Using MACBETH to determine utilities of ... - Semantic Scholar
Using MACBETH to determine utilities of governments to parties in coalition formation? MARC ROUBENS12 , AGNIESZKA RUSINOWSKA??34 , and HARRIE DE SWART5 1
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University of Li`ege, Institute of Mathematics, B37, B-4000 Li`ege Facult´e Polytechnique de Mons, Rue de Houdain 9, B7000 Mons, Belgium [email protected] 3 Radboud University Nijmegen, Nijmegen School of Management P.O. Box 9108, 6500 HK Nijmegen, The Netherlands 4 Warsaw School of Economics, Department of Mathematical Economics Al. Niepodleglosci 162, 02-554 Warsaw, Poland [email protected] 5 Tilburg University, Department of Philosophy, P.O. Box 90153 5000 LE Tilburg, The Netherlands [email protected]
Corresponding author: HARRIE DE SWART Keywords: Game Theory, MACBETH, government, majority coalition, policy on issue Abstract. In the paper, we present an application of the MACBETH approach to a certain model of coalition formation. We apply the MACBETH technique to quantify the attractiveness and repulsiveness of possible governments to parties. We use this method to calculate the utilities of governments to parties. Based on these utilities, stable governments are determined. In the paper, an adequate example is presented.
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Introduction
The literature concerning models of coalition formation is very extensive. A broad overview of the existing coalition formation theories is given, for instance, in [18] or [6]. The problem of coalition formation was also, among others, tackled by [17], [1], [2], [8], [9], [10], [11], and [12]. For further references to coalition formation see [14], which presents one of the recent models of multi-dimensional coalition formation. The main concepts of this model are the concepts of a feasible policy/coalition/government and a stable policy/coalition/government, where a government is defined as a pair consisting of a majority coalition and a policy supported by this coalition. In general, there are several independent ?
??
The authors wish to thank gratefully Jean-Claude Vansnick and Jean-Marie de Corte for very useful discussion and demonstration of the MACBETH software, and an anonymous referee for very valuable suggestions for improvements. This author gratefully acknowledges support by COST Action 274, TARSKI.
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policy issues on which a government will have to decide. Hence, a policy consists of policies on all the issues. Parties are assumed to have preferences regarding each majority coalition and each policy on each given issue. Based on semantic judgements concerning the attractiveness of available alternatives, MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique) is an interactive approach to quantify the attractiveness of each alternative, in such a way that the measurement scale constructed is an interval scale. An overview and some applications of the MACBETH method are presented, for instance, in [3], [5], and on the web site (www.m-macbeth.com), where in particular, one may download the MACBETH software and follow the online tutorial. In this paper, we will apply the MACBETH technique to the model of a stable government ([14]). This method will help to determine the utilities (the values) of governments to parties. Since the notion of a stable government is based on these utilities, the application of this technique to the coalition formation model is very important both from a theoretical and a practical point of view. The paper is organized as follows. In Section 2, the model of a stable government is recapitulated. In Section 3, we apply the MACBETH technique to this model of coalition formation. In Section 4, a simple example illustrating the application of MACBETH to the coalition formation model of a stable government is constructed. In Section 5, we present conclusions.
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The model of coalition formation
In this section, we present a brief description of the model of coalition formation ([14]) to which we apply the MACBETH decision support system. We highlight the notions introduced in [14] that remain the same in the model presented in this paper. However, we also mention which elements of the model introduced in [14] are changed in the present model. Let N be the set of all parties. There are M ≥ 1 independent policy issues on which a government will have to decide. For each policy issue j ∈ {1, ..., M }, there are mj ≥ 1 dependent policy sub-issues j(1), ..., j(mj ). Let P and Pj denote a policy space and a policy sub-space on issue j respectively. A policy is represented by a tuple p = (p1 , ..., pM ) ∈ P , where pj ∈ Pj is a policy on issue j. Only majority coalitions are entitled to form a government. A majority coalition will be denoted by p0 , and the set of all majority coalitions will be denoted by P0 . A government is defined as a pair consisting of a majority coalition and a policy proposed by this coalition, that is, as a pair g = (p0 , p), where p0 ∈ P0 and p = (p1 , ..., pM ) ∈ P . The main difference between the model presented in [14] and the model proposed in this paper concerns the preferences of the parties and the definition of utilities of governments to parties. In [14], parties were supposed to have preferences on all policies on all issues and on all majority coalitions. Each party was allowed to give one of L+2 (L ≥ 1) qualifications to a policy on a certain
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issue and to a majority coalition: desirable of degree l (1 ≤ l ≤ L), acceptable (desirable of degree 0), or unacceptable. The desirability weights d0 , ..., dL were assumed to be given. Based on these weights and on parties preferences, for each party the utility (the value) of a policy on a given issue, of a policy, of a majority coalition, and finally, of a government were defined. In this paper, we define these utilities differently and in a more realistic way. In order to determine the utility (the value) V i (g) of government g to party i ∈ N , we will apply the MACBETH approach. In [14], the notion of feasibility was introduced. A policy was said to be feasible if there was a majority coalition such that the policy on each issue was desirable (of a certain non-negative degree) for each party in that coalition. By a feasible coalition we meant a majority coalition which was desirable for each party in that coalition. Finally, a feasible government consisted of a feasible coalition and a policy feasible for this coalition. Moreover, the utilities of a policy/coalition/government were defined in such a way that a government was feasible if and only if its value was non-negative. In this paper, we do not consider feasibility, since the value of a government to a party will be determined differently. In [14], the concepts of dominance and stability were defined. These notions remain almost the same in the present model. By G∗ we denote the set of all governments. Definition 1 A government g 0 = (p00 , p0 ) ∈ G∗ dominates a government g = (p0 , p) ∈ G∗ (g 0 g) if ∀i ∈ p00 [V i (g 0 ) ≥ V i (g)] ∧ ∃i ∈ p00 [V i (g 0 ) > V i (g)].
(1)
Definition 2 A government g ∈ G∗ is said to be stable if there is no government dominating g, that is, if ¬ ∃g 0 ∈ G∗ [g 0 g]. (2) Let us point out that the relation of dominance of Definition 1 is not necessarily acyclic, because the set of indices in (1) depends on g 0 . Consequently, it is possible that no stable government exists in the sense of Definition 2. This will not be the case with Definitions 3 and 4. In [14], a stable government was defined as a feasible government which was dominated by no feasible government. As mentioned before, we do not incorporate feasibility in the present model. In [7], the notion of stability in a given coalition was considered. We generalize this notion. Definition 3 Let G ⊆ G∗ and p000 ∈ P0 . We say that a government g 0 = g) (p00 , p0 ) ∈ G dominates a government g = (p0 , p) ∈ G in coalition p000 (g 0 G p00 0 if ∀i ∈ p000 [V i (g 0 ) ≥ V i (g)] ∧ ∃i ∈ p000 [V i (g 0 ) > V i (g)]. (3)
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Definition 4 A government g ∈ G is said to be stable in coalition p000 with respect to G ⊆ G∗ if there is no government in G dominating g in p000 , that is, if ¬ ∃g 0 ∈ G [g 0 G g]. p00 0
(4)
For instance, in the example of Section 4 we will consider stability of a government (p0 , p) in the coalition p0 with respect to the set of all governments formed by that coalition. In [7], the notion of internal stability was also considered. Definition 5 A government g = (p0 , p) ∈ G∗ is said to be internally stable if ¬ ∃(p00 , p0 ) ∈ G∗ [(p00 ⊆ p0 ∨ p0 ⊆ p00 ) ∧ ∀i ∈ p0 ∩ p00 [V i (p00 , p0 ) ≥ V i (p0 , p)] ∧ ∃i ∈ p0 ∩ p00 [V i (p00 , p0 ) > V i (p0 , p)]].
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(5)
Applying MACBETH to the model
In our model, the set of all governments G∗ is assumed to be finite. We also allow an infinite number of possible governments if only a finite number of governments is judged differently by each party. To give an example, a policy assuming any amount in the interval [20,30] as a budget for education is, in fact, a continuum of policies, and consequently leads to a continuum of governments. Nevertheless, as long as all parties find each amount from [20,30] equally attractive, we can treat this as one policy. In the present model, each party is assumed to have preferences regarding all governments with respect to certain criteria. The criteria are ‘majority coalition’ and all the policy issues. Let C ∗ be the set of all criteria, that is, C ∗ = {0, 1, ..., M }, where criterion 0 concerns the ‘majority coalition’. ‘Majority coalition’ will be also called a policy on issue 0. For each j ∈ C ∗ , each party orders all policies on issue j taking into account the attractiveness of these policies on the given issue. A policy on issue j ∈ C ∗ may be of a negative, zero, or a positive attractiveness to a party. For each issue j, each party is asked to specify two particular references: neutral, defined as ‘neither satisfying nor unsatisfying’, and good, which is more attractive to a party than neutral, and is defined as ‘undoubtedly satisfying’. A neutral policy on issue j for party i is denoted by neutralji , and a good policy on issue j for party i is denoted by goodij . Both neutral and good policies on a given issue may be fictitious. Nevertheless, they are always incorporated into the model: the set Pj ∪ {neutralji , goodij } will be denoted by Pji . For each j ∈ C ∗ and i ∈ N , based on these two references neutralji and goodij , we distinguish: – an unattractive (or repulsive) policy on issue j to party i if it is less attractive to i than neutralji ,
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– an attractive (or simply attractive) policy on issue j to party i if it is more attractive to i than neutralji , – a very attractive (or outstanding) policy on issue j to party i if it is at least as attractive to i than goodij . For each criterion j ∈ C ∗ , each party i ∈ N is asked to verbally judge the difference of attractiveness between each two policies pj and p0j of Pji , where pj is at least as attractive to i as p0j . When judging, a party has to choose one of the following categories: C0 - no difference of attractiveness C1 - very weak difference of attractiveness C2 - weak difference of attractiveness C3 - moderate difference of attractiveness C4 - strong difference of attractiveness C5 - very strong difference of attractiveness C6 - extreme difference of attractiveness. If a party is unsure about the difference of attractiveness, it may choose the union of several successive categories among these above. Moreover, when comparing two policies on a given issue, an answer ‘I do not know’ is acceptable and appears in the MACBETH software as positive difference of attractiveness. However, the more preference information is provided, the greater is the scale’s level of accuracy. For each party i ∈ N and each criterion j ∈ C ∗ , the M-MACBETH software allows to associate to each policy pj ∈ Pji a real number Uji (pj ) which, in the particular case where there is no hesitation about the difference of attractiveness, satisfies the following rules (see [3] or [5]): ∀i ∈ N ∀j ∈ C ∗ ∀pj , p0j ∈ Pji : Uji (pj ) > Uji (p0j ) ⇔ pj is more attractive to i than p0j
(6)
i 0 ∀i ∈ N ∀j ∈ C ∗ ∀k, k 0 ∈ {1, 2, 3, 4, 5, 6} ∀pj , p0j , p00j , p000 j ∈ Pj with (pj , pj ) ∈ Ck 0 i i 0 i 00 i 000 and (p00j , p000 j ) ∈ Ck0 : k ≥ k + 1 ⇒ Uj (pj ) − Uj (pj ) > Uj (pj ) − Uj (pj ) (7)
This numerical scale is essentially obtained by linear programming and is called the MACBETH basic scale. Of course, the MACBETH scale exists if and only if it is possible to satisfy rules (6) and (7); in such a case the matrix of judgements is called consistent. If it is impossible to satisfy rules (6) and (7), a message appears on the screen (‘inconsistent judgements’), inviting the party to revise the judgements. As we will illustrate in the next section, the MMACBETH software provides tools to obtain a consistent matrix of judgements.
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The basic MACBETH scale, as well as each scale obtained by a positive linear transformation, are pre-cardinal scales. In order to obtain a cardinal scale, a discussion with the party in question around the scale takes place. Apart from presenting the pre-cardinal MACBETH scale in a numerical way, it can also be presented in a graphical way (‘thermometer’). By this, the party has the possibility to modify the positioning of the policies on the given issue such that the relative distances between the policies on the given issue reflect the relative distances of attractiveness that the party deems to exist between these policies on the given issue. In the MACBETH software ([5]), when a party selects with the mouse a policy on a given issue, an interval appears around this policy. By moving the mouse, a party can modify the position of the selected policy on the given issue, but only within this interval. By modifying the position of the policy on the given issue inside the interval, a party obtains a new positioning of all policies on a given issue such that both conditions (6) and (7) are still satisfied. When a party thinks that finally the scale adequately represents the relative magnitude of the judgements, we have the cardinal scale and the (final, agreed) values of all policies on the given issue. Let Vji (pj ) denote the (final) value of policy pj ∈ Pji on issue j ∈ C ∗ to party i ∈ N . We can assume that ∀i ∈ N ∀j ∈ C ∗ : Vji (goodij ) = 100 and Vji (neutralji ) = 0.
(8)
We get then negative values for all repulsive policies on issue j ∈ C ∗ to party i, and values greater than 100 for all outstanding policies on issue j to party i. Finally, we want to measure the global attractiveness of each government, that is, the attractiveness of each government taking all criteria into account. We adopt the following aggregation procedure: V i (g) = V i (p0 , p1 , ..., pM ) =
M X
αji · Vji (pj ),
(9)
j=0
with
i α0i , α1i , ..., αM
≥ 0, and
M X
αji = 1.
(10)
j=0
The coefficients (αji )j∈C ∗ are scaling constants (expressing the relative weights party i assigns to the different criteria) which can also be calculated with the M-MACBETH software. For each party i ∈ N , let us consider the following reference profiles i [neutrali ] = (neutral0i , neutral1i , ..., neutralM ) i [Crit.i0 ] = (goodi0 , neutral1i , ..., neutralM ) i [Crit.i1 ] = (neutral0i , goodi1 , ..., neutralM )
...
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[Crit.iM ] = (neutral0i , neutral1i , ..., goodiM ). Observe that, for each j ∈ C ∗ , the difference in attractiveness between [Crit.ij ] and [neutrali ] corresponds to the added value of the ‘swing’ from neutralji to goodij . Each party ranks the reference profiles in decreasing order of attractiveness and, using categories C0 - C6 , judges the difference of attractiveness between each two reference profiles, where the first one is more attractive than the second one. After the adjustment of the scale proposed by the software (MACBETH scale), we obtain an interval scale V i which measures the overall attractiveness of the reference profiles. We can assume that, for each i ∈ N V i ([neutrali ]) = 0 and
M X
V i ([Crit.ij ]) = 100.
(11)
j=0
Taking into account (8) and (9), we deduce that, for each i ∈ N and j ∈ C ∗ αji =
V i ([Crit.ij ]) . 100
(12)
Finally, using the aggregation procedure given in (9), one may calculate the values of all governments to each party. Using formulae (1) - (5), stable governments may be identified if there are any.
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The Example
In order to illustrate the application of the MACBETH software to the coalition formation model, we construct the following simple example. There are only three parties in parliament: A, B, and C, i.e., N = {A, B, C}, and at least two parties are needed to form a majority coalition. Hence, P0 = {AB, AC, BC, ABC}. There are two issues (M = 2) on which a government will have to decide. Issue 1 is ‘Health Care’, and consists of two sub-issues: ‘Introducing a Reform of the Health Service’ with the set of policies on this sub-issue equal to {yes, no}, and ‘Increase of the Health Expenditure’, with the set of all policies on this sub-issue equal to [0,30]. This set is divided into three intervals: [0,10), [10,20), [20,30], and it is assumed that each party finds two amounts belonging to the same interval equally attractive. These two sub-issues are dependent on each other in the sense that introducing a reform of the health service requires an amount of at least 10 for the increase of the health expenditure. Issue 2 is ‘Foreign Policy’, and there
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are 6 possible foreign policies: x, y, z, r, s, w. More precisely, the following issues are assumed: Issue 1: Health Care: Sub-issue 1(1): Introducing a Reform of the Health Service Sub-issue 1(2): Increase of the Health Expenditure Issue 2: Foreign Policy The policy sub-spaces are (1)
(2)
(3)
(4)
(5)
P1 = {p1 , p1 , p1 , p1 , p1 }, where (1) p1 = (yes, [10, 20)), (3)
(2)
p1 = (yes, [20, 30]) (4)
p1 = (no, [0, 10)), and
(5)
p1 = (no, [10, 20)),
p1 = (no, [20, 30])
P2 = {x, y, z, r, s, w}. In Tables 1 till 4 we assume that each party i ∈ {A, B, C} has indicated a decreasing order of preference with respect to the majority coalitions, to the policies on issue 1 and 2, and with respect to reference profiles [neutrali ], [Crit.i0 ], [Crit.i1 ], [Crit.i2 ]. Table 1 presents a decreasing order of all majority coalitions for all parties. The neutral and good majority coalitions are also included. Table 1: Decreasing order of all majority coalitions party
decreasing order of the majority coalitions
A
AB
goodA 0
B
BC
goodB 0 = AB
C
goodC 0
AC
AC
BC
ABC = neutral0A AC
neutral0B
ABC
neutral0C
BC
AB
ABC
Table 2 presents a decreasing order of all policies on issue 1 for all parties. The neutral and good policies on issue 1 are also included. Table 2: Decreasing order of all policies on issue 1 party A B C
decreasing order of the policies on issue 1 (1)
goodA 1 = p1 goodB 1
(2)
p1
(1)
goodC 1 = p1
(2)
p1
(5)
(4)
neutral1A = p1
p1
(5)
(1)
neutral1B = p1 (4)
p1
(2)
p1
p1 (5)
p1
(4)
p1
(3)
p1
(3)
p1
(3)
neutral1C = p1
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Table 3 presents a decreasing order of all policies on issue 2 for all parties, including the neutral and good policies on issue 2. Table 3: Decreasing order of all policies on issue 2 party
decreasing order of the policies on issue 2 goodA 2
x
y
z
neutral2A
B
goodB 2 =z
r
y
s
neutral2B = x
C
goodC 2 =w
s
r
z
y
A
r
s
w w
neutral2C = x
Additionally, each party i ∈ {A, B, C} will have to judge the difference of attractiveness between each two reference profiles: [neutrali ] = (neutral0i , neutral1i , neutral2i ) [Crit.i0 ] = (goodi0 , neutral1i , neutral2i ) [Crit.i1 ] = (neutral0i , goodi1 , neutral2i ) [Crit.i2 ] = (neutral0i , neutral1i , goodi2 ). Table 4 shows the decreasing orders of these reference profiles for all parties. Table 4: Decreasing order of the reference profiles party decreasing order of the profiles A [Crit.A ] [Crit.A [Crit.A [neutralA ] 0 1] 2] B B B B [Crit.0 ] [Crit.2 ] [Crit.1 ] [neutralB ] C C C C [Crit.1 ] [Crit.2 ] [Crit.0 ] [neutralC ] Let us consider party A, and the policies on issue 1. Party A judges the difference of attractiveness for all the policies on issue 1. Suppose party A proposes the matrix of judgements as presented in Table 5. Table 5: Matrix of judgements of the difference of attractiveness in P1A - inconsistent judgements (1)
p1 = goodA 1 (1)
p1 (2) p1 (4) p1 (5) p1 (3) p1
no
(2)
p1
(4)
p1
very weak moderate no moderate no
(5)
(3)
p1 = neutral1A strong strong weak no
p1
extreme very strong very weak positive no
But then the software immediately sends the information:
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‘Inconsistent judgements! Macbeth has found 2 ways to obtain a consistent matrix of judgements by 1 category change.’ MACBETH proposes either to decrease the difference of attractiveness between (4) (5) p1 and p1 which is now weak, or to increase the difference of attractiveness (4) (3) between p1 and p1 which is now very weak. The software also shows the contradictions (see Table 6). Table 6: Contradictions for matrix presented in Table 5 Problem Difference 1 weak positive 2 extreme weak
Couples Couples (5) (4) (3) − p1 > p 1 − p1 (5) (3) p 1 − p1 > 0 (1) (3) (1) (5) p1 − p1 > p 1 − p1 (4) (5) (4) (3) p1 − p1 > p 1 − p1 (4) p1
(5)
Difference very weak strong very weak
(3)
In problem 1, the first inequality gives p1 −p1 < 0 which is in contradiction (5) (3) to the second inequality p1 − p1 > 0. (5) (3) In problem 2, the first inequality is equivalent to p1 − p1 > 0, but the (5) (3) second inequality is equivalent to p1 − p1 < 0. Suppose that party A follows the second advice of the software, and it changes (4) (3) the difference of attractiveness between p1 and p1 from very weak to weak. The new matrix of judgements is presented in Table 7. Table 7: Matrix of judgements of the difference of attractiveness in P1A - consistent case (1)
(2)
p1 = goodA 1 (1)
p1 (2) p1 (4) p1 (5) p1 (3) p1
no
(4)
p1
(5)
(3)
p1 = neutral1A
p1
very weak moderate no moderate no
strong strong weak no
p1
extreme very strong weak positive no
MACBETH sends then the information ‘Consistent judgements’. Using the ‘thermometer’, party A can adjust the scale. The (final) values of all the policies on issue 1 to party A are presented in Table 8. Table 8: The values of the policies on issue 1 for party A policy p1 = V1A (p1 ) =
(1)
(2)
p1 = goodA 1 p1 100.00
(4)
(5)
p1 p1 = neutral1A
87.50 25
0.00
(3)
p1
−25.00
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Next, party A judges the difference of attractiveness for all the policies on issue 2. Suppose Table 9 presents the matrix of judgements. Table 9: Matrix of judgements of the difference of attractiveness in P2A goodA 2 x y z neutral2A r s w
goodA x y z neutral2A r s w 2 no v. weak v. weak weak strong v. strong v. strong extreme no v. weak weak moderate strong strong v. strong no v. weak moderate moderate strong strong no weak weak weak moderate no v. weak v. weak weak no v. weak v. weak no v. weak no
The values of all the policies on issue 2, calculated by the MACBETH software and adjusted by party A, are presented in Table 10. Table 10: The values of the policies on issue 2 for party A policy p2 = V2A (p2 ) =
goodA 2
x
y
z
100.00 80.77 69.23 38.47
neutral2A 0.00
r
s
w
−15.38 −23.08 −38.47
Finally, suppose that Table 11 represents the way in which party A judges the difference of attractiveness for all the majority coalitions. Table 11: Matrix of judgements of the difference of attractiveness in P0A AB goodA AC ABC = neutral0A BC 0 AB no very weak moderate very strong extreme goodA no moderate moderate very strong 0 AC no moderate moderate ABC no moderate BC no Again, we use MACBETH to calculate the values of all the majority coalitions. The final values are presented in Table 12. Table 12: The values of the majority coalitions for party A coalition p0 = V0A (p0 ) =
AB
A goodA 0 AC ABC = neutral0
125.01 100.00 50.00
0.00
BC −50.00
Table 13 presents the matrix of the difference of attractiveness between the reference profiles.
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Table 13: Matrix of difference of attractiveness between the reference profiles for party A A A [Crit.A [Crit.A 0 ] [Crit.1 ] 2 ] [neutral ] [Crit.A no moderate very strong extreme 0] A [Crit.1 ] no weak moderate [Crit.A no very weak 2] [neutralA ] no
Finally, the values of the reference profiles and the scaling constants are presented in Table 14. Table 14: The scaling constants for party A A A A measure [Crit.A j ] = [Crit.0 ] [Crit.1 ] [Crit.2 ] V A ([Crit.A 60.00 30.00 10.00 j ]) = αjA = 0.60 0.30 0.10
We can apply the same procedure for the remaining parties. Tables 17-24 and 25-32 in the Appendix present the results for party B and party C, respectively. Using the aggregation procedure (see definition (9)), one can calculate for each party the values (the utilities) of all the governments. Since there are 4 majority coalitions, 5 policies on issue 1, and 6 policies on issue 2, we have 4 × 5 × 6 = 120 possible governments. Let us calculate the values of several governments. We consider the following four governments: (1) g 1 = (AB, p1 , x) = (AB, (yes, [10, 20)), x) (2)
g 38 = (AC, p1 , r) = (AC, (yes, [20, 30]), r) (4)
g 75 = (BC, p1 , x) = (BC, (no, [10, 20)), x) (5)
g 104 = (ABC, p1 , y) = (ABC, (no, [20, 30]), y) By virtue of (9), and Tables 8, 10, 12, and 14, we get (1)
V A (g 1 ) = α0A · V0A (AB) + α1A · V1A (p1 ) + α2A · V2A (x) = 113.08 (2)
V A (g 38 ) = α0A · V0A (AC) + α1A · V1A (p1 ) + α2A · V2A (r) = 54.71 (4)
V A (g 75 ) = α0A · V0A (BC) + α1A · V1A (p1 ) + α2A · V2A (x) = −14.42 (5)
V A (g 104 ) = α0A · V0A (ABC) + α1A · V1A (p1 ) + α2A · V2A (y) = 6.92. Let V (g) = (V A (g), V B (g), V C (g)) for each g ∈ G∗ . Table 33 presents the values of all 120 governments. It turns out that no government is stable in the sense of Definition 2. Government g 33 , for instance, dominates most of the governments (it does NOT dominate g 1 , g 5 , g 9 , g 25 , g 29 , g 31 and g 39 only), but g 33 is itself dominated by g 39 . Moreover, g 39 dominates also, among others, g 25 , g 29 , and g 31 . Government g 15 dominates, in particular, g 1 , g 5 , and g 9 . Government g 39 is dominated by g 6 . As mentioned before, the relation of dominance of Definition 1 is not necessarily acyclic. For instance, g 39 g 25 , g 25 g 6 and g 6 g 39 .
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Let us consider stability in a given coalition with respect to the set of all governments formed by that coalition. Table 15 presents the governments stable in coalition p0 with respect to G∗p0 , where p0 ∈ {AB, AC, BC, ABC}, and G∗p0 denotes the set of all governments formed by coalition p0 . Table 15: Governments stable in coalition p0 with respect to G∗p0 , where p0 ∈ {AB, AC, BC, ABC} p0
governments stable in p0 with respect to G∗p0
AB
g 1 , g 5 , g 9 , g 29 , g 33
AC
g 2 , g 6 , g 10 , g 14 , g 18 , g 22
BC
g 11 , g 15 , g 19 , g 23 , g 35 , g 39
ABC
g 4 , g 8 , g 12 , g 16 , g 20 , g 24 , g 32 , g 36 , g 40
Let us consider now internal stability. All governments stable in coalition p0 with respect to G∗p0 , for p0 ∈ {AB, AC, BC} are also internally stable. There are 9 governments stable in the grand coalition with respect to G∗ABC . Nevertheless, none of these governments is internally stable. For instance, both parties A and B find government g 9 more attractive than each of these nine governments (stable in the grand coalition with respect to G∗ABC ). Finally, let us consider stability in a given coalition with respect to G∗ . The set of all governments stable in coalition AB with respect to G∗AB is equal to the set of all governments stable in AB with respect to G∗ . The analogue is not the case for the remaining coalitions. For instance, g 2 , which is stable in AC with respect to G∗AC , is dominated in AC with respect to G∗ (for instance, by g 17 ). Also g 22 formed by AC is stable in coalition BC (and in coalition ABC) with respect to G∗ . Table 16 presents the governments stable in coalition p0 with respect to G∗ , where p0 ∈ {AB, AC, BC, ABC}. Table 16: Governments stable in coalition p0 with respect to G∗ , where p0 ∈ {AB, AC, BC, ABC} p0
governments stable in p0 with respect to G∗
AB
g 1 , g 5 , g 9 , g 29 , g 33
AC
g 1 , g 5 , g 9 , g 13 , g 17 , g 21 , g 22
BC
g 11 , g 15 , g 19 , g 22 , g 23 , g 35 , g 39
ABC
g 1 , g 5 , g 7 , g 9 , g 11 , g 13 , g 15 , g 17 , g 19 g 21 , g 22 , g 23 , g 29 , g 31 , g 33 , g 35 , g 37 , g 39
14
5
Conclusions
There are several advantages of the application of MACBETH to the coalition formation model. In particular, this application increases considerably the applicability of the model. A party can usually order all policies on issues (including all majority coalitions), taking into account their attractiveness to this party. But it seems to be much more difficult to a party to say to which desirability set a given policy on a certain issue belongs, in particular, since these sets are described by precise values of the desirability weights. In general, it will be much easier to a party to judge the difference of attractiveness between each two policies on a given issue. Moreover, the MACBETH software warns when the matrix of judgements of a party becomes inconsistent, and it even gives suggestions to make it consistent. Another problem related to the original coalition formation model concerns determining in reality the desirability weights themselves. We do not have this problem when applying the MACBETH approach. Also assuming the same scale (i.e., the same degrees of desirability) for all issues seems to be a drawback of the original model of coalition formation. In the original model of a stable government, a party had to be very precise in expressing its preferences. In particular, this model could not cover the case of fuzzy preferences. When judging differences of attractiveness, a party does not have to be so sure about its preferences. Hence, in this sense, this new approach allows fuzziness as well. Sometimes many stable governments exist. In this case, negotiations between parties on choosing one stable government may be introduced. For an application of bargaining theory to the coalition formation model defined in [14], see [13]. The notion of absolute judgement has also been used in Saaty’s Analytical Hierarchy Process (AHP). The AHP method is described in [15] and [16]. There are fundamental differences between MACBETH and AHP. In particular, in the MACBETH approach, the absolute judgements concern differences of attractiveness, whereas in Saaty’s method they concern ratios of priority, or of importance. For a critical analysis of the AHP method, see [4].
References [1] R.J. Aumann, Acceptable points in general cooperative n-person games. In: A.W. Tucker and R.D. Luce (eds.), Contributions to the theory of games IV, Princeton University Press, 1959, pp. 287-324. [2] R.J. Aumann, J.H. Dreze, Cooperative games with coalition structure, International Journal of Game Theory, Vol. 3, 1974, pp. 217-237. [3] C.A. Bana e Costa, J.C. Vansnick, The MACBETH approach: basic ideas, software and an application, in: N. Meskens, M. Roubens (Eds.), Advances in Decision Analysis, Kluwer Academic Publishers, Dordrecht, 1999, pp. 131-157. [4] C.A. Bana e Costa, J.C. Vansnick, A fundamental criticism to Saaty’s use of the eigenvalue procedure to derive priorities, LSE OR Working Paper 01.42, 2001. [5] C.A. Bana e Costa, J.M. De Corte, J.C. Vansnick, MACBETH, LSE OR Working Paper 03.56, 2003.
15 [6] A. van Deemen, Coalition Formation and Social Choice, Kluwer Academic Publishers, Dordrecht, 1997. [7] P. Eklund, A. Rusinowska, H. de Swart, A consensus model of political decisionmaking, (www.uvt.nl/faculteiten/fww/onderzoek/skt/pdf/consensus.pdf), 2004. [8] Ana Meca-Martinez, J. Sanchez Soriano, I. Garcia-Jurando and S. Tijs, Strong equilibrium in claim games corresponding to convex games, International Journal of Game Theory, Vol. 27, 1988, pp. 211-217. [9] D.B.G. Meister, K. W. Hipel, Coalition formation, Journal of Scientific and Industrial Research, Vol. 51, 1992, pp. 612-625. [10] R.B. Myerson, Conference structure and fair allocation rules, International Journal of Game Theory, Vol. 9, 1980, pp. 211-217. [11] R.B. Myerson, Graphs and cooperation in games, Mathematics of Operations Research, Vol. 2, 1977, pp. 225-229 [12] G. Owen, Values of games with a priori unions. In: R. Henn and O. Moeschlin (eds.), Mathematical economics and game theory, Springer Verlag, 1977, pp. 76-88. [13] A. Rusinowska, H. de Swart, Negotiating a stable government - an application of bargaining theory to a coalition formation model, (www.uvt.nl/faculteiten/fww/onderzoek/skt/pdf/urbino2003.pdf), 2003. [14] A. Rusinowska, H. de Swart, J.W. van der Rijt, A new model of coalition formation, forthcoming in Social Choice and Welfare, 2004. [15] T.L. Saaty, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977) 234-281. [16] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, 1980. [17] M. Slikker, A. van den Nouweland, Social and economic networks in cooperative game theory, Theory and Decision Library, Series C, Vol. 27, Kluwer Academic Publishers, Dordrecht, 2001. [18] M. de Vries, Governing with Your Closest Neighbour: an Assessment of Spatial Coalition Formation Theories, Print Partners Ipskamp, 1999.
Appendix Table 17: Matrix of judgements of difference of attractiveness in P0B
BC AB AC ABC neutral0B
BC goodB AC ABC neutral0B 0 = AB no very weak strong extreme extreme no moderate moderate moderate no very weak very weak no very weak no
Table 18: The values of the majority coalitions for party B coalition p0 = V0B (p0 ) =
B BC goodB 0 = AB AC ABC neutral0
130.02
100.00
30.01 10.00
0.00
16
Table 19: Matrix of judgements of difference of attractiveness in P1B (2)
goodB 1 (2) p1 (1) p1 (5) p1 (4) p1 (3) p1
(1)
(5)
(4)
(3)
goodB neutral1B = p1 p1 p1 p1 1 p1 no weak moderate moderate strong very strong no moderate moderate strong strong no very weak moderate strong no weak strong no moderate no
Table 20: The values of the policies on issue 1 for party B policy p1 = V1B (p1 ) =
(2)
goodB 1 p1
(1)
neutral1B = p1
100.00 66.68
(5)
p1
(4)
(3)
p1
p1
−16.68 −66.68 −150.00
0.00
Table 21: Matrix of judgements of difference of attractiveness in P2B goodB r y s neutral2B = x w 2 =z z no very weak very weak moderate moderate very strong r no very weak weak moderate strong y no weak weak strong s no very weak moderate x no weak w no
Table 22: The values of the policies on issue 2 for party B policy p2 = V2B (p2 ) =
goodB 2 =z 100.00
r
y
s
neutral2B = x
w
0.00
−66.67
83.33 66.67 16.67
Table 23: Matrix of difference of attractiveness between the reference profiles for party B B B B [Crit.B 0 ] [Crit.2 ] [Crit.1 ] [neutral ] no weak weak strong no weak strong no moderate [neutral ] no
[Crit.B 0 ] [Crit.B 2 ] [Crit.B 1 ] B
17
Table 24: The scaling constants for party B B B B measure [Crit.B j ] = [Crit.0 ] [Crit.2 ] [Crit.1 ] V B ([Crit.B 41.67 33.33 25.00 j ]) = αjB = 0.4167 0.3333 0.25
Table 25: Matrix of judgements of difference of attractiveness in P0C C goodC 0 AC BC neutral0
goodC 0 AC BC neutral0C AB ABC
no
AB
ABC
2 3 5 5 5 no weak moderate moderate moderate no moderate moderate moderate no weak weak no very weak no
Table 26: The values of the majority coalitions for party C goodC 0 AC
coalition p0 = V0C (p0 ) =
BC neutral0C
100.00 61.54 46.15
AB
ABC
−15.38 −23.08
0.00
Table 27: Matrix of judgements of difference of attractiveness in P1C (1)
goodC 1 = p1 (1)
p1 (4) p1 (2) p1 (5) p1 (3) p1
no
(4)
(2)
p1
(5)
p1
(3)
neutral1C = p1
p1
moderate strong very strong no moderate moderate no weak no
extreme very strong moderate weak no
Table 28: The values of the policies on issue 1 for party C policy p1 = V1C (p1 ) =
(1)
goodC 1 = p1 100.00
(4)
p1
(2)
p1
(5)
p1
70.00 40.00 20.00
(3)
neutral1C = p1 0.00
18
Table 29: Matrix of judgements of difference of attractiveness in P2C goodC s r z y neutral2C = x 2 =w w no weak weak strong extreme extreme s no weak weak strong strong r no weak moderate strong z no weak weak y no very weak x no
Table 30: The values of the policies on issue 2 for party C policy p2 = V2C (p2 ) =
goodC 2 =w 100.00
s
r
z
y neutral2C = x
75.00 58.33 37.50 8.33
0.00
Table 31: Matrix of difference of attractiveness between the reference profiles for party C C [Crit.C [Crit.C [neutralC ] 1 ] [Crit.2 ] 0] no very weak very strong extreme no strong very strong no very weak [neutral ] no
[Crit.C 1] [Crit.C 2] [Crit.C 0] C
Table 32: The scaling constants for party C C C C measure [Crit.C j ] = [Crit.1 ] [Crit.2 ] [Crit.0 ] C C V ([Crit.j ]) = 50.00 41.67 8.33 αjC = 0.50 0.4167 0.0833
19
Table 33: Utilities of the governments g (1)
g 1 = (AB, p1 , x) (1) g 4 = (ABC, p1 , x) (1) g 7 = (BC, p1 , y) (1) 10 g = (AC, p1 , z) (1) 13 g = (AB, p1 , r) (1) 16 g = (ABC, p1 , r) (1) g 19 = (BC, p1 , s) (1) g 22 = (AC, p1 , w) (2) 25 g = (AB, p1 , x) (2) 28 g = (ABC, p1 , x) (2) 31 g = (BC, p1 , y) (2) g 34 = (AC, p1 , z) (2) g 37 = (AB, p1 , r) (2) 40 g = (ABC, p1 , r) (2) 43 g = (BC, p1 , s) (2) g 46 = (AC, p1 , w) (3) 49 g = (AB, p1 , x) (3) g 52 = (ABC, p1 , x) (3) g 55 = (BC, p1 , y) (3) 58 g = (AC, p1 , z) (3) g 61 = (AB, p1 , r) (3) 64 g = (ABC, p1 , r) (3) g 67 = (BC, p1 , s) (3) g 70 = (AC, p1 , w) (4) 73 g = (AB, p1 , x) (4) 76 g = (ABC, p1 , x) (4) 79 g = (BC, p1 , y) (4) g 82 = (AC, p1 , z) (4) g 85 = (AB, p1 , r) (4) 88 g = (ABC, p1 , r) (4) 91 g = (BC, p1 , s) (4) g 94 = (AC, p1 , w) (5) 97 g = (AB, p1 , x) (5) g 100 = (ABC, p1 , x) (5) g 103 = (BC, p1 , y) (5) 106 g = (AC, p1 , z) (5) g 109 = (AB, p1 , r) (5) 112 g = (ABC, p1 , r) (5) g 115 = (BC, p1 , s) (5) g 118 = (AC, p1 , w)
V (g)
g
(113.08, 41.67, 48.72) (38.08, 4.17, 48.08) (6.92, 76.4, 57.32) (63.85, 45.84, 70.75) (103.47, 69.44, 73.025) (28.46, 31.94, 72.38) (−2.31, 59.74, 85.1) (56.15, −9.72, 96.8) (109.33, 58.34, 18.72) (34.33, 20.84, 18.08) (3.17, 93.07, 27.32) (60.1, 62.51, 40.75) (99.72, 86.11, 43.03) (24.71, 48.61, 42.38) (−6.06, 76.41, 55.1) (52.4, 6.95, 66.8) (75.58, 4.17, −1.28) (0.58, −33.33, −1.92) (−30.58, 38.9, 7.32) (26.35, 8.34, 20.75) (65.97, 31.94, 23.03) (−9.04, −5.56, 22.38) (−39.81, 22.24, 35.1) (18.65, −47.22, 46.8) (90.58, 25, 33.72) (15.58, −12.5, 33.08) (−15.58, 59.73, 42.32) (41.35, 29.17, 55.75) (80.97, 52.77, 58.03) (5.96, 15.27, 57.38) (−24.81, 43.07, 70.1) (33.65, −26.39, 81.8) (83.08, 37.5, 8.72) (8.08, 0, 8.08) (−23.08, 72.23, 17.32) (33.85, 41.67, 30.75) (73.47, 65.27, 33.03) (−1.54, 27.77, 32.38) (−32.31, 55.57, 45.1) (26.15, −13.89, 56.8)
g 2 = (AC, p1 , x) (1) g 5 = (AB, p1 , y) (1) g 8 = (ABC, p1 , y) (1) 11 g = (BC, p1 , z) (1) g 14 = (AC, p1 , r) (1) 17 g = (AB, p1 , s) (1) g 20 = (ABC, p1 , s) (1) g 23 = (BC, p1 , w) (2) 26 g = (AC, p1 , x) (2) 29 g = (AB, p1 , y) (2) 32 g = (ABC, p1 , y) (2) g 35 = (BC, p1 , z) (2) g 38 = (AC, p1 , r) (2) 41 g = (AB, p1 , s) (2) 44 g = (ABC, p1 , s) (2) 47 g = (BC, p1 , w) (3) g 50 = (AC, p1 , x) (3) g 53 = (AB, p1 , y) (3) g 56 = (ABC, p1 , y) (3) 59 g = (BC, p1 , z) (3) g 62 = (AC, p1 , r) (3) 65 g = (AB, p1 , s) (3) g 68 = (ABC, p1 , s) (3) g 71 = (BC, p1 , w) (4) 74 g = (AC, p1 , x) (4) 77 g = (AB, p1 , y) (4) 80 g = (ABC, p1 , y) (4) g 83 = (BC, p1 , z) (4) g 86 = (AC, p1 , r) (4) 89 g = (AB, p1 , s) (4) 92 g = (ABC, p1 , s) (4) 95 g = (BC, p1 , w) (5) g 98 = (AC, p1 , x) (5) g 101 = (AB, p1 , y) (5) g 104 = (ABC, p1 , y) (5) 107 g = (BC, p1 , z) (5) g 110 = (AC, p1 , r) (5) 113 g = (AB, p1 , s) (5) g 116 = (ABC, p1 , s) (5) g 119 = (BC, p1 , w)
(1)
V (g)
g
(68.08, 12.51, 55.13) (111.93, 63.89, 52.19) (36.92, 26.39, 51.55) (3.85, 87.51, 69.47) (58.46, 40.28, 79.43) (102.7, 47.23, 79.97) (27.69, 9.72, 79.33) (−3.85, 31.96, 95.51) (64.33, 29.18, 25.13) (108.18, 80.56, 22.19) (33.17, 43.06, 21.55) (0.1, 104.18, 39.47) (54.71, 56.95, 49.43) (98.95, 63.9, 49.97) (23.94, 26.39, 49.33) (−7.6, 48.63, 65.51) (30.58, −24.99, 5.13) (74.43, 26.39, 2.19) (−0.58, −11.11, 1.55) (−33.65, 50.01, 19.47) (20.96, 2.78, 29.43) (65.2, 9.73, 29.97) (−9.81, −27.78, 29.33) (−41.35, −5.54, 45.51) (45.58, −4.16, 40.13) (89.43, 47.22, 37.19) (14.42, 9.72, 36.55) (−18.65, 70.84, 54.47) (35.96, 23.61, 64.43) (80.2, 30.56, 64.97) (5.19, −6.95, 64.33) (−26.35, 15.29, 80.51) (38.08, 8.34, 15.13) (81.93, 59.72, 12.19) (6.92, 22.22, 11.55) (−26.15, 83.34, 29.47) (28.46, 36.11, 39.43) (72.7, 43.06, 39.97) (−2.31, 5.55, 39.33) (−33.85, 27.79, 55.51)
g 3 = (BC, p1 , x) (1) g 6 = (AC, p1 , y) (1) g 9 = (AB, p1 , z) (1) 12 g = (ABC, p1 , z) (1) 15 g = (BC, p1 , r) (1) g 18 = (AC, p1 , s) (1) g 21 = (AB, p1 , w) (1) g 24 = (ABC, p1 , w) (2) 27 g = (BC, p1 , x) (2) g 30 = (AC, p1 , y) (2) 33 g = (AB, p1 , z) (2) g 36 = (ABC, p1 , z) (2) g 39 = (BC, p1 , r) (2) 42 g = (AC, p1 , s) (2) g 45 = (AB, p1 , w) (2) 48 g = (ABC, p1 , w) (3) 51 g = (BC, p1 , x) (3) g 54 = (AC, p1 , y) (3) g 57 = (AB, p1 , z) (3) 60 g = (ABC, p1 , z) (3) 63 g = (BC, p1 , r) (3) g 66 = (AC, p1 , s) (3) g 69 = (AB, p1 , w) (3) g 72 = (ABC, p1 , w) (4) 75 g = (BC, p1 , x) (4) g 78 = (AC, p1 , y) (4) 81 g = (AB, p1 , z) (4) g 84 = (ABC, p1 , z) (4) g 87 = (BC, p1 , r) (4) 90 g = (AC, p1 , s) (4) g 93 = (AB, p1 , w) (4) 96 g = (ABC, p1 , w) (5) 99 g = (BC, p1 , x) (5) g 102 = (AC, p1 , y) (5) g 105 = (AB, p1 , z) (5) 108 g = (ABC, p1 , z) (5) 111 g = (BC, p1 , r) (5) g 114 = (AC, p1 , s) (5) g 117 = (AB, p1 , w) (5) g 120 = (ABC, p1 , w)
V (g) (1)
(8.08, 54.18, 53.84) (66.92, 34.73, 58.6) (108.85, 75, 64.35) (33.85, 37.5, 63.7) (−1.54, 81.95, 78.15) (57.69, 18.06, 86.38) (101.16, 19.45, 90.39) (26.15, −18.05, 89.75) (4.33, 70.85, 23.84) (63.17, 51.4, 28.6) (105.1, 91.67, 34.35) (30.1, 54.17, 33.7) (−5.29, 98.62, 48.15) (53.94, 34.73, 56.38) (97.41, 36.12, 60.39) (22.4, −1.38, 59.75) (−29.42, 16.68, 3.84) (29.42, −2.77, 8.6) (71.35, 37.5, 14.35) (−3.65, 0, 13.7) (−39.04, 44.45, 28.15) (20.19, −19.44, 36.38) (63.66, −18.05, 40.39) (−11.35, −55.55, 39.75) (−14.42, 37.51, 38.84) (44.42, 18.06, 43.6) (86.35, 58.33, 49.35) (11.35, 20.83, 48.7) (−24.04, 65.28, 63.15) (35.19, 1.39, 71.38) (78.66, 2.78, 75.39) (3.65, −34.72, 74.75) (−21.92, 50.01, 13.84) (36.92, 30.56, 18.6) (78.85, 70.83, 24.35) (3.85, 33.33, 23.7) (−31.54, 77.78, 38.15) (27.69, 13.89, 46.38) (71.16, 15.28, 50.39) (−3.85, −22.22, 49.75)
2
University of Li`ege, Institute of Mathematics, B37, B-4000 Li`ege Facult´e Polytechnique de Mons, Rue de Houdain 9, B7000 Mons, Belgium [email protected] 3 Radboud University Nijmegen, Nijmegen School of Management P.O. Box 9108, 6500 HK Nijmegen, The Netherlands 4 Warsaw School of Economics, Department of Mathematical Economics Al. Niepodleglosci 162, 02-554 Warsaw, Poland [email protected] 5 Tilburg University, Department of Philosophy, P.O. Box 90153 5000 LE Tilburg, The Netherlands [email protected]
Corresponding author: HARRIE DE SWART Keywords: Game Theory, MACBETH, government, majority coalition, policy on issue Abstract. In the paper, we present an application of the MACBETH approach to a certain model of coalition formation. We apply the MACBETH technique to quantify the attractiveness and repulsiveness of possible governments to parties. We use this method to calculate the utilities of governments to parties. Based on these utilities, stable governments are determined. In the paper, an adequate example is presented.
1
Introduction
The literature concerning models of coalition formation is very extensive. A broad overview of the existing coalition formation theories is given, for instance, in [18] or [6]. The problem of coalition formation was also, among others, tackled by [17], [1], [2], [8], [9], [10], [11], and [12]. For further references to coalition formation see [14], which presents one of the recent models of multi-dimensional coalition formation. The main concepts of this model are the concepts of a feasible policy/coalition/government and a stable policy/coalition/government, where a government is defined as a pair consisting of a majority coalition and a policy supported by this coalition. In general, there are several independent ?
??
The authors wish to thank gratefully Jean-Claude Vansnick and Jean-Marie de Corte for very useful discussion and demonstration of the MACBETH software, and an anonymous referee for very valuable suggestions for improvements. This author gratefully acknowledges support by COST Action 274, TARSKI.
2
policy issues on which a government will have to decide. Hence, a policy consists of policies on all the issues. Parties are assumed to have preferences regarding each majority coalition and each policy on each given issue. Based on semantic judgements concerning the attractiveness of available alternatives, MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique) is an interactive approach to quantify the attractiveness of each alternative, in such a way that the measurement scale constructed is an interval scale. An overview and some applications of the MACBETH method are presented, for instance, in [3], [5], and on the web site (www.m-macbeth.com), where in particular, one may download the MACBETH software and follow the online tutorial. In this paper, we will apply the MACBETH technique to the model of a stable government ([14]). This method will help to determine the utilities (the values) of governments to parties. Since the notion of a stable government is based on these utilities, the application of this technique to the coalition formation model is very important both from a theoretical and a practical point of view. The paper is organized as follows. In Section 2, the model of a stable government is recapitulated. In Section 3, we apply the MACBETH technique to this model of coalition formation. In Section 4, a simple example illustrating the application of MACBETH to the coalition formation model of a stable government is constructed. In Section 5, we present conclusions.
2
The model of coalition formation
In this section, we present a brief description of the model of coalition formation ([14]) to which we apply the MACBETH decision support system. We highlight the notions introduced in [14] that remain the same in the model presented in this paper. However, we also mention which elements of the model introduced in [14] are changed in the present model. Let N be the set of all parties. There are M ≥ 1 independent policy issues on which a government will have to decide. For each policy issue j ∈ {1, ..., M }, there are mj ≥ 1 dependent policy sub-issues j(1), ..., j(mj ). Let P and Pj denote a policy space and a policy sub-space on issue j respectively. A policy is represented by a tuple p = (p1 , ..., pM ) ∈ P , where pj ∈ Pj is a policy on issue j. Only majority coalitions are entitled to form a government. A majority coalition will be denoted by p0 , and the set of all majority coalitions will be denoted by P0 . A government is defined as a pair consisting of a majority coalition and a policy proposed by this coalition, that is, as a pair g = (p0 , p), where p0 ∈ P0 and p = (p1 , ..., pM ) ∈ P . The main difference between the model presented in [14] and the model proposed in this paper concerns the preferences of the parties and the definition of utilities of governments to parties. In [14], parties were supposed to have preferences on all policies on all issues and on all majority coalitions. Each party was allowed to give one of L+2 (L ≥ 1) qualifications to a policy on a certain
3
issue and to a majority coalition: desirable of degree l (1 ≤ l ≤ L), acceptable (desirable of degree 0), or unacceptable. The desirability weights d0 , ..., dL were assumed to be given. Based on these weights and on parties preferences, for each party the utility (the value) of a policy on a given issue, of a policy, of a majority coalition, and finally, of a government were defined. In this paper, we define these utilities differently and in a more realistic way. In order to determine the utility (the value) V i (g) of government g to party i ∈ N , we will apply the MACBETH approach. In [14], the notion of feasibility was introduced. A policy was said to be feasible if there was a majority coalition such that the policy on each issue was desirable (of a certain non-negative degree) for each party in that coalition. By a feasible coalition we meant a majority coalition which was desirable for each party in that coalition. Finally, a feasible government consisted of a feasible coalition and a policy feasible for this coalition. Moreover, the utilities of a policy/coalition/government were defined in such a way that a government was feasible if and only if its value was non-negative. In this paper, we do not consider feasibility, since the value of a government to a party will be determined differently. In [14], the concepts of dominance and stability were defined. These notions remain almost the same in the present model. By G∗ we denote the set of all governments. Definition 1 A government g 0 = (p00 , p0 ) ∈ G∗ dominates a government g = (p0 , p) ∈ G∗ (g 0 g) if ∀i ∈ p00 [V i (g 0 ) ≥ V i (g)] ∧ ∃i ∈ p00 [V i (g 0 ) > V i (g)].
(1)
Definition 2 A government g ∈ G∗ is said to be stable if there is no government dominating g, that is, if ¬ ∃g 0 ∈ G∗ [g 0 g]. (2) Let us point out that the relation of dominance of Definition 1 is not necessarily acyclic, because the set of indices in (1) depends on g 0 . Consequently, it is possible that no stable government exists in the sense of Definition 2. This will not be the case with Definitions 3 and 4. In [14], a stable government was defined as a feasible government which was dominated by no feasible government. As mentioned before, we do not incorporate feasibility in the present model. In [7], the notion of stability in a given coalition was considered. We generalize this notion. Definition 3 Let G ⊆ G∗ and p000 ∈ P0 . We say that a government g 0 = g) (p00 , p0 ) ∈ G dominates a government g = (p0 , p) ∈ G in coalition p000 (g 0 G p00 0 if ∀i ∈ p000 [V i (g 0 ) ≥ V i (g)] ∧ ∃i ∈ p000 [V i (g 0 ) > V i (g)]. (3)
4
Definition 4 A government g ∈ G is said to be stable in coalition p000 with respect to G ⊆ G∗ if there is no government in G dominating g in p000 , that is, if ¬ ∃g 0 ∈ G [g 0 G g]. p00 0
(4)
For instance, in the example of Section 4 we will consider stability of a government (p0 , p) in the coalition p0 with respect to the set of all governments formed by that coalition. In [7], the notion of internal stability was also considered. Definition 5 A government g = (p0 , p) ∈ G∗ is said to be internally stable if ¬ ∃(p00 , p0 ) ∈ G∗ [(p00 ⊆ p0 ∨ p0 ⊆ p00 ) ∧ ∀i ∈ p0 ∩ p00 [V i (p00 , p0 ) ≥ V i (p0 , p)] ∧ ∃i ∈ p0 ∩ p00 [V i (p00 , p0 ) > V i (p0 , p)]].
3
(5)
Applying MACBETH to the model
In our model, the set of all governments G∗ is assumed to be finite. We also allow an infinite number of possible governments if only a finite number of governments is judged differently by each party. To give an example, a policy assuming any amount in the interval [20,30] as a budget for education is, in fact, a continuum of policies, and consequently leads to a continuum of governments. Nevertheless, as long as all parties find each amount from [20,30] equally attractive, we can treat this as one policy. In the present model, each party is assumed to have preferences regarding all governments with respect to certain criteria. The criteria are ‘majority coalition’ and all the policy issues. Let C ∗ be the set of all criteria, that is, C ∗ = {0, 1, ..., M }, where criterion 0 concerns the ‘majority coalition’. ‘Majority coalition’ will be also called a policy on issue 0. For each j ∈ C ∗ , each party orders all policies on issue j taking into account the attractiveness of these policies on the given issue. A policy on issue j ∈ C ∗ may be of a negative, zero, or a positive attractiveness to a party. For each issue j, each party is asked to specify two particular references: neutral, defined as ‘neither satisfying nor unsatisfying’, and good, which is more attractive to a party than neutral, and is defined as ‘undoubtedly satisfying’. A neutral policy on issue j for party i is denoted by neutralji , and a good policy on issue j for party i is denoted by goodij . Both neutral and good policies on a given issue may be fictitious. Nevertheless, they are always incorporated into the model: the set Pj ∪ {neutralji , goodij } will be denoted by Pji . For each j ∈ C ∗ and i ∈ N , based on these two references neutralji and goodij , we distinguish: – an unattractive (or repulsive) policy on issue j to party i if it is less attractive to i than neutralji ,
5
– an attractive (or simply attractive) policy on issue j to party i if it is more attractive to i than neutralji , – a very attractive (or outstanding) policy on issue j to party i if it is at least as attractive to i than goodij . For each criterion j ∈ C ∗ , each party i ∈ N is asked to verbally judge the difference of attractiveness between each two policies pj and p0j of Pji , where pj is at least as attractive to i as p0j . When judging, a party has to choose one of the following categories: C0 - no difference of attractiveness C1 - very weak difference of attractiveness C2 - weak difference of attractiveness C3 - moderate difference of attractiveness C4 - strong difference of attractiveness C5 - very strong difference of attractiveness C6 - extreme difference of attractiveness. If a party is unsure about the difference of attractiveness, it may choose the union of several successive categories among these above. Moreover, when comparing two policies on a given issue, an answer ‘I do not know’ is acceptable and appears in the MACBETH software as positive difference of attractiveness. However, the more preference information is provided, the greater is the scale’s level of accuracy. For each party i ∈ N and each criterion j ∈ C ∗ , the M-MACBETH software allows to associate to each policy pj ∈ Pji a real number Uji (pj ) which, in the particular case where there is no hesitation about the difference of attractiveness, satisfies the following rules (see [3] or [5]): ∀i ∈ N ∀j ∈ C ∗ ∀pj , p0j ∈ Pji : Uji (pj ) > Uji (p0j ) ⇔ pj is more attractive to i than p0j
(6)
i 0 ∀i ∈ N ∀j ∈ C ∗ ∀k, k 0 ∈ {1, 2, 3, 4, 5, 6} ∀pj , p0j , p00j , p000 j ∈ Pj with (pj , pj ) ∈ Ck 0 i i 0 i 00 i 000 and (p00j , p000 j ) ∈ Ck0 : k ≥ k + 1 ⇒ Uj (pj ) − Uj (pj ) > Uj (pj ) − Uj (pj ) (7)
This numerical scale is essentially obtained by linear programming and is called the MACBETH basic scale. Of course, the MACBETH scale exists if and only if it is possible to satisfy rules (6) and (7); in such a case the matrix of judgements is called consistent. If it is impossible to satisfy rules (6) and (7), a message appears on the screen (‘inconsistent judgements’), inviting the party to revise the judgements. As we will illustrate in the next section, the MMACBETH software provides tools to obtain a consistent matrix of judgements.
6
The basic MACBETH scale, as well as each scale obtained by a positive linear transformation, are pre-cardinal scales. In order to obtain a cardinal scale, a discussion with the party in question around the scale takes place. Apart from presenting the pre-cardinal MACBETH scale in a numerical way, it can also be presented in a graphical way (‘thermometer’). By this, the party has the possibility to modify the positioning of the policies on the given issue such that the relative distances between the policies on the given issue reflect the relative distances of attractiveness that the party deems to exist between these policies on the given issue. In the MACBETH software ([5]), when a party selects with the mouse a policy on a given issue, an interval appears around this policy. By moving the mouse, a party can modify the position of the selected policy on the given issue, but only within this interval. By modifying the position of the policy on the given issue inside the interval, a party obtains a new positioning of all policies on a given issue such that both conditions (6) and (7) are still satisfied. When a party thinks that finally the scale adequately represents the relative magnitude of the judgements, we have the cardinal scale and the (final, agreed) values of all policies on the given issue. Let Vji (pj ) denote the (final) value of policy pj ∈ Pji on issue j ∈ C ∗ to party i ∈ N . We can assume that ∀i ∈ N ∀j ∈ C ∗ : Vji (goodij ) = 100 and Vji (neutralji ) = 0.
(8)
We get then negative values for all repulsive policies on issue j ∈ C ∗ to party i, and values greater than 100 for all outstanding policies on issue j to party i. Finally, we want to measure the global attractiveness of each government, that is, the attractiveness of each government taking all criteria into account. We adopt the following aggregation procedure: V i (g) = V i (p0 , p1 , ..., pM ) =
M X
αji · Vji (pj ),
(9)
j=0
with
i α0i , α1i , ..., αM
≥ 0, and
M X
αji = 1.
(10)
j=0
The coefficients (αji )j∈C ∗ are scaling constants (expressing the relative weights party i assigns to the different criteria) which can also be calculated with the M-MACBETH software. For each party i ∈ N , let us consider the following reference profiles i [neutrali ] = (neutral0i , neutral1i , ..., neutralM ) i [Crit.i0 ] = (goodi0 , neutral1i , ..., neutralM ) i [Crit.i1 ] = (neutral0i , goodi1 , ..., neutralM )
...
7
[Crit.iM ] = (neutral0i , neutral1i , ..., goodiM ). Observe that, for each j ∈ C ∗ , the difference in attractiveness between [Crit.ij ] and [neutrali ] corresponds to the added value of the ‘swing’ from neutralji to goodij . Each party ranks the reference profiles in decreasing order of attractiveness and, using categories C0 - C6 , judges the difference of attractiveness between each two reference profiles, where the first one is more attractive than the second one. After the adjustment of the scale proposed by the software (MACBETH scale), we obtain an interval scale V i which measures the overall attractiveness of the reference profiles. We can assume that, for each i ∈ N V i ([neutrali ]) = 0 and
M X
V i ([Crit.ij ]) = 100.
(11)
j=0
Taking into account (8) and (9), we deduce that, for each i ∈ N and j ∈ C ∗ αji =
V i ([Crit.ij ]) . 100
(12)
Finally, using the aggregation procedure given in (9), one may calculate the values of all governments to each party. Using formulae (1) - (5), stable governments may be identified if there are any.
4
The Example
In order to illustrate the application of the MACBETH software to the coalition formation model, we construct the following simple example. There are only three parties in parliament: A, B, and C, i.e., N = {A, B, C}, and at least two parties are needed to form a majority coalition. Hence, P0 = {AB, AC, BC, ABC}. There are two issues (M = 2) on which a government will have to decide. Issue 1 is ‘Health Care’, and consists of two sub-issues: ‘Introducing a Reform of the Health Service’ with the set of policies on this sub-issue equal to {yes, no}, and ‘Increase of the Health Expenditure’, with the set of all policies on this sub-issue equal to [0,30]. This set is divided into three intervals: [0,10), [10,20), [20,30], and it is assumed that each party finds two amounts belonging to the same interval equally attractive. These two sub-issues are dependent on each other in the sense that introducing a reform of the health service requires an amount of at least 10 for the increase of the health expenditure. Issue 2 is ‘Foreign Policy’, and there
8
are 6 possible foreign policies: x, y, z, r, s, w. More precisely, the following issues are assumed: Issue 1: Health Care: Sub-issue 1(1): Introducing a Reform of the Health Service Sub-issue 1(2): Increase of the Health Expenditure Issue 2: Foreign Policy The policy sub-spaces are (1)
(2)
(3)
(4)
(5)
P1 = {p1 , p1 , p1 , p1 , p1 }, where (1) p1 = (yes, [10, 20)), (3)
(2)
p1 = (yes, [20, 30]) (4)
p1 = (no, [0, 10)), and
(5)
p1 = (no, [10, 20)),
p1 = (no, [20, 30])
P2 = {x, y, z, r, s, w}. In Tables 1 till 4 we assume that each party i ∈ {A, B, C} has indicated a decreasing order of preference with respect to the majority coalitions, to the policies on issue 1 and 2, and with respect to reference profiles [neutrali ], [Crit.i0 ], [Crit.i1 ], [Crit.i2 ]. Table 1 presents a decreasing order of all majority coalitions for all parties. The neutral and good majority coalitions are also included. Table 1: Decreasing order of all majority coalitions party
decreasing order of the majority coalitions
A
AB
goodA 0
B
BC
goodB 0 = AB
C
goodC 0
AC
AC
BC
ABC = neutral0A AC
neutral0B
ABC
neutral0C
BC
AB
ABC
Table 2 presents a decreasing order of all policies on issue 1 for all parties. The neutral and good policies on issue 1 are also included. Table 2: Decreasing order of all policies on issue 1 party A B C
decreasing order of the policies on issue 1 (1)
goodA 1 = p1 goodB 1
(2)
p1
(1)
goodC 1 = p1
(2)
p1
(5)
(4)
neutral1A = p1
p1
(5)
(1)
neutral1B = p1 (4)
p1
(2)
p1
p1 (5)
p1
(4)
p1
(3)
p1
(3)
p1
(3)
neutral1C = p1
9
Table 3 presents a decreasing order of all policies on issue 2 for all parties, including the neutral and good policies on issue 2. Table 3: Decreasing order of all policies on issue 2 party
decreasing order of the policies on issue 2 goodA 2
x
y
z
neutral2A
B
goodB 2 =z
r
y
s
neutral2B = x
C
goodC 2 =w
s
r
z
y
A
r
s
w w
neutral2C = x
Additionally, each party i ∈ {A, B, C} will have to judge the difference of attractiveness between each two reference profiles: [neutrali ] = (neutral0i , neutral1i , neutral2i ) [Crit.i0 ] = (goodi0 , neutral1i , neutral2i ) [Crit.i1 ] = (neutral0i , goodi1 , neutral2i ) [Crit.i2 ] = (neutral0i , neutral1i , goodi2 ). Table 4 shows the decreasing orders of these reference profiles for all parties. Table 4: Decreasing order of the reference profiles party decreasing order of the profiles A [Crit.A ] [Crit.A [Crit.A [neutralA ] 0 1] 2] B B B B [Crit.0 ] [Crit.2 ] [Crit.1 ] [neutralB ] C C C C [Crit.1 ] [Crit.2 ] [Crit.0 ] [neutralC ] Let us consider party A, and the policies on issue 1. Party A judges the difference of attractiveness for all the policies on issue 1. Suppose party A proposes the matrix of judgements as presented in Table 5. Table 5: Matrix of judgements of the difference of attractiveness in P1A - inconsistent judgements (1)
p1 = goodA 1 (1)
p1 (2) p1 (4) p1 (5) p1 (3) p1
no
(2)
p1
(4)
p1
very weak moderate no moderate no
(5)
(3)
p1 = neutral1A strong strong weak no
p1
extreme very strong very weak positive no
But then the software immediately sends the information:
10
‘Inconsistent judgements! Macbeth has found 2 ways to obtain a consistent matrix of judgements by 1 category change.’ MACBETH proposes either to decrease the difference of attractiveness between (4) (5) p1 and p1 which is now weak, or to increase the difference of attractiveness (4) (3) between p1 and p1 which is now very weak. The software also shows the contradictions (see Table 6). Table 6: Contradictions for matrix presented in Table 5 Problem Difference 1 weak positive 2 extreme weak
Couples Couples (5) (4) (3) − p1 > p 1 − p1 (5) (3) p 1 − p1 > 0 (1) (3) (1) (5) p1 − p1 > p 1 − p1 (4) (5) (4) (3) p1 − p1 > p 1 − p1 (4) p1
(5)
Difference very weak strong very weak
(3)
In problem 1, the first inequality gives p1 −p1 < 0 which is in contradiction (5) (3) to the second inequality p1 − p1 > 0. (5) (3) In problem 2, the first inequality is equivalent to p1 − p1 > 0, but the (5) (3) second inequality is equivalent to p1 − p1 < 0. Suppose that party A follows the second advice of the software, and it changes (4) (3) the difference of attractiveness between p1 and p1 from very weak to weak. The new matrix of judgements is presented in Table 7. Table 7: Matrix of judgements of the difference of attractiveness in P1A - consistent case (1)
(2)
p1 = goodA 1 (1)
p1 (2) p1 (4) p1 (5) p1 (3) p1
no
(4)
p1
(5)
(3)
p1 = neutral1A
p1
very weak moderate no moderate no
strong strong weak no
p1
extreme very strong weak positive no
MACBETH sends then the information ‘Consistent judgements’. Using the ‘thermometer’, party A can adjust the scale. The (final) values of all the policies on issue 1 to party A are presented in Table 8. Table 8: The values of the policies on issue 1 for party A policy p1 = V1A (p1 ) =
(1)
(2)
p1 = goodA 1 p1 100.00
(4)
(5)
p1 p1 = neutral1A
87.50 25
0.00
(3)
p1
−25.00
11
Next, party A judges the difference of attractiveness for all the policies on issue 2. Suppose Table 9 presents the matrix of judgements. Table 9: Matrix of judgements of the difference of attractiveness in P2A goodA 2 x y z neutral2A r s w
goodA x y z neutral2A r s w 2 no v. weak v. weak weak strong v. strong v. strong extreme no v. weak weak moderate strong strong v. strong no v. weak moderate moderate strong strong no weak weak weak moderate no v. weak v. weak weak no v. weak v. weak no v. weak no
The values of all the policies on issue 2, calculated by the MACBETH software and adjusted by party A, are presented in Table 10. Table 10: The values of the policies on issue 2 for party A policy p2 = V2A (p2 ) =
goodA 2
x
y
z
100.00 80.77 69.23 38.47
neutral2A 0.00
r
s
w
−15.38 −23.08 −38.47
Finally, suppose that Table 11 represents the way in which party A judges the difference of attractiveness for all the majority coalitions. Table 11: Matrix of judgements of the difference of attractiveness in P0A AB goodA AC ABC = neutral0A BC 0 AB no very weak moderate very strong extreme goodA no moderate moderate very strong 0 AC no moderate moderate ABC no moderate BC no Again, we use MACBETH to calculate the values of all the majority coalitions. The final values are presented in Table 12. Table 12: The values of the majority coalitions for party A coalition p0 = V0A (p0 ) =
AB
A goodA 0 AC ABC = neutral0
125.01 100.00 50.00
0.00
BC −50.00
Table 13 presents the matrix of the difference of attractiveness between the reference profiles.
12
Table 13: Matrix of difference of attractiveness between the reference profiles for party A A A [Crit.A [Crit.A 0 ] [Crit.1 ] 2 ] [neutral ] [Crit.A no moderate very strong extreme 0] A [Crit.1 ] no weak moderate [Crit.A no very weak 2] [neutralA ] no
Finally, the values of the reference profiles and the scaling constants are presented in Table 14. Table 14: The scaling constants for party A A A A measure [Crit.A j ] = [Crit.0 ] [Crit.1 ] [Crit.2 ] V A ([Crit.A 60.00 30.00 10.00 j ]) = αjA = 0.60 0.30 0.10
We can apply the same procedure for the remaining parties. Tables 17-24 and 25-32 in the Appendix present the results for party B and party C, respectively. Using the aggregation procedure (see definition (9)), one can calculate for each party the values (the utilities) of all the governments. Since there are 4 majority coalitions, 5 policies on issue 1, and 6 policies on issue 2, we have 4 × 5 × 6 = 120 possible governments. Let us calculate the values of several governments. We consider the following four governments: (1) g 1 = (AB, p1 , x) = (AB, (yes, [10, 20)), x) (2)
g 38 = (AC, p1 , r) = (AC, (yes, [20, 30]), r) (4)
g 75 = (BC, p1 , x) = (BC, (no, [10, 20)), x) (5)
g 104 = (ABC, p1 , y) = (ABC, (no, [20, 30]), y) By virtue of (9), and Tables 8, 10, 12, and 14, we get (1)
V A (g 1 ) = α0A · V0A (AB) + α1A · V1A (p1 ) + α2A · V2A (x) = 113.08 (2)
V A (g 38 ) = α0A · V0A (AC) + α1A · V1A (p1 ) + α2A · V2A (r) = 54.71 (4)
V A (g 75 ) = α0A · V0A (BC) + α1A · V1A (p1 ) + α2A · V2A (x) = −14.42 (5)
V A (g 104 ) = α0A · V0A (ABC) + α1A · V1A (p1 ) + α2A · V2A (y) = 6.92. Let V (g) = (V A (g), V B (g), V C (g)) for each g ∈ G∗ . Table 33 presents the values of all 120 governments. It turns out that no government is stable in the sense of Definition 2. Government g 33 , for instance, dominates most of the governments (it does NOT dominate g 1 , g 5 , g 9 , g 25 , g 29 , g 31 and g 39 only), but g 33 is itself dominated by g 39 . Moreover, g 39 dominates also, among others, g 25 , g 29 , and g 31 . Government g 15 dominates, in particular, g 1 , g 5 , and g 9 . Government g 39 is dominated by g 6 . As mentioned before, the relation of dominance of Definition 1 is not necessarily acyclic. For instance, g 39 g 25 , g 25 g 6 and g 6 g 39 .
13
Let us consider stability in a given coalition with respect to the set of all governments formed by that coalition. Table 15 presents the governments stable in coalition p0 with respect to G∗p0 , where p0 ∈ {AB, AC, BC, ABC}, and G∗p0 denotes the set of all governments formed by coalition p0 . Table 15: Governments stable in coalition p0 with respect to G∗p0 , where p0 ∈ {AB, AC, BC, ABC} p0
governments stable in p0 with respect to G∗p0
AB
g 1 , g 5 , g 9 , g 29 , g 33
AC
g 2 , g 6 , g 10 , g 14 , g 18 , g 22
BC
g 11 , g 15 , g 19 , g 23 , g 35 , g 39
ABC
g 4 , g 8 , g 12 , g 16 , g 20 , g 24 , g 32 , g 36 , g 40
Let us consider now internal stability. All governments stable in coalition p0 with respect to G∗p0 , for p0 ∈ {AB, AC, BC} are also internally stable. There are 9 governments stable in the grand coalition with respect to G∗ABC . Nevertheless, none of these governments is internally stable. For instance, both parties A and B find government g 9 more attractive than each of these nine governments (stable in the grand coalition with respect to G∗ABC ). Finally, let us consider stability in a given coalition with respect to G∗ . The set of all governments stable in coalition AB with respect to G∗AB is equal to the set of all governments stable in AB with respect to G∗ . The analogue is not the case for the remaining coalitions. For instance, g 2 , which is stable in AC with respect to G∗AC , is dominated in AC with respect to G∗ (for instance, by g 17 ). Also g 22 formed by AC is stable in coalition BC (and in coalition ABC) with respect to G∗ . Table 16 presents the governments stable in coalition p0 with respect to G∗ , where p0 ∈ {AB, AC, BC, ABC}. Table 16: Governments stable in coalition p0 with respect to G∗ , where p0 ∈ {AB, AC, BC, ABC} p0
governments stable in p0 with respect to G∗
AB
g 1 , g 5 , g 9 , g 29 , g 33
AC
g 1 , g 5 , g 9 , g 13 , g 17 , g 21 , g 22
BC
g 11 , g 15 , g 19 , g 22 , g 23 , g 35 , g 39
ABC
g 1 , g 5 , g 7 , g 9 , g 11 , g 13 , g 15 , g 17 , g 19 g 21 , g 22 , g 23 , g 29 , g 31 , g 33 , g 35 , g 37 , g 39
14
5
Conclusions
There are several advantages of the application of MACBETH to the coalition formation model. In particular, this application increases considerably the applicability of the model. A party can usually order all policies on issues (including all majority coalitions), taking into account their attractiveness to this party. But it seems to be much more difficult to a party to say to which desirability set a given policy on a certain issue belongs, in particular, since these sets are described by precise values of the desirability weights. In general, it will be much easier to a party to judge the difference of attractiveness between each two policies on a given issue. Moreover, the MACBETH software warns when the matrix of judgements of a party becomes inconsistent, and it even gives suggestions to make it consistent. Another problem related to the original coalition formation model concerns determining in reality the desirability weights themselves. We do not have this problem when applying the MACBETH approach. Also assuming the same scale (i.e., the same degrees of desirability) for all issues seems to be a drawback of the original model of coalition formation. In the original model of a stable government, a party had to be very precise in expressing its preferences. In particular, this model could not cover the case of fuzzy preferences. When judging differences of attractiveness, a party does not have to be so sure about its preferences. Hence, in this sense, this new approach allows fuzziness as well. Sometimes many stable governments exist. In this case, negotiations between parties on choosing one stable government may be introduced. For an application of bargaining theory to the coalition formation model defined in [14], see [13]. The notion of absolute judgement has also been used in Saaty’s Analytical Hierarchy Process (AHP). The AHP method is described in [15] and [16]. There are fundamental differences between MACBETH and AHP. In particular, in the MACBETH approach, the absolute judgements concern differences of attractiveness, whereas in Saaty’s method they concern ratios of priority, or of importance. For a critical analysis of the AHP method, see [4].
References [1] R.J. Aumann, Acceptable points in general cooperative n-person games. In: A.W. Tucker and R.D. Luce (eds.), Contributions to the theory of games IV, Princeton University Press, 1959, pp. 287-324. [2] R.J. Aumann, J.H. Dreze, Cooperative games with coalition structure, International Journal of Game Theory, Vol. 3, 1974, pp. 217-237. [3] C.A. Bana e Costa, J.C. Vansnick, The MACBETH approach: basic ideas, software and an application, in: N. Meskens, M. Roubens (Eds.), Advances in Decision Analysis, Kluwer Academic Publishers, Dordrecht, 1999, pp. 131-157. [4] C.A. Bana e Costa, J.C. Vansnick, A fundamental criticism to Saaty’s use of the eigenvalue procedure to derive priorities, LSE OR Working Paper 01.42, 2001. [5] C.A. Bana e Costa, J.M. De Corte, J.C. Vansnick, MACBETH, LSE OR Working Paper 03.56, 2003.
15 [6] A. van Deemen, Coalition Formation and Social Choice, Kluwer Academic Publishers, Dordrecht, 1997. [7] P. Eklund, A. Rusinowska, H. de Swart, A consensus model of political decisionmaking, (www.uvt.nl/faculteiten/fww/onderzoek/skt/pdf/consensus.pdf), 2004. [8] Ana Meca-Martinez, J. Sanchez Soriano, I. Garcia-Jurando and S. Tijs, Strong equilibrium in claim games corresponding to convex games, International Journal of Game Theory, Vol. 27, 1988, pp. 211-217. [9] D.B.G. Meister, K. W. Hipel, Coalition formation, Journal of Scientific and Industrial Research, Vol. 51, 1992, pp. 612-625. [10] R.B. Myerson, Conference structure and fair allocation rules, International Journal of Game Theory, Vol. 9, 1980, pp. 211-217. [11] R.B. Myerson, Graphs and cooperation in games, Mathematics of Operations Research, Vol. 2, 1977, pp. 225-229 [12] G. Owen, Values of games with a priori unions. In: R. Henn and O. Moeschlin (eds.), Mathematical economics and game theory, Springer Verlag, 1977, pp. 76-88. [13] A. Rusinowska, H. de Swart, Negotiating a stable government - an application of bargaining theory to a coalition formation model, (www.uvt.nl/faculteiten/fww/onderzoek/skt/pdf/urbino2003.pdf), 2003. [14] A. Rusinowska, H. de Swart, J.W. van der Rijt, A new model of coalition formation, forthcoming in Social Choice and Welfare, 2004. [15] T.L. Saaty, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977) 234-281. [16] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, 1980. [17] M. Slikker, A. van den Nouweland, Social and economic networks in cooperative game theory, Theory and Decision Library, Series C, Vol. 27, Kluwer Academic Publishers, Dordrecht, 2001. [18] M. de Vries, Governing with Your Closest Neighbour: an Assessment of Spatial Coalition Formation Theories, Print Partners Ipskamp, 1999.
Appendix Table 17: Matrix of judgements of difference of attractiveness in P0B
BC AB AC ABC neutral0B
BC goodB AC ABC neutral0B 0 = AB no very weak strong extreme extreme no moderate moderate moderate no very weak very weak no very weak no
Table 18: The values of the majority coalitions for party B coalition p0 = V0B (p0 ) =
B BC goodB 0 = AB AC ABC neutral0
130.02
100.00
30.01 10.00
0.00
16
Table 19: Matrix of judgements of difference of attractiveness in P1B (2)
goodB 1 (2) p1 (1) p1 (5) p1 (4) p1 (3) p1
(1)
(5)
(4)
(3)
goodB neutral1B = p1 p1 p1 p1 1 p1 no weak moderate moderate strong very strong no moderate moderate strong strong no very weak moderate strong no weak strong no moderate no
Table 20: The values of the policies on issue 1 for party B policy p1 = V1B (p1 ) =
(2)
goodB 1 p1
(1)
neutral1B = p1
100.00 66.68
(5)
p1
(4)
(3)
p1
p1
−16.68 −66.68 −150.00
0.00
Table 21: Matrix of judgements of difference of attractiveness in P2B goodB r y s neutral2B = x w 2 =z z no very weak very weak moderate moderate very strong r no very weak weak moderate strong y no weak weak strong s no very weak moderate x no weak w no
Table 22: The values of the policies on issue 2 for party B policy p2 = V2B (p2 ) =
goodB 2 =z 100.00
r
y
s
neutral2B = x
w
0.00
−66.67
83.33 66.67 16.67
Table 23: Matrix of difference of attractiveness between the reference profiles for party B B B B [Crit.B 0 ] [Crit.2 ] [Crit.1 ] [neutral ] no weak weak strong no weak strong no moderate [neutral ] no
[Crit.B 0 ] [Crit.B 2 ] [Crit.B 1 ] B
17
Table 24: The scaling constants for party B B B B measure [Crit.B j ] = [Crit.0 ] [Crit.2 ] [Crit.1 ] V B ([Crit.B 41.67 33.33 25.00 j ]) = αjB = 0.4167 0.3333 0.25
Table 25: Matrix of judgements of difference of attractiveness in P0C C goodC 0 AC BC neutral0
goodC 0 AC BC neutral0C AB ABC
no
AB
ABC
2 3 5 5 5 no weak moderate moderate moderate no moderate moderate moderate no weak weak no very weak no
Table 26: The values of the majority coalitions for party C goodC 0 AC
coalition p0 = V0C (p0 ) =
BC neutral0C
100.00 61.54 46.15
AB
ABC
−15.38 −23.08
0.00
Table 27: Matrix of judgements of difference of attractiveness in P1C (1)
goodC 1 = p1 (1)
p1 (4) p1 (2) p1 (5) p1 (3) p1
no
(4)
(2)
p1
(5)
p1
(3)
neutral1C = p1
p1
moderate strong very strong no moderate moderate no weak no
extreme very strong moderate weak no
Table 28: The values of the policies on issue 1 for party C policy p1 = V1C (p1 ) =
(1)
goodC 1 = p1 100.00
(4)
p1
(2)
p1
(5)
p1
70.00 40.00 20.00
(3)
neutral1C = p1 0.00
18
Table 29: Matrix of judgements of difference of attractiveness in P2C goodC s r z y neutral2C = x 2 =w w no weak weak strong extreme extreme s no weak weak strong strong r no weak moderate strong z no weak weak y no very weak x no
Table 30: The values of the policies on issue 2 for party C policy p2 = V2C (p2 ) =
goodC 2 =w 100.00
s
r
z
y neutral2C = x
75.00 58.33 37.50 8.33
0.00
Table 31: Matrix of difference of attractiveness between the reference profiles for party C C [Crit.C [Crit.C [neutralC ] 1 ] [Crit.2 ] 0] no very weak very strong extreme no strong very strong no very weak [neutral ] no
[Crit.C 1] [Crit.C 2] [Crit.C 0] C
Table 32: The scaling constants for party C C C C measure [Crit.C j ] = [Crit.1 ] [Crit.2 ] [Crit.0 ] C C V ([Crit.j ]) = 50.00 41.67 8.33 αjC = 0.50 0.4167 0.0833
19
Table 33: Utilities of the governments g (1)
g 1 = (AB, p1 , x) (1) g 4 = (ABC, p1 , x) (1) g 7 = (BC, p1 , y) (1) 10 g = (AC, p1 , z) (1) 13 g = (AB, p1 , r) (1) 16 g = (ABC, p1 , r) (1) g 19 = (BC, p1 , s) (1) g 22 = (AC, p1 , w) (2) 25 g = (AB, p1 , x) (2) 28 g = (ABC, p1 , x) (2) 31 g = (BC, p1 , y) (2) g 34 = (AC, p1 , z) (2) g 37 = (AB, p1 , r) (2) 40 g = (ABC, p1 , r) (2) 43 g = (BC, p1 , s) (2) g 46 = (AC, p1 , w) (3) 49 g = (AB, p1 , x) (3) g 52 = (ABC, p1 , x) (3) g 55 = (BC, p1 , y) (3) 58 g = (AC, p1 , z) (3) g 61 = (AB, p1 , r) (3) 64 g = (ABC, p1 , r) (3) g 67 = (BC, p1 , s) (3) g 70 = (AC, p1 , w) (4) 73 g = (AB, p1 , x) (4) 76 g = (ABC, p1 , x) (4) 79 g = (BC, p1 , y) (4) g 82 = (AC, p1 , z) (4) g 85 = (AB, p1 , r) (4) 88 g = (ABC, p1 , r) (4) 91 g = (BC, p1 , s) (4) g 94 = (AC, p1 , w) (5) 97 g = (AB, p1 , x) (5) g 100 = (ABC, p1 , x) (5) g 103 = (BC, p1 , y) (5) 106 g = (AC, p1 , z) (5) g 109 = (AB, p1 , r) (5) 112 g = (ABC, p1 , r) (5) g 115 = (BC, p1 , s) (5) g 118 = (AC, p1 , w)
V (g)
g
(113.08, 41.67, 48.72) (38.08, 4.17, 48.08) (6.92, 76.4, 57.32) (63.85, 45.84, 70.75) (103.47, 69.44, 73.025) (28.46, 31.94, 72.38) (−2.31, 59.74, 85.1) (56.15, −9.72, 96.8) (109.33, 58.34, 18.72) (34.33, 20.84, 18.08) (3.17, 93.07, 27.32) (60.1, 62.51, 40.75) (99.72, 86.11, 43.03) (24.71, 48.61, 42.38) (−6.06, 76.41, 55.1) (52.4, 6.95, 66.8) (75.58, 4.17, −1.28) (0.58, −33.33, −1.92) (−30.58, 38.9, 7.32) (26.35, 8.34, 20.75) (65.97, 31.94, 23.03) (−9.04, −5.56, 22.38) (−39.81, 22.24, 35.1) (18.65, −47.22, 46.8) (90.58, 25, 33.72) (15.58, −12.5, 33.08) (−15.58, 59.73, 42.32) (41.35, 29.17, 55.75) (80.97, 52.77, 58.03) (5.96, 15.27, 57.38) (−24.81, 43.07, 70.1) (33.65, −26.39, 81.8) (83.08, 37.5, 8.72) (8.08, 0, 8.08) (−23.08, 72.23, 17.32) (33.85, 41.67, 30.75) (73.47, 65.27, 33.03) (−1.54, 27.77, 32.38) (−32.31, 55.57, 45.1) (26.15, −13.89, 56.8)
g 2 = (AC, p1 , x) (1) g 5 = (AB, p1 , y) (1) g 8 = (ABC, p1 , y) (1) 11 g = (BC, p1 , z) (1) g 14 = (AC, p1 , r) (1) 17 g = (AB, p1 , s) (1) g 20 = (ABC, p1 , s) (1) g 23 = (BC, p1 , w) (2) 26 g = (AC, p1 , x) (2) 29 g = (AB, p1 , y) (2) 32 g = (ABC, p1 , y) (2) g 35 = (BC, p1 , z) (2) g 38 = (AC, p1 , r) (2) 41 g = (AB, p1 , s) (2) 44 g = (ABC, p1 , s) (2) 47 g = (BC, p1 , w) (3) g 50 = (AC, p1 , x) (3) g 53 = (AB, p1 , y) (3) g 56 = (ABC, p1 , y) (3) 59 g = (BC, p1 , z) (3) g 62 = (AC, p1 , r) (3) 65 g = (AB, p1 , s) (3) g 68 = (ABC, p1 , s) (3) g 71 = (BC, p1 , w) (4) 74 g = (AC, p1 , x) (4) 77 g = (AB, p1 , y) (4) 80 g = (ABC, p1 , y) (4) g 83 = (BC, p1 , z) (4) g 86 = (AC, p1 , r) (4) 89 g = (AB, p1 , s) (4) 92 g = (ABC, p1 , s) (4) 95 g = (BC, p1 , w) (5) g 98 = (AC, p1 , x) (5) g 101 = (AB, p1 , y) (5) g 104 = (ABC, p1 , y) (5) 107 g = (BC, p1 , z) (5) g 110 = (AC, p1 , r) (5) 113 g = (AB, p1 , s) (5) g 116 = (ABC, p1 , s) (5) g 119 = (BC, p1 , w)
(1)
V (g)
g
(68.08, 12.51, 55.13) (111.93, 63.89, 52.19) (36.92, 26.39, 51.55) (3.85, 87.51, 69.47) (58.46, 40.28, 79.43) (102.7, 47.23, 79.97) (27.69, 9.72, 79.33) (−3.85, 31.96, 95.51) (64.33, 29.18, 25.13) (108.18, 80.56, 22.19) (33.17, 43.06, 21.55) (0.1, 104.18, 39.47) (54.71, 56.95, 49.43) (98.95, 63.9, 49.97) (23.94, 26.39, 49.33) (−7.6, 48.63, 65.51) (30.58, −24.99, 5.13) (74.43, 26.39, 2.19) (−0.58, −11.11, 1.55) (−33.65, 50.01, 19.47) (20.96, 2.78, 29.43) (65.2, 9.73, 29.97) (−9.81, −27.78, 29.33) (−41.35, −5.54, 45.51) (45.58, −4.16, 40.13) (89.43, 47.22, 37.19) (14.42, 9.72, 36.55) (−18.65, 70.84, 54.47) (35.96, 23.61, 64.43) (80.2, 30.56, 64.97) (5.19, −6.95, 64.33) (−26.35, 15.29, 80.51) (38.08, 8.34, 15.13) (81.93, 59.72, 12.19) (6.92, 22.22, 11.55) (−26.15, 83.34, 29.47) (28.46, 36.11, 39.43) (72.7, 43.06, 39.97) (−2.31, 5.55, 39.33) (−33.85, 27.79, 55.51)
g 3 = (BC, p1 , x) (1) g 6 = (AC, p1 , y) (1) g 9 = (AB, p1 , z) (1) 12 g = (ABC, p1 , z) (1) 15 g = (BC, p1 , r) (1) g 18 = (AC, p1 , s) (1) g 21 = (AB, p1 , w) (1) g 24 = (ABC, p1 , w) (2) 27 g = (BC, p1 , x) (2) g 30 = (AC, p1 , y) (2) 33 g = (AB, p1 , z) (2) g 36 = (ABC, p1 , z) (2) g 39 = (BC, p1 , r) (2) 42 g = (AC, p1 , s) (2) g 45 = (AB, p1 , w) (2) 48 g = (ABC, p1 , w) (3) 51 g = (BC, p1 , x) (3) g 54 = (AC, p1 , y) (3) g 57 = (AB, p1 , z) (3) 60 g = (ABC, p1 , z) (3) 63 g = (BC, p1 , r) (3) g 66 = (AC, p1 , s) (3) g 69 = (AB, p1 , w) (3) g 72 = (ABC, p1 , w) (4) 75 g = (BC, p1 , x) (4) g 78 = (AC, p1 , y) (4) 81 g = (AB, p1 , z) (4) g 84 = (ABC, p1 , z) (4) g 87 = (BC, p1 , r) (4) 90 g = (AC, p1 , s) (4) g 93 = (AB, p1 , w) (4) 96 g = (ABC, p1 , w) (5) 99 g = (BC, p1 , x) (5) g 102 = (AC, p1 , y) (5) g 105 = (AB, p1 , z) (5) 108 g = (ABC, p1 , z) (5) 111 g = (BC, p1 , r) (5) g 114 = (AC, p1 , s) (5) g 117 = (AB, p1 , w) (5) g 120 = (ABC, p1 , w)
V (g) (1)
(8.08, 54.18, 53.84) (66.92, 34.73, 58.6) (108.85, 75, 64.35) (33.85, 37.5, 63.7) (−1.54, 81.95, 78.15) (57.69, 18.06, 86.38) (101.16, 19.45, 90.39) (26.15, −18.05, 89.75) (4.33, 70.85, 23.84) (63.17, 51.4, 28.6) (105.1, 91.67, 34.35) (30.1, 54.17, 33.7) (−5.29, 98.62, 48.15) (53.94, 34.73, 56.38) (97.41, 36.12, 60.39) (22.4, −1.38, 59.75) (−29.42, 16.68, 3.84) (29.42, −2.77, 8.6) (71.35, 37.5, 14.35) (−3.65, 0, 13.7) (−39.04, 44.45, 28.15) (20.19, −19.44, 36.38) (63.66, −18.05, 40.39) (−11.35, −55.55, 39.75) (−14.42, 37.51, 38.84) (44.42, 18.06, 43.6) (86.35, 58.33, 49.35) (11.35, 20.83, 48.7) (−24.04, 65.28, 63.15) (35.19, 1.39, 71.38) (78.66, 2.78, 75.39) (3.65, −34.72, 74.75) (−21.92, 50.01, 13.84) (36.92, 30.56, 18.6) (78.85, 70.83, 24.35) (3.85, 33.33, 23.7) (−31.54, 77.78, 38.15) (27.69, 13.89, 46.38) (71.16, 15.28, 50.39) (−3.85, −22.22, 49.75)
