Optimizing Toll Enforcement in
Transportation Networks: a
Game-Theoretic Approach
Ralf B ORNDÖRFER, Julia B UWAYA,
Guillaume S AGNOL, Elmar S WARAT
Zuse Institut Berlin (ZIB)
INOC 2013, 22 May 2013, Tenerife
Truck toll in Germany
A Huge and original system
Started in 2005
Tax based on distance and truck
category (avg price: 0,17 Euro/km)
Total revenue around
3 Billion Euros / year
No toll barriers
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Truck toll in Germany
A Huge and original system
Started in 2005
Tax based on distance and truck
category (avg price: 0,17 Euro/km)
Total revenue around
3 Billion Euros / year
No toll barriers
Toll Enforcement
Automatic controls: 300 toll checker gantries
(Kontrollbrücke)
Mobile control units: 300 vehicles - 540 officers from BAG
(Federal Office of Goods Transport)
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Optimizing Toll Enforcement
Cooperation project between ZIB and BAG
Development of a tool to generate
feasible duty rosters
optimal tour planning
Specifications
Mobile inspection teams must carry out a network-wide control
whose intensity is proportional to given spatial and time
dependent traffic distributions.
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Outline
1
Problem formulation
Players and Payoffs
Model examples
2
Computation of Equilibria
Nash Equilibria for MAXPROFIT
a MIP formulation for Stackelberg Equilibria
3
Numerical Results
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Outline
1
Problem formulation
Players and Payoffs
Model examples
2
Computation of Equilibria
Nash Equilibria for MAXPROFIT
a MIP formulation for Stackelberg Equilibria
3
Numerical Results
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A simple, general model
w2
a
w1
b
c
w4
w3
xa→d drivers from a to d
d
w5
e
Notation
G(V , E)
K
xk
we
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: Graph of the network
: set of commodities (OD pairs) on the network
: nb of users (trucks) of commodity k ∈ K
: cost for taking e ∈ E (travel charge + toll fare)
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A simple, general model
w2 , σ2 q2
a
w1 , σ1 q1
b
c
w4 , σ4 q4
w3 , σ3 = 0
xa→d drivers from a to d
d
w5 , σ5 q5
e
Notation
G(V , E)
K
xk
we
: Graph of the network
: set of commodities (OD pairs) on the network
: nb of users (trucks) of commodity k ∈ K
: cost for taking e ∈ E (travel charge + toll fare)
qe
σe
: Probability that an inspector is on section e ∈ E
: Probability to be controlled on e,
conditionally to the presence of an inspector
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A network spot-cheking game
Given:
DiGraph G(V , E) with costs and control risks (we , σe )e∈E ,
commodities K = {k1 , . . . , km } with traffic (xk )k ∈K ,
amount P of the penalty if a toll evader is caught.
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A network spot-cheking game
Given:
DiGraph G(V , E) with costs and control risks (we , σe )e∈E ,
commodities K = {k1 , . . . , km } with traffic (xk )k ∈K ,
amount P of the penalty if a toll evader is caught.
Assuming:
stationary regime: users of commodity k have learnt the
probabilities πr to be controlled on each path r ∈ Rk .
selfish, rational drivers: Choose
P their path r ∈ Rk so as to
minimize their expected loss e∈r we + πr P.
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A network spot-cheking game
Given:
DiGraph G(V , E) with costs and control risks (we , σe )e∈E ,
commodities K = {k1 , . . . , km } with traffic (xk )k ∈K ,
amount P of the penalty if a toll evader is caught.
Assuming:
stationary regime: users of commodity k have learnt the
probabilities πr to be controlled on each path r ∈ Rk .
selfish, rational drivers: Choose
P their path r ∈ Rk so as to
minimize their expected loss e∈r we + πr P.
Goal:
Distribute the controls q over the edges of the network, to
maximize the expected revenue (fares + penalties)
OR minimize the number of evaders (or evaded kilometers)
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Players of the spot-checking game
“Player k ” represents all users of commodity k .
• Commits to a mixed strategy p̂ k , i.e. chooses path r with
probability p̂rk .
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Players of the spot-checking game
“Player k ” represents all users of commodity k .
• Commits to a mixed strategy p̂ k , i.e. chooses path r with
probability p̂rk .
• This defines a unit flow p k through commodity k :
X
pek :=
p̂rk .
r 3e
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Players of the spot-checking game
“Player k ” represents all users of commodity k .
• Commits to a mixed strategy p̂ k , i.e. chooses path r with
probability p̂rk .
• This defines a unit flow p k through commodity k :
X
pek :=
p̂rk .
r 3e
“The Inspector” represents the set of controllers.
• Commits to a strategy q ∈ Q ⊂ [0, 1]E .
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Game Theory Background
Player
Inspector
∀k ∈ K, Player k
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Strategy
q∈Q
pk ∈ Pk
Payoff
ΠC (p, q), p := {pk , k ∈ K}
Πk (pk , q)
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Game Theory Background
Player
Inspector
∀k ∈ K, Player k
Strategy
q∈Q
pk ∈ Pk
Payoff
ΠC (p, q), p := {pk , k ∈ K}
Πk (pk , q)
Best response strategies
p∗ ∈ BR(q) ⇐⇒ ∀k ∈ K, ∀p0k ∈ Pk , Πk (p∗k , q) ≥ Πk (p0k , q);
q∗ ∈ BR(p) ⇐⇒ ∀q 0 ∈ Q,
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ΠC (p, q∗ ) ≥ ΠC (p, q0 );
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Game Theory Background
Player
Inspector
∀k ∈ K, Player k
Strategy
q∈Q
pk ∈ Pk
Payoff
ΠC (p, q), p := {pk , k ∈ K}
Πk (pk , q)
Best response strategies
p∗ ∈ BR(q) ⇐⇒ ∀k ∈ K, ∀p0k ∈ Pk , Πk (p∗k , q) ≥ Πk (p0k , q);
q∗ ∈ BR(p) ⇐⇒ ∀q 0 ∈ Q,
ΠC (p, q∗ ) ≥ ΠC (p, q0 );
Nash Equilibrium
(p∗ , q∗ ) is a Nash equilibrium iff
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p∗ ∈ BR(q∗ )
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and
q∗ ∈ BR(p∗ ).
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Game Theory Background
Player
Inspector
∀k ∈ K, Player k
Strategy
q∈Q
pk ∈ Pk
Payoff
ΠC (p, q), p := {pk , k ∈ K}
Πk (pk , q)
Best response strategies
p∗ ∈ BR(q) ⇐⇒ ∀k ∈ K, ∀p0k ∈ Pk , Πk (p∗k , q) ≥ Πk (p0k , q);
q∗ ∈ BR(p) ⇐⇒ ∀q 0 ∈ Q,
ΠC (p, q∗ ) ≥ ΠC (p, q0 );
Nash Equilibrium
(p∗ , q∗ ) is a Nash equilibrium iff
p∗ ∈ BR(q∗ )
and
q∗ ∈ BR(p∗ ).
Stackelberg equilibrium
(p∗ , q∗ ) is a Stackelberg equilibrium iff
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(p∗ , q∗ ) ∈ arg max Πc (p, q).
p∈BR(q)
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Users’ best response strategies
Probability to be controlled on path r
w2 , σ2 q2
a
w1 , σ1 q1
b
c
πr = 1 −
w4 , σ4 q4
w3 , σ3 = 0
xa→d drivers from a to d
d
w5 , σ5 q5
Y
(1 − qe σe )
e∈r
e
'
X
qe σe
e∈r
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Users’ best response strategies
Probability to be controlled on path r
w2 , σ2 q2
a
w1 , σ1 q1
b
c
πr = 1 −
w4 , σ4 q4
w3 , σ3 = 0
xa→d drivers from a to d
d
Y
(1 − qe σe )
e∈r
w5 , σ5 q5
e
'
X
qe σe
e∈r
Expected loss of Player k when choosing path r
!
Πk (r ) = −
X
we + qe σe P
e∈r
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Users’ best response strategies
Probability to be controlled on path r
w2 , σ2 q2
a
w1 , σ1 q1
b
c
πr = 1 −
w4 , σ4 q4
w3 , σ3 = 0
xa→d drivers from a to d
d
Y
(1 − qe σe )
e∈r
w5 , σ5 q5
e
'
X
qe σe
e∈r
Expected loss of Player k when choosing path r
!
Πk (r ) = −
X
we + qe σe P
e∈r
Best strategy for Player k given q ∈ Q: Choose a shortest path
in G(V , E) with augmented weights we0 := we + qe σe P.
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Inspector’ s Payoff
Reward βe each time a user takes edge e
Fraction α of all collected Penalties
!
ΠC (p, q) =
X
k ∈K
xk
X
e∈E
pek (βe + ασe qe P)
Inspector’ s Payoff
Reward βe each time a user takes edge e
Fraction α of all collected Penalties
!
ΠC (p, q) =
X
k ∈K
xk
X
pek (βe + ασe qe P)
e∈E
Particular cases
MAXPROFIT: α = 1, βe = toll fare for edge e.
MAXTOLL:
α = 0, βe = toll fare for edge e.
MINEVADERS: α = 0, βe = 1 when e is an edge
representing the option to pay the toll for the whole trip
through a commodity.
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Model examples: underlying Graph
1. Transit system with fixed access fare T .
a
w2 = c2
σ2 q2
w1 = c1
σ1 q1
b
c
w3 = c3
σ3 q3
w4 = c4
σ4 q4
d
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e
w5 = c5
σ5 q5
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Model examples: underlying Graph
1. Transit system with fixed access fare T .
a
w2 = c2
σ2 q2
w1 = c1
σ1 q1
b
c
w3 = c3
σ3 q3
w[ad] = c1 + c3 + T
σ[ad] = 0
w[ce] = c4 + c5 + T
σ[ce] = 0
w4 = c4
σ4 q4
d
e
w5 = c5
σ5 q5
commodity set: K = {a → d, c → e}
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Model examples: underlying Graph
2. Motorway Network with alternative,
toll-free trunk roads
E
C
A
TOLL EVASION LAYER
D
B
T RANSITION EDGES (σ = 0, transition cost τ )
E
C
A
D
M OTORWAY & S HORTCUTS (σ = 0)
B
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Set of Inspector’s strategies Q
1.Ideal distrubution of γ controllers
Q = {q ∈ [0, 1]E :
X
qe = γ}
e
2.Feasible Duty rosters
Define Q as the set of unit flows in a cyclic duty graph:
E ARLY
MORNING
AC
CD
AB
BD
N IGHT
DE
M ORNING
DE
L ATE
N OON
BD
A FTERNOON
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Outline
1
Problem formulation
Players and Payoffs
Model examples
2
Computation of Equilibria
Nash Equilibria for MAXPROFIT
a MIP formulation for Stackelberg Equilibria
3
Numerical Results
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Nash Equilibria for MAXPROFIT
Theorem
Although non zero-sum, a Nash equilibrium of the MAXPROFIT
spot-checking game can be found by Linear Programming:
max
q,λ,y
s. t.
X
xk λk
(1a)
k
yvs − yus ≤ w(u,v ) + σ(u,v ) q(u,v ) P, ∀s ∈ V , ∀(u, v ) ∈ E;
(1b)
yss = 0,
∀s ∈ V ;
(1c)
∀k ∈ K
(1d)
λk ≤
src(k )
ydst(k )
q∈Q
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(1e)
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Nash Equilibria for MAXPROFIT
Theorem
Although non zero-sum, a Nash equilibrium of the MAXPROFIT
spot-checking game can be found by Linear Programming:
!
max
q,λ,y
s. t.
X
xk λk
k
+
X
k
xk
X
pek (βe
(1a)
yvs − yus ≤ w(u,v ) + σ(u,v ) q(u,v ) P, ∀s ∈ V , ∀(u, v ) ∈ E;
(1b)
yss = 0,
∀s ∈ V ;
(1c)
∀k ∈ K
(1d)
λk ≤
src(k )
ydst(k )
q∈Q
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− we )
e
(1e)
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a MIP for Stackelberg Equilibria
Theorem
A Stackelberg equilibrium of the spot-checking game can be
found by solving the following MIP
max
q,y,λ,µ,ρ
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X
k
αxk λk +
XX
ρse (βe − αwe )
s∈V e∈E
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a MIP for Stackelberg Equilibria
Theorem
A Stackelberg equilibrium of the spot-checking game can be
found by solving the following MIP
max
q,y,λ,µ,ρ
X
αxk λk +
k
XX
ρse (βe − αwe )
s∈V e∈E
0 ≤ w(u,v ) + σ(u,v ) q(u,v ) P − (yvs − yus )
∀s ∈ V ,∀(u, v ) ∈ E;
yss
∀s ∈ S;
= 0,
λk ≤
src(k )
ydst(k )
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∀k ∈ K;
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a MIP for Stackelberg Equilibria
Theorem
A Stackelberg equilibrium of the spot-checking game can be
found by solving the following MIP
max
q,y,λ,µ,ρ
X
αxk λk +
k
XX
ρse (βe − αwe )
s∈V e∈E
0 ≤ w(u,v ) + σ(u,v ) q(u,v ) P − (yvs − yus )
∀s ∈ V ,∀(u, v ) ∈ E;
yss
∀s ∈ S;
= 0,
λk ≤
src(k )
ydst(k )
∀k ∈ K;
q∈Q
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a MIP for Stackelberg Equilibria
Theorem
A Stackelberg equilibrium of the spot-checking game can be
found by solving the following MIP
max
q,y,λ,µ,ρ
X
k
αxk λk +
XX
ρse (βe − αwe )
s∈V e∈E
0 ≤ w(u,v ) + σ(u,v ) q(u,v ) P − (yvs − yus ) ≤ M(1 − µs(u,v ) ),
∀s ∈ V ,∀(u, v ) ∈ E;
yss
∀s ∈ S;
= 0,
λk ≤
src(k )
ydst(k )
∀k ∈ K;
q∈Q
µse ∈ {0, 1},
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∀(s, e) ∈ S × E.
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a MIP for Stackelberg Equilibria
Theorem
A Stackelberg equilibrium of the spot-checking game can be
found by solving the following MIP
max
q,y,λ,µ,ρ
X
αxk λk +
k
XX
ρse (βe − αwe )
s∈V e∈E
0 ≤ w(u,v ) + σ(u,v ) q(u,v ) P − (yvs − yus ) ≤ M(1 − µs(u,v ) ),
∀s ∈ V ,∀(u, v ) ∈ E;
yss
∀s ∈ S;
= 0,
λk ≤
src(k )
ydst(k )
q∈Q
X
ρs(v ,u)
∀k ∈ K;
−
X
{u:(v ,u)∈E}
{u:(u,v )∈E}
s
s
0 ≤ ρe ≤ Mµe ,
µse ∈ {0, 1},
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ρs(u,v )
=
 P

v x(s→v )
−x(s→v )

0
if s = v ;
if (s → v ) ∈ K;
otherwise,
∀(s, v ) ∈ S × V ;
∀(s, e) ∈ S × E;
∀(s, e) ∈ S × E.
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Outline
1
Problem formulation
Players and Payoffs
Model examples
2
Computation of Equilibria
Nash Equilibria for MAXPROFIT
a MIP formulation for Stackelberg Equilibria
3
Numerical Results
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Experimental settings
The Motorway network is divided into control regions. We
solve the problem independently on each region.
Real traffic data (averaged over time).
All instances solved with CPLEX.
We compare the results to the naive solution, where controls
are proportional to the traffic volume on each edge.
All the results are presented for the model with
P one toll-edge
E
for each commodity, and Q = {q ∈ [0, 1] :
e qe = γ}.
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Optimal control strategies
Optimum of the Nash LP,
MAXPROFIT setting, γ = 3 inspectors:
Control rate
Dortmund
Duisburg
Wuppertal
Düsseldorf
Region of North Rhine-Westfalia:
111 nodes, 264 directed edges, 4905 commodities.
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Optimal control strategies
Near optimum of the Stackelberg MIP (gap = 1.5%),
MAXPROFIT setting, γ = 50 inspectors:
Whole Germany:
319 nodes, 2948 edges, 5013 commodities
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Comparison of different equilibria
Revenue (×106 )
Region of Berlin-Brandenburg (|V | = 45, |E| = 130, |K| = 596)
4
3
MAXPROFIT
MAXTOLL
NASH
PROP
2
1
1
2
3
4
5
6
7
γ
Revenue vs. Number of controllers
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Comparison of different equilibria
Profit (in % of MAXPROFIT)
Region of Berlin-Brandenburg (|V | = 45, |E| = 130, |K| = 596)
1
0.9
0.8
MAXPROFIT
MAXTOLL
NASH
PROP
0.7
0.6
1
2
3
4
5
6
7
γ
Revenue vs. Number of controllers
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Comparison of different equilibria
Region of Berlin-Brandenburg (|V | = 45, |E| = 130, |K| = 596)
Toll payment rate
1
0.8
0.6
MAXPROFIT
MAXTOLL
NASH
PROP
0.4
0.2
0
1
2
3
4
5
6
7
γ
Toll Payment rate vs. Number of controllers
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Comparison of different equilibria
Revenue (×106 )
Region of Berlin-Brandenburg (|V | = 45, |E| = 130, |K| = 596)
3
total profit
toll revenue
penalties
2
1
0
0.5
1
α
Effect of α on the fraction of toll in Revenue
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Conclusion and perspectives
Conclusion
A flexible framework to model toll enforcement problems in
transportation networks
LP and MIP formulation to compute Nash and Stackelberg
strategies
The Nash LP remains tractable for very large instances and
seems to approximate the Stackelberg strategy that
optimizes the total revenue.
Future work
Integrate this approach in our duty roster optimizer
Investigate theoretical bounds on the gap between Nash
and Stackelberg Strategies (for MAXPROFIT)
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