Demand monotonicity of a pavement cost function used to determine Aumann-Shapley values in highway cost allocation

Pavement thickness and traffic lanes are two essential requirements affecting the cost of a highway design project. The traffic loadings on a pavement are typically measured in 18-kip equivalent single axle loads (ESALs). In this paper, both ESALs and lanes are treated as two types of players and a pavement cost function is developed to determine the average marginal cost for each type of players. These averages are known as the Aumann-Shapley (A-S) values and are used to allocate the highway cost among all vehicle classes. The proposed pavement cost function is proved to be monotonically increasing as the traffic loadings (ESALs) are increased, a necessary condition for the function to be acceptable for computing Aumann-Shapley values. A severe limitation of the procedure to calculate marginal costs for the traffic-loading players is the extremely large number of permutations since the number of players is enormously high. To overcome this limitation, this article derives a compact form for the discrete A-S values of ESALs and lanes that allows the Aummann-Shapely values to be calculated in a computationally effective manner.