PRICING INCLUDING ROUTE CHOICE AND ELASTIC DEMAND WITH
DIFFERENT USER GROUPS - A GAME THEORY APPROACH
Dusica Joksimovic, Michiel C.J. Bliemer, Piet H.L. Bovy
Delft University of Technology, Faculty of Civil Engineering and Geosciences
Transport and Planning Section
P.O. Box 5048, 2600 GA Delft, The Netherlands
e-mail: [email protected]
1. INTRODUCTION AND BACKGROUND
The optimal toll design problem is a part of a project “A Multi-Disciplinary
Study of Pricing Policies in Transport - A Traffic Engineering Perspective” with
focus on the optimal toll design and dynamic traffic assignment modelling part
of it. This project is being carried out to provide theoretical and practical
evaluation of effects of different road transport pricing policies on behavioural
responses and their consequences. It is part of a larger multi-disciplinary
research program established as a co-operation between four Dutch
universities.
In recent years, researchers have become increasingly interested in the
effects of introducing different road pricing measures on transportation
networks (see more in May and Milne (2000)). The view that pricing can be
one of the strategies to achieve more efficient use of transportation capacity
has led to the expectation that road pricing can relieve congestion on the
roads and improve the use of the transportation system.
Nevertheless, road pricing is a very controversial and complex topic making it
necessary to consider the road-pricing problem from different perspectives.
Who is involved in decision-making and how should decisions be made? How
will the travellers change their travel behaviour after introducing road pricing?
How will travellers interact with each other and how can the road authority
influence or even control travel behaviour of travellers?
To answer these rather complex questions we need a flexible framework for
analysing the behaviour of travellers as well as of the road authority. Game
theory provides such a framework for modelling decision-making processes in
which multiple players are involved with different objectives, rules of the
game, and assumptions. Considering the problem of designing optimal tolls
on the network, there is a need for better insights into the interactions
between travellers and the road authority, their nature, and the consequences
of these interactions. How will different user groups react on different tolls
imposed by the road authority and how they will change their travel
behaviour? The focus of this paper is on assessing the behaviour of different
user groups and their influence on the optimal toll design.
In this paper we analyse in a game theoretic framework a simple route choice
problem with elastic demand where road pricing with different user groups is
introduced. First, the road-pricing problem is formulated using game theory
notions, and different game concepts are described. After that, a gametheoretic approach is applied to formulate the road pricing game as monopoly,
Stackelberg, and Cournot game, respectively. The main purpose of the
experiment reported here is to show the outcomes of different games
established for the optimal toll design problem with different user classes.
Outcomes of different game concepts and optimal toll pattern is influenced by
heterogeneous users.
2. LITERATURE REVIEW
2.1 Transportation problems and game theory
Game theory first appeared in solving transportation problems in the form of
so-called Wardropian equilibrium of route choice, see the work of Wardrop
(1952) which is similar to the Nash equilibrium of an N-player game, see Nash
(1950). The definition of a Nash equilibrium will be presented later in the
paper.
2.2. Optimal traffic control problems and game theory
In Fisk (1984) different problems in transportation systems modelling are
described for the first time in which a game theory approach is proposed as a
potential source of solution algorithms for transportation systems modelling. In
that paper, relationships are drawn between two game theory models (Nash
non-cooperative and Stackelberg game). Examples of the Nash and
Stackelberg games are explained in detail in the cases of carriers competing
for intercity passenger travel and the signal optimisation problem.
In Wie (1995) the dynamic mixed behaviour traffic network equilibrium
problem is formulated as a non-cooperative. N-person, non zero-sum
differential game. A simple network is considered where two types of players
(called user equilibrium (UE)-players and Cournot-Nash (CN)-players,
respectively) interact through the congestion phenomenon. Each of the UE
and CN players attempts to optimise his own objective by making
simultaneous decisions of departure time choice, route choice, and departure
flow rate over a fixed time interval. A procedure to compute system optimal
routings in a dynamic traffic network is introduced by Garcia et al. (2000).
Fictitious play is utilised within a game of identical interests wherein vehicles
are treated as players. All players have a common payoff of average trip times
experienced in the network.
In Bell (2000) a two-player, non-cooperative game between the network user
seeking a path to minimise the expected trip cost on the one hand, and an
“evil entity” choosing link performance scenarios to maximise the expected trip
cost on the other is established. At the Nash mixed-strategy equilibrium, the
user is unable to reduce his expected trip cost by changing his path choice
while the ‘evil entity’ is unable to increase this expected trip cost by changing
the scenario probabilities, without cooperation.
An application of game theory to solve the risk-averse user equilibrium traffic
assignment problem can be found in the work of Bell and Cassir (2002).
Network users have to make route choice decisions in the presence of
uncertainty about route costs. Therefore, uncertainty requires network users
to have a strategy towards risk. Network users “play through” all the possible
states before selecting their best route. In Zhang et al. (2003) a preliminary
model of dynamic multi-layer infrastructure networks is presented in the form
of a differential game. In particular, three network layers (car, urban freight
and data) are modelled as Cournot-Nash dynamic agents. A dynamic game
theoretic model of multi-layer infrastructure networks is introduced to solve the
flow equilibrium and optimal allocation problem for these three layers. In Chen
and Ben-Akiva (1998) the integrated traffic control and dynamic traffic
assignment problem is presented as a non-cooperative game between the
traffic authority and highway users. The objective of the combined controlassignment problem is to find dynamic system optimal signal settings and
dynamic user-optimal traffic flows. The combined control-assignment problem
is first formulated as a single-level Cournot game: the traffic authority and the
users choose their strategies simultaneously. Then, the combined problem is
formulated as a bi-level Stackelberg game in which the traffic authority is the
leader who determines the signal settings in anticipation of the users’
responses.
2.3 Road pricing problems and game theory
The problem of determining optimal tolls in transportation networks is a
complex issue. In the work of Levinson (1988) the question what happens
when jurisdictions have the opportunity to establish tollbooths at the frontier
separating them is examined. If one jurisdiction would be able to set his policy
in a vacuum it is clearly advantageous to impose as high a toll on nonresidents as can be supported. However, the neighbouring jurisdiction can set
a policy in response. This establishes the potential for a classical prisoner’s
dilemma consideration: in this case to tax (cooperate) or to toll (defect). In
Levinson (2003) an application of game theory and queuing analysis to
develop micro-formulations of congestion can be found. Only departure time is
analysed in the context of a two and three-player game respectively where
interactions among players affect the payoffs for other players in a systematic
way. In Joksimovic et al. (2004a) the route choice and elastic demand
problem is considered with focus on different game concepts and different
policy objectives of the optimal toll design problem. A few experiments are
done showing that the Steckleberg is the most promising game between the
road authority and the travellers.
2.4 Different user classes and game theory
An application of game theory to investigate interactions between a private
highway operator and a private transit operator who uses the same highway
can be found in Wang and Verhoef (2004). Heterogeneity of travellers is taken
into account by considering a continuous distribution of values of time of
travellers.
Taking into account the literature study, we can conclude that there is a lack in
the literature considering different user groups in optimal toll design problem
Therefore, different user groups in the optimal toll design problem as well as
different game concepts will be the focus of this paper.
3. PROBLEM STATEMENT (NON-COOPERATIVE GAME THEORY)
The interaction between the travellers and the road authority can be seen as a
non-cooperative, non-zero sum, (N+1) player game between a single traffic
authority on the one side and N motorway users (travellers) on the other. The
objective of the road-pricing problem, which is the combined optimal toll
design and traffic assignment problem, is to find system-optimal tolls and
user-optimal traffic flows. This road-pricing problem is an example of a bi-level
optimisation problem. The user-equilibrium traffic assignment problem (lower
level problem) can be formulated as non-cooperative, N-person, non-zerosum game solved as a Nash game. The upper level problem may have
different objectives depending on what the road authority would like to
achieve. A conceptual framework for the optimal toll design problem is given
in Figure 1.
Trip
Decision
Upper level
The road authority
Change
of travel
behavior
Tolls
Traveler on
the
network
Travel
cost is
high
yes
Travel or
not?
no
Route choice
Travelers
Which route to choose?
Lower level
Figure 1: Conceptual framework for optimal toll design with route and trip
choice
The road authority sets tolls on the network in order to optimise certain
objective while travellers respond to tolls by changing their travel decisions.
Depending on travel costs, they can decide to change routes or decide not to
travel at all. In the road-pricing problem, we are dealing with an N+1-player
game, where there are N players (travellers) making a travel choice decision,
and one player (the road manager) making a control or design decision (in this
case, setting road tolls). Adding the traffic authority to the game is not as
simple as extending an N-player game to an N+1 player game, because the
strategy space and the payoff function for this additional player differs from
the rest of the N players. In fact, there are two games played in conjunction
with each other. The first game is a non-cooperative game where all N
travellers aim to maximise their own utility by choosing the best travel strategy
(i.e. trip choice and route choice), taking into account all other travellers’
strategies. The second game is between the travellers and the road manager,
where the road manager aims to maximise some network performance by
choosing a control strategy, taking into account that travellers respond to the
control strategy by adapting their travel strategies. The two games can be
described as follows:
The outer level game, being the toll design problem, consisting of the
following elements:
1. Players: the government on the one side and N potential travellers on
the other
2. Strategies: Toll levels, possibly restricted.
3. Rule: the government sets the tolls taking the travellers’ behaviour and
responses into account in order to optimise a certain objective.
4. Payoff functions: a) payoff for the government (e.g. social welfare,
revenues), b) payoff for the travellers (travel utilities).
The inner level game, being the network equilibrium problem, consisting of the
following elements:
1. Players: N travellers
2. Strategies: route and trip choice
3. Rule: travellers make optimal trip and route choice decisions as to
maximise their individual subjective utilities given a specific toll pattern.
4. Payoff functions: travel utilities
Our main focus in this paper is to investigate the outer level game between
the road authority and users, although the inner level game between travellers
is part of it. We assume that all game concepts are games of complete and
perfect information. The order in which decisions are made is important. In a
game with complete information every player is fully informed about the rules
of the game, the preferences of each player, and each player knows that
every player knows. In other words, for the road pricing game we assume that
all information about travel attributes (toll levels, available routes, travel times)
are known to all travellers as well as the strategies, rules and outcomes of the
game. A game with perfect information means that decision maker knows the
entire history of the game. Game theory notation used in this paper is adopted
from work of Altman et al. (2003).
4. MODEL STRUCTURE
The objectives of the road authority and the travellers are different and
sometimes even opposite. The upper level objective may be to minimise the
total travel time, to relieve congestion, to improve safety, to raise revenue, to
improve total system utility, etc. The lower level objective may be the
individual travel time, travel cost, or the individual travel utility. In this paper,
we use the total travel utility for individuals as the objective to maximise for
travellers. For road pricing with different objective functions for the road
authority see in Joksimovic et al. (2004a).
Since the purpose of this paper is to gain more insight into the structure of the
optimal toll design problem with different user classes by using game theory,
we restrict ourselves to the case of a very simple network in which only one
origin-destination (OD) pair is considered. Between this OD pair, different nonoverlapping route alternatives are available. The generalized route travel cost
function of traveller i for route p, cip , includes the travel time costs and the toll
costs,
cip = α iτ p + θ p ,
(1)
where τ p is the travel time of route p, Pp is the toll costs of route p, and α i
denotes the value-of-time (VOT) for i user which converts the travel time into
monetary costs. Let U p denote the trip utility for making a trip along route p of
traveller i. This trip utility consists of a fixed net utility U for making the trip (or
arriving at the destination), and a disutility consisting of the generalized route
travel costs c p :
U ip = U − cip .
(2)
According to utility maximization theory, a trip will be made only if the utility of
doing an activity at a destination minus the utility of staying at home and the
disutility of travelling is positive. In other words, if U p ≤ 0 then no trip will be
made. By including a fictitious route in the route choice set representing the
travellers’ choice not to travel, and attaching a utility of zero to this ‘route’
alternative, we combine route choice and trip choice into the model. Travellers
are assumed to respond according to Wardrop’s equilibrium law extended
with elastic demand: At equilibrium, no user can improve his/her trip utility by
unilaterally making another route choice or trip choice decision. It should be
mentioned that we only take into account the deterministic case.
Classical utility maximisation theory uses random utilities taking into account
error term in the expression (1), but for the sake of simplicity we assume this
simpler case. For more elaborate definitions, see the work of Casetta (2001).
5. GAME THEORY APPLIED TO ROAD PRICING
Let us first consider the N-player game of the travellers, where Si is the set of
available alternatives for traveller i, i ∈ {1,… , N }. The strategy si ∈ Si that
traveller i will play depends on the control strategy set by the road manager,
denoted by vector θ , and on the strategies of all other players, denoted by
s− i ≡ ( s1 ,… , si −1 , si +1 ,… , sN ). We assume that each traveller decides
independently and unilaterally seeks the maximum utility payoff, taking into
account the possible rational choices of the other travellers. Let
J i ( si (θ ), s− i (θ ),θ ) denote the utility payoff for traveller i for a given control
strategy θ . The utility payoff can include all kinds of travel utilities and travel
cost. Utility payoff for traveller i can be expressed as follows:
J i ( si (θ ) , s− i (θ ) ,θ ) = U i si
(3)
where U iSi is a travel cost for traveller i if he chooses strategy si , that is
available routes (see Equation (2)). If all other travellers play strategies s−* i ,
then traveller i will play the strategy that maximises his payoff utility, i.e.
si* (θ ) = arg max J i ( si (θ ), s−* i (θ ),θ ) .
si ∈Si
(4)
If Equation (4) holds for all travellers i ∈ {1,… , N }, then s* (θ ) ≡ ( s−* i (θ ), si* (θ )) is
called a Nash equilibrium for the control strategy θ . In this equilibrium, no
traveller can improve his utility payoff by unilateral deviation. Note that this
coincides with the concept of a Wardrop user-equilibrium.
Le us now consider the complete N+1-player game where the road manager
faces the N travellers. The set Θ describes the alternative strategies available
to the road manager. Suppose he chooses strategy θ ∈ Θ. Depending on this
strategy and on the strategies chosen by the travellers, s* (θ ), his utility payoff
is denoted by R ( s* (θ ),θ ), and represent the total system utility or the total
profits made. The road manager chooses the strategy θ * in which he aims to
maximise his utility payoff, depending on the responses of the travellers:
θ * = arg max R ( s* (θ ),θ ) .
θ ∈Θ
(5)
If Equations (4) and (5) are satisfied for all (N+1) players, where θ = θ * in
Equation (4), then this is a Nash equilibrium in which no player can be better
off by unilaterally playing another strategy. Although all equilibria use the
Nash concept, a different equilibrium or game type can be defined in the N+1player game depending on the influence each of the players has in the game.
6. DIFFERENT GAME CONCEPTS
In the following we will distinguish three different types of games between the
road authority and the travellers namely, monopoly, Stackelberg and Cournot
game, respectively.
6.1 Monopoly game
In this case, the road manager not only sets its own control, but also controls
the strategies that the travellers will play. In other words, the road manager
sets θ * as well as s* . This case will lead to a so-called system optimal solution
of the game. A monopoly or solo player game represents the best system
performance and thus may serve as a ‘benchmark’ for other solutions. This
game solution shows what is best for the one player (the road manager),
regardless of the other players. In reality, however, a monopoly solution may
not be realistic since it is usually not in the users’ best interest and is it
practically difficult to force travellers choosing a specific route without an
incentive. From an economic point of view, in this case the road authority has
complete (or full) market power. Mathematically, the problem can be
formulated as follows:
( s* ,θ * ) = arg max R( s,θ ).
(6)
θ ∈Θ , s∈S
6.2 Stackelberg game
In this case, the road manager is the ‘leader’ by setting the control, thereby
directly influencing the travellers that are considered to be ‘followers’. The
travellers may only indirectly influence the road manager by making travel
decisions based on the control. It is assumed that the road manager has
complete knowledge of how travellers respond to control measures. The road
manager sets θ * and the travellers follow by playing s* (θ * ). From an economic
point of view, in a Stackelberg game one player has more market power than
others players in the game (in this case the road authority has more market
power than the travellers). The Stackelberg game is a dynamic, multi-stage
game with the following features:
• the successive moves (implemented strategies) of the two players occur in
sequence,
• all previous moves are observed before the next move is chosen (perfect
history knowledge),
• the players’ payoff from each feasible combination of moves are common
knowledge (complete payoff information).
For more details about complete and imperfect games, see e.g. work of
Ritzberger (2002). The equilibrium is determined by backward induction where
the traffic authority initiates the moves by setting a control strategy. Steps for
the road pricing game are as follows:
1. The road authority chooses toll values from the feasible set of tolls.
2. The travellers react on the route cost (with tolls included) by adapting their
route and/or trip choice.
3. Payoffs for the road authority as well as travellers are computed.
4. The optimal strategy for the road authority including the strategies of
travellers is chosen.
The problem can be mathematically formulated as to find ( s* ,θ * ) such that:
θ * = arg max R ( s* (θ ),θ ), where si* (θ ) = arg max J i ( si , s−* i ,θ ), ∀i = 1,… , N .
θ ∈Θ
si ∈Si
(7)
6.3 Cournot game
In contrast to the Stackelberg game, in this case the travellers are now
assumed to have a direct influence on the road manager, having complete
knowledge of the responses of the road manager to their travel decisions. The
road manager sets θ * ( s* ), depending on the travellers’ strategies s* (θ * ). This
type of a so-called duopoly game, in which two players choose their strategies
simultaneously and therefore one’s player’s response is unknown in advance
to others, is known as a Cournot game. Mathematically the problem can be
formulated as follows. Find ( s* ,θ * ) such that
θ * = arg max R( si* , s−* i ,θ ), and si* = arg max J i ( si , s−* i ,θ * ), ∀i = 1,… , N .
θ ∈Θ
si ∈Si
(8)
The different game concepts will be illustrated in the next section. It should be
pointed out that the Stackelberg game is the most realistic game approach.
For more about Stackelberg game see e.g. Basar and Olsder (1995).
Mathematical bi-level problem formulations can be used for solving more
complex games, see e.g. Joksimovic et al. (2004b).
7. A FEW EXPERIMENTS
Let us now consider the following simple problem to illustrate how the roadpricing problem can be analysed using game theory. Suppose there are two
individuals who want to travel from A to B. There are two alternative routes
available to go to B. The first route is tolled (toll is equal to θ ), the second
route is untolled. Depending on the toll level, the travellers decide to take
either route 1 or route 2, or not to travel at all. The latter choice is represented
by a third virtual route, such that we can consider three route alternatives as
available strategies to each traveller, i.e. Si = {1, 2,3} for traveller i = 1,2.
Figure 2 illustrates the problem.
route 1 (tolled)
A
route 2 (untolled)
B
‘route 3’ (do not travel)
Figure 2: Network description for the road- pricing problem
Each strategy yields a different payoff, depending on the utility to make the
trip, the travel time on the route (that increases whenever more travellers use
it) and a possible route toll. We assume that traveller i aims to maximise
his/her individual travel utility (payoff,) given by
U − α iτ 1 ( s1 (θ ), s2 (θ )) − θ , if si (θ ) = 1,

J i ( s1 (θ ), s2 (θ )) = U − α iτ 2 ( s1 (θ ), s2 (θ )),
if si (θ ) = 2,
0,
if si (θ ) = 3.

(9)
In Equation (9), U represents the trip utility when making the trip to
destination B (in the calculations we assume U =210), τ p (⋅) denotes the route
travel time for route r depending on the chosen strategies, and α i represents
class-specific value of time of traveller. Note that negative net utilities on route
1 and 2 imply that one will not travel, i.e. if the cost (disutility) of making the
trip is larger than the utility of the trip itself. The route travel times are given as
a function of the chosen strategies in the sense that the more travellers use a
certain route, the higher the travel time:
10, if s1 (θ ) = 1 or s2 (θ ) = 1 (e.g. flow on route 1 is 1),
18, if s1 (θ ) = 1 and s2 (θ ) = 1 (e.g. flow on route 1 is 2),
(10)
20, if s1 (θ ) = 2 or s2 (θ ) = 2 (e.g. flow on route 2 is 1),
40, if s1 (θ ) = 2 and s2 (θ ) = 2 (e.g. flow on route 2 is 2).
(11)
τ 1 ( s1 (θ ), s2 (θ )) = 
and
τ 2 ( s1 (θ ), s2 (θ )) = 
For the sake of clarity we will only consider pure strategies in this example,
althought the case may be extended to mixed strategies as well. In pure
strategies, each player is assumed to adopt only one strategy, whereas in
mixed strategies, the players are assumed to adopt probabilities for choosing
each of the available strategies. In our example we are thus consider discrete
flows instead of continuous flows hence Wardrop’s first principle, according to
which all travel utilities are equal for all used alternatives, may no longer hold
in this case. In fact, the more general equilibrium rule applies in which the
minimum trip utility for a traveller is maximised.
Now, let us add the road manager as a player, assuming that he tries to
maximise the total travel utility, i.e.
R ( s* (θ ),θ ) = J1 ( s* (θ )) + J 2 ( s* (θ )).
(12)
7.1 CASE STUDY 1: USERS WITH LOW VALUE OF TIME
Suppose that α = 6 for this group of travellers. The utility payoff table,
depending on the toll θ , is given in Table 1 for both travellers, where the
values between brackets are the payoffs for travellers 1 and 2, respectively.
Route 1
(102 − θ , 102 − θ )
Route 1
(90, 150 − θ )
Traveller 1 Route 2
(0, 150 − θ )
Route 3
Table 1: Utility payoff table for travellers
Traveller 2
Route 2
(150 − θ , 90)
(−30, − 30)
(0, 90)
Route 3
(150 − θ , 0)
(90, 0)
(0, 0)
For example, if traveller 1 chooses route 1 and traveller 2 chooses route 2,
then the travel utility for traveller 1 is J1 (1, 2) = 210 − 6 ⋅10 − θ = 150 − θ . The
strategy set of the road manager is assumed to be Θ = {θ | θ ≥ 0}. The payoffs
for the road manager are presented in Table 2. The payoffs depend on the
strategy θ ∈ Θ that the road manager plays and on the strategies the
travellers play.
Traveller 2
Route 1
Route 2
Route 3
204 − 2θ
240 − θ
150 − θ
Route 1
Traveller 1
240 − θ
-60
90
Route 2
150 − θ
90
0
Route 3
Table 2: Utility payoff table for the road manager if his objective is to maximize
the total travel utility
Let us solve the previously defined payoff tables for different game concepts.
First, we discuss the monopoly game, then the Stackelberg game and finally
the Cournot game.
7.1.1 Monopoly game
In the monopoly game, the road manager sets the toll as well as the travel
decisions of the travellers such that his payoff is maximised. Note that the
travel utility always decreases as θ increases, hence θ * = 0. In this case, the
maximum utility can be obtained if the travellers distribute themselves
between routes 1 and 2, i.e. s* = {(1, 2), (2,1)} . Hence, in this system optimum,
the total travel utility in the system is 240. Note that this optimum would not
occur if travellers have free choice, since θ = 0 yields a Nash-Wardrop
equilibrium for both travellers to choose route 1.
7.1.2 Stackelberg game
Now the travellers will maximize individually their own travel utility, depending
on the toll set by the road manager. Figure 3 illustrates the total travel utility
for different values of θ with the corresponding optimal strategies played by
the travellers. When 0 ≤ θ < 12, travellers will both choose route 1 because
with this strategy combination they will have maximal benefits. According to
Figure 3, if 12 ≤ θ < 150, travellers distribute themselves between route 1 and
2, while for θ ≥ 150 one traveller will take route 2 and another traveller will not
travel at all. Clearly, the optimum for the road manager is θ * = 12, yielding a
total travel utility of 228.
R ( s* (θ ),θ )
228
204
180
90
s* (θ ) = {(1,1)}
s* (θ ) = {(1, 2),(2,1)}
s* (θ ) = {(2,3),(3, 2)}
12
150
θ
Figure 3: Total travel utilities depending on toll value for the user with low
value of time
Let us formulate and solve the Stackelberg game in the following way. The
road authority may choose different strategies for the toll level on route 2
(suppose that strategies θ = {5,12,150} are chosen). After the toll is being set
by the road authority, the travellers react on this toll by reconsidering their
travel choices, maybe choosing different routes. The payoff function for every
strategy combination for the travellers is computed and presented on the
lower level of Figure 4 (e.g. if the road authority set toll θ = 5 then the
travellers can chose strategy (route 1, route 1) and the payoff for that strategy
is 97). The payoff function is computed for all travellers’ combinations as well
as for the road authority’s strategies. The extensive form of the Stackelberg
game is shown in Figure 4.
Player : Road
authority
θ =5
Player:
Travelers
θ = 150
θ = 12
R= 228
R =194
Objective:
Total travel
utility (R)
90
Objective:
individual
utility (J)
0
97
0
90
0
90
-30
0
0
0
90
0
90
0
0
90 -30
0
0
-48
0
0
0
0 -30
0
0
Figure 4: Outcomes of Stackelberg game applied to maximize the total travel
utility.
After that all combinations are computed for the travellers, the payoff for the
road authority is determined and the maximised. From Figure 4 can be seen
that the payoff for the travellers is 90 if they choose to travel on route 1 and
route 2 while the payoff for the road authority is 228. It should be mentioned
that this game is played sequentially.
7.1.3 Cournot game
In order to solve the optimal toll design problem as a Cournot game it is
necessary to construct the payoff table for the road authority as well as
travellers (the payoff table should have all strategy combinations for all
players, because they will choose their strategies simultaneously). Since
Wardrop’s equilibrium law can also be restated as to maximise the minimum
individual utility, the payoff for the travellers combined will be the minimum of
the individual trip utilities. The payoff for the road authority is again the total
system utility excluding the tolls. The formulation of the game is presented in
Table 3.
Road authority (1)
θ =12
θ =170
(1,1)
90,204 (X)
-68,204
97,204 (X)
(1,2)
90,240
90,240 (X)
-20,240
Travellers (1,3)
0,150
0,150
-20,150
(2)
(2,1)
90,240
90,240 (X)
-20,240
(2,2)
0,0
0,0
0,0
(2,3)
0,90
0,90
0,90 (X)
(3,1)
0,150
0,150
-20,150
(3,2)
0,90
0,90
0,90 (X)
(3,3)
0,0
0,0
0,0
Table 3: Cournot solutions of the optimal toll design game (denoted with X)
θ =5
There is however one dominating strategy, being that the travellers both take
route 1 and that the road manager sets zero tolls, yielding a total system utility
of 204. It can be shown that in case the travellers and the road manager have
equal influence on each others strategies, multiple Cournot solutions exist.
Table 3 summarizes the outcomes for the three different games.
Game
θ*
Monopoly
0
Stackelberg
12
si* (θ )
{(1, 2 ) , ( 2,1)}
{(1, 2 ) , ( 2,1)}
{(1,1)}
R
240
228
Ji
{( 90,150 ) , (150,90 )}
{( 90,138) , (138,90 )}
{(102,102 )}
Cournot
0
204
Table 4: Comparison of outcomes of different games for the objective of
maximizing total travel utility
7.2 CASE STUDY 2: USERS WITH HIGH VALUE OF TIME
Suppose that α = 10 for this group of travellers. The utility payoff table,
depending on the toll θ , is given in Table 1 for the two travellers, where the
values between brackets are the payoffs for travellers 1 and 2, respectively.
Traveller 2
Route 1
Route 2
(30 − θ , 30 − θ )
(110 − θ , 10)
Route 1
(10, 110 − θ )
(−30, − 30)
Route 2
Traveller 1
(0, 110 − θ )
(0, 10)
Route 3
Table 5: Utility payoff table for travellers with high value of time
Route 3
(110 − θ , 0)
(10, 0)
(0, 0)
The payoffs for the road manager are presented in Table 6.
Route 1
60 − 2θ
Route 1
Traveller 1
120 − θ
Route 2
110 − θ
Route 3
Table 6: Utility payoff table for the road manager
Traveller 2
Route 2
120 − θ
-60
10
Route 3
110 − θ
10
0
Let us solve the previously defined payoff tables for different game concepts.
7.1.1 Monopoly game
In this case, the maximum utility can be obtained if the travellers distribute
themselves between routes 1 and 2, i.e. s* = {(1, 2), (2,1)} . Hence, in this
system optimum, the total travel utility in the system is 120.
7.1.2 Stackelberg game
Figure 5 illustrates the total travel utility for different values of θ
corresponding optimal strategies played by the travellers.
with the
R ( s* (θ ),θ )
100
60
20
10
s* (θ ) = {(1,1)} s* (θ ) = {(1, 2),(2,1)}
20
s* (θ ) = {(2,3),(3, 2)}
110
θ
Figure 5: Total travel utilities depending on toll value for the users with high
value of time
Clearly, the optimum for the road manager is θ * = 20, yielding a total travel
utility of 100.
7.1.3 Cournot game
The solution of the Cournot game is that road manager sets zero tolls, yielding
a total system utility of 60.
Table 7 summarizes the outcomes for the three different games.
Game
θ*
Monopoly
0
Stackelberg
20
si* (θ )
{(1, 2 ) , ( 2,1)}
{(1, 2 ) , ( 2,1)}
{(1,1)}
R
120
100
Ji
{(10,110 ) , (110,10 )}
{(10,90 ) , ( 90,10 )}
{( 30,30 )}
Cournot
0
60
Table 7: Comparison of outcomes of different games for the users with high
value of time
7.3 CASE STUDY 3: COMBINED USERS (WITH LOW AND HIGH VALUE
OF TIME)
In this case, heterogeneous travellers are present, with high and with low
value of time. Suppose that α = 6 for traveller 1 and α = 10 for traveller 2. The
utility payoff table, depending on the toll θ , is given in Table 8 for both
travellers, where the values between brackets are the payoffs for travellers 1
and 2, respectively.
Traveller 2
Route 1
Route 2
Route 3
(102 − θ , 30 − θ )
(150 − θ , 10)
(150 − θ , 0)
Route 1
(90, 110 − θ )
(−30, − 30)
(90, 0)
Route 2
Traveller 1
(0, 110 − θ )
(0, 10)
(0, 0)
Route 3
Table 8: Utility payoff table for travellers with low and with high value of time
The payoffs for the road manager are presented in Table 9.
Route 1
132 − 2θ
Route 1
Traveller 1
200 − θ
Route 2
110 − θ
Route 3
Table 9: Utility payoff table for the road manager
Traveller 2
Route 2
160 − θ
-60
10
Route 3
150 − θ
90
0
7.3.1 Monopoly game
In this case, the maximum utility can be obtained if the travellers distribute
themselves between routes 1 and 2, i.e. s* = (2,1) Hence, in this system
optimum, the total travel utility in the system is 200.
7.3.2 Stackelberg game
Figure 6 illustrates the total travel utility for different values of θ with the
corresponding optimal strategies played by the travellers. When 0 ≤ θ < 12,
travellers will both choose route 1. Dashed line illustrated the expected payoff
function for the road authority.
R ( s* (θ ),θ )
200
s* (θ ) = {(2,1)}
170
*
150 s (θ ) = {(1,1)}
s* (θ ) = {(2,3),(3, 2)}
100
50
10
s* (θ ) = {(1, 2),(2,1)}
12
15
110
150
θ
Figure 6: Total travel utilities depending on toll value for the users with low
and high value of time
If 12 ≤ θ < 15, travellers distribute themselves between route 1 and 2, because
traveller 1 choose route 2 (value of time for this traveller is lower than for the
other one) while traveller 2 stays on route 1. For tolls between 15 and 110
travellers will distribute themselves over route 1 and 2. Nevertheless, for
110 ≤ θ ≤ 150, one travellers (one with low value of time) will decide to choose
route 3 (not to travel) while the other can afford to stay on route 2. Finally, for
θ ≥ 150 both travellers will be at the same (previous mentioned) position. The
optimum for the road manager is θ * = 12, yielding a total travel utility of 188.
7.3.3 Cournot game
Outcome of the Cournot game is the total system utility of 132.
Table 3 summarizes the outcomes for the three different games.
Game
θ*
Monopoly
0
Stackelberg
12
si* (θ )
{(1, 2 )}
{(1, 2 )}
{(1,1)}
R
200
188
Cournot
0
132
Table 10: Comparison of outcomes of different games
Ji
{( 90,110 )}
{(10,90 )}
{(102,30 )}
8. CONCLUSIONS AND FURTHER EXTENSIONS
The purpose of the paper was to gain more insight into the road-pricing
problem using concepts from game theory as well as different toll designs for
multiple user classes. To that end we presented the notions of game theory
and presented three different game types in order to elucidate the essentials
of the game theoretic approach. These game types were applied to two
different user groups (with low and with high value of time) on a simplistic
demand-supply network system. This clearly revealed differences in design
results in terms of toll levels and payoffs for involved actors, being the road
authority and network users.
For practical use, the presented game-theoretic analysis should be translated
into a modelling system with which tolling designs for real-size road networks
become feasible. For that purpose, the bi-level optimisation framework will be
used (see Joksimovic et al. (2004b)).
The theory presented here can be extended to include other relevant travel
choices such as e.g. departure time choice as well as imperfect information on
the part of the road users.
ACKNOWLEDGMENT
The authors would like to thank to NWO-Connekt for co-financing this project,
as part of its Multi Disciplinary Pricing in Transport (MD-PIT) research
program.
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