Sharing a polluted river through environmental taxes

n agents located along a river generate residues that then require cleaning to return the river to its natural state, which entails some cost. We propose several rules to distribute the total pollutant-cleaning cost among all the agents. We provide axiomatic characterizations using properties based on water taxes. Moreover, we prove that one of the rules coincides with the weighted Shapley value of a game associated with the problem.

charges are mainly a tax instrument used for funding public sewage treatment services (Gago et al. 2006).
An important development in this area is the European Union Water Framework Directive (formally known as Directive 2000/60/EC of the European Parliament and of the Council of 23 October 2000, establishing a framework for Community action in the field of water policy). It is a European Union directive which commits European Union Member States to achieve good qualitative and quantitative status of all water bodies by 2015. This Directive establishes the "polluter-pays" principle, which requires that users pay according to the costs they generate, and determines that Member States will adopt the recovery of costs of water services principle, including environmental and resource costs (see Article 9.1.: recovery of costs for water services).
As such integrated policies among the States of the European Union clarify the link between water use and water pollution, these policies are likely to be more efficient in meeting water management objectives. For example, the costs of cleaning water downstream before it is supplied can be compared with the costs of discouraging pollution upstream. Integrated policies also facilitate cost recovery (OECD 2004). When river-basin authorities have access to the cost of treatment of water supply operators, this provides a wealth of information on the costs of upstream pollution, which can be used to estimate the rates at which the release of pollutants should be charged. Riverbasin management also facilitates water allocation among competing uses within the basin as well as the control of inter-basin transfers. In Spain, river-basin authorities are purchasing water rights for over-exploited water bodies (OECD 2008).
The Communication "Pricing policies for enhancing the sustainability of water resources", from the European Commission to the Council, the European Parliament and the Economic and Social Committee, states that: "A price directly linked to the water quantities used or pollution produced can ensure that pricing has a clear incentive function for consumers to improve water use efficiency and reduce pollution".
This being the case, the use of economic instruments (taxes, duties, financial assistance, negotiable permits) has gained increasing importance and was fully legitimized in the United Nations' Rio Declaration on the Environment and Development in 1992. The central environmental role played by economic instruments is also recognized at Community level. Furthermore, as we have pointed out, the framework Directive on water advocates a boosting of the part played by pricing to improve the sustainability of water resources.
Given the importance of the task of solving water resource management problems, such as the control of water pollution, the problem of sharing the cleaning costs among all the polluters should be studied formally. Ni and Wang (2007) developed a theoretical model to study how to allocate pollutant-cleaning costs among the agents that caused this pollution. They consider a river which is divided into n segments from upstream to downstream. In each segment there is an agent who discharges pollutant substances of some kind into the river. The authorities guarantee the cleansing of the water for public use. The authors propose two rules to divide the total river-polluting responsibility among the polluters, the local responsibility sharing (LRS) rule, which charges the agent in a given segment its own local costs, and the upstream equal sharing (UES) rule, which charges an agent the sum of the equal divisions of all downstream costs, including its own local costs. Littlechild and Owen (1973) studied the airport problem. From a mathematical perspective both models are the same because they both have an order of agents and a cost associated with each agent. We follow the model of Ni and Wang (2007). We propose several ways to define water taxes mainly following the "polluter-pays" principle, which takes into account the different factors that influence the quality of the water. Assuming that the authority knows for sure the agent responsible for the pollution generated in a particular area, then the agent should be charged with the total cost of cleaning that area. However, in most real-life situations, it is not possible to know exactly who is responsible for the pollution generated in each particular area. This fact will be considered throughout the paper.
One of the most important topics in the cost sharing literature is the axiomatic characterization of rules. The idea is to propose desirable properties and determine which of these characterize every rule. Properties often help agents compare different rules and decide which one is preferred for a particular situation. Following this approach Ni and Wang (2007) characterize both rules with different properties. In particular, they prove that the UES rule is the only one satisfying Efficiency, Additivity, Independence of Upstream Costs (ensuring that no agent is responsible for the pollution caused in the upstream segments) and Upstream Symmetry (which states that for any given downstream cost, all upstream polluters share it equally).
Efficiency and Independence of Upstream Costs are very appealing properties. The European Union Water Framework Directive notes that the objectives of water pricing policies include the full recovery of financial costs, this being established under the property of Efficiency. As stated before, one of the main objectives of charging water use is to reduce pollution. Following Ni and Wang (2007) we implicitly assume that agents pollute the river during some period of time and this pollution must be cleaned. However, we are not able to know who is responsible for the particular pollution being generated at each segment. If we were, we would simply charge agents the cleaning cost they are responsible for. In our model, we only know the cost associated with cleaning each area. Given the direction of the water body, the water transfers the pollution from upstream to downstream. Hence, agents are partially responsible for the pollution costs downstream but not those upstream. The property of Independence of Upstream Costs reflects this fact. A property of Independence of Downstream Costs could make sense in some other situations like when you want to assure that the only pollution found at one area was generated only in that area.
Additivity is a standard property in the literature. Mathematically, it is an appealing property because if a rule is additive, then we only need to be concerned with minor problems, usually easier to solve. Moreover, the Additivity axiom has a clear meaning in this context. Assuming that to clean each area different technologies, with associated costs, may be needed, then, the cost of cleaning each area will be the sum of the costs of using each technology in that area. So, it seems reasonable that the amount that an agent should pay to clean any area of the river is equal to the sum of the shares that the agent should pay to use each technology.
Focusing on a real-life situation, the Communication from the Commission to the Council, European Parliament and Economic and Social Committee: Pricing and sustainable management of water resources, COM (2000) 477, reports that: "The overall price P paid by a given user can be computed as F + a.Q + b.Y, with F: an element related to fixed costs, general taxes, etc.; a: a charge per unit of water used; b: a charge per unit of pollution produced; Q: the total quantity of water used; Y: the total pollution produced. A reduction in the quantity of water used (Q) and/or the pollution produced (Y) will then lead to a reduction in the overall water price P paid by the user. Thus, it provides an incentive for users to increase water use efficiency and reduce pollution." Therefore, it is clear that the assumption that a tax should be additive is quite reasonable in our context.
With respect to the property of Upstream Symmetry, as we will explain further on in the paper, many situations exist where this axiom cannot be applied. The main objective of this paper is to change Upstream Symmetry and use other more suitable properties. Thus, we will obtain rules applicable to these situations.
Our first result is a characterization of the set of rules satisfying Efficiency, Additivity and Independence of Upstream Costs. Later on, three more characterizations are provided by adding in each case a different property to those in the first result.
To define the first property we take into account that in many cases all the agents dump the same kind of residues into the water. Moreover, all the residues are biodegradable, hence, they decompose into natural chemical elements through the action of biological processes involving light, air, water, bacteria, plants or animals. In this case, pollution disappears with time. Thus, it seems reasonable to assume that the further away the area is from agent i, the smaller the part of the cost that agent i should pay for cleaning this area. This is the statement of Upstream Monotonicity.
The second property follows from the fact that on several occasions when the residues are biodegradable, it is possible to know their biodegradation rate, that is, the rate of deterioration of materials in question under normal, natural conditions, say δ. Hence, the cost that an agent pays for a polluted area should depend on this biodegradation rate. In many countries, for instance, Belgium, Finland and South Africa, among others, water taxes depend on the quality of pollution (OECD 2006). A way to measure the quality of pollution could be through the biodegradation rate: the smaller the rate is, the less hazardous the pollution, as it disappears quicker, and hence, produces less responsibility for agents downstream. This is the idea of δ−Biodegradation Rate.
The motivation of the third property is that in many countries there are several alternatives in the design of water tax rates. There is a variable component which depends on different factors, such as, volume of water consumed, pollution load, population of the municipality, type of residue, etc. This is the case of Austria, Canada, Finland, France, Germany, Greece, Hungary, Italy, Korea, Spain, Sweden, USA, among many others (Gago et al. 2005(Gago et al. , 2006OECD 2006). Thus, the amount paid by each agent depends proportionally on such a factor, for instance, the pollution load. This idea is collected within the property of Proportional Tax. Ni and Wang (2007) prove that the UES rule coincides with the Shapley value (Shapley 1953b) of a cooperative game c where the value of a coalition S represents the pollutant-cleaning costs of all segments for which agents in S are partially responsible, namely, the segments that are downstream of some agent in S. In this paper we introduce another cooperative game c where the value of a coalition S represents the pollutant-cleaning costs of segments for which only agents in S are responsible, i.e., the segments polluted only by agents belonging to this coalition. We prove that the rule characterized by using Proportional Tax coincides with the weighted Shapley value (Shapley 1953a;Kalai and Samet 1987) of the game c.
The paper is organized as follows. Section 2 introduces the model and the main axioms. Section 3 presents the characterization results for different rules. We also prove that one of the rules coincides with the weighted Shapley value of a particular cooperative game. The concluding remarks and a comparison of the axiomatized rules are presented in Sect. 4 while the proofs appear in the Appendix.

The mathematical model
The model presented by Ni and Wang (2007) was followed.
Consider a river which is divided into n segments indexed in a given order i = 1, 2, . . . , n from upstream to downstream. There are n agents (for instance, firms) located along the river, each located in one of these segments in the above order. We assume that each firm generates a certain amount of pollutants.
In every segment i(i = 1, . . . , n) an environmental authority sets a standard of the degree of pollutants in segment i so that the quality of the water body satisfies the environmental standard.
In this problem we try to find rules that allocate the total costs of cleaning the pollution among all the firms generating this pollution.
Formally, let N = {1, 2, . . .} be the set of all possible agents. Let N ⊂ N be a finite set of agents. Usually we take N = {1, . . . , n}. Let C = (c 1 , . . . , c n ) ∈ R n + be the pollutant-cleaning cost vector, where c i represents the cost incurred by agent i. (c i also captures the costs of using a particular technology for cleaning the segment i).
A pollution cost sharing problem is a pair (N , C). When N is fixed the problem is denoted as C.
A solution to a problem (N , C) is a vector x = (x 1 , . . . , x n ) ∈ R n + such that i∈N x i = i∈N c i , where x i represents the cost share assigned to agent i. A rule is a mapping x that assigns to each problem (N , C) a solution x(N , C). The main axioms used in the characterization results are: A rule is efficient if it distributes the total cost of cleaning the pollution.
Additivity indicates that dividing the total cost among agents is the same as dividing one part of the cost first and then dividing the remaining cost. Independence of upstream costs (IUC) Let l ∈ N and C, C ∈ R n + such that c i = c i for all i > l. Then, The cost paid by an agent only depends on its own pollution cost and all downstream costs, but not on upstream costs for which it has no control or responsibility for.

Main results
Ni and Wang (2007) characterized the UES rule using four axioms: Additivity, Efficiency, Independence of Upstream Costs and Upstream Symmetry. The last one ensures that all the upstream agents have equal responsibility for a given downstream pollution cost. However, situations exist where this axiom does not apply. For instance, in some regions of Spain like Valencia and Catalonia, for water taxes applicable to households, the population size of each municipality where the house is located is taken into account. In this case, it is not possible to assume that all upstream agents are symmetric agents due to the pollution caused. Now, we present the family of rules satisfying Add, Eff and IUC. These rules divide the cost of cleaning each segment among the agents responsible for the pollution at that segment, proportionally to a weight vector. For a particular area, the agents responsible for this pollution are those located upstream. Proof See Appendix.
An example of the family of rules characterized in Theorem 1 is presented: Example E1 Let N = {1, 2, 3}, and c i = 1 for all i ∈ N . Consider that the proportions that agent 1 should pay for each of the costs are the following: p 1 1 = 1, p 2 1 = 0.3, p 3 1 = 0.5; the proportions of agent 2: p 1 2 = 0, p 2 2 = 0.7, p 3 2 = 0.3 and those of agent 3 are given by: p 1 3 = 0, p 2 3 = 0, p 3 3 = 0.2. In that case, the rule introduced in Theorem 1 gives the following assignments: Next we study what happens if we add new properties to those in Theorem 1. These properties are based on possible and real-life taxes over pollution.
Assume that the agents dump the same kind of residues into the water. Moreover, these residues are biodegradable, so the pollution disappears with time. Several biodegradable residues exist, for instance, organic food waste, garden waste, forest residues, some industrial waste, etc. Therefore, it seems reasonable to assume that the further away the area is from agent i, the smaller the part of the cost that agent i should pay for cleaning this area.
A new property in this context is introduced: This property states that the further away the agent is from an area, the less responsibility the agent has for the pollution of that area.
In the next theorem we study what happens if we add UM to the properties in Theorem 1. The family of rules that we characterize shares the cost of cleaning each polluted area among all the agents responsible for this pollution, proportionally to a vector of weights. These weights satisfy that the further away the responsible agent is from an area, the lower the proportion of the cost that the agent has to pay.
Proof See Appendix.

Now an example of the rules introduced in Theorem 2 is presented:
Example E2 Let N = {1, 2, 3}, and c i = 1 for all i ∈ N . Consider that the proportions that agent 1 should pay for each of the costs are the following: p 1 1 = 1, p 2 1 = 0.3, p 3 1 = 0.2; the proportions of agent 2: p 1 2 = 0, p 2 2 = 0.7, p 3 2 = 0.3 and those of agent 3: p 1 3 = 0, p 2 3 = 0, p 3 3 = 0.5. In this case, the rule introduced in Theorem 2 assigns the following: Note that the weight system given in Example E1 is not valid in this context because it does not satisfy the condition: p j i ≤ p j k for any i < k ≤ j. Sometimes, when the agents dump biodegradation residues, it is possible to know their biodegradation rate, say δ. Articles on the computation of biodegradation rate of a particular residue are numerous. We now introduce a new property, δ− Biodegradation Rate, which ensures that the taxes paid by agents depend on this rate. Note that δ = 0 means that the residue of agent i only affects its own area. In this case BR indicates that every agent pays the cost corresponding to its area, namely x i (C) = c i for all C and i ∈ N . Additionally, δ = 1 means that the residue is not biodegradable at all. In this case δ-BR coincides with Upstream Symmetry. δ-BR is stronger than UM (if a rule x satisfies δ-BR, it also satisfies UM).
In the next theorem we study the effects of adding δ-BR to the properties in Theorem 1. We prove that there is a unique rule satisfying these properties. This rule shares the cost of cleaning each segment among the agents responsible for this pollution with respect to a particular vector of weights, such that the proportion that each agent pays for cleaning a particular area takes into account the biodegradation rate of the residue thrown into the water. Proof See Appendix.
Note that the rule characterized in Theorem 3 belongs to the family of rules characterized in Theorem 1.
An example of this rule is presented.
PT−w generalizes Upstream Symmetry because when w i = w j for all i, j ∈ N both properties coincide.
In the next theorem we study the effects of adding this property to the properties in Theorem 1. For each w ∈ R N ++ we prove that there is a unique rule satisfying these properties. This rule, as well as the ones introduced in the previous Theorems, also distributes the cost of cleaning each segment proportionally among all the agents that are responsible for this pollution. However, unlike the others, in this rule, the proportion that each agent pays for cleaning each area considers an exogenous factor that is fixed for each agent. This weight is applied to distribute the cost of cleaning each area the agent is responsible for polluting, whether directly or from the agent's pollution traveling downstream to other areas. For instance, if the agents represent firms, the weight could be as a function of their size, such that the bigger the firm, the higher the weight assigned. Proof See Appendix.

Theorem 4 A rule x satisfies Add, Eff, IUC and PT-w if and only if for
Note that the rule characterized in Theorem 4 belongs to the family of rules characterized in Theorem 1.
To compute the proportions of costs that agents should pay for each area, the condition given in Theorem 4 is taken into account, and therefore, we deduce that those proportions for agent 1 are: p 1 1 = ω 1 ω 1 = 1; p 2 1 = ω 1 ω 1 +ω 2 0.34, p 3 1 = ω 1 ω 1 +ω 2 +ω 3 = 0.25; for agent 2: p 1 2 = 0, p 2 2 = ω 2 ω 1 +ω 2 0.67, p 3 2 = ω 2 ω 1 +ω 2 +ω 3 = 0.5; and for agent 3: ω 1 +ω 2 +ω 3 = 0.25. Then, the final assignment is: Below, the rule characterized in Theorem 4 is related to a well-known solution in transferable utility games. Before presenting the result, some notation: A transferable utility game, T U game, is a pair (N , c) where N ⊂ N is finite and c : 2 N → R satisfies c(∅) = 0. Given a finite subset N ⊂ N , let (N ) denote the set of all orders in N . Given π ∈ (N ), let Pre(i, π) denote the set of elements of N which come before i in the order given by π, i.e., One of the most important values in TU games is the weighted Shapley value (Shapley 1953a). Given a vector of weights w ∈ R N ++ , we denote the weighted Shapley value with weights given by w as φ w i (N , c). It can be expressed (Kalai and Samet 1987) as: . Ni and Wang (2007) prove that the solution they propose is related to some natural T U games they introduce. In particular, the Upstream Equal Sharing rule coincides with the Shapley value (Shapley 1953b)  We now compare the game c of Ni and Wang (2007) with our game c. c (S) takes the pessimistic approach for coalition S and computes the cost of all segments for which agents in S are partially responsible for, namely, the segments that are downstream of some agent in S. In this case, if segment i is downstream of agent j ∈ S, agent j is responsible for part of the pollution in segment i (note that we are assuming that the water could move the pollution from one segment to another). Nevertheless, c (S) takes an optimistic approach for coalition S and computes the cost of the segments for which only agents in S are responsible for. Since we are assuming that pollution could move from one segment to any downstream segment, the pollution of segment i is caused by all its upstream agents (namely, agents in {1, . . . , i}). Thus, agents in S are responsible only for the pollution in segment i if and only if {1, . . . , i} ⊂ S. It is trivial to see that c (S) ≤ c (S) for each S ⊂ N .
In the following table we summarize the results and we compare them with the ones given by c : Theorem 5 Let x w be the solution given by Theorem 4. Then, x w coincides with the weighted Shapley value of (N , c) with weights given by w ∈ R N ++ , φ w (N , c).
Proof See Appendix.
The Upstream Equal Sharing rule coincides with the rule given by Theorem 4 when w i = w k for all i, k ∈ N . By Theorem 5 the Upstream Equal Sharing rule also coincides with the Shapley value of the T U game c.

Concluding remarks
The important task of solving water resource management problems such as the control of water pollution is addressed in this work. It then seems reasonable to study the problem of sharing the cleaning costs among all the polluters in a formal way, this being one of the aims of our model. We simplify the situation by considering a river, divided into many segments, as many polluters are along the river. These polluters may represent a town, a firm, a household, a region or even a country, depending on the particular situation studied. We propose several ways to define water taxes, mainly following the "polluter-pays" principle, which takes into account different factors that influence the quality of the water. Assuming that an environmental authority knows exactly who the agent responsible for the pollution generated in a particular area is, then the agent should be charged with the total cost of cleaning that area, however, in most real-life situations, this is not possible. Thus, we propose several rules that reduce the incentive for agents to cause pollution, which is the main objective of water taxes in most countries.
In this paper we characterize several rules that have a common structure: all proportionally distribute the cost of cleaning each area among all the agents that are responsible for this pollution. However, the weights assigned to the agents are different in each case.
In Theorem 1, we characterize a family of rules that distribute the costs with respect to a general vector of weights, such that the cost of a particular area is totally shared among all the agents that are responsible for the pollution present in that segment, i.e., the agents located upstream.
In Theorem 2, the family of rules characterized distributes the costs with respect to a vector of weights that satisfies the following condition: the further away an agent is from a polluted area (for which it is responsible), the lower the proportion of the cost of cleaning that area.
The rule characterized in Theorem 3 could be applied when the agents throw the same kind of residue and it is biodegradable. The biodegradation rate of the residue represents the amount of pollution that is transferred from one area to the following ones. There, the weights assigned to each agent depend on the biodegradation rate of the residue. Therefore, for a specific biodegradation rate, there will be a specific vector of weights.
The rule characterized in Theorem 4 takes into account a weight system in which the weight assigned to each agent is fixed a priori. Then, the cost of cleaning each area is distributed among all the agents responsible for the pollution there, proportionally to these weights.
Finally, in Theorem 5 we relate the last rule characterized with one of the most important values in TU games. We prove that this rule coincides with the weighted Shapley value of a particular game.

Appendix
In this section we include the formal proofs of the results presented previously. Proof of Theorem 1 Let x be a rule defined as above. We first prove that x satisfies Eff, Add and IUC.
x satisfies Eff: x satisfies Add: Let C and C ∈ R n + and i ∈ N . Thus, x satisfies IUC: Let l ∈ N and C, C ∈ R n + such that We now prove the reciprocal. Assume that x is a solution satisfying Eff, Add and IUC. For each j ∈ N , let 1 j = (y 1 , . . . ., y n ) ∈ R n + such that y j = 1 and y i = 0 when i = j. We define p j = x(1 j ). It is trivial to see that p j j∈N is a weight system. Let x p be the rule induced by the weight system p j j∈N . We will prove that x = x p . We first prove two claims following Bergantiños and Vidal-Puga (2004). Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.