1) Did the PTW complete the overtake attempt? (0/1)
Please cite this article in press as: Barmpounakis, E.N., et al. Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials. International Journal of Transportation Science and Technology (2017), http://dx.doi.org/ 10.1016/j.ijtst.2016.11.001
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Fig. 4. Illustration of variables that are measured in each overtaking attempt.
Table 2 Goodness of fit assessment for SEM models. Fit statistic
Estimated values for Player 1
Estimated values for Player 2
Likelihood ratio x2_ms(7) x2 (p > x2) – Saturated x2_bs(18) x2 (p > x2) – Baseline
Criteria
5.701 0.575 197.963 0.000
5.196 0.268 526.410 0.000
Population error RMSEA 90% CI, lower bound Upper bound pclose
0.000 0.000 0.037 0.993
0.019 0.000 0.058 0.891
<0.05 <0.05 <0.10 >0.05
Information criteria AIC BIC
30106.381 30187.050
31053.808 31134.477
– –
Baseline comparison CFI TLI
1.000 1.019
0.998 0.991
>0.90 <3.00
Size of residuals SRMR CD
0.011 0.614
0.09 0.396
<0.10
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Table 3 Results of MIMIC Model for Player 1. Structural
Coefficient
Std. Err.
z
p > |z|
CaO PTW_lane Veh_lane Passenger Helmet Opening
5.74756 5.23472 3.67808 5.11538 0.11394
0.89046 1.57036 2.23441 1.65496 0.01354
6.450 3.330 1.650 3.090 8.410
0.000 0.001 0.100 0.002 0.000
Vm CaO Veh_lane Constant
1 4.20860 77.66768
(Constrained) 1.77279 1.86331
2.370 41.680
0.018 0.000
Distance CaO Opening Constant
0.06518 0.01676 6.40407
0.03173 0.00569 0.23018
2.050 2.940 27.820
0.040 0.003 0.000
0.25616 4.74621 4.09514 12.12033 0.70234 49.41105 9.19457
6.210 2.090 11.910 141.501 13.008 796.460 9.256
0.000 0.037 0.000 189.179 15.766 990.532 49.692
Back_dis CaO 1.59063 Helmet 9.90969 Constant 48.75891 e.Vm 163.61200 e.DistanceXY 14.32069 e.bd 888.21130 e. CaO 21.44651 LR test of model vs. saturated: x2(10) = 5.70, Prob > x2 = 0.5751
Please cite this article in press as: Barmpounakis, E.N., et al. Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials. International Journal of Transportation Science and Technology (2017), http://dx.doi.org/ 10.1016/j.ijtst.2016.11.001
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Fig. 5. MIMIC model for Player 1.
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The CaO for P2 is estimated by the model described in Table 4 and illustrated in Fig. 6. It is seen that the placement of the vehicles and specifically whether the PTW is moving in the same lane as the vehicle (lane_same) as well as the lane the PTW tried to overtake P2 (Over_RL). Moreover, as far as the indicators are concerned, the coefficient of d1 is similar to opening, which shows that all vehicles have the same effect on a driver’s CaO. The above mentioned parameters clearly depict the desire of P2 not being surrounded by other vehicles while the strong relationship between the space headway of P1 and P2 (Distance) and the speed difference between them (Diff) represent the a TTC equivalent measurement. Using the results from the two MIMIC models, we calculate the CaO for each driver per overtaking attempt. Αll values are normalized in 0–1 range so that the worst case will stand for a CaO equal to 0, while the best for a CaO equal to 1. The average value of the CaO of P1 (n = 850) is 0.51 (±0.20) and for P2 0.42 (±0.24). To better illustrate the meaning of the CaO measure using the MIMIC models developed two examples follow: An overtaking attempt when P1 is wearing a helmet and no passenger is on the PTW, the two players are on the same lane, their distance is about 13 m and the opening space is more than
Table 4 Results of MIMIC Model for Player 2. Structural
Coefficient
Std. Err.
z
p > |z|
CaO Lane_Same Over_RL Distance Platoon
26.78280 5.54144 0.82980 4.44533
5.78190 2.97884 0.26588 3.16298
4.630 1.860 3.120 1.410
0.000 0.063 0.002 0.160
Opening CaO Lane_Same Over_RL Constant
1 15.27592 7.82455 33.69178
(Constrained) 5.64972 2.66714 2.44310
2.700 2.930 13.790
0.007 0.003 0.000
Diff CaO Over_RL DistanceXY Constant
0.10804 6.99010 0.41246 21.74381
0.02332 1.04108 0.12618 1.08515
4.630 6.710 3.270 20.040
0.000 0.000 0.001 0.000
0.20709 2.96258 128.56300 9.37399 143.96470 135.52300
5.020 30.120 142.654 173.244 613.696 421.385
0.000 0.000 701.611 210.046 1188.352 968.072
Measurement d1 CaO 1.04002 _cons 89.22012 e.Opening 316.36580 e.Diff 190.75950 e.d1new 853.98320 e.CaO 638.69450 LR test of model vs. saturated: x2(10) = 1.70, Prob > x2 = 0.8892
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Fig. 6. MIMIC model for Player 2.
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20 m will give a CaO equal to 0.60. As far as P2 is concerned, a CaO around 0.50 stands for an overtaking attempt when the distance between the two players is more than 5.5 m and no other vehicles are present in a radius of 20 m.
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4.3. Game theoretic model
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In order to implement the game theoretic model that was described in Section 2, the payoffs for each player should be estimated for each terminal node. First, we separate all 850 overtaking attempts depending on the strategy each player followed. Thus, four clusters are created; both being cooperative (C, C), both being non-cooperative (NC, NC), one player being cooperative and the other non-cooperative (NC, C) and (C, NC). Using the average of CaO values for each cluster as payoff, the extensive game is formed as in Fig. 7. For example, the average of CaO values when both players are Cooperative is equal to 0.58 for P1 and 0.50 for P2. Both average values of CaO when both players are cooperative, stand for proper conditions for a safe and comfortable pass, while payoffs are reduced for Non Cooperative strategies. Black color stands for Player 1 and grey for Player 2.
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Fig. 7. The game in extensive form.
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As can be observed, the Dominant Strategy for both players is to be cooperative. However, from the collected data and based on the definition of cooperativeness, a PTW driver (P1) chooses to be cooperative 82% and the lead vehicle’s driver (P2) chooses to be cooperative at 68%. The question that arises is why a driver would follow another strategy than the dominant one, one more paradox to be added to the many that have been documented in social experience and actual experiments (Axelrod, 1984). The explanation for this inconsistency is not as simple as it may seem. Someone could argue that lead vehicles’ drivers may not leave adequate opening for the PTW to overtake, since they are not interested in the gain they would have or that a PTW driver may follow a Non-Cooperative strategy in order to be ready for overtaking, thinking that it may lead to higher speeds and shorter travel times. As seen in Table 5, which depicts the number of total cases recorded – grouped by strategy set – as well as the number and percentage of successful overtake attempts are depicted, a driver may choose to be NC, but complete the overtake. This percentage is higher when compared to the successful overtakes produced by Cooperative PTW drivers. As can be observed, when the PTW driver follows a NC strategy, he is most likely to complete an overtake than when he follows a C strategy. An overtake attempt is considered as successful when the PTW did complete the overtake and passed in front of the lead vehicle, as opposed to an unsuccessful one, when the PTW did not pass the lead vehicle, but stayed behind it. This finding suggests that PTW drivers may undertake an aggressive behavior while overtaking probably because, even if they conduct risky maneuvers, they aim at pursuing short-term gains during driving. As expected, the lowest percentages occur when the PTW driver is Cooperative and the other driver is not. Evidently, overtaking is a high-risk maneuver and timid PTW drivers will most likely seek cooperation in order to complete an overtake. Therefore, a PTW driver may choose a NC behavior because he may enjoy the risky driving, sharp passing, extreme acceleration and adrenaline while overtaking in comparison to a timid driver that will prefer safety over enjoyment and higher speeds. Similar findings can be documented in a research that focuses on driving risk perception (Dionne et al., 2007), where authors conclude that drivers are more cautious when they have an improved risk perception in contrast to those who underestimate it and tend to be more at risk. The risk taking behavior by some users has also been identified in (Andersson and Lundborg, 2007; Rubinfeld and Rodgers, 1992). What is interesting is that most of the times, a player tends to cooperate with the other one. It is seen that in 91% of total overtaking attempts, at least one player was cooperating and in 59% of total overtaking attempts, both followed a cooperative behavior. These figures show that, although both players may frequently seek for the long-term gain, they may exhibit interchangeably shifts to non-cooperative (aggressive) behavior while driving. This finding has been emphasized in many research efforts that studied microscopic behavior from various angles (Laval, 2011; Orfanou et al., 2012; Papacharalampous and Vlahogianni, 2014). From a game theoretic perspective, literature states: ‘‘very often game theory predicts behavior that can be criticized as unnatural and non-rational” (Papadimitriou and Yannakakis, 1994). Evidently, regardless of the predictive equilibrium of the proposed game, the evaluation with real data underlines that drivers may not always choose to exhibit a selfishly antisocial way, but often give up a short-term gain in order to behave in a cooperative manner. Finally, another parameter that could affect driving behavior is fuel efficiency since acceleration and deceleration modes increase fuel consumption than steady-driving modes (Tong et al., 2000). Some drivers may prefer to drive in a more economical way, which corresponds to Cooperative driving in a way of keeping steady speeds and gaps. Consequently, including accelerative information in the payoff function based on the above statements could lead in different results in comparison to real data.
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5. Rational versus irrational drivers’ expectations: a general discussion
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These inconsistencies between the solution produced by the game theoretic model and the real data may be also understood using the theory of bounded rationality (Simon, 2000). In this framework, for explaining PTW driver’s choices when overtaking, as well as the interactions with the rest of traffic, focus should be shed to both the process of decision and the quality of the outcome. The process of decision is understood through studying the psychology of the decision maker. At an overtaking game it is questionable as to what extent the drivers use other abilities, such as to have the necessary knowledge of the phenomena and evoke it when relevant, to work out the consequences of their actions, to conjure up possible courses of action to cope with uncertainty, rather than maximizing or optimizing their utilities. Similar findings can be found concerning the use of safety equipment in (Blomquist, 1991). Thus, it is of interest to investigate the dynamic changes in the players/decision makers driving behavior during overtaking, providing a better insight at defining rationality. For
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Table 5 Summary table of overtake attempts by strategy. Strategy (P1, P2)
Overtake attempts
Successful overtakes
Percentage (%)
NC, NC NC, C C, NC C, C
73 76 198 503
57 67 96 306
78% 88% 49% 61%
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Fig. 8. Quantal response equilibrium for both players.
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example, is the assumption of independent decisions made accurate? Does the driver have a long or short memory that evokes when asked to conduct an overtake? To address the questions raised above, we use the Quantal Response Equilibrium (QRE) concept introduced by McKelvey and Palfrey (Mckelvey and Palfrey, 1995), which replaces the perfectly rational expectations with imperfect, or noisy, rational expectations equilibrium. An interesting property of QRE is that systematic deviations from Nash Equilibrium may be predicted without introducing systematic features to the error structure. Detailed definitions and notations of QRE for extensive form games can be found in (McKelvey and Palfrey, 1998). QRE theory can be summarized as follows; better responses are more likely to be chosen than worse responses, but that does not mean that best responses are played with certainty. Theory suggests that as a player gains experience playing a specific game and makes repeated observations about the actual payoffs received from different action choices, he can be expected to make more precise estimates of the expected payoffs from different strategies. This experience gained by repeated observations can be modeled through the parameter k. Although the QRE theory does not describe the origins of the parameter k, it can be treated as a rationality parameter, where, if k ! 0, the player becomes completely irrational and plays each strategy with equal probability, while if k ! 1 the player becomes completely rational. Using the previously described database and utilizing the algorithm described in (Turocy, 2009), both players learn to be cooperative after a number of iterations as seen in Fig. 8 (McKelvey et al., 2016). The two cases that should be stood out are the ones that are illustrated with dashed lines. Concerning P1, it is seen that when he plays first (dashed), the learning process to cooperativeness is faster than when he plays second (straight). At the same time, when P1 chooses to cooperate and P2 plays second (dashed), the learning process to cooperativeness is slower due to the small difference between the two payoffs (0.50 and 0.45).
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6. Conclusion
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The present paper implemented concepts from Game Theory to describe the PTW overtaking phenomena in urban arterials. A Non-cooperative Game with perfect information is proposed which consists of two players; i. the PTW driver and ii. the lead’s vehicle driver. Both drivers can either be Cooperative or Non-Cooperative, at the emergence of an PTW driver overtake attempt. A PTW driver is considered to be cooperative, when he keeps adequate distance from the lead vehicle, and the
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latter is considered to be cooperative when he leaves adequate opening for the PTW to pass. The pay-off of each player is defined by the latent variable Comfort at Overtake (CaO), which is calculated by a MIMIC model. It is found that the game has the optimum strategy for both players to cooperate at the occurrence of an overtake attempt. Game theoretic results are further compared to real data. Interestingly, the evaluation with real world data showed that although both drivers exhibit – mainly – a cooperative behavior, overtaking can sometimes be conducted through aggressive driving and a Non Cooperative behavior. Then a discussion arises as to examining the reasons that both players do not always follow the cooperative strategy concerning the Quantal Response Equilibrium (QRE) concept, player rationality and payoff function. It is suggested that this game theoretic approach can be extended to various forms of games to overcome the underlined limitations, such as the independence in players’ decision and the memory properties of driver’s behavior. Further, different payoff functions should be considered. Overall, the present paper shows that Game Theory has a significant advantage for modelling the interplay between drivers on the road, since drivers’ behavior is dependent on their environment. In future studies, more game formations can be examined that will be able to describe more adequately the overtaking procedure. Latest technological advances in Vehicle Communication Systems can implement findings from such models by leveraging the models’ ability to explain interactions among two or more drivers on a road to improve existing Advanced Driver Assistance Systems (ADAS) or develop novel ones. Communication between drivers could significantly improve modeling accuracy and promote cooperation among them, since, as theory suggests, cooperation can evolve even without foresight or friendship between players (Axelrod, 1984). If cooperation among drivers becomes a reality in the future, driver’s view of commuting will have to expand towards new territories. Cooperation could be translated to information exchange between the drivers, either by situation awareness, or by giving assisting directions to drivers. The identified irrationality of some drivers could be alleviated by the use of such an ADAS context, for example to assist driving to complex traffic conditions (stop-and-go conditions, moving bottleneck conditions etc.), to dis/encourage overtakes in certain traffic conditions or to control spatial characteristics among traffic for optimization or emergency purposes, as an additional mean of traffic management.
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Contents lists available at ScienceDirect
International Journal of Transportation Science and Technology journal homepage: www.elsevier.com/locate/ijtst 5 6
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Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials
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Emmanouil N. Barmpounakis ⇑, Eleni I. Vlahogianni, John C. Golias
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National Technical University of Athens, 5 Iroon Polytechniou Str, Zografou Campus, 157 73 Athens, Greece
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Article history: Received 2 September 2016 Received in revised form 26 November 2016 Accepted 28 November 2016 Available online xxxx Keywords: Powered two-wheelers Overtaking Urban arterials Game theory Structural equation modeling
a b s t r a c t A difference between Powered Two Wheelers (PTW) drivers’ behavior and other drivers’ behavior in urban arterials is the frequency of overtaking. The present paper focuses on PTW overtaking and models the specific behavior using concepts of Game Theory. Both the PTW driver and the lead vehicle’s driver are assumed rational decision-makers that develop strategies, trying to maximize their payoffs. These strategies may be cooperative or not with respect to the distances and safety gaps and other behavioral aspects. The payoff function is formulated based on a novel latent statistically determined driving indicator, which quantifies both the driving risk and comfort. The proposed model is evaluated using trajectory data from video recordings on an urban arterial. Results show that both drivers have maximized gains by following a cooperative strategy. Findings also reveal that the successful overtaking rate is higher, when the PTW driver is non-cooperative, whereas lower overtaking rates occur, when the driver of the lead vehicle is non-cooperative. Finally, the concepts of Dominant Strategies, bounded rationality and the construction of the optimum payoff function are further discussed. Ó 2016 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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One of the players that got vigorously into the game of city commuting is Power-Two Wheelers (PTW). PTW can take advantage of their smaller width and horse power to weight ratio to navigate through city traffic more efficiently through overtaking, filtering, or lane sharing (Barmpounakis et al., 2016a; Correa et al., 2016; Wong and Lee, 2015). Moreover, with the increasing popularity of PTW in European roads, their effect on multimodal urban environments has been magnified (ACEM, 2015). The interactions with the rest of the vehicles on urban road networks may be significant to traffic operations and safety, and may lead to increased delays at intersections, reduced level of service at arterials and increased accident risk for both PTW drivers and other road users (Barmpounakis et al., 2016b). Therefore, the manner PTW interact with the rest of the traffic could underlie a key issue of traffic operations management. Most studies model PTW macroscopic and microscopic traffic characteristics during normal conditions or attempt to address safety issues using macroscopic or individual based Intelligent Transportation Systems (ITS) applications (Barmpounakis et al., 2016b; Vlahogianni et al., 2012). Literature has emphasized on the need to address special cases of traffic flow conditions with emphasis on overtaking phenomena that, until recently, have been rarely addressed in the context of
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Peer review under responsibility of Tongji University and Tongji University Press. ⇑ Corresponding author. Fax: +30 210 772 1454. E-mail addresses: [email protected] (E.N. Barmpounakis), [email protected] (E.I. Vlahogianni), [email protected] (J.C. Golias). http://dx.doi.org/10.1016/j.ijtst.2016.11.001 2046-0430/Ó 2016 Tongji University and Tongji University Press. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: Barmpounakis, E.N., et al. Modeling cooperation and powered-two wheelers short-term strategic decisions during overtaking in urban arterials. International Journal of Transportation Science and Technology (2017), http://dx.doi.org/ 10.1016/j.ijtst.2016.11.001
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urban road networks, as well as PTW circulation (Lee et al., 2012; Vlahogianni et al., 2013, 2012). Most of these studies suggest that there is a strong interaction between PTW and surrounding vehicles, affecting how they move, which is still an ongoing research topic, characterized by complex traffic phenomena and trajectories (Barmpounakis et al., 2016a; Mallikarjuna and Kuzhiyamkunnath, 2014; Minh et al., 2005; Nguyen and Hanaoka, 2013; Theofilatos and Yannis, 2015; Wong and Lee, 2015). The principles of Game Theory can be implemented in overtaking conditions, since both PTW and the rest of the road users (drivers, pedestrians etc.) are continuously challenged to make decisions in order to move fast and safely on urban arterials. Given that drivers of different means of transportation have specific characteristics and different way of moving than others, the study of how they make decisions of conflict and cooperation should be studied under the prism of driving behavior. Thus, a PTW driver and the driver of the vehicle being overtaken are considered two different rational decision makers (players), both developing strategies trying to get the best outcome for their decisions. Each one has his own set of actions depending on his driver profile. This game theoretic concept is distinctive to drivers following specific driving patterns while overtaking. It is assumed that each driver has his own strategy when commuting, affecting the surrounding vehicles and traffic macroscopically, for example when it comes to special maneuvers, this strategic thinking can have a considerable effect on traffic flow and traffic volume, by changing the placement of the vehicles, density or distance and time headway between them. Therefore, the study and modeling of drivers’ strategies can be the square one for designing systems that would not only optimize traffic flow but also enhance safety in an automated or semi-automated vehicles environment. Especially for the future roadway conditions and Vehicle to Vehicle (V2V) environments, understanding and controlling the strategies of the drivers is of crucial importance for designing efficient communication protocols. The aim of this paper is to present a game theoretic approach to describe PTW overtaking phenomena and interactions with the rest of the traffic to set the framework of a Cooperative ITS design. First, a concise theoretical analysis on Game Theory and its solution concepts are presented in the context of traffic overtaking conditions. Next, a definition of cooperativeness among drivers is provided and the specifications of a novel payoff function are extracted using multivariate structural equation statistical modeling. Following, the study area and the driving experiment are described, and the payoffs for each player are estimated based on real data extracted from video recordings. The Game Theoretic overtaking model is, then, formed and evaluated. Finally, conclusions are summarized and future research directions are discussed.
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2. Game theory: a short introduction
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Game Theory includes mathematical tools and models for describing conflict and cooperation conditions between intelligent rational decision makers that will affect their welfare (Myerson, 2013). In transportation and traffic engineering, game theory applications refer mostly to logistics, urban networks and public transportation, mostly connected with economics (Hollander and Prashker, 2006; Zhang et al., 2010). Concepts of Game theory have been previously applied to traffic analysis and modeling with the aim to explain driving patterns and behavioral characteristics, for example the merging or lane changing maneuvers (Kita, 1999; Kita et al., 2002; Liu et al., 2007; Talebpour et al., 2015), the decrease in safety due to the selfish actions of drivers (Rass et al., 2008), or the formulation of control strategies for automated vehicles, while obeying safety and comfort requirements (Wang et al., 2015). In most of the above studies, the use of strategic decision making of explaining drivers behavior has acknowledged. Nevertheless, game theory applicability to driving behavior has been rarely supported by real world data. Towards this direction, in a recent study, a strategic game was formed to explain the PTW overtaking procedure in urban arterials, in which each player considers his plan of actions at the beginning of the game without evaluating the other player’s action (Barmpounakis et al., 2015). This study extends previous research and proposes an extensive-form game to model PTW overtaking behavior, which is an explicit description of sequential structure of the decision problems encountered by the players in a strategic situation, where a player can consider his plan of actions at any point of time at which he has to make a decision (Osborne and Rubinstein, 1994). As overtaking phenomena are common in everyday commuting, it is assumed that both players know the strategy that the other player has chosen. Therefore, the game formed is an extensive game with perfect information. A finite extensive game consists of a finite set of i = 1, 2, . . ., I players and a set X of decision points (nodes), which forms a tree, with Z X being the terminal nodes. Moreover, a set of functions that describe for each x R Z, the player i(x) who moves at x, the set of A(x) of possible actions at x and the successor noden(x, a) resulting from action a. Finally, there is a payoff function ui:Z ? assigning payoffs to players as a function of the terminal node reached and an information partition, that for each x, let h(x) denote the set of nodes that are possible given what player i(x) knows. Thus, if x0 e h(x), then i(x0 ) = i(x), A(x0 ) = A(x) and h(x0 ) = h(x) (Levin, 2002). The situation described above is could be represented by a ‘‘Stackelberg game” where one player is a ‘‘leader” and chooses an action from a set A1 and the other a ‘‘follower”, who informed of the leader’s choice chooses his action from an action set A2. A more realistic representation of the game would be to insert ‘‘nature” as a third player, who decides at the initial point of the game who plays first. Therefore, in our case the game will start with ‘‘nature” (N) deciding randomly who plays first. Next, depending the choice of ‘‘nature”, Player 1 (P1) or Player 2 (P2) will choose his/her action following the choice of the second player to end the game (Fig. 1). Although Fig. 1 describes a rather abstract representation of the game to describe the
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Fig. 1. The game theoretic representation of PTW overtaking phenomena.
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sequential decisions of the players, in the following paragraphs it will be enriched with the players’ strategies and payoffs for its full form. Every game has a number of solution concepts, depending on the application and desired outcome. One of the most widely used one is the Nash Equilibrium solution, in which no player has anything to gain by deviating from his strategy after considering the opponent’s choice, assuming that his opponent will also not deviate. However, the Nash equilibrium treats the strategies as choices that are made once and for all, therefore in an extensive-form game it ignores the sequential structure of the game; thus, detecting the Dominant Strategy for each player is also very important in a game. A strategy is dominant when the payoff is maximized regardless the actions of the other players. This means that, using backwards induction, every player chooses the action that gives the best outcome, while only keeping actions that are available to a player from reachable nodes.
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3. Modeling interactions between vehicles during overtaking
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3.1. Cooperativeness among drivers revisited
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How one can define the cooperativeness of a driver during overtaking? Cooperativeness is a latent concept, which cannot be measured, but needs to be estimated using observed variables (Talebpour et al., 2015). Cooperation between vehicles approaching an intersection or a stopped vehicle has been previously considered to occur when no collision is achieved (Li and Wang, 2006; Sengupta et al., 2007; van Arem et al., 2006; Yang et al., 2004). In another study, cooperativeness has been treated as being univariately defined based on speeds or spatial factors (Barmpounakis et al., 2015). In the framework of lane changing maneuvers, a previous research attempt defined cooperative driving when no critical situation arises or when the efficiency of the maneuver can be maximized with respect to fuel or time consumption (Heesen et al., 2012). Since overtaking is one of the most complex maneuvers, in this study we focus on safety and risk free driving conditions. Let Player 1 (P1) be the subject PTW and Player 2 (P2) the lead vehicle. Moreover, consider that cooperativeness of a driver during overtaking can be defined by the following measures: i. the spacing between the two players, ii. the available space for P1 to pass P2, iii. the lane P1 is using in order to overtake, and iv. the lead vehicle’s acceleration (Fig. 2). Specifically, the cooperativeness of the PTW (P1) can be described by the following rules: If the PTW driver keeps a low distance from P2 (tailgating), then his behavior is non-cooperative due to safety reasons. Based on recent statistical findings, this minimum safety gap is set to 4.95 m (Nguyen and Hanaoka, 2013). Moreover, the PTW driver has a cooperative behavior
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Fig. 2. The basic measures for defining cooperativeness.
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in relation to the lead vehicle (P2), when driving in the adjacent lane at the beginning of the overtake, since no lateral or other risky maneuvers are required and less conspicuity issues may occur. Concerning the cooperativeness of the lead vehicle’s driver, to our knowledge, there is no previous research on the safe gap for the PTW to complete an overtake, therefore, we extend the use of the safety gap of 4.95 m. Consider the case of a two lane arterial stream where P2 is the lead vehicle that is been overtaken by the PTW and V3 a vehicle in the adjacent lane in front of the PTW. Let the minimum safety gap of PTW and the lead vehicle (P2) equal to 4.95 m. Moreover, let the same minimum gap apply between the PTW and V3 at the time which the PTW overtakes. Then, the opening during overtake between the lead vehicle and V3 should be at least two times the safety gap, which equals to 9.90 m. If the average length of motorcycles and scooters (2.1 m) is also added, then, the opening required for a safe and comfortable pass is increased from 9.90 m to at least 12 m. Therefore, if the opening space is less than 12 m P2 is considered to be non-cooperative. In addition, P2 is considered non-cooperative, when he accelerates during a PTW overtake attempt.
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3.2. Comfort at overtake as a game’s payoff
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In order to quantify the gain of a driver during overtaking, the critical factor, which may significantly affect overtaking, should be identified and modeled. Although recent approaches focus on the effect of speeds during overtaking (Barmpounakis et al., 2015), literature has been systematically underlined the effect of other supplementary factors, such as Time To Collision (TTC), perceived safety of the individual driver-vehicle system, joy of driving, social welfare that should be taken into consideration (Altendorf and Flemisch, 2014; Arbis et al., 2016; Barmpounakis et al., 2015). The above show that the proposed payoff function should have a certain multivariate nature to account for the entire spatio-temporal aspects of interactions between PTW and other vehicles during overtaking. To account for the multivariate aspects of overtaking, a novel measure termed as Comfort at Overtake (CaO) is defined, which takes into consideration both safety measures (TTC and crash avoidance with surrounding vehicles) and the desire of PTW to move faster than other vehicles. The CaO is not observable (latent variable), but needs to be estimated in a certain consistent manner taking into consideration real world collected vehicle trajectories and is quantified using advanced statistical modeling. In order to quantify the unobserved variable CaO for both Player 1 and 2 at an overtaking attempt, a Multiple-InputsMultiple-Outputs (MIMIC) structural equation model is constructed. The MIMIC model is a special case of latent variable statistical modeling, a thorough technique for testing hypotheses for the relations between observed and unobserved (latent) variables. The model consists of two components: a measurement model which defines the relations between a latent variable and its indicators and a structural model which specifies the casual relationships among latent variables and explains the casual effects. MIMIC models consider the latent variable g to be scalar and relates the vector of indicators y and the
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observed exogenous variables x that cause y by the following system of equations
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?? ¼ Cx þ ??
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y ¼ K?? þ ??
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where C and K are matrices of unknown parameters to be estimated and e and f; are the error terms (Washington et al., 2010). These type of models has been previously implemented in (Barmpounakis et al., 2016a; Chung et al., 2012; Hassan and Abdel-Aty, 2011; Karlaftis et al., 2001). The goodness of fit of the models developed are assessed by implementing likelihood ratio tests for comparing the proposed model to the saturated one (the model that fits the covariances perfectly) and baseline models (model that includes the means and variances of all observed variables plus the covariances of all observed exogenous variables). In addition, other indicators that were examined were the root mean squared error of approximation (RMSEA) along with the probability of RMSEA being below 0.05, the standardized root mean squared residual SRMR and the coefficient of determination of the various models.
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4. Implementation and findings
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4.1. The study area
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In order to avoid complex overtaking phenomena, the proposed approach is implemented on data from a divided arterial with two lanes per directions (Fig. 3). Except from offering the ability to observe and monitor the upstream traffic, the spot is ideal for not affecting drivers’ behavior. More information on the database used for this research can be found in a previous study where the same database was used (Barmpounakis et al., 2016a). Data collected through trajectory tracking, consists of 850 cases of PTW overtaking. Table 1 and Fig. 4 depict the variables that are measured in each case and will be used in the formation of the game. All speeds and headways are measured at the start of the overtaking maneuver.
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Fig. 3. Camcorder setup on Calatrava’s pedestrian bridge in Athens (Greece).
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4.2. Quantifying comfort at overtake
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Two distinct models are developed to estimate the latent CaO parameter for players 1 and 2, the PTW and the lead vehicle respectively. For the case of PTW (P1), three variables are defined as indicators in the MIMIC model; i. The speed of the PTW (Vm) expresses the desire of the PTW driver to move faster, ii. the variable DistanceXY expresses the desire to avoid a crash with the preceding vehicle, and iii. Back_dis the desire to avoid a crash with the vehicle that is coming from the back. For the lead vehicle (P2), the indicators of comfort at overtake are: i. speed difference between the two vehicles (Diff) expresses the desire of Player 2 not to be overtaken with excessive speeds, ii. the distance d1 and iii. the Opening which expresses the desire of not having many vehicles around him. Table 2 shows the goodness of fit for the best-fitted models (Washington et al., 2010). The CaO for P1 is calculated using the model that is described in Table 3 and illustrated in Fig. 5. All variables depicted are significant in 5% significance level, while passenger is significant in 10% significance level. The coefficients of the variables show that the speed of the PTW (Vm) and the distance from the vehicle in the back (Back_dis) are critical parameters for defining CaO. As far as the predictors of the models are concerned, the placement of the vehicles concerning the lane each one is present (PTW_lane and Veh_lane) and whether the PTW driver is wearing a helmet or not are also significant parameters. It should be noted that the opening space for the PTW (Opening) is also of importance when calculating CaO.
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Table 1 Description of variables measured. Variable
Description
Vm V1 Diff d2 d3 Opening s_x s_y Distance d1 Back_dis Passenger Platoon Helmet PTW_lane Veh_lane Over_RL Lane_same Over
Subject PTW’s [Player 1] speed (km/h) Vehicle 1 [Player 2] speed (km/h) Speed difference between Player 1 and Player 2, diff = Vm V1 (km/h) Oblique distance between Player 2 and front vehicle (m) Lateral distance between Player 2 and front vehicle (m) Opening space in front of Player 2, Opening = (d22 + d23)0.5 (m). Lateral distance between Player 1 and Player 2 (m) Oblique distance between Player 1 and Player 2 (m) Distance between Player 1 and Player 2, (s_x2 + s_y2)0.5 (m) Distance between Vehicle 2 and 3 (m) Distance between Player 1 and Vehicle 2 (m) Is there a passenger on the PTW? (0/1) Existence of platoon, subject PTW is the leader (0/1) Does the PTW driver use a helmet? (0/1) Left or right lane in which PTW is moving (0/1) Right or left lane in which vehicle that is being overtaken is moving (0/1) Player 1 tried to overtake Player 2 from Right or Left? (0/1) Are Player 1 and Player 2 in the same lane? (0/1) Did the PTW complete the overtake attempt? (0/1)
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Fig. 4. Illustration of variables that are measured in each overtaking attempt.
Table 2 Goodness of fit assessment for SEM models. Fit statistic
Estimated values for Player 1
Estimated values for Player 2
Likelihood ratio x2_ms(7) x2 (p > x2) – Saturated x2_bs(18) x2 (p > x2) – Baseline
Criteria
5.701 0.575 197.963 0.000
5.196 0.268 526.410 0.000
Population error RMSEA 90% CI, lower bound Upper bound pclose
0.000 0.000 0.037 0.993
0.019 0.000 0.058 0.891
<0.05 <0.05 <0.10 >0.05
Information criteria AIC BIC
30106.381 30187.050
31053.808 31134.477
– –
Baseline comparison CFI TLI
1.000 1.019
0.998 0.991
>0.90 <3.00
Size of residuals SRMR CD
0.011 0.614
0.09 0.396
<0.10
– –
Table 3 Results of MIMIC Model for Player 1. Structural
Coefficient
Std. Err.
z
p > |z|
CaO PTW_lane Veh_lane Passenger Helmet Opening
5.74756 5.23472 3.67808 5.11538 0.11394
0.89046 1.57036 2.23441 1.65496 0.01354
6.450 3.330 1.650 3.090 8.410
0.000 0.001 0.100 0.002 0.000
Vm CaO Veh_lane Constant
1 4.20860 77.66768
(Constrained) 1.77279 1.86331
2.370 41.680
0.018 0.000
Distance CaO Opening Constant
0.06518 0.01676 6.40407
0.03173 0.00569 0.23018
2.050 2.940 27.820
0.040 0.003 0.000
0.25616 4.74621 4.09514 12.12033 0.70234 49.41105 9.19457
6.210 2.090 11.910 141.501 13.008 796.460 9.256
0.000 0.037 0.000 189.179 15.766 990.532 49.692
Back_dis CaO 1.59063 Helmet 9.90969 Constant 48.75891 e.Vm 163.61200 e.DistanceXY 14.32069 e.bd 888.21130 e. CaO 21.44651 LR test of model vs. saturated: x2(10) = 5.70, Prob > x2 = 0.5751
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Fig. 5. MIMIC model for Player 1.
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The CaO for P2 is estimated by the model described in Table 4 and illustrated in Fig. 6. It is seen that the placement of the vehicles and specifically whether the PTW is moving in the same lane as the vehicle (lane_same) as well as the lane the PTW tried to overtake P2 (Over_RL). Moreover, as far as the indicators are concerned, the coefficient of d1 is similar to opening, which shows that all vehicles have the same effect on a driver’s CaO. The above mentioned parameters clearly depict the desire of P2 not being surrounded by other vehicles while the strong relationship between the space headway of P1 and P2 (Distance) and the speed difference between them (Diff) represent the a TTC equivalent measurement. Using the results from the two MIMIC models, we calculate the CaO for each driver per overtaking attempt. Αll values are normalized in 0–1 range so that the worst case will stand for a CaO equal to 0, while the best for a CaO equal to 1. The average value of the CaO of P1 (n = 850) is 0.51 (±0.20) and for P2 0.42 (±0.24). To better illustrate the meaning of the CaO measure using the MIMIC models developed two examples follow: An overtaking attempt when P1 is wearing a helmet and no passenger is on the PTW, the two players are on the same lane, their distance is about 13 m and the opening space is more than
Table 4 Results of MIMIC Model for Player 2. Structural
Coefficient
Std. Err.
z
p > |z|
CaO Lane_Same Over_RL Distance Platoon
26.78280 5.54144 0.82980 4.44533
5.78190 2.97884 0.26588 3.16298
4.630 1.860 3.120 1.410
0.000 0.063 0.002 0.160
Opening CaO Lane_Same Over_RL Constant
1 15.27592 7.82455 33.69178
(Constrained) 5.64972 2.66714 2.44310
2.700 2.930 13.790
0.007 0.003 0.000
Diff CaO Over_RL DistanceXY Constant
0.10804 6.99010 0.41246 21.74381
0.02332 1.04108 0.12618 1.08515
4.630 6.710 3.270 20.040
0.000 0.000 0.001 0.000
0.20709 2.96258 128.56300 9.37399 143.96470 135.52300
5.020 30.120 142.654 173.244 613.696 421.385
0.000 0.000 701.611 210.046 1188.352 968.072
Measurement d1 CaO 1.04002 _cons 89.22012 e.Opening 316.36580 e.Diff 190.75950 e.d1new 853.98320 e.CaO 638.69450 LR test of model vs. saturated: x2(10) = 1.70, Prob > x2 = 0.8892
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Fig. 6. MIMIC model for Player 2.
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20 m will give a CaO equal to 0.60. As far as P2 is concerned, a CaO around 0.50 stands for an overtaking attempt when the distance between the two players is more than 5.5 m and no other vehicles are present in a radius of 20 m.
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4.3. Game theoretic model
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In order to implement the game theoretic model that was described in Section 2, the payoffs for each player should be estimated for each terminal node. First, we separate all 850 overtaking attempts depending on the strategy each player followed. Thus, four clusters are created; both being cooperative (C, C), both being non-cooperative (NC, NC), one player being cooperative and the other non-cooperative (NC, C) and (C, NC). Using the average of CaO values for each cluster as payoff, the extensive game is formed as in Fig. 7. For example, the average of CaO values when both players are Cooperative is equal to 0.58 for P1 and 0.50 for P2. Both average values of CaO when both players are cooperative, stand for proper conditions for a safe and comfortable pass, while payoffs are reduced for Non Cooperative strategies. Black color stands for Player 1 and grey for Player 2.
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Fig. 7. The game in extensive form.
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As can be observed, the Dominant Strategy for both players is to be cooperative. However, from the collected data and based on the definition of cooperativeness, a PTW driver (P1) chooses to be cooperative 82% and the lead vehicle’s driver (P2) chooses to be cooperative at 68%. The question that arises is why a driver would follow another strategy than the dominant one, one more paradox to be added to the many that have been documented in social experience and actual experiments (Axelrod, 1984). The explanation for this inconsistency is not as simple as it may seem. Someone could argue that lead vehicles’ drivers may not leave adequate opening for the PTW to overtake, since they are not interested in the gain they would have or that a PTW driver may follow a Non-Cooperative strategy in order to be ready for overtaking, thinking that it may lead to higher speeds and shorter travel times. As seen in Table 5, which depicts the number of total cases recorded – grouped by strategy set – as well as the number and percentage of successful overtake attempts are depicted, a driver may choose to be NC, but complete the overtake. This percentage is higher when compared to the successful overtakes produced by Cooperative PTW drivers. As can be observed, when the PTW driver follows a NC strategy, he is most likely to complete an overtake than when he follows a C strategy. An overtake attempt is considered as successful when the PTW did complete the overtake and passed in front of the lead vehicle, as opposed to an unsuccessful one, when the PTW did not pass the lead vehicle, but stayed behind it. This finding suggests that PTW drivers may undertake an aggressive behavior while overtaking probably because, even if they conduct risky maneuvers, they aim at pursuing short-term gains during driving. As expected, the lowest percentages occur when the PTW driver is Cooperative and the other driver is not. Evidently, overtaking is a high-risk maneuver and timid PTW drivers will most likely seek cooperation in order to complete an overtake. Therefore, a PTW driver may choose a NC behavior because he may enjoy the risky driving, sharp passing, extreme acceleration and adrenaline while overtaking in comparison to a timid driver that will prefer safety over enjoyment and higher speeds. Similar findings can be documented in a research that focuses on driving risk perception (Dionne et al., 2007), where authors conclude that drivers are more cautious when they have an improved risk perception in contrast to those who underestimate it and tend to be more at risk. The risk taking behavior by some users has also been identified in (Andersson and Lundborg, 2007; Rubinfeld and Rodgers, 1992). What is interesting is that most of the times, a player tends to cooperate with the other one. It is seen that in 91% of total overtaking attempts, at least one player was cooperating and in 59% of total overtaking attempts, both followed a cooperative behavior. These figures show that, although both players may frequently seek for the long-term gain, they may exhibit interchangeably shifts to non-cooperative (aggressive) behavior while driving. This finding has been emphasized in many research efforts that studied microscopic behavior from various angles (Laval, 2011; Orfanou et al., 2012; Papacharalampous and Vlahogianni, 2014). From a game theoretic perspective, literature states: ‘‘very often game theory predicts behavior that can be criticized as unnatural and non-rational” (Papadimitriou and Yannakakis, 1994). Evidently, regardless of the predictive equilibrium of the proposed game, the evaluation with real data underlines that drivers may not always choose to exhibit a selfishly antisocial way, but often give up a short-term gain in order to behave in a cooperative manner. Finally, another parameter that could affect driving behavior is fuel efficiency since acceleration and deceleration modes increase fuel consumption than steady-driving modes (Tong et al., 2000). Some drivers may prefer to drive in a more economical way, which corresponds to Cooperative driving in a way of keeping steady speeds and gaps. Consequently, including accelerative information in the payoff function based on the above statements could lead in different results in comparison to real data.
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5. Rational versus irrational drivers’ expectations: a general discussion
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These inconsistencies between the solution produced by the game theoretic model and the real data may be also understood using the theory of bounded rationality (Simon, 2000). In this framework, for explaining PTW driver’s choices when overtaking, as well as the interactions with the rest of traffic, focus should be shed to both the process of decision and the quality of the outcome. The process of decision is understood through studying the psychology of the decision maker. At an overtaking game it is questionable as to what extent the drivers use other abilities, such as to have the necessary knowledge of the phenomena and evoke it when relevant, to work out the consequences of their actions, to conjure up possible courses of action to cope with uncertainty, rather than maximizing or optimizing their utilities. Similar findings can be found concerning the use of safety equipment in (Blomquist, 1991). Thus, it is of interest to investigate the dynamic changes in the players/decision makers driving behavior during overtaking, providing a better insight at defining rationality. For
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Table 5 Summary table of overtake attempts by strategy. Strategy (P1, P2)
Overtake attempts
Successful overtakes
Percentage (%)
NC, NC NC, C C, NC C, C
73 76 198 503
57 67 96 306
78% 88% 49% 61%
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Fig. 8. Quantal response equilibrium for both players.
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example, is the assumption of independent decisions made accurate? Does the driver have a long or short memory that evokes when asked to conduct an overtake? To address the questions raised above, we use the Quantal Response Equilibrium (QRE) concept introduced by McKelvey and Palfrey (Mckelvey and Palfrey, 1995), which replaces the perfectly rational expectations with imperfect, or noisy, rational expectations equilibrium. An interesting property of QRE is that systematic deviations from Nash Equilibrium may be predicted without introducing systematic features to the error structure. Detailed definitions and notations of QRE for extensive form games can be found in (McKelvey and Palfrey, 1998). QRE theory can be summarized as follows; better responses are more likely to be chosen than worse responses, but that does not mean that best responses are played with certainty. Theory suggests that as a player gains experience playing a specific game and makes repeated observations about the actual payoffs received from different action choices, he can be expected to make more precise estimates of the expected payoffs from different strategies. This experience gained by repeated observations can be modeled through the parameter k. Although the QRE theory does not describe the origins of the parameter k, it can be treated as a rationality parameter, where, if k ! 0, the player becomes completely irrational and plays each strategy with equal probability, while if k ! 1 the player becomes completely rational. Using the previously described database and utilizing the algorithm described in (Turocy, 2009), both players learn to be cooperative after a number of iterations as seen in Fig. 8 (McKelvey et al., 2016). The two cases that should be stood out are the ones that are illustrated with dashed lines. Concerning P1, it is seen that when he plays first (dashed), the learning process to cooperativeness is faster than when he plays second (straight). At the same time, when P1 chooses to cooperate and P2 plays second (dashed), the learning process to cooperativeness is slower due to the small difference between the two payoffs (0.50 and 0.45).
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6. Conclusion
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The present paper implemented concepts from Game Theory to describe the PTW overtaking phenomena in urban arterials. A Non-cooperative Game with perfect information is proposed which consists of two players; i. the PTW driver and ii. the lead’s vehicle driver. Both drivers can either be Cooperative or Non-Cooperative, at the emergence of an PTW driver overtake attempt. A PTW driver is considered to be cooperative, when he keeps adequate distance from the lead vehicle, and the
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latter is considered to be cooperative when he leaves adequate opening for the PTW to pass. The pay-off of each player is defined by the latent variable Comfort at Overtake (CaO), which is calculated by a MIMIC model. It is found that the game has the optimum strategy for both players to cooperate at the occurrence of an overtake attempt. Game theoretic results are further compared to real data. Interestingly, the evaluation with real world data showed that although both drivers exhibit – mainly – a cooperative behavior, overtaking can sometimes be conducted through aggressive driving and a Non Cooperative behavior. Then a discussion arises as to examining the reasons that both players do not always follow the cooperative strategy concerning the Quantal Response Equilibrium (QRE) concept, player rationality and payoff function. It is suggested that this game theoretic approach can be extended to various forms of games to overcome the underlined limitations, such as the independence in players’ decision and the memory properties of driver’s behavior. Further, different payoff functions should be considered. Overall, the present paper shows that Game Theory has a significant advantage for modelling the interplay between drivers on the road, since drivers’ behavior is dependent on their environment. In future studies, more game formations can be examined that will be able to describe more adequately the overtaking procedure. Latest technological advances in Vehicle Communication Systems can implement findings from such models by leveraging the models’ ability to explain interactions among two or more drivers on a road to improve existing Advanced Driver Assistance Systems (ADAS) or develop novel ones. Communication between drivers could significantly improve modeling accuracy and promote cooperation among them, since, as theory suggests, cooperation can evolve even without foresight or friendship between players (Axelrod, 1984). If cooperation among drivers becomes a reality in the future, driver’s view of commuting will have to expand towards new territories. Cooperation could be translated to information exchange between the drivers, either by situation awareness, or by giving assisting directions to drivers. The identified irrationality of some drivers could be alleviated by the use of such an ADAS context, for example to assist driving to complex traffic conditions (stop-and-go conditions, moving bottleneck conditions etc.), to dis/encourage overtakes in certain traffic conditions or to control spatial characteristics among traffic for optimization or emergency purposes, as an additional mean of traffic management.
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