**Models for Highway Cost Allocation**

Alberto Garcia-Diaz and Dong-Ju Lee

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53927

## **1. Introduction**

28 Will-be-set-by-IN-TECH

[20] Zhao, Q., Pan, Z. & Fernandez, E. [2009]. Convergence analysis for reduced-order adaptive controller design of uncertain siso linear systems with noisy output

measurements, *International Journal of Control* 82(11): 1971–1990.

Historically, *equity* has been one of the most important principles applied to formulate tax policy. It has been considered when raising revenues and allocating funds for maintenance, capital improvements, operating programs, and services to the public. The problem of determining how the total cost of a shared facility or service should be divided fairly and rationally is common both in public and private enterprises. The theory of cooperative games is widely used for allocating these costs. Examples of this include but are not limited to public utilities providing telephone services, electricity, water, and transport; public works projects designed to serve different constituencies; access fees or user charges for airports, highways, bridges or waterways; internal accounting rules to allocate overhead costs in private companies [1-4].

The purpose of a *Highway Cost Allocation* (HCA) study is to determine the fair share that each class of road user (vehicle class) should pay for the construction, maintenance, operation, improvement, and related costs of highways, roads, bridges, and streets in a highway network, such as those managed by state Departments of Transportation in the U.S.A. Particular emphasis should be placed on criteria and methods for allocating costs among vehicle classes using a common highway facility (road or bridge, for example) in a just, equitable, fair, and reasonable manner. Cost allocation is ultimately concerned with fairness. Through a comparison of revenues (user fees paid) and cost responsibilities, this study will estimate current equity and recommend alternatives to bring about a closer match between payments and cost responsibilities for each vehicle class.

A significant objective of HCA studies is to analyze highway-related costs attributable to different highway users as a basis for evaluating the equity and efficiency of user charges. Ideally, the costs incurred by the various user groups should be in proportion to the damage they contribute to the highway system. The cost of supporting a highway infrastructure may be deemed fair if there is an equitable distribution of costs and revenues among the various groups of highway users. With this assumption, equity is achieved when each group's

percentage of total assigned costs is equal to the percentage of the revenues contributed by that group. This chapter focuses exclusively on highway cost allocation, specifically the allocation of pavement and bridge costs.

Models for Highway Cost Allocation 137

several state Departments of Transportation. In the **Incremental Method** a highway facility is initially designed to accommodate only the vehicles with lowest axle weight, and then it is sequentially redesigned as the additional vehicle classes are included in increasing order of axle weights. As the process of adding vehicle classes continues, after each inclusion the marginal or incremental cost is charged to the most recently included class. This method satisfies two of the three fundamental properties: completeness and marginality, sometimes marginality but this not guaranteed. Furthermore, this method is not *consistent* because the cost allocated to each vehicle class depends on the *order* in which vehicle classes are included in the analysis. As the name suggests, the **Proportional Method** distributes costs proportionally among vehicle classes according to a specified measure. The cost allocator could be vehicle-miles of travel (VMTs), 18,000 lb. *equivalent single-axle loads* (ESALs), or some other measure. While this procedure may not satisfy marginality and rationality, it

Several non-traditional allocation methods have been developed based on concepts from the theory of cooperative games by Neumann and Morgenstern [7].The application of *nonatomic game theory* to cost allocation was proposed by Castaño-Pardo and Garcia-Diaz [4]. This approach is different from the analysis of the game in which entire vehicle classes are considered as players; instead, each vehicle passage is considered as a player. Such a game obviously has a large number of players, and the decisions of a single player are irrelevant to the total outcome of the game. The value of this non-atomic game is utilized to find the

The **Generalized Method** is based on concepts from the theory of cooperative games [7], and was proposed for conducting highway cost allocation by Villarreal and Garcia-Diaz [8]. The method satisfies completeness, marginality, and rationality because these principles are forcibly satisfied due to constraints in its mathematical formulation. In essence the method guarantees that every vehicle class will be allocated a lower cost in the *grand coalition* (consisting of all vehicle classes), as compared to any other *smaller coalition* (one with fewer vehicle classes than the grand coalition). This method is known in the game theory literature as the *Nucleolus Method*. Its conditions are considered of primary importance in a large number of applications (as in public utility pricing, for example). The solution procedure is actually an application of *Linear Programming* (*LP*). Sometimes the linear programming

solution may not be unique and then there is the need to introduce a tie-breaker rule.

The **Shapley Value** [9] is the average marginal cost for a vehicle class considering all possible permutations of the vehicles in the grand coalition. For example, if there are three vehicle classes, represented by 1, 2, 3, the following permutations are possible: 123, 132, 213, 231, 321, and 312. If we calculate the marginal cost for each vehicle and the compute the average for the six permutations, this average marginal cost is known as the Shapley value. The Shapley value, primarily due to its simplicity and mathematical properties, is one of the most widely studied and used joint cost allocation solution concepts. It represents the

does satisfy the completeness principle.

**2.2. Non-traditional HCA methods** 

solution to the problem of pavement cost allocation.

Highway users are concerned about the fairness of road-use charges and demand that these be allocated equitably among the various vehicle classes occasioning the total cost. Although the word *equity* conveys the general intent of any cost allocation procedure, there are many possible ways to formulate a cost allocation objective to measure equity. In general, there exists no perfect cost allocation method. This is why there is a rich menu of cost allocation methods each intended to reflect the problem-specific logical, historical, political, economic, as well as mathematical analysis.

Costs associated with highway construction, maintenance, and operation can be divided into several categories. Because the impact of different vehicle classes on the costs is different, each of the cost categories should be allocated among the various user groups or vehicle classes in a different manner. These cost categories are:


In addition to this Introduction, this chapter is organized according to six additional sections. Section 2 briefly describes several traditional and non-traditional procedures for highway cost allocation and outlines some important properties of game-theory-based procedures. Section 3 presents a conceptual framework for conducting a highway cost allocation study for a transportation agency, such as a U.S. state Department of Transportation. Section 4 discusses the application of the *nucleolus method* in highway cost allocation combining it with the concept of statistical cost effect to determine a unique solution from multiple optimal solutions. Section 5 describes a new procedure for allocating highway costs having one component due to pavement thickness and another one due to traffic capacity (measured in terms of lanes). Section 6 develops a procedure for bridge cost allocation that integrates both game theory concepts and the traditional incremental approach. Two numerical examples are designed to illustrate the proposed procedures.

### **2. Highway cost allocation procedures and properties**

#### **2.1. Traditional HCA Methods**

During the last three decades, several methods have been developed for the purpose of allocating the total cost of a transportation facility among all the vehicle classes using it. Most procedures that can be used to achieve this goal can be grouped as either *incremental* or *proportional* allocation procedures, or a combination of these two. The proportional and incremental methods have been used by the Federal Highway Administration [5][6] and by several state Departments of Transportation. In the **Incremental Method** a highway facility is initially designed to accommodate only the vehicles with lowest axle weight, and then it is sequentially redesigned as the additional vehicle classes are included in increasing order of axle weights. As the process of adding vehicle classes continues, after each inclusion the marginal or incremental cost is charged to the most recently included class. This method satisfies two of the three fundamental properties: completeness and marginality, sometimes marginality but this not guaranteed. Furthermore, this method is not *consistent* because the cost allocated to each vehicle class depends on the *order* in which vehicle classes are included in the analysis. As the name suggests, the **Proportional Method** distributes costs proportionally among vehicle classes according to a specified measure. The cost allocator could be vehicle-miles of travel (VMTs), 18,000 lb. *equivalent single-axle loads* (ESALs), or some other measure. While this procedure may not satisfy marginality and rationality, it does satisfy the completeness principle.

#### **2.2. Non-traditional HCA methods**

136 Game Theory Relaunched

allocation of pavement and bridge costs.

as well as mathematical analysis.

f. Other highway-related costs.

**2.1. Traditional HCA Methods** 

percentage of total assigned costs is equal to the percentage of the revenues contributed by that group. This chapter focuses exclusively on highway cost allocation, specifically the

Highway users are concerned about the fairness of road-use charges and demand that these be allocated equitably among the various vehicle classes occasioning the total cost. Although the word *equity* conveys the general intent of any cost allocation procedure, there are many possible ways to formulate a cost allocation objective to measure equity. In general, there exists no perfect cost allocation method. This is why there is a rich menu of cost allocation methods each intended to reflect the problem-specific logical, historical, political, economic,

Costs associated with highway construction, maintenance, and operation can be divided into several categories. Because the impact of different vehicle classes on the costs is different, each of the cost categories should be allocated among the various user groups or

In addition to this Introduction, this chapter is organized according to six additional sections. Section 2 briefly describes several traditional and non-traditional procedures for highway cost allocation and outlines some important properties of game-theory-based procedures. Section 3 presents a conceptual framework for conducting a highway cost allocation study for a transportation agency, such as a U.S. state Department of Transportation. Section 4 discusses the application of the *nucleolus method* in highway cost allocation combining it with the concept of statistical cost effect to determine a unique solution from multiple optimal solutions. Section 5 describes a new procedure for allocating highway costs having one component due to pavement thickness and another one due to traffic capacity (measured in terms of lanes). Section 6 develops a procedure for bridge cost allocation that integrates both game theory concepts and the traditional incremental approach. Two numerical examples are designed to illustrate the proposed procedures.

During the last three decades, several methods have been developed for the purpose of allocating the total cost of a transportation facility among all the vehicle classes using it. Most procedures that can be used to achieve this goal can be grouped as either *incremental* or *proportional* allocation procedures, or a combination of these two. The proportional and incremental methods have been used by the Federal Highway Administration [5][6] and by

b. Costs associated with pavement maintenance, rehabilitation, and reconstruction.

d. Costs associated with bridge maintenance, rehabilitation, and reconstruction.

vehicle classes in a different manner. These cost categories are:

**2. Highway cost allocation procedures and properties** 

a. Costs associated with new pavement construction.

c. Costs associated with new bridge construction.

e. Costs associated with system enhancement.

Several non-traditional allocation methods have been developed based on concepts from the theory of cooperative games by Neumann and Morgenstern [7].The application of *nonatomic game theory* to cost allocation was proposed by Castaño-Pardo and Garcia-Diaz [4]. This approach is different from the analysis of the game in which entire vehicle classes are considered as players; instead, each vehicle passage is considered as a player. Such a game obviously has a large number of players, and the decisions of a single player are irrelevant to the total outcome of the game. The value of this non-atomic game is utilized to find the solution to the problem of pavement cost allocation.

The **Generalized Method** is based on concepts from the theory of cooperative games [7], and was proposed for conducting highway cost allocation by Villarreal and Garcia-Diaz [8]. The method satisfies completeness, marginality, and rationality because these principles are forcibly satisfied due to constraints in its mathematical formulation. In essence the method guarantees that every vehicle class will be allocated a lower cost in the *grand coalition* (consisting of all vehicle classes), as compared to any other *smaller coalition* (one with fewer vehicle classes than the grand coalition). This method is known in the game theory literature as the *Nucleolus Method*. Its conditions are considered of primary importance in a large number of applications (as in public utility pricing, for example). The solution procedure is actually an application of *Linear Programming* (*LP*). Sometimes the linear programming solution may not be unique and then there is the need to introduce a tie-breaker rule.

The **Shapley Value** [9] is the average marginal cost for a vehicle class considering all possible permutations of the vehicles in the grand coalition. For example, if there are three vehicle classes, represented by 1, 2, 3, the following permutations are possible: 123, 132, 213, 231, 321, and 312. If we calculate the marginal cost for each vehicle and the compute the average for the six permutations, this average marginal cost is known as the Shapley value. The Shapley value, primarily due to its simplicity and mathematical properties, is one of the most widely studied and used joint cost allocation solution concepts. It represents the average marginal cost contribution each vehicle class *i* would make to the grand coalition if it were to form one vehicle class at a time. Thus the average or expected cost assessment is

$$\mathcal{X}\_{i} = \sum\_{\substack{i \in S \\ S \subseteq N}} \frac{\{\lfloor S \rfloor - 1\}! \{\lfloor N \rfloor - \lfloor S \rfloor\}!}{\lfloor N \rfloor !} \mathcal{C}^{i} \{\mathcal{S}\} \tag{1}$$

Models for Highway Cost Allocation 139

be designed and for which the cost is available. **Marginality** means that each vehicle class should pay at least the incremental cost incurred by including it in the grand coalition. D**emand monotonicity** is a property that implies that the cost-share of a player does not decrease when the player increases its level of demand. **Additivity** means that the allocated costs can be divided into two corresponding components if a cost function can be divided into two distinct and independent cost components. The **dummy** property means that a cost allocation should be equal to zero for a player that does not contribute to any coalition.

Figure 1 outlines a typical framework of a highway cost allocation study for a transportation agency, such as a State Department of Transportation. Instead of directly allocating a total cost at the state level, a more equitable approach is to divide the total cost on the basis of three *classification attributes* known as climatic region, highway system, and highway location. For each of these three attributes several choices must be identified. As an example, a state may be divided into one to four climatic regions depending on the climatic factors affecting pavement performance, the highways may be classified into at least two highway systems to include state and federal highways as a minimum, and the locations may be

classified into at least two major classes to accommodate urban and rural highways.

For any *cost classification*, i.e. one choice of each region, highway system and location, the corresponding total cost to be allocated among vehicle classes is first calculated or estimated by dividing the state total among all classifications according to well-known cost allocators, such as vehicle miles of travel (VMTs) or vehicle loadings measured in terms of *18,000 lb* 

Some of these properties will be further addressed in Sections 4 and 5.

**3. Overview of a highway cost allocation study** 

**Figure 1.** Framework for HCA Study.

where |S| and |N| represent the cardinality of sets *S* and *N*, *Ci* (*S*) represents the marginal cost contribution of *i* relative to *S*, which can readily be computed using *Ci* (*S*) = *C*(*S*)-C(*S-i*) if *i* **S** , and where the sum is computed over all subsets *S* containing vehicle class *i*. For example, for the cost game given by *C(1)=7, C(2)=8, C(3)=8, C(1,2)=10, C(1,3)=10, C(2,3)=15* and *C(1,2,3)=17*, the Shapley value allocation is calculated as shown below:

$$\begin{aligned} \mathbf{x}\_1 &= \frac{0! 2!}{3!} (7 - 0) + \frac{1! 1!}{3!} (10 - 8) + \frac{1! 1!}{3!} (10 - 8) + \frac{2! 10!}{3!} (17 - 15) = \frac{11}{3} = 3.67 \\\\ \mathbf{x}\_2 &= \frac{0! 2!}{3!} (8 - 0) + \frac{1! 1!}{3!} (10 - 7) + \frac{1! 1!}{3!} (15 - 8) + \frac{2! 0!}{3!} (17 - 10) = \frac{20}{3} = 6.67 \\\\ \mathbf{x}\_3 &= \frac{0! 2!}{3!} (8 - 0) + \frac{1! 1!}{3!} (10 - 7) + \frac{1! 1!}{3!} (15 - 8) + \frac{2! 0!}{3!} (17 - 10) = \frac{11}{3} = 6.67 \end{aligned}$$

The **Aumann-Shapley Value** [10,11] is a procedure that considers two types of costs. The first cost is for ESALs (pavement thickness) and the second cost is for highway-lanes (traffic capacity). The total cost allocated to a vehicle class is the sum of these two costs. This procedure allows the consideration of the number of lanes as being variable and depending on the composition of the traffic using a highway. In particular, it addresses two seemingly conflicting objectives: lighter vehicles require less pavement thickness and more lanes while heavier vehicles require fewer lanes but thicker pavements. This method calculates a cost per ESAL and a cost per lane. Then it allocates the number of available lanes among the vehicle classes using the Shapley value (which is the average incremental number of lanes over all possible orderings of the vehicle classes). Since the ESALs are given as data, then the cost allocated to a vehicle class can be calculated as the sum of the *ESALs cost* plus the *lanes cost.*

#### **2.3. Desirable HCA properties**

In order to explain some desirable properties of Highway Cost Allocation (HCA) procedures we will consider a highway facility such as a pavement or a bridge. First, **completeness** is the property that highway costs (construction, rehabilitation, maintenance) are fully paid for by all participating vehicle classes. Second, **rationality** is the property that each vehicle class is guaranteed a lower cost by participating in the *grand coalition* (group consisting of all vehicle classes). The fundamental observation is that if a highway facility is designed for the grand coalition, the cost share of each vehicle class would be smaller than the share paid by the vehicle class in a smaller coalition for which an alternative facility can be designed and for which the cost is available. **Marginality** means that each vehicle class should pay at least the incremental cost incurred by including it in the grand coalition. D**emand monotonicity** is a property that implies that the cost-share of a player does not decrease when the player increases its level of demand. **Additivity** means that the allocated costs can be divided into two corresponding components if a cost function can be divided into two distinct and independent cost components. The **dummy** property means that a cost allocation should be equal to zero for a player that does not contribute to any coalition. Some of these properties will be further addressed in Sections 4 and 5.

#### **3. Overview of a highway cost allocation study**

138 Game Theory Relaunched

1

*x*

2

*x*

3

**2.3. Desirable HCA properties** 

*x*

*cost.*

average marginal cost contribution each vehicle class *i* would make to the grand coalition if it were to form one vehicle class at a time. Thus the average or expected cost assessment is

> *S NS x C S*

if *i* **S** , and where the sum is computed over all subsets *S* containing vehicle class *i*. For example, for the cost game given by *C(1)=7, C(2)=8, C(3)=8, C(1,2)=10, C(1,3)=10, C(2,3)=15*

0!2! 1!1! 1!1! 2!0! 11 7 0 10 8 10 8 17 15 3.67 3! 3! 3! 3! 3

0!2! 1!1! 1!1! 2!0! 20 8 0 10 7 15 8 17 10 6.67 3! 3! 3! 3! 3

0!2! 1!1! 1!1! 2!0! 11 8 0 10 7 15 8 17 10 6.67 3! 3! 3! 3! 3

The **Aumann-Shapley Value** [10,11] is a procedure that considers two types of costs. The first cost is for ESALs (pavement thickness) and the second cost is for highway-lanes (traffic capacity). The total cost allocated to a vehicle class is the sum of these two costs. This procedure allows the consideration of the number of lanes as being variable and depending on the composition of the traffic using a highway. In particular, it addresses two seemingly conflicting objectives: lighter vehicles require less pavement thickness and more lanes while heavier vehicles require fewer lanes but thicker pavements. This method calculates a cost per ESAL and a cost per lane. Then it allocates the number of available lanes among the vehicle classes using the Shapley value (which is the average incremental number of lanes over all possible orderings of the vehicle classes). Since the ESALs are given as data, then the cost allocated to a vehicle class can be calculated as the sum of the *ESALs cost* plus the *lanes* 

In order to explain some desirable properties of Highway Cost Allocation (HCA) procedures we will consider a highway facility such as a pavement or a bridge. First, **completeness** is the property that highway costs (construction, rehabilitation, maintenance) are fully paid for by all participating vehicle classes. Second, **rationality** is the property that each vehicle class is guaranteed a lower cost by participating in the *grand coalition* (group consisting of all vehicle classes). The fundamental observation is that if a highway facility is designed for the grand coalition, the cost share of each vehicle class would be smaller than the share paid by the vehicle class in a smaller coalition for which an alternative facility can

*N*

*i i S S N*

where |S| and |N| represent the cardinality of sets *S* and *N*, *Ci*

cost contribution of *i* relative to *S*, which can readily be computed using *Ci*

and *C(1,2,3)=17*, the Shapley value allocation is calculated as shown below:

(| | 1)! (| | | |)! ( ) | |!

*i*

(1)

(*S*) represents the marginal

(*S*) = *C*(*S*)-C(*S-i*)

Figure 1 outlines a typical framework of a highway cost allocation study for a transportation agency, such as a State Department of Transportation. Instead of directly allocating a total cost at the state level, a more equitable approach is to divide the total cost on the basis of three *classification attributes* known as climatic region, highway system, and highway location. For each of these three attributes several choices must be identified. As an example, a state may be divided into one to four climatic regions depending on the climatic factors affecting pavement performance, the highways may be classified into at least two highway systems to include state and federal highways as a minimum, and the locations may be classified into at least two major classes to accommodate urban and rural highways.

For any *cost classification*, i.e. one choice of each region, highway system and location, the corresponding total cost to be allocated among vehicle classes is first calculated or estimated by dividing the state total among all classifications according to well-known cost allocators, such as vehicle miles of travel (VMTs) or vehicle loadings measured in terms of *18,000 lb* 

**Figure 1.** Framework for HCA Study.

*Equivalent Single-Axle Load* applications (ESALs). To divide the cost for any *cost classification* among vehicle classes, we need to find a cost function, known in game theory as the *characteristic function*, that provides a cost in \$/mile for any specified number of ESALs. The characteristic function can be determined by statistical regression analysis using data on expenditures and traffic volumes extracted from several representative highway projects. The characteristic function allows the use of game-theoretic procedures that require costs estimates for coalitions or groups of vehicle classes. In particular, the Shapley value, Generalized Method, and A-S value methods require the use of a characteristic function.

Models for Highway Cost Allocation 141

For each of the resulting classifications or combinations of climactic region, highway system, and location, at least three (or four) projects are extracted from the database and used to estimate cost relationships (characteristic functions) that can be used to estimate costs for

different levels of ESALs. Typically traffic data available will include the following:

b. Required number of lanes for various combinations of vehicle classes

lifespan. This cost is calculated for the following highway work activities:

for high traffic level roads and thus the cost will be also higher.

trucks, etc.)

information can be used:

a. Vehicle Miles Traveled (VMT)

sealing, chip sealing, etc.

and surface layer.

**4. Generalized method** 

the cost paid by vehicle class *i*

*rationality* property can be formulated as

a. Annual Average Daily Traffic (AADT) and Equivalent Single-Axle Loads (ESALs). b. The distribution of vehicles on the road (proportion of passenger cars, single-axle

In order to generate data for a more detailed level of classification, the following

Since each treated or constructed pavement has a specific service life and all the vehicles traveled in its service life should pay the maintenance or construction cost, the *Equivalent Annual Cost* (*EAC*) of the project in its service life is calculated and used as the cost of that specific project. *EAC* is the cost per year of owning and operating an asset over its entire

1. *Pavement maintenance*: typically both routine and preventive maintenance activities are included in this cost component. Routine maintenance activities are needed to repair cracks of different types, fill pot holes and correct other signs of pavement distress. Preventive maintenance is done mostly applying thin seal coats, micro surfacing, fog

2. *Pavement rehabilitation*: pavement rehabilitation activities include conventional hot mixed asphalt overlay with or without milling. Generally, thicker overlays will be used

3. *Pavement construction*: new pavement construction includes the subgrade, base layer

Let *N* be the set (grand coalition) of all vehicle classes using a highway. Let *C*(*N*) be the cost per mile of this highway (construction, rehabilitation or maintenance). Furthermore, let *Ri*

> *i i N*

*R C(N)*

Now, let us consider a subset (coalition) of vehicle classes, *SN*, and let *C*(*S*) be the cost per mile of a highway designed specifically to accommodate only the vehicle classes in *S*. The

> *R C(S) for all S N <sup>i</sup> i S*

*N*. The *completeness* property can be formulated as

(2)

(3)

#### **3.1. Vehicle classes**

Vehicle classes are viewed as players in a cooperative game. The object of a highway cost location procedure is to fairly divide the construction, rehabilitation or maintenance cost of a transportation facility, such as a highway or a bridge, among these users or players. The following vehicle classes are typically included in highway cost allocation studies:


#### **3.2. Database description**

The database includes the information of traffic levels and costs of relevant pavement maintenance or rehabilitation projects for different data classifications. Typically the database has data for all classifications formed with the following attributes:


For each of the resulting classifications or combinations of climactic region, highway system, and location, at least three (or four) projects are extracted from the database and used to estimate cost relationships (characteristic functions) that can be used to estimate costs for different levels of ESALs. Typically traffic data available will include the following:


In order to generate data for a more detailed level of classification, the following information can be used:

a. Vehicle Miles Traveled (VMT)

140 Game Theory Relaunched

characteristic function.

**3.1. Vehicle classes** 

1. Motorcycles 2. Passenger cars

4. Buses

*Equivalent Single-Axle Load* applications (ESALs). To divide the cost for any *cost classification* among vehicle classes, we need to find a cost function, known in game theory as the *characteristic function*, that provides a cost in \$/mile for any specified number of ESALs. The characteristic function can be determined by statistical regression analysis using data on expenditures and traffic volumes extracted from several representative highway projects. The characteristic function allows the use of game-theoretic procedures that require costs estimates for coalitions or groups of vehicle classes. In particular, the Shapley value, Generalized Method, and A-S value methods require the use of a

Vehicle classes are viewed as players in a cooperative game. The object of a highway cost location procedure is to fairly divide the construction, rehabilitation or maintenance cost of a transportation facility, such as a highway or a bridge, among these users or players. The

The database includes the information of traffic levels and costs of relevant pavement maintenance or rehabilitation projects for different data classifications. Typically the

1. *Climactic Regions.* Since the performance of a pavement is affected by climatic conditions, it is customary to divide a large geographic area into smaller homogeneous

2. *Highway Systems.* In a number of studies two to three highway systems are included when defining the scope of the study. In a number of U.S. states at least Interstate

3. *Highway Locations.* There are two primary types of locations considered in a number of

database has data for all classifications formed with the following attributes:

Highways, US highways, and State highways/roads are included.

following vehicle classes are typically included in highway cost allocation studies:

3. Other Two-Axle, Four-Tire Single Unit Vehicles

5. Two-Axle, Six-Tire, Single-Unit Trucks

7. Four or More Axle Single-Unit Trucks 8. Four or Fewer Axle Single-Trailer Trucks

10. Six or More Axle Single-Trailer Trucks 11. Five or fewer Axle Multi-Trailer Trucks

13. Seven or More Axle Multi-Trailer Trucks

6. Three-Axle Single-Unit Trucks

9. Five-Axle Single-Trailer Trucks

12. Six-Axle Multi-Trailer Trucks

**3.2. Database description** 

climatic regions.

studies: urban and rural areas.

b. Required number of lanes for various combinations of vehicle classes

Since each treated or constructed pavement has a specific service life and all the vehicles traveled in its service life should pay the maintenance or construction cost, the *Equivalent Annual Cost* (*EAC*) of the project in its service life is calculated and used as the cost of that specific project. *EAC* is the cost per year of owning and operating an asset over its entire lifespan. This cost is calculated for the following highway work activities:


### **4. Generalized method**

Let *N* be the set (grand coalition) of all vehicle classes using a highway. Let *C*(*N*) be the cost per mile of this highway (construction, rehabilitation or maintenance). Furthermore, let *Ri* the cost paid by vehicle class *iN*. The *completeness* property can be formulated as

$$\sum\_{i \in N} R\_i = \mathbf{C}(\mathbf{N}) \tag{2}$$

Now, let us consider a subset (coalition) of vehicle classes, *SN*, and let *C*(*S*) be the cost per mile of a highway designed specifically to accommodate only the vehicle classes in *S*. The *rationality* property can be formulated as

$$\sum\_{\substack{i \in S \\ i \in S}} R\_{\substack{j \\ l}} \le \mathbb{C}(S) \quad \text{for all} \quad S \subset N \tag{3}$$

Furthermore, the *marginality* property implies that

$$\sum\_{\substack{i \sum \ R\_j \ge C(N) - C(N-S) \\ i \in S}} R\_{\frac{i}{N}} \ge C(N) - C(N-S) \quad \text{for all} \quad S \subset N \tag{4}$$

It can be proved that if the completeness property (2) is held then the rationality and marginality properties (3) and (4) are equivalent. From (3) it is concluded that the savings enjoyed by a coalition *S* when joining the grand coalition are given by

$$\{C(S) - \sum\_{\vec{i} \in S} R\_{\vec{j}}\} \tag{5}$$

Models for Highway Cost Allocation 143

loadings, measured in ESALs, for a specified design period (typically 20 years). The highway cost per mile should be strictly increasing as the number of ESALs increases. Under this assumption, Constraints (8)-(14) define a feasible region called the *core of the game*  when *t=0.* If *W1* and *W2* are measured in ESALS then the core exists if C1, C2, C3, C12, C13, C23

It can be proved that a typical non-decreasing cost function satisfying (16) is the one represented in Figure 2. In this figure, *W* is the total number of standard loads (ESALs) for the grand coalition and *C(W)* is the cost to be allocated*.* In a number of highway cost allocation studies functions like the one shown in this figure are found using regression analysis from cost data for a set of highway projects available in the database of the study.

Figure 3(a) shows the feasible region for the above formulation when *t = 0.* Figure 3(b) shows the effect of increasing the value of the variable *t.* It is noted in this figure that as the value of *t* increases, the feasible region gets smaller, becoming either a point or a line when *t* reaches its maximum value. A solution represented by one point indicates a unique solution. The line represents infinitely many optimal solutions, a case already indicated in

When the model formulated in (7)-(15) has infinitely many optimal solutions an additional condition must be considered to select a unique solution. The solution procedure can, therefore, be divided into two phases, with the second one needed only to break the tie

12 1 2 *CW W CW CW* (16)

and C123 satisfy the following condition

**Figure 2.** Cost function.

among multiple solutions in the first phase.

Section 2.

To maximize these savings, we maximize *t*, where

$$\mathcal{C}(\mathcal{S}) - \sum\_{\vec{l} \in \mathcal{S}} \mathcal{R}\_{\vec{l}} \ge t^\*$$

which can be rewritten as

$$\sum\_{\substack{i \in S \\ i \in S}} R\_{\frac{i}{2}} \le C(S) \cdot t \tag{6}$$

As an illustration, for *N* = {1,2,3}, the *LP* model for the Generalized Method is formulated in (7)-(15).

$$\text{Maximize } \mathbf{t}$$

Subject to

$$R\_1 \le C\_1 - t \tag{8}$$

$$R\_2 \le \mathcal{C}\_2 - t \tag{9}$$

$$R\_3 \le C\_3 - \ t \tag{10}$$

$$R\_1 + R\_2 \le C\_{12} - t \tag{11}$$

$$R\_1 + R\_3 \le \mathcal{C}\_{13} - t \tag{12}$$

$$R\_2 + R\_3 \le C\_{23} - t \tag{13}$$

$$R\_1 + R\_2 + R\_3 = \mathcal{C}\_{123} \tag{14}$$

$$R\_{1'} \ R\_{2'} \ R\_{3'} \ t \ge 0 \tag{15}$$

Constraints (8)-(10) correspond to highways (pavements) designed to accommodate singlevehicle-class coalitions. Constraints (11)-(13) correspond to two-vehicle-class coalitions. Constraint (14) corresponds to the grand coalition. Each coalition has a level of traffic loadings, measured in ESALs, for a specified design period (typically 20 years). The highway cost per mile should be strictly increasing as the number of ESALs increases. Under this assumption, Constraints (8)-(14) define a feasible region called the *core of the game*  when *t=0.* If *W1* and *W2* are measured in ESALS then the core exists if C1, C2, C3, C12, C13, C23 and C123 satisfy the following condition

$$\mathbb{C}\left(\mathcal{W}\_1 + \mathcal{W}\_2\right) \le \mathbb{C}\left(\mathcal{W}\_1\right) \,+ \,\mathbb{C}\left(\mathcal{W}\_2\right) \tag{16}$$

It can be proved that a typical non-decreasing cost function satisfying (16) is the one represented in Figure 2. In this figure, *W* is the total number of standard loads (ESALs) for the grand coalition and *C(W)* is the cost to be allocated*.* In a number of highway cost allocation studies functions like the one shown in this figure are found using regression analysis from cost data for a set of highway projects available in the database of the study.

**Figure 2.** Cost function.

142 Game Theory Relaunched

Furthermore, the *marginality* property implies that

To maximize these savings, we maximize *t*, where

which can be rewritten as

(7)-(15).

Subject to

enjoyed by a coalition *S* when joining the grand coalition are given by

*R C(N) C(N S) for all S N <sup>i</sup> i S* 

It can be proved that if the completeness property (2) is held then the rationality and marginality properties (3) and (4) are equivalent. From (3) it is concluded that the savings

> *C(S) Ri i S*

*C(S) R t <sup>i</sup> i S* 

> *R C(S) - t <sup>i</sup> i S*

As an illustration, for *N* = {1,2,3}, the *LP* model for the Generalized Method is formulated in

Constraints (8)-(10) correspond to highways (pavements) designed to accommodate singlevehicle-class coalitions. Constraints (11)-(13) correspond to two-vehicle-class coalitions. Constraint (14) corresponds to the grand coalition. Each coalition has a level of traffic

(4)

(5)

(6)

*Maximize t* (7)

1 1 *RC t* – (8)

2 2 *RC t* – (9)

3 3 *RC t* – (10)

1 2 12 *R RC t* – (11)

1 3 13 *R RC t* – (12)

2 3 23 *R RC t* – (13)

1 2 3 123 *RRR C* (14)

<sup>123</sup> *RRRt* , , , 0 (15)

Figure 3(a) shows the feasible region for the above formulation when *t = 0.* Figure 3(b) shows the effect of increasing the value of the variable *t.* It is noted in this figure that as the value of *t* increases, the feasible region gets smaller, becoming either a point or a line when *t* reaches its maximum value. A solution represented by one point indicates a unique solution. The line represents infinitely many optimal solutions, a case already indicated in Section 2.

When the model formulated in (7)-(15) has infinitely many optimal solutions an additional condition must be considered to select a unique solution. The solution procedure can, therefore, be divided into two phases, with the second one needed only to break the tie among multiple solutions in the first phase.

#### **4.1. Phase 1 of generalized method**

$$\text{Maximize } \mathbf{t}$$

Models for Highway Cost Allocation 145

(23)

(27)

(31)

(32)

(21)

(22)

, 0 *R t for all i N <sup>i</sup>* (24)

(25)

(28)

for all *ii ii r L H e iN* (26)

(29)

, 0 *R t for all i N <sup>i</sup>* (30)

A unique solution is obtained from the solution to the *non-linear* model formulated in (21)-

*r e*

*R CN*

*R CS t S N* ( ) - \* for all *<sup>i</sup> i S* 

where *t\** is the optimal value obtained for *t* in Phase 1. The model formulated in (21)-(24) can

*L H*

*R CS t S N* ( ) - \* for all *<sup>i</sup> i S* 

*R CN*

*R r R iN*

; ,

*i i*

*i i*

*r e*

; .

*r e*

0; , *ii i i*

0; ,

*iii i*

*re r e*

 

*erre*

*i N*

*i N*

*i N*

*i N*


() *<sup>i</sup>*

() *i i*

() *<sup>i</sup>*

( ) for all *ii i*

It is noted that by *LP* optimality conditions,

be *linearized* as shown in (25)-(30).

*i N*

*i*

*i*

*H*

*L*

(24).

Minimize

Subject to

Minimize

Subject to

Subject to

$$\sum\_{i \in N} \mathcal{R}\_i = \mathcal{C}(N) \tag{18}$$

$$\sum\_{\substack{j \in S \\ j \in S}} R\_j \le \mathcal{C}(\mathcal{S}) \cdot t \quad \text{for all} \ \mathcal{S} \subset \mathcal{N} \tag{19}$$

$$R\_{i'} \ t \ge 0 \ for \ all \ i \in N \tag{20}$$

#### **4.2. Phase 2 of generalized method**

Villarreal-Cavazos and Garcia-Diaz [13] proposed to break the tie among multiple solutions using the concept of statistical *cost effect* of vehicle classes. This is defined as the difference in average cost between all coalitions *including* a given vehicle class and all coalitions *not including* the class. If *Ei* is the cost effect of vehicle class *i*, the *relative effect* is defined as

$$e\_i = \frac{E\_i}{\sum\_{i \in N} E\_i} \quad \text{for all } i \in N$$

Also, the *relative cost allocated* to vehicle class *i* is defined as

$$r\_i = \frac{R\_i}{\sum\_{\iota \in N} R\_\iota} \quad \text{for all} \quad i \in N$$

A unique solution is obtained from the solution to the *non-linear* model formulated in (21)- (24).

Minimize

144 Game Theory Relaunched

**Figure 3.** Feasible region.

Subject to

**4.1. Phase 1 of generalized method** 

**4.2. Phase 2 of generalized method** 

*Maximize t* (17)

(18)

, 0 *R t for all i N <sup>i</sup>* (20)

(19)

( ) *<sup>i</sup>*

*R CN*

*R CS t S N* ( )- for all *<sup>i</sup> i S* 

Villarreal-Cavazos and Garcia-Diaz [13] proposed to break the tie among multiple solutions using the concept of statistical *cost effect* of vehicle classes. This is defined as the difference in average cost between all coalitions *including* a given vehicle class and all coalitions *not including* the class. If *Ei* is the cost effect of vehicle class *i*, the *relative effect* is defined as

for all

for all

*r i N R*

*e i N E*

 *i*

*R*

*i i N*

 *i*

*E*

*i i N*

*i*

*i*

Also, the *relative cost allocated* to vehicle class *i* is defined as

*i N*

$$\sum\_{i \in N} \|r\_i - e\_i\| \tag{21}$$

Subject to

$$\sum\_{l \in \mathcal{N}} R\_l = \mathcal{C} \{ \mathcal{N} \} \tag{22}$$

$$\sum\_{\substack{i \in S \\ i \in S}} R\_{\frac{i}{2}} \le C(S) \cdot t^\* \quad \text{for all } S \subset N \tag{23}$$

$$R\_{i'} \ t \ge 0 \ for \ all \ i \in N \tag{24}$$

where *t\** is the optimal value obtained for *t* in Phase 1. The model formulated in (21)-(24) can be *linearized* as shown in (25)-(30).

Minimize

$$\sum\_{i \in \mathcal{N}} \left( L\_i + H\_i \right) \tag{25}$$

Subject to

$$r\_i + L\_i - H\_i = e\_i \quad \text{for} \quad \text{all} \quad i \in \mathcal{N} \tag{26}$$

$$\sum\_{\substack{\sum \ R\_j \leq C(S)-t^\bullet \\ i \in S}} R\_j \leq C(S) \cdot t^\bullet \quad \text{for all} \ S \subset N \tag{27}$$

$$\sum\_{l \in N} R\_l = \mathbf{C} \{ \mathbf{N} \} \tag{28}$$

$$(\sum\_{i \in N} R\_i) \times r\_i = R\_i \quad \text{for} \quad \text{all} \quad i \in N \tag{29}$$

$$\mathcal{R}\_{i'} \ t \ge 0 \ for \ all \ i \in \mathcal{N} \tag{30}$$

It is noted that by *LP* optimality conditions,

$$L\_i = \begin{cases} \mathbf{e}\_i - r\_i \; ; \; r\_i < \mathbf{e}\_i, \\ \quad \mathbf{0} \; ; \; r\_i \ge \mathbf{e}\_i, \end{cases} \tag{31}$$

$$H\_i = \begin{cases} 0 \ \vdots \ r\_i \le \mathbf{e}\_i, \\ r\_i - \mathbf{e}\_i \ \vdots \ r\_i > \mathbf{e}\_i. \end{cases} \tag{32}$$

#### **4.3. Statistical cost effects**

A grand coalition consisting of the set of vehicle classes {1,2,3} is considered again to illustrate the calculation of the relative cost effects of the classes. First we regard each vehicle class as a *two-level factor*. The levels can be represented by the signs – and +, where – means that the vehicle class is not in a coalition and + indicates that it is in the coalition. Moreover, the number of level combinations for three two-level factors is equal to 23 = 8. These eight combinations are listed in Table 1. Now, it is noted that combinations 2-8 represent the 7 coalitions that can be formed with the three vehicle classes being considered. The last column in the table shows the highway cost for each coalition. Combination 1 corresponds to an *empty coalition.* Its cost can be viewed as the environmental cost, that is, the cost needed to have a facility able to withstand the impact of climatic conditions alone, not considering the impact of vehicle loadings. In HCA studies this cost can be regarded as a specified fraction of C123.

Models for Highway Cost Allocation 147

and we can formulate the tie-breaking constraints (26) needed in the second phase of the

The proposed approach [12] distributes traffic-related costs in a more fair way than any other method by considering both traffic loads and traffic capacity. Furthermore, the development of a new cost allocation methodology considering allows us to analyze the impact of traffic capacity costs. The two concepts used in the proposed methodology to allocate costs among vehicle classes, according to traffic load and capacity requirements, are known as the Shapley value and the Aumann-Shapley value. In essence the Aumann-Shapley value determines an *average cost per ESAL* and an *average cost per lane* (per mile). The Shapley value allocates the total number of lanes of a highway among the vehicle classes. With these results, it is then possible to calculate costs per mile for each vehicle class by adding the cost due to ESALs (pavement thickness) and the cost due to lanes (capacity).

Two types of players will be considered. **E** = {*1,2,…,q*1} and **L =** {*1,2,…,q*2} are sets of players of type 1 and type 2, respectively. Thus, **M** = E **L** is the set of all players. Now, let *P(***M***)* be the set of all subsets or *coalitions* formed with the elements of **M**. Furthermore, let **N** be the set of natural numbers and **R+** be the set of positive real numbers. Let *C*: *P*(**M**) **R+** be a real-valued cost function known as the *characteristic function*. Finally, let *x*(*q1,q2;C*) be allocated costs yielded by a cost allocation method, *x1*(*q*1*,q*2*;C*) be the cost allocated to player 1 and *x2*(*q*1*,q*2*;C*) be the allocated cost to player 2. With these conventions, four important

If *x*(*q*1*,q*2;*C1*+*C2* ) = *x*(*q*1*,q*2;*C1*) + *x*(*q*1*,q*2;*C2* ), where *C1* and *C2* are non-decreasing cost functions, then the method *x* is called *additive*. If a cost function can be divided into two distinct and independent cost components, then the *allocated costs* can be divided into two corresponding

If *C*(*S*) – *C*(*S*\{*i*}) = 0 for any *i S, i N ,* and *S N*, then *xi*(*q*1*,q*2;*C*) = 0. In this case, the method *x* is called *dummy*. If any player does not contribute to any coalition, then the cost

If *x*1(*q*1*,q*2*;C*) ≥ *x*1(*q*1*-*1*,q*2*;C*), then the cost allocation method *x* is called *demand monotonic* for any *q*1*>2.* Similarly, if *x2*(*q*1*,q*2*;C*) ≥ *x*2(*q*1*,q*2-1*;C*), then the cost allocation method *x* is called

**5. Separation of pavement thickness and traffic capacity costs** 

If *x1*(*q*1*,q*2*;C*) + *x2*(*q*1*,q*2*;C*) = *C*(**M**), then the method *x* is called *complete*.

generalized method.

definitions are given below.

**5.1. Definitions** 

**Definition 1** 

**Definition 2** 

components.

**Definition 3** 

**Definition 4** 

allocated to it is zero.


**Table 1.** Level combinations

The effect of factor Xi, for example, as previously indicated, is the difference in average cost between the coalitions including vehicle class i and those not including it. Based on this definition, the cost effects of the three vehicle classes are obtained as follows using the results shown in Table 1:

$$E\_1 = \frac{C\_1 + C\_{12} + C\_{13} + C\_{123}}{4} - \frac{C\_o + C\_2 + C\_3 + C\_{23}}{4}$$

$$E\_2 = \frac{C\_2 + C\_{12} + C\_{23} + C\_{123}}{4} - \frac{C\_o + C\_1 + C\_3 + C\_{13}}{4}$$

$$E\_3 = \frac{C\_3 + C\_{13} + C\_{23} + C\_{123}}{4} - \frac{C\_o + C\_1 + C\_2 + C\_{12}}{4}$$

Once *E1, E2, …, En* are calculated their values are used to define the relative cost effects

$$e\_i = \frac{E\_i}{\sum\_{i \in N} E\_i} \text{ for all } i \in N$$

and we can formulate the tie-breaking constraints (26) needed in the second phase of the generalized method.

#### **5. Separation of pavement thickness and traffic capacity costs**

The proposed approach [12] distributes traffic-related costs in a more fair way than any other method by considering both traffic loads and traffic capacity. Furthermore, the development of a new cost allocation methodology considering allows us to analyze the impact of traffic capacity costs. The two concepts used in the proposed methodology to allocate costs among vehicle classes, according to traffic load and capacity requirements, are known as the Shapley value and the Aumann-Shapley value. In essence the Aumann-Shapley value determines an *average cost per ESAL* and an *average cost per lane* (per mile). The Shapley value allocates the total number of lanes of a highway among the vehicle classes. With these results, it is then possible to calculate costs per mile for each vehicle class by adding the cost due to ESALs (pavement thickness) and the cost due to lanes (capacity).

Two types of players will be considered. **E** = {*1,2,…,q*1} and **L =** {*1,2,…,q*2} are sets of players of type 1 and type 2, respectively. Thus, **M** = E **L** is the set of all players. Now, let *P(***M***)* be the set of all subsets or *coalitions* formed with the elements of **M**. Furthermore, let **N** be the set of natural numbers and **R+** be the set of positive real numbers. Let *C*: *P*(**M**) **R+** be a real-valued cost function known as the *characteristic function*. Finally, let *x*(*q1,q2;C*) be allocated costs yielded by a cost allocation method, *x1*(*q*1*,q*2*;C*) be the cost allocated to player 1 and *x2*(*q*1*,q*2*;C*) be the allocated cost to player 2. With these conventions, four important definitions are given below.

#### **5.1. Definitions**

#### **Definition 1**

146 Game Theory Relaunched

fraction of C123.

**Table 1.** Level combinations

results shown in Table 1:

**4.3. Statistical cost effects** 

A grand coalition consisting of the set of vehicle classes {1,2,3} is considered again to illustrate the calculation of the relative cost effects of the classes. First we regard each vehicle class as a *two-level factor*. The levels can be represented by the signs – and +, where – means that the vehicle class is not in a coalition and + indicates that it is in the coalition. Moreover, the number of level combinations for three two-level factors is equal to 23 = 8. These eight combinations are listed in Table 1. Now, it is noted that combinations 2-8 represent the 7 coalitions that can be formed with the three vehicle classes being considered. The last column in the table shows the highway cost for each coalition. Combination 1 corresponds to an *empty coalition.* Its cost can be viewed as the environmental cost, that is, the cost needed to have a facility able to withstand the impact of climatic conditions alone, not considering the impact of vehicle loadings. In HCA studies this cost can be regarded as a specified

> Combination X1 X2 X3 Cost 1 - - - Co 2 + - - C1 3 - + - C2 4 + + - C12 5 - - + C3 6 + - + C13 7 - + + C23 8 + + + C123

The effect of factor Xi, for example, as previously indicated, is the difference in average cost between the coalitions including vehicle class i and those not including it. Based on this definition, the cost effects of the three vehicle classes are obtained as follows using the

1 12 13 123 2 3 23

2 12 23 123 1 3 13

3 13 23 123 1 2 12

1 4 4 *CC C C CCCC <sup>o</sup> <sup>E</sup>* 

2 4 4 *CC C C CCCC <sup>o</sup> <sup>E</sup>* 

3 4 4 *CC C C CCCC <sup>o</sup> <sup>E</sup>* 

Once *E1, E2, …, En* are calculated their values are used to define the relative cost effects

*i i N*

*E*

*i*

for all *<sup>i</sup>*

*e i N E* 

If *x1*(*q*1*,q*2*;C*) + *x2*(*q*1*,q*2*;C*) = *C*(**M**), then the method *x* is called *complete*.

#### **Definition 2**

If *x*(*q*1*,q*2;*C1*+*C2* ) = *x*(*q*1*,q*2;*C1*) + *x*(*q*1*,q*2;*C2* ), where *C1* and *C2* are non-decreasing cost functions, then the method *x* is called *additive*. If a cost function can be divided into two distinct and independent cost components, then the *allocated costs* can be divided into two corresponding components.

#### **Definition 3**

If *C*(*S*) – *C*(*S*\{*i*}) = 0 for any *i S, i N ,* and *S N*, then *xi*(*q*1*,q*2;*C*) = 0. In this case, the method *x* is called *dummy*. If any player does not contribute to any coalition, then the cost allocated to it is zero.

#### **Definition 4**

If *x*1(*q*1*,q*2*;C*) ≥ *x*1(*q*1*-*1*,q*2*;C*), then the cost allocation method *x* is called *demand monotonic* for any *q*1*>2.* Similarly, if *x2*(*q*1*,q*2*;C*) ≥ *x*2(*q*1*,q*2-1*;C*), then the cost allocation method *x* is called *demand monotonic* for any *q*2*>2.* The cost-share of a player should not decrease when the player increases its demand.

Friedman [13] shows that the A-S value is *complete*, *additive*, and *dummy.* In addition, Friedman and Moulin [14] show that the A-S value does not satisfy the demand monotonicity property for general non-decreasing cost functions. Lee & Garcia-Diaz [15] show that demand monotonicity will be held in the following cases.

#### **5.2. Pavement and capacity costs allocation**

Assume the *log concave* cost function formulated in (33):

$$C(e,l) = l\left(a+b\ e^{r}\right) \tag{33}$$

Models for Highway Cost Allocation 149

(35)

(37)

(38)

1 2

*<sup>q</sup>* (36)

12 12

12 1 2

(39)

[0, ] 0

*q q*

*q q*

*t t*

*<sup>e</sup> t t*

1 2 12 11 22

*q q tt q t q t*

0 1 1 2 12 1 1 2 2

1 2 12 11 22

*qq t t qtqt*

2

*t*

0 1 1 2 1 2 11 22

*t q t*

*i*

calculated as follows, where i = e or l:

and (38).

*l*

*Step 2. Lane assignment* 

…, *n*

  1

*j i j i i j*

> *<sup>Q</sup> <sup>T</sup> T*

*q q t t*

( ) <sup>1</sup>

*i*

(; ) [ ( ) ( , ,..., 1,..., )].

*i i i m*

The cost per ESAL and the cost per lane are calculated by averaging since all the players of the same type are identical. Thus, the cost per ESAL (*Ce*) and the cost per lane (*Cl*) can be

*x qC <sup>C</sup>*

There are two types of players, namely, ESALs and lanes. Furthermore, let *q*1 be the total number of players for ESALs, and *q2* be the total number of players for lanes. Then, the cost per lane and the cost per ESAL can be calculated from Redekop's formula as shown in (37)

(; ) *<sup>i</sup>*

*i*

!( 1)! ( 1)! ( )!{ ( , ) ( , 1)} ( )! !( 1)! ( )!( )!

( 1)! ! ( 1)! ( )!{ ( , ) ( 1, )} ( )! ( 1)! ! ( )!( )!

1

1

(40)

1

*t*

0 1 (, ) ( ,1) <sup>1</sup> *q*

*q q tt qtqt <sup>C</sup> Ct t Ct t*

If the cost increment remains the same when *t1* (or *t2*) is fixed and *t2* (or *t1*) is increased by 1 the A-S value can be determined using the simplified compact form formulated in (39).

2

Since the A-S value satisfies the *completeness property*, the sum of costs for traffic capacity and traffic load for the grand coalition equals the total cost for that coalition. The sum of ESALs over all vehicle classes is equal to the number of ESALs for the grand coalition (*q*1), but the sum of the lanes required for each vehicle class is greater than or equal to the lanes required for the grand coalition (*q*2). Hence, to calculate cost responsibilities for each vehicle class, the number of lanes for the grand coalition should be assigned to the vehicle classes. The Shapley value will be used to determine the number of lanes assigned to vehicle class *i* (*Li*). The *i*th Shapley value for *n* players is determined using (1), with *i =* 1,

1 :

*n*

*<sup>i</sup> <sup>s</sup> S Ni S*

*n*


*s ns <sup>L</sup> FS FS i*

*S s*

( 1)!( )! () ( ) !

*<sup>q</sup> x qC Ct q* 

*qq t t q t q t <sup>C</sup> Ct t Ct t*

*x qC q Ct Ct t t t*

where *e* is the number of *ESAL*s, *l*  **N** is the number of lanes, *C*(*e,l*) is the cost in dollars per lane-mile, and *a*, *b,* and *r* are non-negative parameters. For this function the following results can be proved:


In this chapter we use a *compact form* developed for the discrete A-S value [15]. This compact form allows the use of the A-S value in realistic applications with a large number of players, where the computational work becomes excessive without using the form. This section states some fundamental results regarding the demand monotonicity of the log concave characteristic function. The proposed approach [12] is composed of the following three steps.

#### *Step 1. Traffic-related pavement cost separation*

To separate traffic-related pavement costs into the costs for traffic load and the costs for traffic capacity, the discrete A-S value is used. Suppose that there are *m* types of players and *qi* players of a type *i*. Further, let

$$Q = \sum\_{i} q\_{i'} \ T = \sum\_{i} t\_{i'} \ T' = \sum\_{i} t'\_{i'}$$

and

$$t'\_{\ i} = |q\_i - t\_i|.$$

There are two formulas for the discrete A-S value. A formula by Moulin [11] is shown in (34), where *i =* 1,…, *m*:

$$\text{tr}\_i(q; \mathbb{C}) = \frac{q\_1! \dots q\_m!}{Q} \sum\_{t \in \{0, \rho\}} \frac{T!}{t\_1! \dots t\_m!} \frac{T'!}{t'\_1! \dots t'\_m!} (\frac{t\_i}{T} - \frac{t'\_i}{T'}) \text{ C(t)}.\tag{34}$$

Another formula by Redekop [16] is given in (35):

$$\mathbf{x}\_{i}(q; \mathbf{C}) = \sum\_{\substack{t \in [0, q] \\ t\_{i} > 0}} q\_{i} \times \frac{\begin{pmatrix} q\_{i} - 1 \\ t\_{i} - 1 \end{pmatrix} \text{(II)} \begin{pmatrix} q\_{j} \\ t\_{j} \end{pmatrix}}{T \begin{pmatrix} \mathcal{Q} \\ T \end{pmatrix}} [\mathbf{C}(t) - \mathbf{C}(t\_{1}, t\_{2}, \dots, t\_{i} - 1, \dots, t\_{m})].\tag{35}$$

The cost per ESAL and the cost per lane are calculated by averaging since all the players of the same type are identical. Thus, the cost per ESAL (*Ce*) and the cost per lane (*Cl*) can be calculated as follows, where i = e or l:

$$\mathbf{C}\_{i} = \frac{\mathbf{x}\_{i} \{q; \mathbf{C}\}}{q\_{i}} \tag{36}$$

There are two types of players, namely, ESALs and lanes. Furthermore, let *q*1 be the total number of players for ESALs, and *q2* be the total number of players for lanes. Then, the cost per lane and the cost per ESAL can be calculated from Redekop's formula as shown in (37) and (38).

$$\mathbf{C}\_{l} = \frac{q\_{1}!(q\_{2}-1)!}{(q\_{1}+q\_{2})!} \sum\_{t\_{1}=0}^{q\_{1}} \frac{(t\_{1}+t\_{2}-1)!}{t\_{1}!(t\_{2}-1)!} \frac{(q\_{1}-t\_{1}+q\_{2}-t\_{2})!}{(q\_{1}-t\_{1})!(q\_{2}-t\_{2})!} [\mathbf{C}(t\_{1},t\_{2}) - \mathbf{C}(t\_{1},t\_{2}-1)] \tag{37}$$

$$\mathbf{C}\_{\varepsilon} = \frac{(q\_1 - 1)! \, q\_2!}{(q\_1 + q\_2)!} \sum\_{t\_2 = 0}^{q\_1} \sum\_{t\_1 = 1}^{q\_2} \frac{(t\_1 + t\_2 - 1)!}{(t\_1 - 1)! t\_2!} \frac{(q\_1 - t\_1 + q\_2 - t\_2)!}{(q\_1 - t\_1)! (q\_2 - t\_2)!} \mathrm{[C(t\_1, t\_2) - C(t\_1 - 1, t\_2)]} \tag{38}$$

If the cost increment remains the same when *t1* (or *t2*) is fixed and *t2* (or *t1*) is increased by 1 the A-S value can be determined using the simplified compact form formulated in (39).

$$\text{tr}\_{t\_2}(q, \mathbf{C}) = \frac{q\_2}{q\_1 + 1} \sum\_{t\_1=0}^{q\_1} \mathbf{C}(t\_1, 1) \tag{39}$$

#### *Step 2. Lane assignment*

148 Game Theory Relaunched

player increases its demand.

results can be proved:

steps.

and

*demand monotonic* for any *q*2*>2.* The cost-share of a player should not decrease when the

Friedman [13] shows that the A-S value is *complete*, *additive*, and *dummy.* In addition, Friedman and Moulin [14] show that the A-S value does not satisfy the demand monotonicity property for general non-decreasing cost functions. Lee & Garcia-Diaz [15]

where *e* is the number of *ESAL*s, *l*  **N** is the number of lanes, *C*(*e,l*) is the cost in dollars per lane-mile, and *a*, *b,* and *r* are non-negative parameters. For this function the following

In this chapter we use a *compact form* developed for the discrete A-S value [15]. This compact form allows the use of the A-S value in realistic applications with a large number of players, where the computational work becomes excessive without using the form. This section states some fundamental results regarding the demand monotonicity of the log concave characteristic function. The proposed approach [12] is composed of the following three

To separate traffic-related pavement costs into the costs for traffic load and the costs for traffic capacity, the discrete A-S value is used. Suppose that there are *m* types of players and

> ' – . *i ii t qt*

There are two formulas for the discrete A-S value. A formula by Moulin [11] is shown in

[0, ] 1 1 !... ! ! '! ' (; ) ( ) ( ). !... ! ' !... ' ! ' *m i i*

*t q m m q q T T t t x qC C t Q t tt t TT*

(34)

, ,' ' *ii i ii i Q qT tT t*

( ,) *<sup>r</sup> C e l l a be* (33)

show that demand monotonicity will be held in the following cases.

**5.2. Pavement and capacity costs allocation** 

Assume the *log concave* cost function formulated in (33):

a. Demand monotonicity for the number of lanes.

*Step 1. Traffic-related pavement cost separation* 

*qi* players of a type *i*. Further, let

(34), where *i =* 1,…, *m*:

b. Demand monotonicity for the number of *ESAL*s *r* 0.32 .

1

*i*

Another formula by Redekop [16] is given in (35):

Since the A-S value satisfies the *completeness property*, the sum of costs for traffic capacity and traffic load for the grand coalition equals the total cost for that coalition. The sum of ESALs over all vehicle classes is equal to the number of ESALs for the grand coalition (*q*1), but the sum of the lanes required for each vehicle class is greater than or equal to the lanes required for the grand coalition (*q*2). Hence, to calculate cost responsibilities for each vehicle class, the number of lanes for the grand coalition should be assigned to the vehicle classes. The Shapley value will be used to determine the number of lanes assigned to vehicle class *i* (*Li*). The *i*th Shapley value for *n* players is determined using (1), with *i =* 1, …, *n*

$$L\_{i} = \sum\_{s=1}^{n} \frac{(s-1)!(n-s)!}{n!} \sum\_{\substack{S \subseteq N: i \in S \\ \|S\| = s}} \left( F(S) - F(S - i) \right) \tag{40}$$

#### *Step 3. Cost allocation*

Costs are allocated to each vehicle class in proportion to the number of ESALs and the number of lanes, that is

$$\mathbf{x}\_i \begin{pmatrix} E\_{i'} \ L\_i \end{pmatrix} = \begin{array}{c} E\_i \mathbf{C}\_e \ + L\_i \mathbf{C}\_l \end{array} \tag{41}$$

Models for Highway Cost Allocation 151

**Sequences** Inclusion Sequences

**Table 3.** All possible inclusion sequences for the A-S value

**Table 4.** All possible inclusion sequences for the Shapley value

The Shapley values for the three vehicle classes are:

**1** L E E E E L L 2 L E E E L E L 3 L E E E L L E 4 L E E L E E L 5 L E E L E L E 6 L E E L L E E 7 L E L E E E L 8 L E L E E L E 9 L E L E L E E 10 L E L L E E E 11 L L E E E L E 12 L L E E E E L 13 L L E E L E E 14 L L E L E E E 15 L L L E E E E

Sequences Including sequences Marginal number of lanes **1** A P T 1 1 0 2 A T P 1 1 0 3 P A T 1 1 0 4 P T A 1 0 1 5 T A P 0 2 0 6 T P A 0 1 1

1

1

1

the base lane cost has been allocated proportionally according to ESALs.

(1 1 1 1 2 1) 1.67

(1 0 1 1 0 1) 0.67

(0 1 0 0 0 0) 0.16 <sup>6</sup> *TL* .

<sup>6</sup> *AL*

<sup>6</sup> *<sup>P</sup> <sup>L</sup>*

The cost for the base lane is 2. This cost may be allocated proportionally by ESALs or, perhaps more appropriately, by vehicle miles of travel (VMT), since this cost is a non-loadrelated cost. Cost responsibilities for the three vehicle classes are shown in Table 5, where

where

*xi(Ei ,Li)* : Cost allocated to vehicle class i *Ei* : ESALs for vehicle class i *Ce*: Cost per ESAL *Li* : Number of lanes assigned to vehicle class i *Cl*: Cost per lane

#### **5.3. An example**

column is for the *base lane*.

The proposed approach is now illustrated using a simple example. Suppose that there are 3 vehicles: two automobiles (A), one pickup truck (P), and one 5-axle-trailer truck (T). Furthermore, there is 1 *base lane*, 2 additional lanes, and a total of 4 ESALs. These loads are divided into 1 ESAL for two automobiles, 1 ESALs for one pickup truck, and 2 ESALs for one 5 axle-trailer truck. The numbers of additional lanes required by each vehicle coalition are in shown in Table 2.


**Table 2.** Number of additional lanes required by each vehicle coalition

The cost in \$/mile as a function of the number of ESALs and the number of lanes is assumed to be*Cel l e* ( ,) 2 3 . To calculate the A-S value for cost per ESAL (*Ce*) and cost per lane (*Cl*), Table 3 will be used. All possible 6!/2!4! = 15 inclusion sequences are shown in this table, where an E stands for one unit of ESALs and an L for one unit of lanes. The gray-colored

A *base lane* is first included in any possible sequence, and then either E or L is included. The average marginal costs, *Ce* and *Cl*, for including E or L in each sequence can be calculated from Table 3. The A-S values (*Ce* and *Cl*) can be also calculated by using the formulas shown in Step 1. The calculated values for *Ce* and *Cl* are 2.66 and 5.68, respectively.

To calculate number of lanes assigned to each vehicle class by the Shapley value, we first determine the total number of possible sequences as 3! = 6. The average marginal number of lanes, *Li*, for including A, P, or T in each sequence is calculated from Table 4. The Shapley value for *Li* can be also calculated by using formulas shown in Step 2.


**Table 3.** All possible inclusion sequences for the A-S value

where

*Step 3. Cost allocation* 

number of lanes, that is

*Ei* : ESALs for vehicle class i

*Ce*: Cost per ESAL

*Cl*: Cost per lane

**5.3. An example** 

are in shown in Table 2.

column is for the *base lane*.

*xi(Ei ,Li)* : Cost allocated to vehicle class i

*Li* : Number of lanes assigned to vehicle class i

Costs are allocated to each vehicle class in proportion to the number of ESALs and the

The proposed approach is now illustrated using a simple example. Suppose that there are 3 vehicles: two automobiles (A), one pickup truck (P), and one 5-axle-trailer truck (T). Furthermore, there is 1 *base lane*, 2 additional lanes, and a total of 4 ESALs. These loads are divided into 1 ESAL for two automobiles, 1 ESALs for one pickup truck, and 2 ESALs for one 5 axle-trailer truck. The numbers of additional lanes required by each vehicle coalition

**COALITION** {A} {P} {T} {A,P} {A,T} {P,T} {A,P,T}

Number of additional lanes 1 1 0 2 2 1 2

The cost in \$/mile as a function of the number of ESALs and the number of lanes is assumed to be*Cel l e* ( ,) 2 3 . To calculate the A-S value for cost per ESAL (*Ce*) and cost per lane (*Cl*), Table 3 will be used. All possible 6!/2!4! = 15 inclusion sequences are shown in this table, where an E stands for one unit of ESALs and an L for one unit of lanes. The gray-colored

A *base lane* is first included in any possible sequence, and then either E or L is included. The average marginal costs, *Ce* and *Cl*, for including E or L in each sequence can be calculated from Table 3. The A-S values (*Ce* and *Cl*) can be also calculated by using the formulas shown

To calculate number of lanes assigned to each vehicle class by the Shapley value, we first determine the total number of possible sequences as 3! = 6. The average marginal number of lanes, *Li*, for including A, P, or T in each sequence is calculated from Table 4. The Shapley

in Step 1. The calculated values for *Ce* and *Cl* are 2.66 and 5.68, respectively.

value for *Li* can be also calculated by using formulas shown in Step 2.

**Table 2.** Number of additional lanes required by each vehicle coalition

, *i i i ie il x E L EC LC* (41)


**Table 4.** All possible inclusion sequences for the Shapley value

The Shapley values for the three vehicle classes are:

$$\begin{aligned} L\_A &= \frac{1}{6}(1+1+1+1+2+1) = 1.67 \\\\ L\_P &= \frac{1}{6}(1+0+1+1+0+1) = 0.67 \\\\ L\_T &= \frac{1}{6}(0+1+0+0+0+0) = 0.16 \end{aligned}$$

The cost for the base lane is 2. This cost may be allocated proportionally by ESALs or, perhaps more appropriately, by vehicle miles of travel (VMT), since this cost is a non-loadrelated cost. Cost responsibilities for the three vehicle classes are shown in Table 5, where the base lane cost has been allocated proportionally according to ESALs.


Models for Highway Cost Allocation 153

*Step 1. Traffic-related pavement cost separation* 

functions for this example are formulated below:

The following results are obtained in each step.

*Step 1. Bridge construction cost separation* 

*Step 2. Traffic-load cost allocation* 

(ADT).

groups.

*Step 3. Lane assignment* 

*Step 4. Cost allocation* 

**6.2. An example** 

This step is identical Step 1 in the methodology described in Section 5 of this chapter.

The cost per unit of weight (*Ce*) was obtained in *Step 1*. The traffic-load cost can be allocated to each weight group in vehicle class by using the incremental method, as indicated below:

a. The lightest vehicle group is first considered. The unit of weight (*Ce*) is allocated to each vehicle class in this group and all heavier groups according to average daily traffic

b. The next light group is considered. The marginal cost equal to *Ce* is allocated to each

d. If a vehicle class *i* has several weight groups, then sum up the cost for those weight

Again, this procedure is identical to the Step 2 of the methodology described in Section 5.

This procedure is also identical to the Step 3 of the methodology described in Section 5.

A simple hypothetical numerical example is presented in this section to illustrate and clarify the application of the proposed method. It is assumed that there are 3 vehicles: automobile {A}, pickup truck {P}, and 5-ax-trailer truck {T}. Also, it is assumed that 1 *base* lane is required. The number of *additional* lanes is the same in Table 1. The total vehicle weight is distributed along four intervals: 0-10 kips, 11-20 kips, and 21-30 kips. The percentages of total ADT due to vehicles of each class, for the given weight intervals, are: {A} belongs to the 0-10 kip interval with 65 % of ADT; {P} belongs to the 0-10 kip interval with 20 % of ADT and to the 11-20 kip interval with 5 percent of ADT; {T} belongs to the11-20 kips interval with 5 percent of ADT and to the 21-30 kip interval with 5 percent of ADT. The cost

*Ckl l k l* ( , ) (1 2 ) 1

*Ckl l k l* ( , ) (2 3 ) 2

To calculate the A-S value for the cost per unit of weight (10 kips in this example) and the cost per lane the sequences shown in Table 7 can be used. It is noted that the total number of sequences is 5!/(3!2!) = 10. In Table 7 letter K represents one unit of weight (10 kips) and

vehicle class in this group and all heavier groups according to ADT. c. If the heaviest group is considered, then go to d. Otherwise, continue to b.

**Table 5.** Cost responsibility calculation for each vehicle class

## **6. Separation of bridge construction and traffic capacity costs**

A cost function is needed to estimate the bridge construction cost for the gross vehicle weight associated with any coalition of vehicle classes. This cost function can be developed by determining the cost of the bridge required by a coalition as a percentage of the cost of a *baseline bridge.* To accommodate all possible coalitions, the range of gross vehicle weight can be divided into an adequate number of intervals or categories. Results for nine categories of gross weight ranging from 5,000 lb to 108,000 lb are shown in Table 6. This table was built using a study by Moses [17] and the 1997 Federal Highway Cost allocation Study [6]. The table shows the required bridge cost for each gross vehicle weight category as a percentage of the cost of a baseline HS20 bridge which has a weight carrying capacity of 72,000 lb. The results for each gross vehicle weight category are the coordinates of one point of the bridge cost function.


**Table 6.** Bridge cost percentages considering a baseline HS20 bridge

#### **6.1. Bridge cost allocation procedure**

The proposed model for the relationship between cost per lane-mile and the gross vehicle weight to be applied is formulated as

$$Y = l\_{\iota} \ (a\_{\iota} + b\_{\iota} X) \tag{42}$$

where *Y* is the cost in dollars per lane-mile, *li* is the number of lanes of bridge type *i*, *X* is the gross vehicle weight in kips, and *ai* and *bi* are known parameters (to be estimated using regression analysis). Depending on the required number of lanes, more than one cost function can be formulated to determine accurate bridge construction cost estimates. A short-span structured bridge may be proper for a bridge with one lane in each direction, while a longer-span structured bridge may be so for a bridge with more lanes

The bridge construction cost allocation procedure is outlined below [18]. The procedure is essentially the same one developed in Section 5. In the case of bridges, however, there is an additional step (referred to as Step 2 below) to apply the incremental method of highway cost allocation.

#### *Step 1. Traffic-related pavement cost separation*

This step is identical Step 1 in the methodology described in Section 5 of this chapter.

#### *Step 2. Traffic-load cost allocation*

152 Game Theory Relaunched

cost function.

Gross Vehicle

Bridge Cost

cost allocation.

Vehicle Classes Load costs

(*EiCe)*

**Table 5.** Cost responsibility calculation for each vehicle class

**Table 6.** Bridge cost percentages considering a baseline HS20 bridge

**6.1. Bridge cost allocation procedure** 

weight to be applied is formulated as

Capacity costs (*LiCl*)

**6. Separation of bridge construction and traffic capacity costs** 

Automobile 12.66 1.175.68 20.50 10.30 Pickup truck 12.66 0.675.68 20.25 6.97 5-ax-trailer truck 22.66 0.165.68 20.25 6.73

A cost function is needed to estimate the bridge construction cost for the gross vehicle weight associated with any coalition of vehicle classes. This cost function can be developed by determining the cost of the bridge required by a coalition as a percentage of the cost of a *baseline bridge.* To accommodate all possible coalitions, the range of gross vehicle weight can be divided into an adequate number of intervals or categories. Results for nine categories of gross weight ranging from 5,000 lb to 108,000 lb are shown in Table 6. This table was built using a study by Moses [17] and the 1997 Federal Highway Cost allocation Study [6]. The table shows the required bridge cost for each gross vehicle weight category as a percentage of the cost of a baseline HS20 bridge which has a weight carrying capacity of 72,000 lb. The results for each gross vehicle weight category are the coordinates of one point of the bridge

Weight (1000 lb) 5 10 20 30 40 54 **72** 90 108

Percentage 80.78 82.61 86.52 90.43 95.80 94.59 **100** 105 110

The proposed model for the relationship between cost per lane-mile and the gross vehicle

where *Y* is the cost in dollars per lane-mile, *li* is the number of lanes of bridge type *i*, *X* is the gross vehicle weight in kips, and *ai* and *bi* are known parameters (to be estimated using regression analysis). Depending on the required number of lanes, more than one cost function can be formulated to determine accurate bridge construction cost estimates. A short-span structured bridge may be proper for a bridge with one lane in each direction,

The bridge construction cost allocation procedure is outlined below [18]. The procedure is essentially the same one developed in Section 5. In the case of bridges, however, there is an additional step (referred to as Step 2 below) to apply the incremental method of highway

while a longer-span structured bridge may be so for a bridge with more lanes

*Y l* (*a b X*) *<sup>i</sup> <sup>i</sup> <sup>i</sup>* (42)

Costs for base lane (proportional)

Cost responsibilities

> The cost per unit of weight (*Ce*) was obtained in *Step 1*. The traffic-load cost can be allocated to each weight group in vehicle class by using the incremental method, as indicated below:


#### *Step 3. Lane assignment*

Again, this procedure is identical to the Step 2 of the methodology described in Section 5.

#### *Step 4. Cost allocation*

This procedure is also identical to the Step 3 of the methodology described in Section 5.

#### **6.2. An example**

A simple hypothetical numerical example is presented in this section to illustrate and clarify the application of the proposed method. It is assumed that there are 3 vehicles: automobile {A}, pickup truck {P}, and 5-ax-trailer truck {T}. Also, it is assumed that 1 *base* lane is required. The number of *additional* lanes is the same in Table 1. The total vehicle weight is distributed along four intervals: 0-10 kips, 11-20 kips, and 21-30 kips. The percentages of total ADT due to vehicles of each class, for the given weight intervals, are: {A} belongs to the 0-10 kip interval with 65 % of ADT; {P} belongs to the 0-10 kip interval with 20 % of ADT and to the 11-20 kip interval with 5 percent of ADT; {T} belongs to the11-20 kips interval with 5 percent of ADT and to the 21-30 kip interval with 5 percent of ADT. The cost functions for this example are formulated below:

$$\mathcal{C}(k,l) = l(1+2k) \quad l=1$$

$$\mathcal{C}(k,l) = l(2+3k) \quad l \ge 2$$

The following results are obtained in each step.

#### *Step 1. Bridge construction cost separation*

To calculate the A-S value for the cost per unit of weight (10 kips in this example) and the cost per lane the sequences shown in Table 7 can be used. It is noted that the total number of sequences is 5!/(3!2!) = 10. In Table 7 letter K represents one unit of weight (10 kips) and letter L represents one unit of lanes. A gray-shaded column is used for the base lane. A base lane is first included in any possible sequence, and then either a K or an L is included. The average marginal costs (or the A-S value) *Ck* and *Cl* can be calculated by using Table 7. The calculated values are *Ck* = 170/30 = 5.67 and *Cl* = 150/20 = 7.5.

Models for Highway Cost Allocation 155

**Author details** 

Alberto Garcia-Diaz

Dong-Ju Lee

**7. References** 

NY, pp. 3-29.

University Press.

Press, Princeton, NJ.

Review; 46: 303-332.

Texas A&M University.

Economic Theory. 87: 275-312.

658.

*Working Paper.*

*Department of Industrial & Information Engineering, The University of Tennessee, USA* 

*Department of Industrial & Systems Engineering, Kongju National University, South Korea* 

[1] Young H P (1985) Methods and Principles of Cost Allocation In: Young, H P (Eds.), Cost Allocation: Methods, Principles, Applications. Elsevier Science Ltd., North Holland,

[2] Dror M (1990) Cost Allocation: The Traveling Salesman, Binpacking, and the Knapsack.

[3] Hartman, B C, Dror M (1996) Cost Allocation in Continuous-Review Inventory Models.

[4] Castano-Pardo A, Garcia-Diaz A (1995) Highway Cost Allocation: An Application of the Theory of Nonatomic Games. Transportation Research Part A 29: 187-203. [5] Federal Highway Administration (1982) Final Report on the Federal Highway Cost

[6] Federal Highway Administration (U.S. Department of Transportation) (1997) 1997

[7] Neumann J, Morgenstern O (1944) Theory of Games and Economic Behavior, Princeton

[8] Villarreal-Cavazos A, Garcia-Diaz A (1985) Development and Application of New Highway Cost Allocation Procedures. Transportation Research Record 1009: 34-41. [9] Shapley L S (1953) A Value for n-person Games. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contributions to the Theory of Games, Vol. II, Annual Mathematics Study 28. Princeton

[10] Aumann R J, Shapley L S (1974) Values of Non-Atomic Games. Princeton University

[11] Moulin H (1995) On Additive Methods to Share Joint Costs. The Japanese Economic

[12] Lee D (2002) Game-Theoretic Procedures for Determining Pavement Thickness and Traffic Lane Costs in Highway Cost Allocation. Ph. D. Dissertation. College Station, TX:

[13] Friedman E J (2004) Strong Monotonicity in Surplus Sharing. Economic theory. 23: 643-

[14] Friedman E, Moulin H (1999) Three Methods to Share Joint Costs or Surplus. Journal of

[15] Lee D, Garcia-Diaz A, Lee C (2012) "Demand Monotonicity Analysis of the Discrete Aumann-Shapley Value Based on a Compact Form Used in Highway Cost Allocation"

Allocation Study, U.S. Department of Transportation, Washington, D.C.

Applied Mathematics and Computation 35: 191-207.

Federal Highway Cost Allocation Study Final Report.

University Press, Princeton, NJ, pp. 307-317.

Naval Research Logistics 43: 549-561.


**Table 7.** Sequences and marginal cost for calculation of A-S value

#### *Step 2. Traffic-load cost allocation:*

$$E\_T: \quad 5.67 \times \frac{5+5}{65+20+5+5+5} + 5.67 \times \frac{5+5}{5+5+5} + 5.67 \times \frac{5}{5} = 10.5$$

$$E\_A: \quad 5.67 \times \frac{65}{65+20+5+5+5} = 3.7$$

$$E\_P: \quad 5.67 \times \frac{20+5}{65+20+5+5+5} + 5.67 \times \frac{5}{5+5+5} = 3.3$$

#### *Step 3. Lane assignment:*

See Table 3. *LA*=1.17, *LP*=0.67, *LT*=0.16

#### *Step 4. Cost allocation:*

The value (cost) of parameter *a* for the base lane is 2. This cost is allocated proportionally by ADTs in this example. The total cost allocations for the three vehicle classes are shown in Table 8.


**Table 8.** Cost allocations for vehicle classes

## **Author details**

#### Alberto Garcia-Diaz

*Department of Industrial & Information Engineering, The University of Tennessee, USA* 

Dong-Ju Lee

154 Game Theory Relaunched

letter L represents one unit of lanes. A gray-shaded column is used for the base lane. A base lane is first included in any possible sequence, and then either a K or an L is included. The average marginal costs (or the A-S value) *Ck* and *Cl* can be calculated by using Table 7. The

1 L K K K L L 1 2 2 2 15 11 2 L K K L K L 1 2 2 11 6 11 3 L K K L L K 1 2 2 11 8 9 4 L K L K K L 1 2 7 6 6 11 5 L K L K L K 1 2 7 6 8 9 6 L K L L K K 1 2 7 5 9 9 7 L L K K K L 1 3 6 6 6 11 8 L L K K L K 1 3 6 6 8 9 9 L L K L K K 1 3 6 5 9 9 10 L L L K K K 1 3 2 9 9 9

> 5 5 55 5 : 5.67 5.67 5.67 10 65 20 5 5 5 5 5 5 5 *<sup>T</sup> <sup>E</sup>*

> > <sup>65</sup> : 5.67 3.7 65 20 5 5 5 *<sup>A</sup> <sup>E</sup>*

20 5 <sup>5</sup> : 5.67 5.67 3.3 65 20 5 5 5 5 5 5 *<sup>P</sup> <sup>E</sup>*

The value (cost) of parameter *a* for the base lane is 2. This cost is allocated proportionally by ADTs in this example. The total cost allocations for the three vehicle classes are shown in

> Costs for base lanes (proportional)

Cost responsibilities

Capacity costs (*LiCl*)

Automobile 3.7 1.177.5 10.65 13.12 Pickup truck 3.3 0.677.5 10.25 8.58 5-ax-trailer truck 10 0.167.5 10.10 11.30

Sequence Including Sequence Marginal Cost

calculated values are *Ck* = 170/30 = 5.67 and *Cl* = 150/20 = 7.5.

**Table 7.** Sequences and marginal cost for calculation of A-S value

*Step 2. Traffic-load cost allocation:* 

*Step 3. Lane assignment:* 

*Step 4. Cost allocation:* 

Table 8.

See Table 3. *LA*=1.17, *LP*=0.67, *LT*=0.16

Vehicle Classes Load costs

**Table 8.** Cost allocations for vehicle classes

(*Ei)*

*Department of Industrial & Systems Engineering, Kongju National University, South Korea* 

## **7. References**


[16] Redekop J (2000) Increasing marginal cost and the monotonicity of Aumann-Shapley pricing. Working Paper, Department of Economics, University of Waterloo, Waterloo, Ontario, Canada.

**Chapter 0**

**Chapter 7**

**A Game Theoretic Analysis of Price-QoS Market**

Mohamed Baslam, Rachid El-Azouzi, Essaid Sabir,

Additional information is available at the end of the chapter

cited.

Loubna Echabbi and El-Houssine Bouyakhf

http://dx.doi.org/10.5772/54380

**1. Introduction**

delay and jitter, etc.

**Share in Presence of Adversarial Service Providers**

Recently, the selfish behavior of customers and Service Providers (SPs) in telecommunications systems has been widely analyzed using game theory with all its powerful solution concepts. It was shown in several works that customer's selfish behavior leads to a network collapse, where a typical prisonerís dilemma situation arises. Despite of the bounty of works and efforts investigated in analyzing market share game, this filed is still an ideal tool to understand interaction among SPs and customers. Indeed, it is common in the literature to assume a single decision action (e.g., cost) through which an equilibrium would be computed. Yet, in order to take into account Quality of Service (QoS), it is necessary to incorporate into the model more than one decision parameter. A simple example is to include both price and some measure of QoS (e.g., delay, throughput, loss probability, etc.). Other multi-criteria models may incorporate, for example, delay and reliability, the latter representing the QoS, price or

The competition in terms of prices and QoS among SPs entails the formation of non-cooperative games. We consider multiple SPs (players of the game), where each one seeks to maximize its own revenue, whereby the whole system of SPs would have no incentive to deviate from the Nash equilibrium1 point, i.e., the vector of equilibrium strategies. Yet, such equilibrium point should first mathematically exist. In this chapter, we present a general model for computing a bi-criteria Nash equilibrium for multiple SPs. We shall then analyze the interactions between SPs who won't attract more clients and maximize their respective profits. We address the important problem of Nash Equilibrium characterization with two-component action, when the two components of each provider are the service price and a measure of QoS. Our model is mainly inspired from, [6], where the authors studied a

<sup>1</sup> A Nash equilibrium is a strategy profile where no player has sensitive to deviate unilaterally from its current strategy.

©2013 El-Azouzi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2013 El-Azouzi et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

