Towards a collaborative connected, automated driving environment: A game theory based decision framework for unprotected left turn maneuvers

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Abstract—Connectivity and automation provide the
opportunity to enhance safety and mitigate congestion in
transportation systems. In fact, these technologies can enhance
the efficiency of drivers/vehicles’ decision-making by managing
and coordinating the interactions among human-driven and
connected, automated vehicles. Such management and
coordination can lead to developing a collaborative connected,
automated driving environment. Game theory, as a
methodology to model the outcome of the interactions among
multiple players, is a perfect tool to characterize the interaction
between these vehicles. One of the most challenging maneuvers
to model is drivers/vehicles’ tactical decisions at intersections.
Focusing on unprotected left turn maneuvers, this study aims
at developing a game theory based framework to characterize
driver behavior in unprotected left turn maneuvers in a
connected, automated driving environment. A two-person non-
zero-sum non-cooperative game under complete information is
selected to model the underlying decision-making. NGSIM data
is used to calibrate the payoff functions based on Maximum
Likelihood Estimation. Validation results indicate that this
framework can effectively capture vehicle interactions when
performing conflicting turning movements while achieving a
relatively high accuracy in predicting vehicles' real choice.
I. INTRODUCTION
Connected Vehicles Technology (CVT) and vehicle
automation provide the opportunity to enhance the efficiency
of interactions in a connected, automated driving
environment. The additional information from Vehicle-to-
Vehicle (V2V), Vehicle-to-Infrastructure (V2I), and on-board
sensors can potentially enhance drivers/automated vehicles’
decision-making and facilitate the collaboration among
human-driven and connected, automated vehicles (CAVs) by
coordinating the decisions.
Characterizing the interactions among human-driven and
CAVs is the key to achieve such coordination. Game theory
provides a basis for modeling interactions (both cooperation
and conflicts) between intelligent and rational decision
makers. Therefore, it provides all the necessary tools to
characterize and model the interactions in a connected,
automated driving environment.
Tactical decisions at intersections (e.g., left-turn
maneuvers) are among the most challenging driving
maneuvers. The approach that drivers utilize to handle such
Yalda Rahmati is a graduate student in the Zachry Department of Civil
Engineering, Texas A&M University, College Station, TX, 77843 (email:
yalda.rahmati@tamu.edu)
Alireza Talebpour (Corresponding Author) is an assistant professor in
the Zachry Department of Civil Engineering, Texas A&M University,
College Station, TX, 77843 (phone: 979-845-0875; email:
atalebpour@tamu.edu)
conflicting movements can significantly increase delays and
reduce efficiency at the intersection. These are particularly
due to human drivers’ lack of awareness, slow reaction,
and/or decision errors. These factors contribute to unsafe and
unreliable turns at intersections and drivers usually decrease
their speed significantly to ensure safety. CAVs, however, do
not have these restrictions. A CAV can accurately measure
distance and speed, monitor its surroundings, and react
almost instantly (excluding vehicle actuation time) to various
driving situations. Therefore, a behavioral based modeling
approach in a connected, automated driving environment is
expected to result in an efficient operation of CAVs at
intersections.
Accordingly, the main objective of this study is to
develop a game theory based framework to characterize
driver behavior in unprotected left turn maneuvers in a
connected driving environment. Unprotected left turn
maneuvers are considered as one of the major conflicting
movements at urban intersections. The order in which
drivers perform a left turn maneuver in the presence of
conflicting through vehicles is usually described based on
the provided gap between through vehicles and the left-
turning vehicles’ gap acceptance behavior. However, real
world observations indicate that actual behaviors can be
more complicated (e.g., the through vehicle can accelerate to
prevent the left turn maneuver or can decelerate to provide
more gap for the turning vehicle). The present study applies
a game-theoretical approach to capture the dynamic
interactive behavior between these conflicting vehicles in a
connected environment. The results of this study can be
directly applied by CAVs to predict the behavior of other
vehicles and form the basis for a collaborative decision-
making at signalized and unsignalized intersections.
The remainder of this paper is organized as follows.
First, related works are briefly discussed in section II. The
model formulation including the definition of the game and
payoff functions is presented in section III. This is followed
by introducing a maximum likelihood based method for
calibrating the game, and presenting the results of the model
calibration and validation in section IV. The study concludes
with some summary remarks and future research directions
in section V.
II. BACKGROUND
In 2015, 35,092 people died in motor vehicle crashes on
US roads [1]. Researches show that 94 percent of crashes are
tied to human decisions and errors [1]. Safety and efficiency
are among the main factors that derive the rapid
development of connected, automated driving environment
in recent years. Excluding human component from driving
motor vehicles not only creates safer roads, but also
introduces new degrees of freedom in operation, control, and
Towards a Collaborative Connected, Automated Driving Environment: A Game
Theory Based Decision Framework for Unprotected Left Turn Maneuvers
Yalda Rahmati and Alireza Talebpour

management of transportation systems [2]. Security and
privacy of telecommunications (with focus on connected
vehicles) have been studied extensively in the literature [3
and 4], however, there are still limitations on how to capture,
specify, and model tactical and operational driving behaviors
(i.e. lane changing, acceleration, turning, gap acceptance and
merging decisions) in a connected, automated environment
and translate such behaviors into efficient/practical V2V and
V2I communication logic [5]. Conflicting turning
movements at intersections, in particular, have not been
studied extensively in a connected environment. When
dealing with these maneuvers, many adopt a one-way
approach by focusing on the effect of opposing through
traffic on left-turning vehicles' gap acceptance behavior [6],
ignoring the interactions among the conflicting vehicles.
This study adopts a game-theoretical approach to capture
drivers' interactive behavior and model their operational
decisions when performing unprotected left turn maneuvers
at intersections.
Game theory has been adopted multiple times in
transportation analysis in order to capture and model human
drivers' tactical and operational behaviors in recent years [6].
According to Zhang et. al. [7], applications of game theory
in transportation analysis can be classified into two main
categories; macro-policy analysis, and micro-behavior
simulation. While many of these studies have focused on
macro-policy level such as road/parking tolls policy [8-11],
vehicle routing problems [12 and 13], transportation network
reliability [14], urban traffic demand [15], and transport
modes competition [16], limited research efforts have
studied games played between travelers in micro level
operational decision makings, in particular, conflicting
movements at intersections. Adopting a game-theoretical
approach, the present study introduces realistic rules in
formulating players' payoff functions in order to capture the
interactive decision making behavior of conflicting vehicles
at intersections. The resulting pair of the players' actions is
then formulated as an equilibrium solution. Finally, a
calibration procedure based on maximum likelihood method
is proposed to estimate the model parameters. Applicability
of the proposed model is also examined and validated using
the well-known NGSIM trajectory data.
III. MODEL FORMULATION
A. Game Definition
This study considers a game when a vehicle (player A)
attempts to perform an unprotected left turn maneuver in the
presence of an opposing through movement (player B) at an
intersection. Figure 1 illustrates the scheme of such
maneuver.
The study models the game for a situation where both of
the conflicting vehicles approach the intersection during the
green indication. As illustrated in Figure 1, player A is the
vehicle in the left most lane on the northbound approach
trying to make a left turn, and player B is the nearest vehicle
to the intersection on a southbound approach that wants to
continue its through movement and pass the intersection.
The lag vehicle is the closest vehicle to player B on the
Figure 1. Schematic of typical unprotected left-turn maneuver
southbound direction, which also attempts to perform a
through movement. It is assumed that the two players will
start the game by deciding on a set of actions to maximize
their respective awards as soon as player A enters the
exclusive left turn lane and no other vehicle exists between
player A and the northbound direction stop line of the
intersection. It should be noted that albeit the lag vehicle is
not directly considered as a player, its influence is implicitly
accounted for by incorporating the spacing between this
vehicle and player B into the payoff functions, which will be
discussed in detail in forthcoming sections.
Table I shows the structure of the game between player
A and B in the normal form. Considering a green indication
during the game, each of the players can choose between
two actions as follows:
 Player A's options are either to perform the left turn
maneuver (A1) or to wait behind the stop line until
player B passes the conflicting point (A2).
 Player B’s options are either to continue without any
significant change in speed (B1) or to perform a
courtesy yield (B2).
B. Graphical Representation of the Best Response
Considering a game with mixed strategies, the strategy of
each player can be defined as a set of probabilities assigned
to each action [17]. Generally, the strategy that produces the
most favorable outcome for a player, given other player's
strategies, is the best response for that player. In a mixed
strategy game, the best response for each player would be
the probability that maximizes his/her expected payoff given
the probabilities chosen by other players [17].
Let p denote the probability that the left-turning vehicle
chooses A1, and q denote the probability of choosing B1 by
the through vehicle. Assuming independence among the
pure strategies, (1-p) and (1-q) denote the probability of
choosing A2 and B2, respectively.
TABLE I. STRUCTURE OF THE GAME BETWEEN PLAYER A AND B
Player B (Through vehicle)
Player A (Left-turning
vehicle)
B1 (Keep driving)
B2 (Decelerate)
A1 (Turn)
P11A, P11B
P12A, P12B
A2 (Wait)
P21A, P21B
P22A, P22B

Figure 2. Graphical illustration of the best response
Assuming the general payoff matrix shown in Table I,
the expected payoff for players A and B can be given by (1)
and (2), respectively.
E(PayoffA) = p[q(P11A+(1–q)p12A] + (1–p)[qP21A+(1–q)P22A]
E(PayoffB) = q[p(P11B+(1–p)p12B] + (1–q)[pP21B+(1–p)P22B]
The best response can be determined by maximizing the
expected payoff for each player.
From (1):
If q < (P22A–P12A)/[(P22A–P12A)+( P11A–P21A)] → p = 1
If q > (P22A–P12A)/[(P22A–P12A)+( P11A–P21A)] → p = 0
If q = (P22A–P12A)/[(P22A–P12A)+( P11A–P21A)] → p + q = 1
From (2):
If p < (P22B–P12B)/[(P22B–P12B)+( P11B–P21B)] → q = 1
If p > (P22B–P12B)/[(P22B–P12B)+( P11B–P21B)] → q = 0
If p = (P22B–P12B)/[(P22B–P12B)+( P11B–P21B)] → p + q = 1
The best response of the player 1 and 2 can be shown by
line I and II in Figure 2, respectively. The intersection points
of these lines represent the Nash equilibria for the game
which is achieved when each player is best responding to
what the other player is choosing.
C. Payoff Function Formulation
Payoffs are assumed as a function of certain factors that
characterize the environment surrounding the player's
decision making and reflect the motivation of each player to
choose a strategy. In games that have been applied in the
domain of traffic engineering, it is usually assumed that the
players' ultimate goal is to minimize the risk of collision and
thus, payoff functions are formulated based on time to
collision [18 and 19]. However, since collision risks affect
both players, this assumption may lead to unrealistic Nash
equilibria [20]. To model the conflicting movements at an
intersection in a more general context, the present study
considers more realistic behavioral rules incorporated into
corresponding payoff functions of conflicting vehicles.
It is assumed that the left-turning vehicle evaluates the
outcome of performing a left turn maneuver by calculating
the deceleration/acceleration rate required to avoid a
collision, comfortable acceleration/deceleration rate, and the
spacing between the through and lag vehicles. On the other
hand, the through vehicle either ignores the left-turning
vehicle by keeping on his/her way or decides to decelerate at
a comfortable rate to let the left turn movement take place.
Prior to detailing these payoff functions, it should be
noted that: 1) through and lag vehicle are assumed to be in a
car following mode when approaching the intersection, 2)
there is no vehicle in front of the left-turning vehicle at the
time of decision-making, 3) both players can accurately
calculate the variables in the payoff functions.
Player A (the left-turning vehicle): At the decision time,
player A needs to decide whether to perform a turning
maneuver or wait behind the stop line until the player B
passes the point of conflict within the intersection.
Generally, drivers' tendency to perform a safe maneuver has
a significant effect on their decisions. Thereby, player A
calculates the acceleration/deceleration rate required to
avoid collisions before performing a left turn maneuver.
Drivers also prefer to maintain a comfortable level of
acceleration/deceleration rate when executing a turning
maneuver. In addition to the desire for performing a safe and
comfortable maneuver, left-turning vehicles usually compare
the current distance of the nearest opposing through vehicle
to the intersection (h0 in Figure 1) and the next available gap
(h1 in Figure 1). Incorporating this factor into player A's
payoff function will endogenously take into account the
effect of lag vehicle on decisions made by player A.
First, consider a situation where player A decides to
perform a left turn maneuver (A1). If player B chooses to
ignore player A and keep his/her way, player A needs to
calculate the acceleration/deceleration rate required to avoid
the collision. He/she also considers a comfortable
acceleration/deceleration rate to perform the left-turn
maneuver. On the other hand, if player B decides to wait and
allow the left-turn maneuver to take place, player A does not
need to consider acceleration/deceleration rate required to
avoid the collision; however, he/she still takes into account
the comfortable acceleration/deceleration rate. Player A's
payoff functions are presented by (3) and (4). An error term
is introduced to capture the effects of unobserved factors.
P11A = 11 +11AccACollision +11AccComf + 11 (3)
P12A = 12 +12AccComf + 12 (4)
where AccACollision is the acceleration/deceleration rate
required for player A to avoid the collision, AccComf is the
comfortable acceleration/deceleration rate, 11 and 12 are
error terms to capture the effect of unobserved factors, and
11, 11, 11, 12, and 12 are parameters to be estimated.
Second, consider a situation where player A decides to
wait until player B passes the conflicting point at the
intersection (A2). In this case, the comfortable deceleration
rate will be a major consideration for player A, regardless of
the action taken by player B. In addition to maintaining a
comfortable deceleration rate to stop behind the stop line,
player A will also conduct a comparison between the
distance of player B from the intersection and the next
available gap (h0 and h1 in Figure 1, respectively); higher
h1/h0 indicates a higher likelihood for player A to wait for
the next available gap (i.e., the gap between player B and the
lag vehicle). The corresponding payoff functions for player
A are presented by equations 5 and 6.

P21A = 21 +21 AccComf +21 h1/h0 + 21 (5)
P22A = 22 +22 AccComf +22 h1/h0 + 22 (6)
where h0 is the distance of player B from the intersection, h1
is the distance between player B and the nearest vehicle
behind it, 22 and 21 are error terms to capture the effect of
unobserved factors, and 21, 21, 21, 22, 22, and 22
are parameters to be estimated.
Player B (the through vehicle): At the decision time,
player B decides whether to ignore player A and continue
his/her way to pass through the intersection (keep driving) or
to perform a courtesy yield and let the left turn maneuver
take place (decelerate). As for player A,
acceleration/deceleration rate required to avoid collision still
plays an important role in player B's decision making.
He/she also takes into account a comfortable
acceleration/deceleration rate in making any maneuver.
Unlike player A, player B does not consider other vehicles
behind player A waiting to make a left turn. Indeed, player B
plays a game with the first vehicle in the left turn lane and
decides whether to pass through the intersection or
decelerate to let the left-turning vehicle perform its
maneuver.
First, consider the case where player B chooses to ignore
player A and keeps driving. If player A also decides to
perform its left turn maneuver, player B needs to calculate
the acceleration/deceleration rate required to avoid a
collision, and also wants to perform a comfortable maneuver
in terms of acceleration/deceleration rate. However, if player
A waits behind the stop line, player B only needs to maintain
the comfortable acceleration/deceleration rate.
Corresponding payoff functions for player B in both
situations are presented by (7) and (8).
P11B = 11 +11 AccBCollision +11 AccComf + 11 (7)
P21B = 21 +21 AccComf + 21 (8)
where AccBCollision is the acceleration/deceleration rate
required for player B to avoid the collision, AccComf is
comfortable acceleration/deceleration rate, 11 and 21 are
error terms to capture the effect of unobserved factors,
and11, 11, 11, 21, and 21 are parameters to be
estimated.
Alternatively, consider the case where player B chooses
to decelerate and let player A make a left turn (action B2). In
this situation, if player A decides to make a left turn
maneuver, player B needs to account for a comfortable
deceleration rate which will also avoid colliding with player
A. On the other hand, if player A chooses to wait, player B
only considers a comfortable deceleration rate. Equations (9)
and (10) present the associated payoff functions for player
B.
P12B = 12+12 AccBCollision + 12 AccComf + 12 (9)
P22B = 22 +22 AccComf + 22 (10)
Where 12 and 22 are error terms to capture the effect of
unobserved factors, and 12, 12, 12, 22, and 22 are
parameters to be estimated. Table II and III summarize the
resulting payoff functions for player A and player B,
respectively.
TABLE II. PAYOFF MATRIX OF PLAYER A
Player B (Through vehicle)
B1 (Keep driving)
B2 (Decelerate)
Player A (Left-
turning vehicle)
A1 (Turn)
11 +11AccACollision
+11AccComf + 11
12 +12AccComf
+ 12
A2 (Wait)
21 +21 AccComf
+21 h1/h0 + 21
22 +22 AccComf
+22 h1/h0 + 22
TABLE III. PAYOFF MATRIX OF PLAYER B
Player B (Through vehicle)
B1 (Keep driving)
B2 (Decelerate)
Player A (Left-
turning vehicle)
A1 (Turn)
11 +11 AccBCollision
+11 AccComf + 11
12+12AccBCollision
+12 AccComf + 12
A2 (Wait)
21 +21 AccComf
+ 21
22 +22 AccComf +
22
IV. MODEL CALIBRATION
A. Calibration Method
The present study proposes a maximum likelihood
method for calibrating the game and estimating the
parameters. An outline of this calibration method is
introduced here, and readers are referenced to Kita et al. [17]
for further details.
As discussed before, payoffs for each player are defined
as a utility function (U) which consists of a deterministic
term (v) and an error term () to capture the effect of
unobserved factors. The deterministic term is a function of
the parameter vector (), and the vector of factors which
affect the payoff functions (X).
U = v (, X) +  
This study assumes a normal distribution for the error terms.
Let sk denote the best response of player k representing a
pure strategy in the game. The probability assigned to each
action in a mixed strategy game (et) will then be the direct
product of each player's best response:
et = kϵN sk 
If Psk(U) denote the probability that sk is the best
response. The probability that et is an equilibrium point can
be given by:
P(et) = kϵN Psk (U) 
Let P(ot|et) denote the probability that outcome 𝑜𝑡 is
designated as equilibrium by et, then the probability that
outcome ot is observed can be defined as:
P(ot) = kϵN P(ot|et). P(et) 
Finally, suppose that n(ot) is the number of observations
of outcome ot; in other words, the number of observations
for each combination of pure strategies in the real world.
The likelihood function L can then be given by:
L = ot ϵ O P(ot) n(ot) 

where O denotes the set of outcomes.
The maximum likelihood estimate for parameter vector 
will be the values of parameters that maximize the likelihood
function L. Once the parameters are estimated; the payoff
functions can be identified.
B. Calibration and Validation Results
The well-known NGSIM data [21 and 22] is used to
calibrate and validate the proposed model. The dataset
represents vehicle trajectories on a segment of Peachtree
Street in Atlanta, Georgia, and a segment of Lankershim
Boulevard, located in Los Angeles, California, which have
been collected in November 2006 and June 2007,
respectively.
Peachtree Street: Complete vehicle trajectories in this
segment were transcribed for two 15-minute periods; one
from 12:45 PM to 01:00 PM, and the other from 04:00 to
04:15 PM. The segment is approximately 2100 feet in length
with three to four lanes in each direction through the section.
There were five signalized intersections in this segment
among which northbound and southbound left turn
movements were unprotected in only two intersections.
Therefore, northbound and southbound left turns, and their
associated opposing through movements at these two
intersections are chosen for the aim of this study.
Lankershim Boulevard: A total of 32 minutes vehicle
trajectories were processed during a period from 08:28 AM
to 09:00 AM. This period is divided into one 17-minute and
one 15-minute period for analysis. The segment is
approximately 1600 feet in length with five intersections and
two/three lanes in each direction throughout the section.
Northbound and southbound unprotected left turn
movements with their associated opposing through
movements in two intersections are chosen for this study.
From these two datasets, a total of 40 unprotected left
turn maneuvers, where both conflicting vehicles approach
the associated intersection during the green indication, are
identified. For each case, trajectories of the involved
vehicles are traced and their actual choices are determined
by examining their respective speed and acceleration
profiles.
TABLE IV. MAXIMUM LIKELIHOOD ESTIMATION RESULTS
Parameter
Calibrated value
Parameter
Calibrated value
11
-2.99e+01
11
1.36e+01
11
5.47e-01
11
1.22e+01
11
-4.82e+00
11
-2.23e+00
12
-2.61e+00
12
5.10e+01
12
-3.20e-01
12
1.22e+01
21
-2.00e+01
12
-1.39e+00
21
-1.10e+00
21
-8.68e+00
21
-4.34e+00
21
-5.90e+00
22
-6.71e+00
22
9.29e-03
22
-8.17e-01
22
-5.45e+01
22
1.08e-02
From the total of 40 cases, the extracted trajectories of 30
cases are used for calibration and numeric parameter
estimation. Table IV presents the obtained results. The log
likelihood ratio (2) for the proposed model is calculated to
be 0.976.
The model is also validated to see if it is a good
representation of the real-world system. A comparison is
made based on how accurately the outcomes of the model
represent the outcomes of the actual system. To this end, the
rest of 10 cases in the selected dataset, which correspond to
10 games, are used to validate the proposed model. Using
the calibrated values for the model parameters, the values of
the payoff functions are calculated for each player and the
Nash Equilibria are in turn identified in each game. The
equilibrium with the most occurrence probability calculated
by the maximum likelihood estimator is chosen to be the
predicted outcome of the game by the proposed model.
These results are then compared to the real-world behavior
of the drivers (players). Results indicate that 80 percent of
the model predictions reflect the real outcome of the game.
Indeed, the model correctly predicts the actions chosen by
both players at 80 percent of the cases.
Considering the safety issues in performing conflicting
vehicular movement, one can conclude that 80% is not a
particularly high prediction power. The analysis of the
incorrectly predicted cases indicates that the proposed model
performs more conservatively when the approaching speed
of the through vehicle (player B) is relatively high. It is
expected that different criteria in defining the collision risk
and comfortable acceleration/deceleration rates in payoff
formulations can improve the model performance in
presenting the real-world behavior of the drivers. Calibration
and validation with larger data sets are also required to
provide more variation in the value of payoff function and
hence a better evaluation of model capability in predicting
and representing the various cases happening in real world.
To further evaluate the model calibration results, Root
Mean Square Error (RMSE) is also calculated according to
(16).
RMSE = [1/n (1:n(1(x'i-xi))2)]0.5 (16)
where n is the number of observations, and x'i and xi are the
predicted and actual values, respectively. 1(x'i-xi) is equal to
zero if the predicted and actual values are identical (x'i = xi),
and one otherwise. The model evaluation results are
presented in Table V. It should be noted that, although the
calculated value of RMSE is not very low, it still indicates a
relatively high prediction performance of the proposed
model. In evaluating the results of RMSE test, one should
notice that the result of this test is highly affected by the low
number of the validated cases; and larger number of
validated cases can lead to more accurate evaluations of the
model performance by RMSE test.
TABLE V. MODEL EVALUATION RESULTS
2
0.979
Percent correct
80%
RMSE
0.447

V. CONCLUSION
Intersections are a major source of delay and collisions
and can adversely affect the level of service in the entire
system if not designed properly. One of the major reasons is
the fact that human drivers are not aware enough and quick
enough to decide on the best response when they are
required to perform conflicting movements at intersections.
The connected vehicle technology and vehicle automation,
however, can solve these issues by improving
drivers/vehicles’ awareness of their surrounding
environment, which in turn leads to improved strategic,
tactical, and operational decisions by drivers/vehicles. The
present paper investigates unprotected left turn maneuvers at
intersections. Most of the studies in this domain have
focused on the available gap between the left turning vehicle
and the opposing through vehicles. This study, however,
adopts a game-theoretical approach to capture the dynamic
interactions among the left turning and the opposing through
vehicles using the flow of information in a connected
environment. A two-person non-zero-sum non-cooperative
game under complete information is selected to model this
behavior. Payoff functions are defined in terms of utility
functions which include an error term to capture the effect of
unobserved factors. Finally, a calibration process is proposed
based on the maximum likelihood method to estimate the
game parameters. The model parameters are numerically
estimated using NGSIM trajectory data. Results indicate that
this framework can effectively capture vehicle interactions
when performing conflicting turning movements while
achieving a relatively high accuracy in predicting vehicles'
real choice. Limited data source was the major challenge in
this study. Further calibration and validation based on larger
data sets are required to accurately determine the prediction
capability of the proposed approach.
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