Taxonomy approach to a cabinet formation problem

~,ialhematical Social Sciences 12 (1986) 159-167

159

-,~r ~h-Holland

TAXONOMY PROBLEM

APPROACH

TO A CABINET FORMATION

Jacek W. M E R C I K a n d W a l d e m a r K O L O D Z I E J C Z Y K Institute o f Production Engineering and Management, Tec/mical University (~/ IVroclaw, 5 0 3 7 0 ~t'roclaw, 25 Smoluchov:ski Str, Poland ( o m m u n i c a t e d bv S.J. Brams Received 7 April 1985 Revised 21 October 1985 In this paper the problem of choosing a winning coalition in simple (voting) games is considercd. It is assumed that voters (playerst have different numbers of votes and definite preferences describing their willingness to enter into a coalition with other players. A function describing the willingness (preferation degree) of a given player to unite with each possible coalition is presented. Then, the method of an econometric template as a base for choosing the winning coalition most acceptable to all players is proposed as well as taxonomy approach leading to an order over all winning coalitions with regard to all players. An application of the proposed method is given bv an example of the 1972 Italian Assembly. We obtain results in close agreement with reality.

k e y wordf." Cabinet formation; voting game; coalition formation: taxonomy ~olution.

1. The description of a given player preferation (degree of preferment) function related to each possible coalition To describe the p r o b l e m one may use the s t a n d a r d simple g a m e n o t a t i o n (Owen, 1978). Let a set .Q = { 1, 2, ..., n} describe the set of all players and the g a m e be given by a characteristic f u n c t i o n over subsets (coalitions) o f .(2, with value given by the sum of votes o f the players o f a given coalition. The winning coalitions are all those coalitions for which the characteristic f u n c t i o n values exceed a given value specified in advance. We will use as initial i n f o r m a t i o n for all the following considerations the n u m b e r of votes at the disposal o f a giw_m player and linear order over all players a c c o r d i n g to distinguishing characteristic(s). As a g o o d example o f such order we can consider the left-right p a r l i a m e n t a r y positions of political parties. We p r o p o s e to rank all players generated by a so-called policy scale (after the Swaan, 1973). =(7r(1), ..., 7r(n)) - we will lreat as ranking of the set f2 according to the policy scale. Further we p r o p o s e that on the basis o f rr we m a y generate a ranking o f the set L! with regard to each ith player, 7 z i = ( l r i ( l ) , . . . , T r i ( t T ) ) , where 7ri(jt is the range <~t65-48t~6 86/$3.50 :: 1986, Elsevier Science Publishers B.V. (North-Holland)

d. [4". Mercik, l'I'. Aoh)dzie)cz, y k

160

(~d.ffnct tortnatt(m

given by player i to p l a y e r j . This re, is such an ordering if the following condit are fulfilled for all i-

7ri(i)--- 1 rci(j)<_rr,(k)

I°

9° for iTr(i)-Tr(j}!~ITt(i)-~(t,'l: i,k : l," tz 3 ~" 7r~(Itl)Gtl if m is the most distant player from player / For all i,j 7h(j) takes values from the closed inier,,al [l,tl]. Let S[ .... ,S~ cover the set .C2 such that, S/N,S7=O f o r / a / . Elements o f the se~ for j = l, 2, ... , we wilt describe as follows:

1 ° s l - - {i} "~° S!= {k E U2" player k is j t h (in regard to distance from player i) element o f ' for j = " : In this paper we a s s u m e the following m e t h o d of describing values o f 7r,(.,/ t all i , j = l , 2 , . . . , n for kc_ Sj 7ri(k ) -

1 :

Ill

E

~S'~

It is easy to see that the generated values of 7r,(jl fulfill the above condition lO_3 ° " The a s s u m e d ordering enables us to construct an i n t r o d u c t o r y preference relatio~ o f the ith ( i = 1,2 .... ,n) player uniting with an o t h e r player to build a coalition (nol necessarily a winning one). This relation is induced by values o f the following func. tion u(i,j)=l

hi(J)

for,/~i

(1t

/7

and l~(i, i) = l. T h e induced relation is a n t i s y m m e t r i c , transitive and reflexive a n d the value ot satisfies the following inequaliiv

/a(i,j) for all i , j = l , 2 , . . . , n O<_]afi, j)<_ 1.

12)

T h e n , we define an i n t r o d u c t o r y preference relation of the iih player (i = 1, 2, ..., ,~ uniting with any already existing coalition So_ ( 2 - {i1 to build a new coalition ln,,i necessarily a winning one). This relation is induced by, values o f the following func tion 1 -.(i,S)= 1 - -hiS}

j~.s

n,(j),

ill

Inequality (2) is true f o r / a ( i , S ) too. There are m a n y different principles used in the description of the unity proccs, o f players in coalitions for the p u r p o s e o f the f o r m a t i o n o f a cabinet. A complelc discussion o f the p r o p o s i t i o n s m a y be f o u n d , for example, in de Swaan (1973). Nol

.I. H. Merci/:, H'. Kegod.~iejcz.vk

Cabinet./brmatmt~

l~S1

:,.~ving u p - t o - d a t e results in this field we want to p r o p o s e a n o t h e r principle which :. ~d a posleriori gave us an e x t r a o r d i n a r y c o n f o r m i t y to realitx. To achieve lhis goal ,.,. m o d i f y f u n c t i o n s (1) and (3). \ r e a s s u m e that the value of the f u n c t i o n s d e p e n d s on, in addition to the linear ,r~ter over all the players, the chances o f a newly created coalition to ~orm a cabinet. e, ,, lalk about 'chances' because the newly created coalition m a y be ~oo ' w e a k ' to ,,~m a cabinet. We think the indexes of Shapley ~after Owen. 1968) and Banzhaf ,,~Iter Brains, 1975) are an accurate m e a s u r e of such chances. I;efinilion. The simple g a m e with mixed structure (SGMS) defined bv the game . '~aracteristic f u n c t i o n is a g a m e in which all players c o n n e c t e d in coalition are '~cated as a single player and the characteristic function o f such a player has a value c,lual to the s u m o f the n u m b e r o f votes a f f o r d e d to individual players f r o m this ~alition. ,

The simple g a m e is a special case o f SGMS. l e t the set {SGMS} be the set o f the players from SGMS. Following ()wen (1968) ,~nd Brains (1975) we define the Shapley and B a n z h a f indexes for players f r o m the 'q(;MS: the Shapley index for the ith player from the S G M S is Sh:[,l-- E U- 1)!(:l-:)! 7 n!

(4)

.~here the s u m m a t i o n runs o',.er all winning coalitions T, such that Twithou~ player belonging to the S G M S is no longer a ,,',,inning coalition. We rnav define the t;anzhaf Index for the ith player f r o m the S G M S by g,[v], in a similar form as for he Shapley Index, (4), treating all coalitions T - { i } as one coalition if they differ rely one from a n o t h e r by the p e r m u t a t i o n o f players m a k i n g up this coalition. Having the defined indexes Shi[v], g,[u] we can m a k e the following m o d i f i c a t i o n ,I f u n c t i o n / a ( i , j ) , where ie..Q a n d j c {SGMS}, to take into c o n s i d e r a t i o n their influence on the values o f this f u n c t i o n . We think that the m o d i f i e d f u n c t i o n / ~ l i . j ) = • (/;ti,/), Bi[~ ],Sh i[v]) should have the following properties ! ~i(i,./) is n o n d e c r e a s i n g according to p(i,./) • 1 <_l~(i,j)<_l ~, ~)(i,/) is n o n d e c r e a s i n g according to Sh/[t:], B:p,]. ~' /~!i,j) is nonincreasing according to Shi[,o ], B,[t:]. In this paper we a s s u m e d the l o l l o w i n g form of thc function /~(/,,~1

!-*(i,S)=u(i,S)+{

~ =:(J) (Sh~l~4 -Sh/[~:] + B~[~,]--B:lv]).

{5)

~)nc m a y notice that !~(i,j), where i ¢ D a n d j e {SGMS}, fulfills the above four properties. The q u a n t i t y I was selected by experiment in order to fulfill lhe second properl> which ',re interpret as follows:

J.W. Mercik, W. Kolodziejczyk

162

('abtne~ ~,
if fi(i,j)= 1 then player i¢.O has the highest degree oI p~eference tunclion to unil with player j e {SGMS} in coalition {i} U{./{. f i ( i , j ) = - 1 then player i e D in the highest degree rejecls the possibility or unitml with player j e {SGMS} in coalition ti} O {~il All medial values tell us about the degree of willingness or unwillingness of playc! i e ( 2 to unite with player j e {SGMS} into coalition { i l / J {j}.

2. C h o o s i n g a w i n n i n g coalition accepled in the highest degree by all players from IJ When calculating values of the function fi(i,j) for i~ (2 and j ¢ {SGMS} we o b rain the so-called matrix of estimates [flU, J)] with dinaensions n × ( 2 " - 1 } --- ~,e, omit the empty set. Matrix [fi(i,j)] permits us to establish the ( 2 " - 1 ) × ( 2 " - 1) matrix D = [d0], i.¢~ the matrix of coalition distances of each element of the {SGMS} from all the re, maining elements of the {SGMS}. In this paper we use the following method of calculating the coalition distances. t?

dij= Z

ifi(k,i)-fi(k,J)i,

{A)

k=l

where k represents the numbers of players from (2 and t,,j are the sequential numbers of players from the {SGMS}. The as-defined coalition distance is a metric in R 2'' ~. One may also use other favourite metrics in substitution of the above metrics. Choosing a winning coalition accepted in the highest degree by all players is based upon the idea of art econometric template (Hellwig, 1968). According to the conditions of the function flU, J) the most ideal coalition i, coalition Z for which fi(i,Z)=l

for i = 1 , 2 ..... n.

{?)

D e f i n i t i o n . The coalition Z, which fulfills condition (7) w.e call the template c()ah. tion. D e f i n i t i o n . Coalition S* e {SGMS} fulfilling conditions 1 `~

v(S*) = ',~ w/,._> w,

I~ill

keS*

where: o(S*) is the value of the characteristic function of coalition S*, wk is the number of votes of player k e S * , w is a given level of characteristic function: sin. passing this value means that the coalitions are winning ones. and 2'

ds*z =

min S e {SGMS}

dsz

~

J. ~t. ,.'vlerci::, II ~. Kolodziejc=yl, ;' Cabtr~et l b r m a t i o ~

16

we call the coalition accepted in the highest degree by all players - C A L P (Coalition .\ccepted with the Largest Preference). In this case the second c o n d i t i o n is as tollows /I

d s+z:

min ./e{S(;MS}

~ ]/~(k,j)-l.

¢10)

k: 1

In a n o t h e r words, C A L P is the coalition which is winning and the least distant from ~he template coalition in the sense of i n t r o d u c e d metrics in R 2

3. T a x o n o m i c order over winning coalilions

The preference f u n c t i o n is u n d e r s t o o d as a distance in defined metrics, so it seems to be c o m p l e t e l y natural when describing the ranking of the set {SGMS} by means of graph t h e o r y language and subsequently the W r o d a w T a x o n o m y M e t h o d m a y be used as a m e t h o d o f ranking. Wroclaw T a x o n o m y (Florek et al., 1951) is based upon the idea o f a s p a n n i n g tree which is an undirected, acyclic and c o n n e c t e d graph. In applying the m e t h o d nodes o f a graph represent units of the set under investigation and the tree which is s p a n n e d over then is d e n o t e d by T = (1, E), where F = {1,2 . . . . . 2 " - 1 } is the set of nodes, E c { { i , j } : i , j • F} is the set of edges. Coalition distance, d,), for {i,j} • E is called the length of edge {i, ,/} of spanning tree. T i s t h e r e f o r e in general a n o n l i n e a r order over coalitions from {SGMS]. i)efinition. T h e sum of the length of edges of s p a n n i n g tree T is called the length of the order.

l)efinilion. T h e best order is that one for which the length of the order is the minireal one. The a l g o r i t h m for c h o o s i n g the best order T*=(I:,E*) is as follows: Step one. 1, 2 .... , 2 " are closest in the first

Substitute E : = {{ij, 1 }, {i 2,2},..., {/,,, n} } where e!:,a -~ min:~k (t:a., k - 1, which means connect by the edges all nodes with the nodes which to t h e m , eliminate repeated j u n c t i o n s according to the rule of choosing t u r n edges already belonging to E.

Steptwo. I f g r a p h ( I : , E ) is c o n n e c t e d then substitute E*"

E, T * - = = ( I K*). If not,

then ._.ooto step three.

Step t/wee. Let ( V~, E+ ), ( V2,/721>, ..., ( V,, E:) be tile s u b s p a n n i n g trees received up to this s t e p ( l / F ' l V , = 0 f o r i ~ j ) . S u b s t i t u t e E : = E U { l s , [ ) } } , \~hered,:, mini,]~..\.d,i]; V = {{i,j}" i e 1~{,.,j e I:,., x = 1,2 . . . . . /, 3,= 1,2 . . . . . r; x ¢ 3 ' } , which means unite two s u b s p a n n i n g trees by the shortest edge. T h e n go to step two.

164

J.W. Mercik, 14". Kolodziejczyk / Cabinet cbrmatton

Using the algorithm presented above it is possible to order (generally non-linear} winning coalitions relatively with regard to each other and with regard to the prefer~ ences of all the players at the same time. We eliminate the relativity of such an order by the method of choosing an initial point of the order. We propose CALP to be this initial point. Choice of the initial point lets us order all coalitions from the mosl to the least preferred coalition by all players in the sense of an order generated by order T* =(V,E*>.

4. The example We used the above approach to find the most probable cabinet after the general election in Italy in t972. In the result of this election the following assembly and cabinet (after de Swaan, 1973) were formed: Italy 1972, Assembly Majority larger than 315 Actors in Assembly PCI PSI PSDI DCHR PLI MS! Policy scale order 1 2 3 4 ~ 6 Seats in Assembly 179 61 29 267 20 56 Andreotti's Cabinet (7,72) PSDI + DCHR + PLI Using only the information given above we performed the calculation accordinl,l to the proposed method. The calculations presented in this paper were computer supported but because ol their large size they are only partially given here. Table I Values o f f i ( P S D I , . ) Coalitions

fi(PSDI, . )

DCHR + PLI PSI+PCI+MSI

0.573) 0.41 / the w i n n i n g c o a l i t o n s

DCHR PSI

0.684 0.599

PSI + P L I

0.46 ~4

PSI + PCI

0.467

PSI + P C I + PI_I

0.466

PSI + PLI + MSI

(}.41

P S D I + PSI + MSI

0.347

P C I ~- P L !

0.312

P C I + PLI + MSI

0.309

F'CI

0.284

PI.I

(/.25

P L I + MSI

0.198

P C I + MS1

0.195

MSI

0.045

J.W. Mercik, g. Kolodziejczyk ,, Cabinet formatiotl

165

In the first step we calculated the function/2(i, j ) for all parties of the 1972 Italian Parliament reflecting their individual preferences against all possible coalitions. For example, the policy scale order for PSDI generated by the general policy scale order looks as follows PSDI ;> PSI ) D C H R > PCI ) PLI > MSI 1 2.5 2.5 4.5 4.5 6

The above order gives the following results for/~(PSDI, • ) as show.n in Table 1. The results of the function ,Oti, j) m a k e up the 6 × ( 2 6 - 1) matrix of estimates [.,5(i, j ) ] , which are partially present in Table 2 together with the value d s / ( t h e star means that this coalition is a winning one - majority cabinet). According to the definition of C A L P the coalition most accepted by all players and which may form a majority cabinet is PSDI + D C H R + PLI, which is surprising on account of our accidentally chosen example, in agreement with the 1972 Andreotti's Cabinet. The matrix of estimates fi(i, j ) enable us to form the matrix of coalition distances

D= [d i/]. Because of the large dimensions (63 × 63) of this matrix we omit it. We present in Fig. I the spanning tree being the best ordered T * - ( V , E * ) (according ~t~ the rules of Wroclaw T a x o n o m y ) over all the coalitions which can possibly be tormed by given in the example parties. This spanning tree enables the generation of an order over two subsets: the subset of winning coalitions and the subset of nonwinning coalitions. We present in Table 3 the first twenty winning coalitions, which are capable of forming majority cabinets (the n u m b e r of a given coalition corresponding with its n u m b e r in Fig. 1 is given in parenthesis).

Table 2 I'he f r a g m e n t o f t h e m a t r i x o f e s t i m a t e s MSI

dsz

0.601

0

4.355

0.505

0.5

5.297

0.323 0.29

0 0

0.689 (1

4.988 5.71

0.466

0

0.309

0

4.345

0.41

0

0

0.17

4.64

0.25

0.41

0

0.476

(t.39

4.474

0

0

0

()

5.798

C o a l i t i o n s IS)

PCI

PSI

PSDI

*PSD1 + D C H R + P L I

0

0

0.573

0.471

0

0

0.198

0

0 0

0 0

0 0

PCI -~ PS[ + P S D I + P L I

0.44

0.44

*PCI + PSI + P S D I + MSI

0.39

0.39

PSI4 PSDI +PLI + MSI

0

PCI + D C H R

0.466

PSDI + PLI + MSI *DCHR + PLI + MSI * P C I + [)5;I + P S D I + D C H R

DCHR

-- 0.204

PLI

J . W . Mercik, W. Kolodziejczyk / ('ahinet q~rmat~on

166

,£, UD

.

_J

CB

) _ 2D_ _

Q~'

C_3

C2

a

C)"

C~ C3

c~

(23

~D

;L

(3_

g

m.

c~

d 32 C~

{3-

f3 L,'3 13-

~,

13..

+

~5

c'c cj C3

(3'?' ~ O

÷

~

)-

C

C3

IX.

r~

~

r~

© ~

rid

CD"

(3. +

CD

-J

D_

+

EL

Fig. 1. The best order T* over all coalitons (coalitions given in the fragment of matrix [~(i, j)} are signed additively). The black arrow means CALP, the white arrow means the most accepted non-winning cw~h tion. Full line plus dashed line means a junction of subspanning trees (Step 3 ¢)f the algorithm).

J. IlL Mercik,

1I. lxolodz.iejc=yk .... Cabinet./brmaticm

167

~id~ 3 n,.. lirst t w e n t y w i n n i n g coalitions

D i s t a n c e f r o m the CAI_P

,,alilions !",E}I + D C H R + P L I (38) !",'~ + P S D I + D C H R + P L I + M S I

(62'.)

1.523

i",! , D C H R + P L I + MSI (55)

1.583

!",i ~ PSDI + D C f t R + M S I

1.583

(53)

~'t I ~ P S D I + I ) C H R + P L I + M S I

(61)

t .605

!"~l)I + D C H R + P L I + MSI (56)

1.6O5

~'~

+ PSI + P S D I + D C H R + PLI (57)

1.6(15

' ,

+ PSI + D C H R + MSI (46}

1.646

!,

-~ P S I + D C H R + PLI (45)

1.665

'~

+ P S I + P S D I + D C H R (42)

1.665

i',

+ PSD[ a D C H R + MSI (49)

l .665

!',

~ I)CHR+PLI+MSI

1.665

¢',

+ P S D I + D C H R + P L I 148/

(511

1.667

~"~l ~ D C H R + MSI (36)

1.767

!"q ~ D C H R + P L I (35)

1.872

!', I -~ PSI + D C H R (23)

1.903

I', I + D C H R + MS1 (30)

1. 9 0 3

!",1 = PSI)I + D C H R (32)

2.027

~',l - D C H R

2.118

(13)

!'~ 1~- I ) ( ' H R + PI.I (29)

2.231

References I. Brains, G a m e T h e o r y and Politics (Free Press, New York, 1975). Florek, J. l . u k a s z e w i c z , J. P e r k a l , H. S t e i n h a u s a n d S. Z u b r z y c k i , T h e W r o c l a w I a x o n o m y (in Polish), P r z e g l a d A n t r o p o l o g i c z n y XVII (1951). Hellwig, Using a t a x o n o m i c m e t h o d in the typological dividing o f c o u n t r i e s with regard k~ their level of d e v e l o p m e n t , r e s o u r c e s and s t r u c t u r e o f q u a l i f i e d s t a f f (in Polish), P r z e g l a d S t a t y s t y c z n y 4 (1968). , ( ) w e n , G a m e T h e o r y ( S a u n d e r s C o m p a n y , P h i l a d e l p h i a - L o n d o n - T o r o n l o , 19681. , de S ~ a a n , C o a l i t i o n T h e o r i e s and C a b i n e t F o r m a t i o n s (Elsevier, A m s t e r d a m , I o 7 2 )