A three-level mathematical programming model of road pricing
Paolo Ferrari
University School of Engineering of Pisa
Via Diotisalvi 2 – 56126 Pisa – Italy
ABSTRACT
At present the Public Agencies of many countries are examining the opportunity of
imposing tolls on some links of their road networks (generally on links of motorways or roads
having characteristics very similar to those of motorways) in order to recoup, at least in part,
the costs of road improvement, maintenance and management. The toll imposition provokes
the transfer of part of traffic from roads on which tolls are imposed to other roads, which
often pass through towns or areas having particular environmental value. This fact often
produces a conflict between the Public Agency, whose financial needs require the toll
imposition, and the local communities which suffer the environmental damages consequent to
toll imposition. A solution of this conflict calls for the evaluation and the compensation of
these damages, which can be performed by the transfer of a part of the toll revenue to the
local communities.
The definition of the compensation is the result of a bargaining between two parts – the
Public Agency and the local communities – which must follow such rules as to prevent each
part from having unfair behaviour. This paper proposes a model of the bargaining
management which gives rise to a hierarchical three-level game, whose actors are the Public
Agency, the local communities, and the drivers travelling the network.
The model supposes that the Public Agency defines a compensation function which
provides the compensation as a function of the toll. On the other hand the latter is chosen by
the local communities, while the network users distribute themselves among the various
itineraries of the network in dependence of the toll, thus produce the damages consequent to
toll imposition.
Let f ( ) be the vector of equilibrium flows on the network, which is function of  toll,
g  f ( ) a measure of the environmental damages consequent to toll imposition, g  f ( )
the monetary value attributed by the local communities to the environmental damages, where
 is a parameter which is known to the local communities, whereas the Public Agency
knows only its probability law.
Let  ( ,  ) the compensation paid by the Public Agency to the local communities: the
paper supposes that it is a quadratic function of  , whereas  is a parameter vector defined
Public Agency. Toll  is function of  , because it is chosen by the local
by the
communities in such a way that utility U consequent to the compensation, i.e. the difference
between compensation  ( ,  ) and the monetary value of damages g  f ( ) , is maximised
under the constraint U  0 .
Since the Public Agency knows only the probability law of  , it chooses the vector  in
such a way that the expected net toll revenue is maximised. Let I be the set of network links
on which a  toll per km is imposed, l i the length of link i  I , f i ( ) the flow on link
i  I , R (  ), f ( )     (  ) f i ( )l i the toll revenue, f  (  ) the probability density
iI
function of  , 1 ,  2  the interval on which f  (  ) is defined. The Public Agency solves
the following problem:
max 

2
1
R (  ), f ( )    ,  (  )f  (  )d
(1)
where  (  ) is implicitly defined as solution of the problem that faces the local communities:
max U  ,  ,  , g  f ( )   ( ,  )  g  f ( )

s.t. U  0
(2)
where vector f ( ) is solution of the equilibrium variational inequality of the network, or of
an equivalent minimisation problem if the cost functions are separable, when the toll per km
on links i  I is  .
Thus we deal with a three-level mathematical programming problem, for which this paper
proposes a solution method based on a polinomial approximation of the equilibrium function
f ( ) . The paper proves that the introduction of the compensation function  ( ,  ) induces
the local communities to reveal the true value of parameter  , while it leads in its turn the
Public Agency to choose a probability law of  which represents its actual knowledge degree
of the monetary value the local communities attribute to environmental damages.
The paper presents an example of application of the model to a real case.
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