metadata: authors: 0: author: Parrachino,Irene
1: author: Patrone,Fioravante
2: author: Zara,Stefano
count: World
docna: 0: docna: Cooperative game theory and its application
            to natural, environmental, and water resource issues : 1.
            basic theory
docty: Policy Research Working Paper
subtopic: Judicial System Reform,Hydrology,Coastal and Marine Resources,Energy and Natural Resources,Global Environment
teratopic: Macroeconomics and Economic Growth,Environment,Agriculture,Education
abstracts: cdata!: Game theory provides useful insights
            into the way parties that share a scarce resource may plan
            their use of the resource under different situations. This
            review provides a brief and self-contained introduction to
            the theory of cooperative games. It can be used to get
            acquainted with the basics of cooperative games. Its goal is
            also to provide a basic introduction to this theory, in
            connection with a couple of surveys that analyze its use in
            the context of environmental problems and models. The main
            models (bargaining games, transfer utility, and non-transfer
            utility games) and issues and solutions are considered:
            bargaining solutions, single-value solutions like the
            Shapley value and the nucleolus, and multi-value solutions
            such as the core. The cooperative game theory (CGT) models
            that are reviewed in this paper favor solutions that include
            all possible players and ignore the strategic stages leading
            to coalition building. They focus on the possible results of
            the cooperation by answering questions such as: Which
            coalitions can be formed? And how can the coalitional gains
            be divided to secure a sustainable agreement? An important
            aspect associated with the solution concepts of CGT is the
            equitable and fair sharing of the cooperation gains.
repnme: repnme: Cooperative game theory and its application
            to natural, environmental, and water resource issues : 1.
            basic theory
display_title: Cooperative game theory and its
            application to natural, environmental, and water resource
            issues : 1. basic theory
listing_relative_url: /research/2006/11/7216745/cooperative-game-theory-application-natural-environmental-water-resource-issues-1-basic-theory
topicv3: Justice m1326106 840,Natural Resources m1326446 1979,Water m1326522 137,Environment, Natural Resources and Blue
            Economy m1327391 566
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Environment,Agriculture,Education
no_of_pages: 30
author_keys: 0: author_key: 713366
1: author_key: 713359
2: author_key: 714044
model_extractions: WPS4072

# **COOPERATIVE GAME THEORY AND ITS APPLICATION TO NATURAL, ENVIRONMENTAL AND WATER RESOURCE ISSUES:**

# **1. Basic Theory**

Irene Parrachino, Stefano Zara and Fioravante Patrone

## **Abstract**

Game theory provides useful insights into the way parties that share a scarce resource may plan their utilization of the resource under different situations. This review provides a brief and self-contained introduction to the theory of cooperative games. It can be used to get acquainted with the basics of cooperative games. Its goal is also to provide a basic introduction to this theory, in connection with a couple of surveys that analyze its use in the context of environmental problems and models. The main models (bargaining games, transfer utility and non transfer utility games) and issues and solutions are considered: bargaining solutions, single-value solutions like the Shapley value and the nucleolus, and multi-value solutions such as the core. The cooperative game theory (CGT) models that are reviewed in this paper favor solutions that include all possible players and ignore the strategic stages leading to coalition building. They focus on the possible results of the cooperation by answering questions such as: Which coalitions can be formed? And how can the coalitional gains be divided in order to secure a sustainable agreement? An important aspect associated with the solution concepts of CGT is the _equitable and fair_ sharing of the cooperation gains.

World Bank Policy Research Working Paper 4072, November 2006 _The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they represent. Policy Research Working Papers are available online at http://econ.worldbank.org._ This Working paper is a product of the first phase of the study “Cooperative Arrangements for Allocation of Water among Production and Environmental Uses under Stochastic Supply Conditions”, funded by the Italian Trust Fund and the Development Economics Research Group of the World Bank. Members of the study team include: Ariel Dinar (Task Manager), Fioravante Patrone, Irene Parrachino, Stefano Zara of the University of Genoa, Carlo Carraro, Carmen Marchiori, and Alessandra Sgobbi of the University of Venice. Comments by Rashid Sumaila, University of British Columbia, Canada and Kim Hang Pham Do, Massey University, New Zealand, and participants of the 6<sup>th</sup> meeting on Game Theory and Practice Dedicated to Development, Natural Resources and the Environment, July 10-12, Zaragoza, Spain, are much appreciated. Two additional Working Papers are also devoted to some special topics, which are chosen having in mind their use in applications to environmental issues: 2. Application to Natural and Environmental Resources (PRWP 4073), and 3. Application to Water Resources (PRWP 4074).
# **CONTENTS**

|**A SEMI-TECHNICAL BACKGROUND.............................................................................................................3**|
|---|
|WHAT ISGAMETHEORY?...........................................................................................................................................3|
|**THE COOPERATIVE BARGAINING PROBLEM...........................................................................................4**|
|THE MAIN BARGAINING SOLUTIONS............................................................................................................................6|
|_The Nash solution.............................................................................................................................................6_|
|_The Kalai-Smorodinski solution.......................................................................................................................7_|
|**_N_-PERSON COOPERATIVE GAMES ................................................................................................................7**|
|SUBSET SOLUTION CONCEPTS...................................................................................................................................10|
|_The Core .........................................................................................................................................................11_|
|_The bargaining set..........................................................................................................................................14_|
|_The Kernel ......................................................................................................................................................15_|
|_The Least Core ...............................................................................................................................................16_|
|_Unique (One-point) solution concepts...........................................................................................................16_|
|The Nucleolus..............................................................................................................................................................16|
|The Shapley value........................................................................................................................................................17|
|Theτ - value...............................................................................................................................................................19|
|Modifications to the above solution concepts.............................................................................................................20|
|**NON TRANSFERABLE UTILITY (NTU)-GAMES ........................................................................................21**|
|**COST GAMES .......................................................................................................................................................21**|
|COST-SHARING RULES..............................................................................................................................................22|
|_The Alternate Cost Avoided (ACA) and the Separable Cost Remaining Benefits (SCRB) methods ............23_<br>Monotonicity................................................................................................................................................................24|
|,<br>α β **AND**γ **-CHARACTERISTIC FUNCTIONS...........................................................................................25**|
|**CONCLUDING REMARKS ................................................................................................................................26**|
|**REFERENCES.......................................................................................................................................................28**|

2
# **A SEMI-TECHNICAL BACKGROUND**

Resource scarcity is a growing concern to individuals as well as to governments. Competition over smaller amounts of water and other environmental amenities of deteriorating quality is increasingly becoming common in many parts of the world. As more have to share less, the question of strategic behavior becomes imminent. In such a situation, Game Theory (GT) may provide useful insights into the way parties that share a scarce resource should plan their utilization of the resource under different situations.

The purpose of this working paper is twofold. First it provides the basic building blocks of Cooperative Game Theory (CGT), presented in simple and applicable terms. Second, it reviews the literature dealing with application of cooperative game theory in environmental and water resources and explores the possibilities of expanding its use. The paper does not provide a general introduction to GT, or even to CGT: it is just a sketchy introduction, focusing on the tools that have been used in applications to water and environmental problems. The goal is to offer a self-contained introduction, instrumental to the reading and understanding of companion surveys, for people who do not have a general GT background.

## **What is Game Theory?**

Game Theory (hereafter GT) is the study of mathematical modeling of strategic behavior of decision makers (players), in situations where one player’s decisions may affect the other players. The basic assumption of Game Theory is that decision makers are rational players, that they are intelligent, so, while pursuing well-defined objectives, players take into account other decision-makers’ rationality and, accordingly, build expectations on their behavior.

GT consists of a modeling part and a solution part. Mathematical models of conflicts and of cooperation provide strategic behavioral patterns, and the resulting payoffs to the players are determined according to certain _solution concepts_ .

There are two main branches of GT.<sup>1</sup> The first is _non-cooperative Game Theory_ (hereafter NCGT) and the second is _cooperative Game Theory_ (hereafter CGT). The main distinction between the two is that NCGT models situations where players see only their own strategic objectives and thus binding agreements among the players are not possible, while CGT actually is based mainly on agreements to allocate cooperative gains ( _solution concepts_ ).

Therefore, while NCGT models describe and take into account the strategic interaction among the players, CGT ignores the strategic stages leading to coalition building and focuses on the possible results of the cooperation. CGT attempts answering questions such as which coalitions can be formed? And how can the coalitional gains be divided in order to secure a sustainable agreement? In particular, CGT favors solutions that include all possible players ( _Grand Coalition_ ), and thus most CGT solution concepts refer to the Grand Coalition.

An important aspect associated with the solution concepts of CGT is the _equitable and fair_ sharing of the cooperation gains. Young (1994) notices that equity is something dealt with in everyday life. One can refer to equity in a comprehensive framework, that is, social justice: a proper distribution of resources, welfare, rights, duties, opportunities, or in the narrow framework, for example, how to solve everyday distributive problems. This second case is the one more frequently addressed by GT, which provides the tools to examine equity in a

> 1 Additional typologies hold as well, e.g.,  static and dynamic games, one-shot games and repeated games.

3
rigorous way, and the problem turns out to be a choice between rules under an axiomatic perspective. But, as Young underlines, the axiomatic approach has two weaknesses: first, the axioms, reasonable by themselves, may lead to “impossibility theorems”; second, the axiomatic method may result in a solution that is too far from the practical problem dealt with: the perceived equity always depends on the particulars of the case. Furthermore, the _empirical rules_ of equity, that one can see applied in real situations, are usually more complex than a single normative principle, and often represent a balance or compromise between competing principles.

The term fairness in the literature is sometimes used as a synonym for equity, but some authors often mean something different: their idea of fairness coincides with the acceptability and stability of the cost-benefit apportionment among the players.

As was indicated above, the following background is not intended to provide a comprehensive technical basis in GT. Readers who wish to widen their familiarity with the field of GT could refer to Owen (1995), Myerson (1991), Osborne and Rubinstein (1994), Driessen (1988), Peters (1997), and Aumann and Hart (1992, 1994, 2002).

# **THE COOPERATIVE BARGAINING PROBLEM**

A GT approach to _bargaining_ was first introduced in Nash (1950), who developed a cooperative and static game model. In a _two-person bargaining problem_ , two players have access to a set of alternatives, which is called the _feasible set_ . It is assumed that each player has preferences over the alternatives, and that the preferences are represented by a couple of functions<sup>_u_</sup> 1<sup>and</sup><sup>_u_</sup> 2<sup>. In the original approach by Nash, as in many later developments, it has</sup> been assumed that these utility functions are von Neumann-Morgenstern utilities<sup>2</sup> .

The utility functions,<sup>_u_</sup> 1<sup>and</sup> _u_ 2 , define a subset _S_ of R<sup>2</sup> , which is the image of the set of feasible alternatives. Each point of _S_ represents a solution for the bargaining problem that is an agreement between the players. Note that the solution is defined in the ‘image’ space, so that in principle it could correspond to more different ‘equivalent’ agreements.

Within the feasible set there is also a point, called _threat point_ or _disagreement point_ , which is where “the game ends” should no agreement be reached. We will call _S_ the feasible set and _d_ the disagreement point.

While defining his axiomatic solution, Nash wanted it to be a rule that associates a point of _S_ with each problem ( _S_ , _d_ ). To develop it, we will introduce the formal model (and an example), then we will report the Nash axiomatic solution and finally we will present some of the other solution concepts proposed after Nash.

Definition: _A two-person bargaining problem_ is a pair ( _S, d_ ) such that _S_ is convex, closed (it contains its boundary) and a comprehensive<sup>3</sup> subset of R<sup>2</sup> ; 4 _d_ ∈ _S_ , and there exists at least one _x_ ∈ _S_ such that _x_ > _d_ ;

> 2 von Neumann-Morgenstern utility: An axiomatic extension of the ordinal concept of utility to uncertain payoffs. An agent characterized by a von Neumann-Morgenstern utility function ranks uncertain payoffs according to (higher) expected value of the utility of the individual outcomes that may occur.

> 3 ∀∈ _x S_ ⊆ R 2, ∀∈ _y_ R 2, _y_ ≤ _x_ ⇒ _y_ ∈ _S_

4
a r J <!-- Start of picture text --> (0,9)<br>u,<br>d<br>(3,0): u,<br><!-- End of picture text -->
# **The main bargaining solutions**

# **_The Nash solution_**

As mentioned before, the Nash bargaining solution is based on several axioms: <u>1. Individual Rationality.</u> Each player receives at least what he would receive in the threat point, that is, for all ( _S_ , _d_ ) ∈ _B_ 2 we have φ( _S_ , _d_ ) ≥ _d_ ; <u>2. Pareto optimality. This is a standard efficiency condition and means that all gains from</u> cooperation should be shared. Formally, it is expressed as follow: φ satisfies the Pareto optimality if, for each ( _S_ , _d_ ) ∈ _B_ 2 and for each _x_ ∈ _S_ , _x_ ≥φ ( _S_ , _d_ ) ; then _x_ =φ ( _S_ , _d_ ) ; <u>3. Independence from Irrelevant Alternatives. If an alternative is judged to be the solution to</u> a problem, then it should still be considered the most suitable for any sub-problem containing it. Formally, it is expressed as follows: φ satisfies this property if ( φ _S_ , _d_ ) ∈ _T_ implies ( φ _T_ , _d_ ) =φ ( _S_ , _d_ ) , for each ( _S_ , _d_ ) ∈ _B_ 2 and for each _T_ ⊂ _S_ such that ( _T_ , _d_ ) ∈ _B_ 2; <u>4. Covariance under positive affine transformations.</u> The solution is required to be independent of any particular members in the families of von Neumann and Morgenstern’s utility functions (von Neumann and Morgenstern, 1944) representing the agents’ preferences chosen to describe the problem. Formally, it is expressed as follows: A positive affine transformation is a map A: R<sup>2</sup> → R<sup>2</sup> such that, for every _i_ ∈{ 1,2 } , _Ai_ ( _x_ ) = _a xi i_ + _bi_ , where _a_ 1, _a_ 2, _b_ 1, _b_ 2 ∈ R ,<sup>_a_</sup> 1<sup>and</sup><sup>_a_</sup> 2<sup>being positive (</sup><sup>_A_</sup> _i_ (<sup>_x_</sup> )<sup>denotes the</sup><sup>_i_-th component of</sup> _A_ ( _x_ ) ). φ satisfies covariance with positive affine transformation if, for each ( _S_ , _d_ ) ∈ _B_ 2 and for each positive affine transformation _A_ , ( φ _A_ ( _S_ ), _A_ ( _d_ )) = _A_ ( φ ( _S_ , _d_ )) ; <u>5. Symmetry. If the bargaining problem is invariant with interchanging the agents, the</u> solution should give the same to both. Formally, it is expressed as follows: φ satisfies symmetry if, for each ( _S_ , _d_ ) ∈ _B_ 2 such that _S_ is symmetric (i.e. ( _x_ 1, _x_ 2 ) ∈ _S_ then ( _x_ 2, _x_ 1) ∈ _S_ ) and _d_ 1 = _d_ 2 , it holds that φ 1( _S_ , _d_ ) =φ 2 ( _S_ , _d_ ) .

Nash proved that there is a unique solution, Ω , satisfying all these axioms, that is expressed in the following formula: No axiom is universally applicable; for a critique to the set of these 5 axioms see Thomson (1994).

The Nash solution has been generalized by Harsanyi (1959) for _n_ players with _n_ > 2. The Nash-Harsanyi solution, _Z_ , to a n-person bargaining game is given below.

6
## **_The Kalai-Smorodinski solution_**

Kalai and Smorodinski (1975) thought that the Nash solution does not take sufficiently into account the aspirations of the players: they believed that if the preferences and the utility values of the players change, the compromise point between the agents should not vary. They proposed a solution taking into account the aspiration levels of the players. The property introduced is the following: <u>Individual monotonicity. It requires that an expansion of the feasible set “in a direction</u> favorable to a particular agent”, always benefits the agent. Formally it is expressed as follows: Let<sup>_u_</sup> _i_ (<sup>_S_</sup> ,<sup>_d_</sup> )<sup>be</sup> max { _xi_ | _x_ ∈ _Sd_ } (for _n_ = 2 ): if _T_ ⊇ _S_ , and _u j_ ( _T_ ) = _u j_ ( _S_ ) , then φ _i_ ( _T_ , _d_ ) ≥φ _i_ ( _S_ , _d_ ) , for _i_ ≠ _j_ .

By simply replacing independence of irrelevant alternatives by individual monotonicity in the list of axioms proposed by Nash, Kalai and Smorodinski obtained the characterization of their proposed solution.

The Kalai-Smorodinski solution (a unique solution satisfying the above axioms) is: Although there are several solutions to bargaining problems; the two solutions presented here are the most important ones. Other solution concepts in the game theoretical literature can be found in Peters (1992). For example, Armstrong (1994) applies the Salukvadze solution (Yu and Leitman, 1974) to a fishery management problem.

# **_N_ -PERSON COOPERATIVE GAMES**

Definition: An _n-person cooperative game in the characteristic function form_ is an ordered pair _G_ = ( _N_ , _v_ ) where _N_ = { 1,2,..., _n_ } is a finite set with _n_ elements.

_N_ = { 1,2,... _n_ } is the set of players. A subset _S_ of the player set _N_ (notation: _S_ ⊆ _N_ ) is called _coalition_ , and the collection of 2<sup>_n_</sup> coalitions of _N_ (i.e. _P_ ( _N_ ) ) is denoted by 2<sup>_n_</sup> , including _N_ itself, the empty set and all the one-element subsets. _v_ : _P N_ → R is real-valued function defined over all the subsets of N and such that ( ) _v_ (∅=) 0 , where ∅ is the empty set.

_N_ is the _grand coalition_ and it consists of the set of all the players. The number of the players of a coalition _S_ is denoted by _S_ or, sometimes, _s_ . _v_ is the _characteristic function_ , or coalitional function, and assigns a “worth” _v_ ( _S_ ) to each coalition _S_ .<sup>_5_</sup>

> 5 It is necessary to note that the characteristic function is defined considering only the players within coalition _S_ , without considering those players which are outside of the coalition and that, specifically in

> environmental issues, could affect _v_ ( _S_ ) . For a better explanation, see Section 6.

7
> In many cases the elements of the players’ set _N_ represent real people, e.g., landowners and peasants, traders, creditors or voters, but the player set can also consist of more abstract membership, such as sectors, as in the well-known Tennessee Valley Authority case (Straffin and Heaney), or (in the “airport game”) of landing by planes, or agricultural associations and city water services.

It is often assumed that the coalitional/characteristic function is expressed in units of an infinitely divisible commodity which “stores” utility, and which can be transferred without loss to the players. If a subset of players (coalition) can obtain a total utility _v_ , this utility can be divided among the members of the coalition in any possible way. It is also assumed that there exist some transferable commodity that enables the transfers among players in order to reallocate the benefits gained through the coalition, and that there is a common scale to compare the players’ utilities.

> The theory developed based on these assumptions is called the _side-payments_ theory.  We

> will call games satisfying the above presumptions transferable utility games, or _TU-games_ .

> We will often refer to ( _N_ , _v_ ) simply as _v_ , and will denote by _G_ ( _N_ ) the class of TU-games with set of players _N_ .  The coalitional function is often called a _game_ (with transferable

> utility) _in the characteristic form_ .

Definition: In a TU-game _v_ ∈ _G_ ( _N_ ) _v_ is _superadditive_ if, for all _S_ , _T_ ⊆ _N_ , with _S_ ∩ _T_ = ∅ ,

> Definition: Let _v_ ∈ _G_ ( _N_ ) . _v_ is said to be _constant-sum_ if, for all _S_ ⊆ _N_ ,

If a game is superadditive, then players have real incentives for cooperation, in the sense that the union of any two disjoint groups of players shall never diminish the total benefits; the merger of the disjoint coalitions can only improve their prospects. The superadditivity of the game is guaranteed in case _v (S)_ is defined as the total amount of utility that coalition _C_ can guarantee for itself, irrespective of the strategies adopted by the remaining players (see the definition of the α - characteristic function given in the last section). It is not always sensible to assume superadditivity, and this assumption may be questioned in some environmental applications, due to the presence of externalities (again, see the last section for some comments on this issue). Assuming superadditivity means that the grand coalition is efficient, thus the problem may turn to be the sharing of the overall utility _v_ ( _N_ ) among all the players.

In this paper we focus on the class of superadditive games, explicitly specifying if a particular game doesn’t fulfill the above condition. A stronger notion than superadditivity is convexity. Definition: Let _v_ ∈ _G_ ( _N_ ) . _v_ is _convex_ if for all coalitions S and T, An equivalent definition is that for any player _i_ , and any coalitions _S_ ⊃ _T_ not containing _i_ , _v_ ( _S_ ∪ _i_ ) − _v_ ( _S_ ) ≥ _v_ ( _T_ ∪ _i_ ) − _v_ ( _T_ ) . The larger the coalition, the greater the marginal contribution<sup>6</sup> of new members.

> 6 For the concept of Marginal Contribution see the section of the Shapley value

8
With the characteristic function at hand and supposing that the players agree to work together on a certain objective, they will have to divide the total payoff _v_ ( _N_ ) of the grand coalition. “An allocation problem arises whenever a bundle of resources, rights, burdens, or costs is temporarily held in common by a group of individuals and must be allotted to them individually. An allocation or distribution is an assignment of the objects to specific individuals.” (Young, 1994: p.7) An allocation, usually, requires two different types of decisions: (a) the choice of the total amount of the good to be distributed and (b) the formula or principle of the allocation of that amount. The focus in this review is on the second choice. “An _allocation rule_ is a method, process, or formula, which allocates any given _supply_ of goods among any potential _group_ of claimants according to the salient characteristics of these claimants.” (Young, 1994) A distribution of the amount of _v_ ( _N)_ among the players will be represented by a real value function _x_ on the player set _N_ satisfying the efficiency principle: Here _x_ ( _i_ ), also denoted _xi_ , represents the payoff to player _i_ , according to the involved payoff function _x._ We usually identify the function _x_ ∈ R<sup>_N_</sup> on _N_ with the corresponding _n_ -tuple _x_ = ( _x_ 1<sup>,</sup> K , _xn_ ) ∈ R<sup>_N_</sup> of real numbers, called _allocation_ . The vectors _x_ ∈ R<sup>_n_</sup> satisfying the efficiency principle are called _efficient payoff vectors_ or _pre-imputations_ .

Most of the proposed solution concepts meet also the individual rationality principle, requiring that the payoff to any player _i_ by the payoff vector _x_ be at least the amount that player _i_ can attain by himself.

This determines the set of imputations: Definition: Let _v_ ∈ _G_ ( _N_ ) . An _imputation_ of _v_ is a vector _x_ ∈ R<sup>_n_</sup> such that: _I_ ( _v_ ) denotes the set of imputations of _v_ . It is the set of allocations of _v_ ( _N_ ) among the players satisfying efficiency and individual rationality (that is, as we already mentioned, one player joining a coalition should gain at least what he/she would get playing alone).

Applying superadditivity repeatedly (i.e.: adding the players one by one), it is clear that one imputation and the game is called _inessential_ . In this situation, there isn’t any incentive to form coalitions because they don’t get more than the sum of the stand-alone payoffs. The other games are called _essential_ .

9
An important approach, developed by von Neumann and Morgenstern (1944) deals with the concepts of _dominance_ , _D-Core_ and _Stable Sets_ . We couldn’t find, in our literature review of CGT and environmental and water resources, any explicit applications of this approach, but still, we have to mention it because of its relevance in GT evolution.

In a game we may find many allocations satisfying the requisites above, namely being imputations, but what we might need is a criterion to make a distinction between different possible imputations. _Dominance_ is such a criterion.

Definition: Given _v_ ∈ _G_ ( _N_ ) , a non-empty _S_ ⊆ _N_ , and _x_ , _y_ ∈ _I_ ( _v_ ) , then _x dominates y_ _through S_ if: _xi_ > _yi_ for all _i_ ∈ _S_ . This states that all of the members of S prefer _x_ to _y_ ; and ∑ _xi_ ≤ _v_ ( _S_ ) . This states that the players of _S_ are able, via cooperation by themselves, to _i_ ∈ _S_ obtain the allocation _x_ .

As we will better observe later, CGT tools are used not only to share benefits from cooperation, but also to allocate costs. In this case it is useful to consider a _cost function c_ instead of _v_ , and this change doesn’t introduce substantial differences in the model. In such a cost allocation situation it’s also possible to develop a model dealing with the function _v_ , which describes the savings of the players that are the result of cooperation.

The main goal of the theory of TU-games is to select, for every TU-game, an allocation, or a set of allocations, which is admissible to the players.

In the next section we will present some of the most important CGT solution concepts, which have been associated with the applications that we found in the literature on CGT and environmental and water resources.

As we indicated earlier, the questions addressed in a CGT model are:

1. Which coalitions form?

2. If the grand coalition forms, how do the players divide _v_ ( _N_ ) (or allocate costs)?

The solution concepts give many answers to these questions.

One can find two groups of solution concepts: (a) Subset solution concepts: _Core_ (Gillies, 1953; Shapley, 1971), _D-Core_ , _Stable sets_ (von Neumann and Morgenstern, 1944), _bargaining set_ (Aumann and Maschler, 1964), _Kernel_ , _Least Core_ , and (b) One-point (unique) solution concepts: _Nucleolus_ (Schmeidler, 1969), _Shapley value_ (Shapley, 1953); τ _- value_ (Tijs, 1981).

Apart from those quoted above, in the GT literature other solutions can be found, or variants. In this paper, we’ll confine ourselves to an overview of the CGT solution concepts that have been applied in the literature on environmental and water resources issues.

# **Subset solution concepts**

The solution concepts in this group refer to a ‘range’ of values that fulfill certain conditions as will be seen below. The flexibility subset solution concepts provide is very convenient in practice as they allow the decision maker consideration of various policy interventions and their evaluation.

10
# **_The Core_**

This concept was introduced by Gillies (1953). An allocation satisfying the two properties (efficiency and individual rationality) stated before is an imputation. From a “practical” point of view, an imputation is a distribution of gains that satisfies only the individual rationality. A similar condition for coalitions should also be taken into account.

So, we can introduce another condition, called _coalitional rationality_ that extends the concept of individual rationality to each of the coalitions.

Definition: Let _v_ ∈ _G_ ( _N_ ) . A _Core element_ of a game _v_ is an allocation _x_ ∈ R<sup>_N_</sup> such that: The _Core_ of _v_ , that we denote _C_ ( _v_ ), is the following set of vectors: Core allocations provide incentives for cooperation. If the Core exists, however, there are usually an uncountable number of allocations. Which is the most equitable? Thus, two problems may arise: first, the Core may have more than one allocation; second the Core might be empty. The two following examples underline these problems.

**EXAMPLE** : a game with a nonempty Core.

The simplex in R 3 , with vertexes (12, 0, 0), (0, 12, 0), (0, 0,12) represents the part of the _pre-_ 3 _imputation_ set ( ∑ _xi_ = 12 ) which satisfies _x_ i ≥ 0. Individual rationality is already satisfied _i_ = 1 for players 1 and 2, but to get the imputation set we have to consider the condition for player 3 also (see the line _x_ 3 ≥ _v_ ( 3 ) = 1 in the figure below).

11
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A typical example of a game with an empty Core is the _simple majority game_ . A coalition wins if it is composed by more than half of the participants, _N_ = { 1,2,...., _n_ } . Then: The Core conditions for this game provide no solutions.

For example, if we consider a three-person simple majority game we will have: _N_ = { 1,2,3 } and The Core will be defined by the following system:

> There is no ( _x_ 1, _x_ 2, _x_ 3 ) satisfying the above conditions. In fact, if we sum the third, fourth and fifth inequality we get:

and this is impossible because of the last Core condition (efficiency).

Bondareva (1963) and Shapley (1967) provided necessary and sufficient conditions for a game to have a nonempty Core in terms of balanced conditions.

Theorem: (Bondareva, 1963 and Shapley, 1967) A TU-game has nonempty Core if and only if it is _balanced_<sup>_7_</sup> _._ Another well-known relation between convexity of games and the existence of Core elements is as follows:

> Theorem: (Shapley, 1971) If a game ( _N_ , _v_ ) is convex, then _C_ ( _v_ ) is a nonempty set.

> 7 Entering into a detailed analysis of the concept of balancedness, is not useful in this paper.

13
Note that convexity implies that the game has a nonempty Core, but the reverse is not true.

# **_The bargaining set_**

The _bargaining set_ was introduced by Aumann and Maschler (1964). Here we follow Maschler (1992). It is necessary to start form this concept, in order to better understand two other game theoretic concepts: the _Kernel_ and the _Nucleolus_ .

The idea is that it is necessary to have a process through which it is possible to allocate the gains from cooperation when the grand coalition has not been formed, but we have a certain _coalition structure_ :

> Definition: Let ( _N_ , _v_ ) be a TU-game.  A _coalition structure_ is a partition _B_ ={ _B_ 1,..., _B_ k} of

_k_ the n players, i.e.,<sup>U</sup> _i_ = 1<sup>_B_i =</sup><sup>_N_and for all</sup><sup>_i_</sup> ≠ _j_ , _B_ i ∩ _B_ j = ∅ .

When a particular coalition structure is formed, it is necessary to define an appropriate payoff configuration, satisfying individual and group rationality.

We will use the notation _x_ ( _B_ i) = ∑ _h_<sup>_x_</sup> _h_<sup>, with</sup><sup>_h_∈</sup><sup>_B_i, to indicate the sum of the payoffs of the</sup>

> players in a certain coalition _B_ i.

Definition: An _imputation_ for a coalition structure _B_ is a payoff vector _x_ = ( _x_ 1, _x_ 2,..., _xn_ ) satisfying: _x_ ( _B_ i) = _v_ ( _B_ i), for all _B_ i in _B_ (coalitional rationality);

> <sup>_x_</sup> _i_ ≥ _v_ ({ } _i_ ) , for all _i_ in _N_ (individual rationality).

We denote by _X_ ( _B_ ) the space of all the imputations for the coalition structure _B_ i.

The players belonging to each coalition try to reach their best outcome advancing proposals about their needs, according with the imputation.

> Definition: Let _x_ be an imputation in a game ( _N_ , _v_ ) for a coalition structure _B_ . Let _k_ and _l_

> be two distinct players in a coalition _B_ i of _B_ . An _objection_ of _k_ against _l_ at _x_ is a

> pair ( _S_ , _y_ ) , satisfying:

> This is a way for _k_ to inform _l_ that he could gain more someplace else, and to say to _l_ that he should transfer some of its gains to _k_ .

But _l_ can answer with a counter-objection:

> Definition: Let ( _S_ , _y_ ) be an objection of _k_ against _l_ at _x_ , _x_ ∈ _X_ ( _B_ ), _k_ , _l_ ∈ _B_ i ∈ _B_ .  A

> counter-objection to this objection is a pair ( _T_ , _z_ ) , satisfying:

_T_ ⊂ _N_ , _l_ ∈ _T_ , _k_ ∉ _T_ ; _T z_ ∈ R , _z_ ( _T_ ) = _v_ ( _T_ ) ; _zi_ ≥ _yi_ , all in T _i_ ∩ _S_ ; 14
In the counter-objection, player _l_ claims that he can assure to himself his gain by forming _T_ . It is important to note that _k_ can object against _l_ only if they belong to the same coalition of the coalition structure. The objection is justified if it has no counter-objection; otherwise, the objection is unjustified.

The bargaining set will be defined as follows:

> Definition: Let ( _N_ , _v_ ) be a cooperative game with side payments. The _bargaining set M_ 1 _i_ ( _B_ )

> for a coalition structure _B_ is:

_M_ 1 _i_ ( _B_ ) := { _x_ ∈ _X_ ( _B_ ): every objection at x can be countered}

**=** { _x_ ∈ _X_ ( _B_ ): there exists no justified objection at _x_ }.

When a particular coalition structure has formed, the aim is to find a particular imputation in order to create some “stability” among the coalitions in the coalition structure.

To compute, at least partially, the bargaining set, the Kernel has been introduced. If the game is superadditive, then the Kernel is non-empty and it is a subset of the bargaining set.

# **_The Kernel_**

For the _n_ -person game _v_ , let _S_ be a coalition ( _S_ ⊆ _N_ ) and _x_ = ( _x_ 1<sup>,...,</sup> _xn_ ) a payoff vector (not necessarily an imputation).

We define _excess_ of _S_ with regards to the imputation _x_ as the quantity corresponding to the difference between the value of the coalition _S_ (given by the characteristic function) and the utility received according to _x_ .

If we consider two different players ( _i_ ≠ _j_ ) the _surplus_ of _i_ against _j_ can be defined as: where the maximum is taken over all coalitions _S_ such that _x_ ∈ _S_ and _j_ ∉ _S_ .

Thus,<sup>_s_</sup> _ij_<sup>represents the maximum gain that player</sup><sup>_i_can hope to get without the cooperation</sup> of _j_ .

Now, if we consider an individually rational payoff configuration _x_ , we can say that _i outweighs j_ (notation _i_ ff _j_ ) if and only if:

> _sij_ ( _x_ ) > _s ji_ ( _x_ ) and<sup>_x_</sup> _j_ > _v_ ( _j_ ) .

If _i_ ff _j_ there is a certain instability, because _i_ can make _j_ a demand that he cannot counter.

The Kernel is the set of individually rational payoff configurations for which such instability does not occur, that is ∀⊆ _S N_ , there are no _i_ , _j_ ∈ _S_ , such that _i_ ff _j_ .

15
# **_The Least Core_**

Given an _n_ -person game _v_ , we can consider any coalition _S_ ⊆ _N_ and any imputation _x_ = ( _x_ 1<sup>,...,</sup> _xn_ ) ∈ _I_ ( _v_ ) ; the _excess_ of _S_ with regard to the imputation _x_ is described by the quantity Let<sup>_e_</sup> 1 ( _x_ ) be the largest excess of any coalition relative to _x_ ,<sup>_e_</sup> 2 ( _x_ ) the second largest excess,<sup>_e_</sup> 3 ( _x_ ) the next and so on.

The _Least Core_ is the set _X_ 1 of all _x_ that minimize<sup>_e_</sup> 1 ( _x_ ) .

# **_Unique (One-point) solution concepts_**

Under this group of solution concepts we find several solution concepts such as the Nucleolus, the Kernel, the Shapley Value, and the τ -value.

## **<u>The Nucleolus</u>**

Starting from the definition of the Least Core, let _X_ 2 be the set of all _x_ in _X_ 1 that minimize<sup>_e_</sup> 2 ( _x_ ) , _X_ 3 the set of all _x_ in _X_ 2 that minimize<sup>_e_</sup> 3 ( _x_ ) , ...

This process will eventually lead to an _X k_ consisting of a single imputation _x_ (as Schmeidler proved), called the _Nucleolus_ .

We can better understand and formalize this concept defining the 2<sup>_n_</sup> -vector θ( _x_ ) , as the vector whose components are the excesses of the 2<sup>_n_</sup> subsets _S_ ⊆ _N_ , arranged in decreasing order, i.e., EXAMPLE: For the three person game _v_ , where _v_ ( _S_ ) = 1, if _S_ has two or three players and _v_ ( _S_ ) = 0 otherwise, the payoff vectors _x_ = (0.3, 0.5, 0.2) and _y_ = (0.1, 0.5, 0.4) give the following excesses:

|S|_e_(_S_,_x_)|_e_(_S_,_y_)|
|---|---|---|
|∅|0|0|
|1|- 0.3|- 0.1|
|2|- 0.5|- 0.5|
|3|- 0.2|- 0.4|
|1, 2|0.2|0.4|
|1, 3|0.5|0.5|
|2, 3|0.3|0.1|
|N|0|0|

16
We can order the several vectors θ( _x_ ) by the lexicographic order. The lexicographic order can be applied comparing the first components of two vectors: if they are equal, compare the seconds and so on, until we find two different components. The (lexicographically) bigger vector will be the one containing the higher of these two different components. Formally, given two vectors α = (α 1<sup>,...,</sup> α _q_ ) and β = (β 1<sup>,...,</sup> β _q_ ) , we say that α is lexicographically smaller than β if there is some integer _k_ , 1 ≤ _k_ ≤ _q_ , such that: α _l_ = β _l_ for 1 ≤≤ _l k_ , α _k_ < β _k_ and we can write α < _L_ β . (In the above example θ( _x_ ) < _L_ θ( _y_ ) .) The _Nucleolus_ is the imputation that minimizes the function θ( _x_ ) (in the lexicographic order). If the Core is non-empty, then it can be proved that the Nucleolus lies in the Core.

# **<u>The Shapley value</u>**

Several solution concepts are introduced in an _axiomatic_ way (as we have seen in Nash and Kalai-Smorodinski): it means that the solution is built as a consequence of some properties that must be satisfied. Usually these properties are applied to a class of games. For this reason, we consider _G_ ( _N_ ) , the class of superadditive games with a given set _N_ of players. We shall assume the _N_ is a finite set containing _n_ elements.

Shapley proposed to consider a mapping φ : _G_ ( _N_ ) → R<sup>_n_</sup> , satisfying the following properties: <u>Efficiency:</u> φ satisfies efficiency if, for all _v_ ∈ _G_ ( _N_ ) **,** ∑φ _i_ ( _v_ ) = _v_ ( _N_ ) **.** This property _i_ ∈ _N_ means that all the available amount must be allocated by the solution;

> <u>The Dummy player property:</u> φ satisfies this property if, for every _v_ ∈ _G_ ( _N_ ) and every dummy player _i_ ∈ _N_ , φ _i_ (<sup>_v_</sup> ) = _v_ ( _i_ ) .

A player is a _dummy player_ if _v_ ( _S_ ∪{ } _i_ ) = _v_ ( _S_ ) + _v_ ( _i_ ) for all coalition _S_ such that _i_ ∉ _S_ , that is its contribution to S is only _v_ ( _i_ ) . (A special case of dummy player is a _null player_ : a null player does not generate any benefits and should receive nothing); <u>Anonymity:</u> φ satisfies anonymity if, for every _v_ ∈ _G_ ( _N_ ) and every , _i j_ ∈ _N_ that are interchangeable<sup>8</sup> players in _v_ , it holds that φ _i_ ( _v_ ) =φ _j_ ( _v_ ) . This property means that the value must treat equally strong players in an equal way; 8 If two players equivalently contribute in the game, we say that they are interchangeable. Formally, _i_ , _j_ ⊂ _N_ are said to be interchangeable players in the game _v_ if, for every coalition _S_ ⊂ _N_ \ { _i_ , _j_ } , _v_ ( _S_ ∪{ } _i_ ) = _v_ ( _S_ ∪{ } _i_ ) 17
> <u>Additivity:</u> φ satisfies additivity if, for every _v_ , _w_ ∈ _G_ ( _N_ ) , it holds that φ( _v_ + _w_ ) =φ( _v_ ) +φ( _w_ ) .  It is difficult to justify this axiom with regards to fairness, but it is an obviously useful property for the solution.

> Shapley proved that there exists a unique value φ : _G_ ( _N_ ) → R<sup>_n_</sup> that satisfies those four properties:

for all _v_ ∈ _G_ ( _N_ ) and all _i_ ∈ _N_ , when _s_ and _n_ denote the cardinality of _S_ and _N_ , respectively.

We can interpret the Shapley value as follows: given a game _v_ ∈ _G_ ( _N_ ) , we consider any permutation π of the set _N_ and any player _i_ ∈ _N_ . If _P_ ( _i_ , π) is the set of players that precede _i_ in the permutation π , _M_ ( _i_ , π) = _v_ ( _P_ ( _i_ , π) ∪{ } _i_ ) − _v_ ( _P_ ( _i_ , π)) is the _marginal contribution_ of _i_ to the coalition _P_ ( _i_ , π) . The Shapley value φ : _G_ ( _N_ ) → R<sup>_n_</sup> will be: 1 φ _i_ (<sup>_v_</sup> ) = ∑ _M_ ( _i_ , π) . _n_ ! π In order to better understand this concept, consider a situation with _n_ players agreeing to meet in a certain room; imagine the _n_ players entering one at a time into that room in a random order (specified by the permutation π , that is, all possible arrival orderings are equally probable) and that each player, as soon as he enters and reaches the coalition _S_ created by the players arrived before him, receives a reward equal to _v_ ( _S_ ∪{ } _i_ ) − _v_ ( _S_ ) , that is his marginal contribution.

The Shapley value is the mean marginal contribution, averaged on all of the _n!_ permutations π .

**EXAMPLE:** Consider the three-person game:

The marginal contributions are given in the following table:

|π|_M_(1,π)|_M_(2,π)|_M_(3,π)|
|---|---|---|---|
|(1, 2, 3)|0|4|16|
|(1, 3, 2)|0|13|7|
|(2, 1, 3)|4|0|16|
|(2, 3, 1)|5|0|15|
|(3, 1, 2)|7|13|0|
|(3, 2, 1)|5|15|0|

18
1 The average of the sum of the marginal contributions will be ( 21,45,54 ) , that is, by 6 definition, the Shapley value for this game.

> Theorem: Let ( _N_ , _v_ ) be a TU-game. Then the following statements are equivalent:

The game is convex (that is, for any _S_ , _T_ ⊆ _N_ , _v_ ( _S_ ) + _v_ ( _T_ ) ≤ _v_ ( _S_ ∩ _T_ ) + _v_ ( _S_ ∪ _T_ ) ); For each permutation π , the marginal contribution vector is in the Core.

> The Shapley value, being the mean marginal contribution averaged on the permutation π in all of the _n!_ possible cases, is in the Core of a convex game.

# **<u>The</u>** τ **- value**

The τ _- value_ is a feasible compromise between two vectors: _M_ ( _v_ ), that is the _marginal vector_ ( _utopia vector_ ) and _m_ ( _v_ ), that is the _minimum right vector_ (see below for further explanation).

For player _i_ we define: (Marginal contribution of player _i_ to the grand coalition. It is the best payoff player _i_ can hope to obtain: if he wants more, then it is preferable for the other players in _N_ to throw him away.) (Minimum right for player _i_ , called the remainder for _i_ in the coalition _S_ , if all the other players in _S_ obtain their respective utopia payoffs.) One can prove that ∀∈ _x C_ ( _v_ ) , _mi_ ( _v_ ) ≤ _xi_ ≤ _M i_ ( _v_ ) , for each _i_ ∈ _N_ , that is _m_ and _M_ are upper and lower bounds, respectively, for the Core of the game.

As we saw in the section about the Core, games with a nonempty Core are balanced.

A game is called _Quasi Balanced_ (QB) if:

> _mi_ ( _v_ ) ≤ _M i_ ( _v_ ) ∀∈ _i N_ ∑ _mi_ ( _v_ ) ≤ _v_ ( _N_ ) _i_ ∈ _N_ ∑ _M i_ ( _v_ ) ≥ _v_ ( _N_ ) _i_ ∈ _N_

One can prove that each balanced game is quasi balanced.

For a QB game, τ( _v_ ) is the unique efficient payoff vector on the line segment ⎡⎣ _m_ ( _v_ ) , _M_ ( _v_ )⎤⎦<sup>,</sup> lying in the _hyperplane E_ of efficient payoff _n_ ⎧ _N_ ⎫ vectors _E_ = ⎨ _x_ ∈ R ∑ _xi_ = _v_ ( _N_ )⎬ . ⎩ _i_ = 1 ⎭ 19
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Whinston (1976) , Shapley (1981) , Kalai and Samet (1987), Young (1994). For a review of the literature on weighted Shapley values see Kalai and Samet (1988).

# **NON TRANSFERABLE UTILITY (NTU)-GAMES**

> An NTU-game is a pair ( _N_ , _V_ ) where _N_ = { 1,..., _n_ } and where _V_ is a map assigning to each _S_ ∈ 2 _N_ \ {∅} a subset _V_ ( _S_ ) of R<sup>_S_</sup> such that the following three properties hold for each _S_ (Tijs, 2003):

> _V_ ( _S_ ) is a non-empty closed proper subset of R<sup>_S_</sup> ;

> _V_ ( _S_ ) is comprehensive, i.e. if _x_ ∈ _V_ ( _S_ ) and _y_ ∈ R<sup>_S_</sup> such that _y_ ≤ _x_ , then _y_ ∈ _V_ ( _S_ ) . { _x_ ∈ R _n_ | _xi_ ≥ _v_ ( _i_ ) , _x_ ∈ _V_ ( _N_ )} is bounded, where _v_ ( _i_ ) is the maximum of _V_ ({ } _i_ ) .

Note that R<sup>_S_</sup> is defined as follows: _S_ R = { _f_ : _S_ → R } .

{ 1,2 } { 1,3 } This notation is useful in order to distinguish, for example, between R and R . It is possible to turn R<sup>_S_</sup> into R<sup>_n_</sup> setting the coordinates not belonging to _S_ equal to zero. The elements of _N_ are players, who can cooperate. If coalition _S_ forms, then each of the payoff vectors _x_ ∈ _V_ ( _S_ ) is attainable, giving reward (utility)<sup>_x_</sup> _i_<sup>to player</sup> _i_<sup>∈</sup> _S_ .

Each _n_ -person TU-game ( _N_ , _v_ ) gives rise to an _n_ -person NTU-game ( _N_ , _V_ ) , where

> A two-person bargaining game ( _S_ , _d_ ) can be seen as a 2-person NTU-game ({ 1,2 } , _V_ ) where

# **COST GAMES**

In this section we will underline the link between cooperative Game Theory and the analysis of the allocation of costs (and gains) within a group of players.

By _cost-sharing_ problem, we usually mean a problem arising when a group has to decide how to allocate costs of a joint enterprise. A cost-sharing problem can be modelled in CGT using a _cost-game_ .

To introduce cost games we use a very typical situation of multipurpose joint project. Let _N_ = { 1,2,..., _n_ } be a set of projects, products, or services (or agents). The projects can be provided jointly or separately by some organization. Let _c_ ( _i_ ) be the cost of providing _i_ by itself, and for each subset _S_ ⊆ _N_ let _c_ ( _S_ ) be the cost of providing the items to _S_ jointly. For convenience, we set _c_ (∅=) 0 . (We assume that if the project is not undertaken, then the cost 21
is zero.) The function _c_ ( _S_ ) from the subsets _S_ of _N_ to the real numbers is called _cost function_ on _N_ , or, sometimes, _cost-sharing game_ .

On the other hand, the _cost-savings function_ (also identified with the corresponding _costsavings game)_ is _v_ ( _S_ ) = ∑ _i_ ∈ _S c_ ( _i_ ) − _c_ ( _S_ ) and it represents the gain from carrying the projects in S jointly, rather than separately by _i_ . If _v_ ( _N_ ) ≥ _v_ ( _S_ ) , ∀⊆ _S N_ , then the multipurpose project is efficient and the problem is to allocate the cost _c_ ( _N_ ) in an equitable way among the projects.

This property is known as _subadditivity_ (equivalent to superadditivity for _v_ ).

A second “natural” property of a cost function is that costs increase as the number of projects increase, that is Such a cost function is called _monotonic_ .

If _c_ is subadditive, then _v_ is nonnegative and monotonically increasing in _S_ .

In fact, ∀ _S_ , _S_ ' and _i_ ∉ _S_ , subadditivity implies that _c_ ( _S_ + _i_ ) ≤ _c_ ( _S_ ) + _c_ ( _i_ ) , and it follows that _v_ ( _S_ ) ≤ _v_ ( _S_ + _i_ ) . Since _v_ (∅=) 0 , _v_ is nonnegative and monotonic.

It also follows that _v_ ( _N_ ) is the largest among all _v_ ( _S_ ). In this case, from a formal point of view, _N_ is the efficient set of projects to undertake.

Usually in the cost games the terms _allocation_ or _solution_ for the problem ( _c_ , _N_ ) stand for the vector _x_ = ( _x_ 1<sup>,</sup> K , _xn_ ) ∈ R<sup>_N_</sup> such that ∑ _i_ ∈ _N_<sup>_x_</sup> _i_ = _c_ ( _N_ ) where _xi_ is the amount charged to project _i_ , while in general when we require the efficiency condition, we talk about preimputations.

A _cost allocation rule_ or _cost sharing rule_ is a function Φ ( _c_ ) that associates a unique solution to every cost-sharing game.

# **Cost-sharing rules**

“A _cost sharing rule_ allocates the total cost of a project among the members of a group that participate in that project for every possible specification of the cost function. An allocation is in the _Core_ of the cost-sharing game if no participant, or group of participants, pays more than its stand-alone cost.” (Young, 1994).

As we can see, two of the CGT concepts we introduced in the previous sections immediately come out: the _imputation set_ and the _Core_ .

<u>The Core</u> can be interpreted, in such situations, as an incentive for the parties to cooperate. A first requirement for a cost allocation is that no participant or group of participants be charged more than their stand-alone costs.

Formally: 22
How to select an equitable allocation in the Core? And how to find out a cost allocation if the cost-game has an empty Core?

The above reported solution concepts can obviously be useful in this framework also, but some problems still remain and many studies’ results are that no cost-sharing method can be universally preferred.

<u>The Shapley value</u> can be interpreted from a cost-sharing point of view in the following way: “Given a cost-sharing game on a fixed set of players, let the players join the cooperative enterprise one at a time in some predetermined order. As each player joins, the number of players to be served increases. The player’s _cost contribution_ is his net addition to cost when he joins, that is, the incremental cost of adding him to the group of players that has already joined. The _Shapley value_ of a player is his average cost contribution over all possible orderings of the players.” (Young, 1994). Here the player’s _cost contribution_ has to be referred to as his _marginal contribution_ .

Then, the Shapley value can also be formulated as Following is an overview on the main cost sharing rules in the literature (the “classical” ones plus some more recent methods). Almost all the methods charge a minimum cost called _separable cost_ , that is, in a multipurpose project, a change is imposed due to adding a project to the set of the others: If the savings game is convex it follows that and the remaining allocation problem turns to be the allocation of the _non-separable costs_ : A possibility is to equally allocate the NSC among the projects (Egalitarian Non separable Cost Method, or ENSC)

# **_The Alternate Cost Avoided (ACA) and the Separable Cost Remaining Benefits (SCRB) methods_**

Another method proposed by a TVA consultant, Martin Glaeser, in 1938, assigns to each project the following charge 23
This method attributes to a project its separable cost and a rate of the _NSC_ proportional to _ci_ − _SCi_ . This difference is the _Alternate Cost Avoided_ .

A modification of this method is the allocation method, which has gained widest acceptance among water resource engineers and is still in use.

In this modification _c_ ( _i_ ) is replaced by min [ _b_ ( _i_ ), _c_ ( _i_ )] in the following equation This is called _Separable Cost Remaining Benefits method_ (James and Lee, 1971) and it incorporates benefits as follows. Let _b_ ( _i_ ) be the benefit from undertaking project _i_ by itself. Then the _maximum justifiable expenditure_ for _i_ is min[ _b_ ( _i_ ), _c_ ( _i_ )].

The ACA method allocates cost savings in proportion to each project’s marginal contribution to savings (Straffin and Heaney, 1981).

This solution was proposed independently in the game theory literature as a means of _minimizing players’ “propensity to disrupt”_ (Gately (1974), Littlechild and Vaidya (1976), Charnes et al. (1979)).

Gately’s definition of propensity to disrupt was: This equation represents the ratio of what the members of coalition _N_ - _i_ would loose if player _i_ disrupted the grand coalition, to what player _i_ himself would lose.

Straffin and Heaney showed that Gately’s apportionment method, based on minimizing the maximum propensity to disrupt, is exactly the ACA method.

There is no reason to think that the ACA method yields a solution in the Core, and indeed it does generally not. However when there are three projects at most, and the cost function is subadditive, then it is in the Core, provided that the Core is nonempty.

Dickinson and Heaney have proposed a modification to the SCRB (1982): the _Minimum Costs Remaining Savings_ ( _MCRS_ ) _method_ .

Driessen and Tijs (1986) proposed another cost allocation method, deriving from the τ _-_ value, that is the _Non Separable Cost Gap_ ( _NSCG_ ) _method_ .

# **<u>Monotonicity</u>**

Up to this point we have implicitly assumed that all cost information is at hand, and the agents need only to reach an agreement on the final allocation.

In practice, however, the parties may need to make an agreement before the actual costs are known, and they commit themselves to a rule for allocating costs rather than to a single cost allocation.

This turns our attention to the type of rule on which the players will agree. For example, if total costs were higher than anticipated, it would be unreasonable for any player’s charge to 24
go down. Similarly, if costs were lower, it would be unreasonable for any player’s charge to go up.

Formally an allocation rule Φ is _monotonic_ (in the aggregate) if for any set of projects _N_ , and for any cost functions _c_ and _c’_ on _N_ : implies We say that the cost allocation method Φ is _coalitionally monotonic_ if an increase in the cost of any particular coalition implies, ceteris paribus, no decrease in the allocation to any member of that coalition. That is, for every set of projects _N_ , every two cost functions _c_ and _c’_ on _N_ , and every _T_ ⊆ _N_ .

' In the same way _c_ ( _T_ ) ≥ _c_ ( _T_ ) and ' _c_ ( _S_ ) = _c_ ( _S_ ) for all _S_ ≠ _T_ implies There is a theorem (Young, 1985) stating that: for ⏐ N ⏐≥ 5 there exists no Core allocation method that is coalitionally monotonic.

# α , β **AND** γ **-CHARACTERISTIC FUNCTIONS**

As we saw in the previous sections CGT models require defining a real-valued function (the characteristic function) assigning a real number to every coalition of players.

The worth of a coalition is what it can achieve on its own without the cooperation of nonmembers. If there are no externalities, i.e., if the payoffs of the members of a coalition do not depend on the actions of non-members, the model doesn’t have to take into account the actions of the non-members. But if there are externalities, then it is a must.

There are several examples of externalities such as the market externalities and the environmental externalities (Finus, 2003).

If we admit the possibility of binding agreements among a subset _S_ of players, there are different ways to get _v_ ( _S_ ) and, in general, the characteristic function _v_ of the game, depending on the assumptions about the behavior of the players outside the coalition. Thus, in the literature, we find for example the α , β and γ -characteristic functions. (As a consequence, α − and β− Core were defined by Aumann (1959).) In the α - characteristic function it is assumed that when a coalition _S_ forms the players outside the coalition ( _N_ \ _S_ ) choose a strategy implying the worst outcome for _S_ ; it is the socalled _maximin_ outcome. Formally: _v_ α ( _S_ ) = max _xS_ ∈ _X S xN_ min\ _S_ ∈ _X N_ \ _S_ π _S_ ( _xS_ , _xN_ \ _S_ ) , where _X S_ is the set of possible strategies for coalition _S_ , _X N_ \ _S_ is the set of possible strategies for coalition _N_ \ _S_ , .

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and π _S_ is the payoff function of coalition _S_ (equal to the sum of payoff functions of players belonging to _S_ ).

In the β - characteristic function it is assumed that the players in _N_ \ _S_ give more chances to the players in _S_ , and let them reach at least _v_ β ( _S_ ) , that is the minimum among the maximum payoffs that _S_ can obtain once fixed the strategies of the other players.

This is called the _minimax_ outcome: Somehow, the α - characteristic function represents a pessimistic point of view (of the players in _S_ ), while the β - characteristic function represents an optimistic perception.

There is also another important interpretation of the behavior of the players outside _S_ , leading to the γ -characteristic function, developed by Tulkens (1979), and Chander and Tulkens (1992 and 1993). In this case it is assumed that players outside _S_ do not play against it choosing the worst strategy for _S_ , but they try to reach the Nash equilibrium of the game, with _S_ considered as a single player. It is important to underline that, in their framework, Chander and Tulkens prove that the Nash equilibrium reached is unique; this result is not extendible to all the strategic interactions, since the Nash equilibrium might not be unique, and, furthermore, different equilibria lead to different payoffs for the players. Formally:

> where _x_ * _N_ \ _S_ is the Nash equilibrium of the non-cooperative game above mentioned.

# **CONCLUDING REMARKS**

With increased competition over natural resources and environmental amenities, decision makers face strategic decisions in various management and use aspects. Cooperation approaches have been used in many cases, and were proven to be useful under certain conditions. In this review we provide information on various Cooperative Game Theory (CGT) concepts, their theoretical basis and application difficulties.

Several points have to be highlighted. CGT focuses on answering questions such as which coalitions can be formed? And how can the coalitional gains be divided in order to secure a sustainable agreement? In particular, CGT favors solutions that include all possible players (Grand Coalition), and thus most CGT solution concepts refer to the Grand Coalition. While this may not be always a desired or even feasible goal, in relatively small groups of interested parties (players) the coalitional transaction costs are relatively low and could be ignored. Thus, for relatively small environments, the promotion of the Grand Coalition is desirable.

Another important aspect associated with the solution concepts of CGT is the equitable and fair sharing of the cooperation gains. Equity is something dealt with in everyday life. It is implicitly on the agenda of agencies that are responsible to setting rules for the allocation of natural resources among users. Thus, equity should be viewed in a comprehensive way, that is, social justice: a proper distribution of resources, welfare, rights, duties, opportunities. Equity can also be viewed in a narrow framework, for example, how to solve everyday distributive problems. While this second view is the one addressed more frequently by CGT, it is still able at providing a wider look at such issues.

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Acknowledging the shortcoming of CGT is important in appreciating its ability. As an axiomatic approach it has two weaknesses: first, the axioms, reasonable by themselves, may lead to “impossibility theorems”; second, the axiomatic method may result in a solution that is way too far from the practical problem dealt with: the perceived equity always depends on the particulars of the case. The various approaches reviewed in this paper and the examples used to demonstrate their application were selected to reflect both the opportunity and the difficulties embedded in CGT.

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