236 S.Hart,andM

Coalitions and Collective Action @ Physica- Verlag, Wuerzburg (Germany) 1984

Stable Coalition Structures!)

S. Hart, Tel Aviv, and M. Kurz, Stanford

By a "coalition" one means a set of players who decide to act together, as onegroup, relative to the rest of the players. By the term "coalition structure" we mean apartition of the set of players into a number of coalitions each aiming to enhance theinterests of its members. The typical coalition structure assumed in the literature is thecollection of singletons. However, in many real situations individuals act throughsocial organizations like political parties, unions, trade groups and others. Thus onenotes that at any moment of time society organizes itself into a coalition structure andthe outcome of any game calls for a division of gains among coalitions as well as amongthe members of each coalition. This means that the existence of coalition structuresimplies that the interactions among the players are conducted on two levels: First,among the coalitions, and second, within each coalition. In most of game theory it isassumed that the coalition structure is given exogenously. In contrast, the theory whichwe presented in Hart/Kurz [1983] addresses the problem of why do coalitionstructures form, and predicts, as an endogenous outcome, which one will indeed form.

This theory is based on two concepts. First, a coalition structure value (CS-value,for short) is defined; it is an evaluation of the players' prospects for any coalitionstructure. Second, based on this value, one finds which coalition structure is stable, inthe sense that no player or group of players can change the coalition structure to theiradvantage.

One of the main properties which we postulate the CS-value to have is overallefficiency. This means that our analysis does not aim to characterize that organizationof society which is needed in order to achieve social efficiency. Rather, we consider theformation of coalitions as one among many strategic acts used by the players, withinthe bargaining process, in order to increase as much as possible their share of the totalsocial "pie". More specifically, we do not assume that each one of the groups receiveswhat it can assure itself (in technical terms - its worth) but rather, that all the coalitionsbargain for the division of the total, which is the worth of the grand coalition.2)

1) This work was supported by National Science Foundation Grant SES80-O6654 at the Institute for

Mathematical Studies in the Social Sciences, Stanford University. The authors thank R.J. Aumann and

LS. Shapley for helpful comments.2) Since one of the possible cooperative decisions is to act separately, cooperation can never decrease

the total available resources (i.e., one has "super-additivity").

~

236 S. Hart, and M. Kurz

As for stability, the concept of strong equilibrium is used. Thus, a coalitionstructure is stable if no group of players can reorganize itself in such a way that they allexpect to be better off (according to the CS-value).

The reader is referred to Hart/ Kurz [1983] for detailed discussions of theseconcepts. To assist the reader we briefly review in Section 2 the main ideas and recall theprecise definitions.

This paper is devoted to the application of our theory to various situations.Section 3 includes some general stability results together with a complete analysis of allthree-person games and aU four-person symmetric game~. Two classes of games arethen studied: Apex games (in Section 4) and symmetric majority games (in Section 5).

2. The Model

The mathematical model is presented in full detail in Hart/Kurz [1983]. We willreview here the definitions and the main results.

The universe of players is an infinite set U. A game v is a real function defined on allsubsets of U, and satisfying v(~) = 0; we call v(S) the worth of S (for S c U). A setN c U is a carrier of v if v(S) = v(S n N) for all S c U; we will consider only gameswith finite carriers.

A coalition structure ~ is a finite partition ~ = {Bl>B2,...,Bm} of U (i.e.,

Uk'= t Bt = U and Bt n B, = cjJfor k :F I). For a subset of players N (usually taken to be

a carrier of some game), we will denote by ~N the restriction of ~ to N; namely,~N = {Bt n Nlk = 1,2,...,m}, which is a partition of N (empty sets Bt n N arediscarded).

A Coalition Structure Value (CS-value) is an operator cjJwhich assigns to everygame v with finite carrier, to every coalition stn,lcture~, and to every player i e U, a realnumber cjJ'(v, ~). Equivalently, one may think of cjJ(v,~) as a (finitely) additive measureon U, defined by

cjJ(v,aJ)(S) = L cjJi(v, ~),ieS

for aU S c U.We consider the following axioms on cjJ(assumed to hold for all games v and v'

and aU coalition structures ffI and ffI').

Axiom 1: Carrier. Let N be a carrier of v, then

(i) cjJ(v,aJ)(N) == .' cjJi(V, ffI) = v(N).I~

(ii) If fflN = fflN, then cjJ(v,~) = cjJ(v,~').

Let 1t be a permutation of the players; i.e., a one-to-one mapping of U onto itself.For S c U, we write 1tSfor the image of Sunder 1t;given a game v, we define a new

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Stable Coalition Structures 237

game 1tV by

(1tv)(S) = v(1tS)

for all S c: u.

Axiom II: Symmetry. Let 1t be a permutation of the players. Then

q,(1tv,1td/) = 1tq,(v,5I).

The sum v + v' of two games v and v' is the game given by

(v + v')(S) = v(S) + v'(S),

for all S c: U.

Axiom III: Additivity.

q,(v + v', 51) = q,(v,5I) + q,(v',5I).

Given a game v and a coalition structure 51 = {BI>B2,...,BIII}' we say that the gameamong coalitions is inessential if

V(UBt)= L v(Bt)hA: hA:

for all subsets K of {1, 2, ..., m}; i.e., v restricted to the field generated by 51 is additive.

Axiom IV: Inessential Game. Let v and 51= {Bi, B2,..., Bill} be such that thegame among coalitions is inessential. Then

q,(v,5I)(Bt) = v(Bt)

forallk= 1,2,...,m.

Theorem 2.1: There is a unique CS-value q, satisfying Axioms I-IV.

The proof of this result, together with a discussion on the interpretations andalternative formulations of these axioms can be found in Hartl Kurz [1983]. It is alsoshown there that the CS-value coincides with Owen's [1977] "value for games with apriori unions".

We now bring a number of results from HartlKurz [1983].Let N be a finite set of players, and 51 = {B I>B 2, . . . , Bill} a coalition structure. A

complete (linear) order on N is 51-consistent if, for all k = 1, 2,. . . , m and all i,j E Bto al!


elements of N between i andj also belong to Bt. A £i-consistent random order on N is arandom variable whose values are the orders on N that are £i-consistent, all equallyprobable (i.e., each with probability (m!b1!b2!...bm!)-t, where bt is the number ofelements of Bt n N). The interpretation is as follows: The players arrive randomly, butsuch that all members of the same coalition do so successively. This is the same asrandomly ordering first the coalitions and then the members within each coalition.

Proposition 2.2: Let v be a game with a finite carrier N, and let £i be a coalitionstructure. The CS-value cjJ satisfies,for all i E N,

cjJl(V,aJ)= E[V({lt1 u {i}) - v({It!)], (2.3)

where the expectation E is taken over all £i-consistent random orders on a carrier N ofv, and (It! denotes the (random) set of predecessors of i.

Corollary 2.4: For all Bt E £i,

cjJ(v,£i)(Bk) = (Sh vLe)(Bk),

where Sh denotes the Shapley value operator and (VLe'£i) is the game v restricted to thefield generated by £i (i.e., each Bt E £i is a "player").

By Axiom I(ii), only the partition of a carrier N of v needs to be specified. We willslightly abuse our notation by writing cjJ( v, £iN) for cjJ(v,£i) (recall that £iN was defined asthe restriction of £i to N). We also introduce the following notation: The partition of aset S = {i1,i2,..., is} into singletons, namely, {{id, {i2},..., {is}}, will be denoted by(S) (in contrast, {S} corresponds to all the members of S forming one coalition).

Corollary 2.5: Let N be a finite carrier of v, then

cjJ(v,{N}) = cjJ(v,(N») = Shv.

We mention an additional remarkable property of the CS-value: its consistency;namely, the two bargaining processes - among coalitions and within each coalition -are both reflected in a consistent way in the CS-value.

We come now to the notion of stability of coalition structures. The CS-valuecjJl(V, aJ) is interpreted as the utility of player i of participating in the game v, when theplayers are organized according to £i. Based on this, each player is able to compare thevarious coalition structures. Stability will thus occur whenever no group of players canreorganize itself in such a way that, in the new coalition structure, they are all better off(according to the CS-value). Depending on the reaction of the other players, twonotions of stability are obtained. In the y-model, it is assumed that coalitions which areleft by even one member break apart into singletons; in the c5-model, the remainingplayers form instead one smaller coalition.

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Let v be a game with a finite carrier N = {I, 2,..., n}.The games r ==r(v, N) and~ ==~(v, N) are given in normal form as follows:

Model y: The game r consists of:

(2.6) The set of players is N.(2.7) For each i E N, the set L:i of strategies of i consists of all coalitions S that contain

i, namely, L:i = {S c: Nli E S}.(2.8) For each n-tuple of strategies u = (S1,s2,...,sn)EL:l X L:2 X ... X L:n and

each i E N, the payoff to i is rjJi(V,~~», where

.

{

Si ifSj=Si for all j'E SiT' = '

,a {i}, otherwise,

and~~) = {T~liEN}.

Model (): The game ~ consists of (2.6), (2.7) and

(2.9) For each n-tuple of strategies u = (SI,S2,..., sn)E L:l X L:2 X ... x L:n andeach i EN, the payoff to i is rjJi(V,~~», where ~~) = {T c: N Ii,j E T if and onlyif Si = Sj}.

For a coalition structure ~ and a player i E N, let S;. be that element of ~ towhich i belongs: i E S;. E ~ (this defines S;. uniquely); put Uill= (S;')ieN' If the playerschoose UilI,then in both r and ~ the coalition structure that results is clearly~.

Definition 2.10: The coalition structure ~ is y-stable (resp., ().stable) in the game(v,N) if Uillis a strong equilibrium in r(v, N) (resp., ~(v,N»); i.e., if there exists no non-empty T c: N and no ai E L:i for all i E T, such that rjJi(V,~) > rjJi(V,~) for all i E T,where ~~orresponds to ((ai)ieT,(ufe)jeN\T) by (2.8) (resp., (2.9»).

.

Again, we refer the reader to Hartl Kurz [1983] for detailed discussions of theseconcepts.

3. First App6cations

This section includes some general results on stability, and also the completeanalysis of all three-person games and all four-person symmetric games.

It is well known that each game3) (v, N) can be uniquely represented as a linearcombination of unanimity games:

v = L: asus,S<=N

3) From now on, we will write (v, N) for a game v with a finite carrier N.


where us(T) = 1 if T ::I Sand us(T) = 0 otherwise, and 4)

(1.s= L (_l)lsl-ITlv(T),TeS

In the following two propositions, "stable" stands for both "y-stable" and "0-stable". S)

Proposition 3.1: Let (v, N) and (v', N) be two games, v = Ls e N(1.sUsandvi = Ls e N(1.sus'If (1.s= (1.1.for all S with IS I ~ 3, then ~ is stable in v if and only if ~ isstable in v'.

Proof: By formula (2.3), q,(',~) is linear on the space of games; therefore,q,(v,&f)= LseN(1.sq,(US'~)' If ISI = 1, say S = {I} without loss of generality, thenq,(us,&f)= (1,0,...,0) for all ~. If ISI= 2, say S = {1,2}, then q,(us,~) =(1/2,1/2,0,...,0) for all &f.Therefore

q,i(V,~) = (1.li)+_21L (1.li,J) + L (1.sq,(us'~),

J"'i Se N. ISI2:3

~ence all the conditions for stability, which are of the form q,i(V,j) > q,i(V,~) (for some,&f),do not depend on (1.sforIS I = lor ISI = 2. Since(1.s = (1.s-forISI ~ 3, the conclusionfollows. Q.E.D.

Given a game (v, N), its dual game (v*, N) is given by v*(S) = v(N) - v(N\S) forall SeN. We will denote by (ii,N) the game ii = (1/2)(v + v*) (a self-d~al game).

Proposition 3,2: The following three statements are equivalent:

(i) ~ is stable in v.(ii) ~ is stable in v*,

(Hi) ~ is stable in ii,

Proof: For any order on N that is ~-consistent, the "reverse" order (i.e., from"last" to "first") is also ~-consistent, and they are equally probable, Formula (2.3) nowimpliesq,(v,~) = q,(v*,&f),hencealso q,(v,~) = q,(ii,~). Q.E.D.

To shorten notation, from now on we will write coalition structures in a "compactway": [1213] will mean {{1,2}, {3}}, and so on. We analyze now the three-persongames; in view of Proposition 3.1, they are the smallest non-trivial games.

4) The number of elements of a finite set A is denoted by'IAI.') Or, for any other concept based on the CS-value(e.g.,see Definition 5.1).

. ,p(v',.)

[11213] G~N,j~N,j~N)

[1213] ell)4~N'4~N'2~N

[1 312] ell)4~N'2~N'4~N

[2311] ell)2~N'4~N'4~N

[1 23] ell)3~N"3~N'3~N


Proposition 3.3: Let (v,N) bea three-person game, N = {1,2,3}. The stablecoalition structures are as follows:

(1) if aN= 0: all coalition structures are "1- and b-stable;(2) if aN > 0: [11213] is "1- and b-stable,and [123] is "I-stablebut not b-stable.(3) if aN < 0: [1213], [1312] and [2311] are "1- and b-stable.

Proof: By Proposition 3.1, v is equivalent (in terms of the stability of coalitionstructures) to v' = aNuN' The CS-values of v' for the various coalition structures are

If aN = 0, v' == 0 and q,(v',!t) == o for all !t. If aN > 0, then (1/3)aN > (1/4)aN'hence [11213] is stable, whereas [1213], [1312] and [2 3

1

1] are not. Asfor [12 3], it isnot b-stable since 1is better off in [2311] (note that (112 3] iscS-stable,since 1annotforce 2 and 3 to form a coalition). If aN < 0, the reverse holds. Q.E.D.

Note that aN = v(l2 3) - v(12) - v(13) - v(23) + v(l) + v(2) + v(3). Considerthe three-person majority game; the stable coalition structures are those whereprecisely two players join to form a "winning" coalition (here, all) = 0, all,})= 1,aN =- 2). On the other extreme, in the unanimity game, no coalitions form (all) = OCli.}) =0,aN = 1).This is due to the fact that, when each player has veto power, they are allequal; however, when two players, say 1 and 2,join together, the number of "vetoers" isreduced to two, hence 1 and 2 together have half the power, and each - only 1/41

This is a particular case of the following:

Theorem 3.4: Let (v, N), v = Is c:NaSuS be such that as ~ 0 for all S withISI ~ 3, with at least one strict inequality.6) Then!t = (N) (i.e., every player being asingleton) is both "1- and b-stable, and !t = {N} is y-stable but not b-stable.

Proof: Without loss of generality, assume that as = o for all ISI :S;2 (cf.Proposition 3.1). By Corollary 2.5, we have q,'(US'{S}) = q,'(US'(S») = 1/(\SI) for

6)Otherwise, all coalition structures are both "/- and c5-stable. .


all i E S, and = 0 for all i r1S. Therefore, for all i E N,

cJi(v, (N») = cJi(v, {N}) = J;'I~I. as.

Now assume by contradiction that (N) is not ')I-stable;let j be the new coalitionstructure, obtained from (N) by a non-empty set T c: N such that, for all i e T,

c/>'(v,bf) > c/>'(v,(N»). (3.5)

For every S c: N, denote ms= Ijsl = I{Be~IB n S #: c/>}I(i.e., the number ofcoalitions in bf, restricted to S);then ms ~ ISI.

.

Letj ~ T; he is a singleton in j (as he was in (N»), hence also in js for all S 3 j.Therefore,

J A J Allc/> (us,~) = c/>(us,~s) =

ms~

1ST

for all S 3 j; hence

A-.JA

)'1 J

'I'(v,~) ~

/';J 1STas = c/>(v,~). (3.6)

But (3.5) for all i e T and (3.6) for allj rJ T contradicts the efficiency of c/>(Axiom I(i)).The same argument shows the t5-stability of (N) and the ')I-stability of {N}. To

show that {N} is not t5-stable, assume without loss of generality that as > 0 for some Scontaining player 1. Then

c/>1(V,[1123... n]) = ~;';1as >

);11~1as = c/>1(V,{N})

(recall that a1 = 0). Q.E.D.

Remarks:(1) It is easy to check that if v = Ls c:NaSuS, then v* = Ls c:Na;uS and v =

Ls c:Niisus, where

a; = (-I)ISI-1 L aTTc:S

and

- 1( *)as = '2 as + as ,


for all S c: N. By Proposition 3.2, we can obtain additional suffiCient conditions forthe conclusion of Theorem 3.4: Either a! ~ 0 for all IS I ~ 3, or as ~ 0 for allISI ~ 3.

(2) The condition as ~ 0 for alII S I ~ 3 is clearly not necessary. From the proof one canderive weaker sufficient conditions (however, they are not necessary either). Forexample,

S~j (~S -1~I)as ~ 0,

ISI2:3

for all.i and all playersj that are singletons in.i. Moreover, the use of Proposition3.2 generates additional conditions.

(3) It is interesting to point out that the new solution concept of Harsanyi [1979] co-incides with the Shapley value (which, by Corollary 2.5, is precisely 4J(v,(N» =4J(v, {N}») when as ~ 0 for all S.

In order to further understand Theorem 3.4, note that IXs~ 0 implies that S has anon-negative net contribution with respect to all its subcoalitions (indeed,as = v(S) - LT~SaT; following Harsanyi [1977], as is the "total net dividend" ofcoalition S). Therefore, "combining forces" ought to be rewarding, and this seems to

. suggest that non-trivial coalitions (i.e., non singletons) should form in such a game.However, this reasoning is based on the view that the players' objective is to make thetotal payoff as large as possible. In contrast, we regard v(N) as being "available", andeach player's objective in forming coalitions is to obtain the best possible share of it.Hence, it is preCisely in the case that all "dividends" are non-negative, that the best aplayer can do is to be a singleton - since all bargaining units, whether individuals orgroups, receive equal shares. This underscores again the difference between our notionof coalition formation and earlier such concepts [see the discussion in Section 2 ofHart/ Kurz].

Finally, we analyze the four-person symmetric games. A game is symmetric if theworth of any coalition depends only on the number of its elements; equivalently, if thegame is invariant under all permutations of the players. For a symmetric game v, wewill usually denote by Vkthe worth of any coalition of size k.

Proposition 3.7: Let (v,N) be a symmetric four-person game, with N ={I, 2, 3,4}. Put

A = V4 - 2V3 + 2Vi'

then the coalition structures that are stable are as follows:

(1) if A = 0: all coalition structures are y-and <5-stable;(2) if A > 0: [1121314] is y- and <5-stable,and [1234] is y-stable but not <5-stable;(3) if A < 0: [12314], [12413], [1 3412] and [234/1] are y- and (i-stable.

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!JI q,1(V,!JI) q,2(V,!JI) q,](v,!JI) q,4(V,!JI)

[1121314]1 1 1 141X4 + IX] 41X4 + IX] 41X4 + IX] 41X4 + IX]

[121314]1 S 1 S 1 7 1 761X4 + 61X] 61X4 + 61X] )1X4 + 6"'] )1X4 + 61X]

[12134]1 1 1 141X4 + IX] 41X4 + IX] 41X4 + IX] 41X4 + IX]

[12314]1 S 1 S 1 S 1 361X4 + 61X] 61X4 + 61X] 61X4 + 61X] 21X4 + 21X]

1 1 1 1[1234] 41X4 + IX] 41X4 + IX] 41X4 + IX] 41X4 + IX]

The critical quantity is clearly

(~Ot4 + Ot)) - (~Ot4 + ~Ot)) = 1~(Ot4 + 2Ot)) = 112A.


Proof: By Proposition 3.1, we can assume without loss of generality that

v = Ot3 L Us + Ot4uN,

Isl"'3

where we write Otlslfor Ots(because of symmetry). The following table contains thevarious coalition structures and the corresponding CS-value (we omit those that arepermutations of the ones given). -

In case A = 0, the CS-valueis the same for all coalitionstructures.If A > 0, someunions are "harmful". [121314] and [12134] can be improved upon by {I, 2} (in[1121314] and [112134],respectively),and [12314] by {1,2,3}. Moreover, [1234] isnot l5-stablesince 4 is better off in [12314]. When A < 0, some unions are desirable;indeed, [12314] is better for {I, 2,3} than [1121314], [12134] and [1234], while[12134] is preferred by 3 and 4 to [121314]. Q.E.D.

4. Apex Games

This section is devoted to the study of stability in the so-called Apex games. Theseare simple games (i.e.,v(S) equals o or 1for all S), consisting of one "major" player (the"apex"), and n - 1 minor players. The winning coalitions (namely, those S withv(S) = 1) are:The major player together with at least one minor player, and all minorplayers together. Thus, if N = {I, 2,..., n} and 1 is the apex, then

v(S) = {

I, if 1 E S.and S\{I} :F~, or S = N\{I},0, otherwIse.

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Equivalently, we can represent v as a homogeneous weighted majority game, as follows

v = [n - 1;n - 2,1,1,...,1]., ,

n-1

These games were already considered by van Neumann/Morgenstern [1944], whocharacterized their solutions [see Section 55 and Section 54.3 there, including thediscussion of the reasons for studying them]. Note that the first interesting case isn = 4, since for n = 3 we obtain the three-player majority game (which is symmetric).

Proposition 4.1.. Let v be the n-player Apex game. For n ~ 5, the unique y-stablecoalition structure is [1123... n], and there are no t5-stable coalition structures. Forn = 4, [112 3 4] is y- but not t5-stable, and [121314], [1 31214] and [141213] are bothy- and l5-stable.

It is interesting to note that the Apex games (for n ~ 5, at least) result in all theminor players forming a (winning) coalition. This is similar to the Case (I) solutions ofvon N eumann/ Morgenstern [1944, Section 55.3], which they interpret as a "segrega-tion" of the major player.

Proof: We first compute the CS-value for a given coalition structure ~ ={BI>B2,..., B",}. Without loss of generality, it will be assumed throughout this proofthat 1e B

l' and k willdenote the number of elements of B l' It is clear that player 1 is apivot (i.e., his marginal contribution is 1) if and only if he is neither first nor last in a(random) order of all players. The probability of B1 being first is 11m, and theprobability of 1 being first among the members of B1 is 1/k{see(2.3) and the discussionpreceding the statement of Proposition 2.2). Similarly for being "last", and we obtain

1 2<P(v,~) = 1 - mk'

(4.2)

We distinguish now two cases. If k ~ 2, i.e., B1 contains other players beside 1, then B1is winning and all Bj for j :F 1have CS-value 0 (and so do all their members). In this case

i 2<P(v,~) = mk(k - 1)'

(4.3)

for all i e Bt \ {I} (by symmetry). If k = 1, ie., B1 = {I}, each Bj for j :F 1 will be a pivotwhenever the order of the coalitions is either (B I> B j, . . .), or (. . . ,BJ'Bd. Therefore,

2<p(v,~)(Bj) = m(m - 1)'

(4.4)

and each of its members will get 1/IBjl of this amount.

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We will now consider separately each of the following four cases for fJI:

(i) 91 = [1123...n],(ii) k = 1, fJI :;l:[1123... n],

(iii) k ~ 2, mk > 6,(iv) k ~ 2, mk ~ 6.

Case (i): fJI = [1123... n]. We have

( 1 1 1 )q,(v,fJI)= °'- 1'- 1""'- 1.

n- n- n-

To s~e that fJI is not c5-stable,consider T = {1,2} and iJ = [12134.. .n], withq,(v,fJI) = (1/2,1/2,0,0,...,0).

Now assume fJIis not y-stable; let TeN be a non-empty coalition, and let iJ bethe coalition structure it induces according to the game r, such that all members of Tare ~tter 2fT. We will denote the elements of iJ by Bi (for j = 1,2,..., Iii),with 1 E B.and k = IB.I.

If B. = {I}, then Tmustcontainatleastonei:;l: 1(otherwiseiJ = fJI).Moreover,all i ~ T, i :;l:1 must be singletons in iJ (according to the rule y). Let i E T, i :;l:1, theni E Bi for somej:;l: 1,and we have by (4.4)

2 ~ 1~ = A./(v fJI) > A./(v fJI) = -1ii(1ii- 1)IBil '1', '1', n - 1

.

This implies

2 11ii(1ii- 1) >

n - 1'

therefore we have q,'(v,£i) > q,/(v,fJI) not only for i E T, but also for all i ~ T, i :;l: 1 (by

(4.4) again, since they are singletons). Hence q,(v;iJ)(N \ {I}) > q,(v, fJI) (N\ {I}) = 1,which is a contradiction.

If k~2, then q,'(v,£i)=o for alli~£i.. and we must have Tc B., henceT = B. (otherwise £i cannot be obtained by T from fJI). By (4.3), we get

2 1~~ >-.

mk(k - 1) n - 1

Since all i ~ T = B. are singletons, Iii = n - k + 1, therefore

2(n - 1)> k(k - 1)(n- k + 1).

The right-hand side is a polynomial of degree 3 in k,with roots 0, 1 and n + 1. In theregion 2 ~ k ~ n, it is positive, and its minimal value is attained at one of the two

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extremes. In this case, at k = 2, where it equals 2(n - 1).Therefore the above inequalityis not possible, which proves the y-stability of 91.

Case (ii): k = 1 and 91 # [1123... n]. First let 91= < N >; it is dominated,both y-and <5-,by [1123., .n] (with T = N\{I}). Second, let 91 # <N>. Without lossof generality assume B2 is the largest coalition in 91, and also that 2 E B2. By (4.4),ljJ2(V,91) ~ ljJl(V,91) for all i # 1. Since 91 # [1\23,.. n], ljJ1(v, 91) > 0, therefore

2 1 ~ I 1ljJ (v, 91) ~

_1 L. ljJ(v,9I) < _

I 'n- 1..1 n-(4.5)

Let T = {1,2},ql =q2

= {1,2}. The resulting coalition structure a will havek = 2 andm ~ m (according to <5,m = m; according to y, m = m + IB21- 2). Substituting this in(4.2)and (4.3)gives

1 - 2 1 1 2 1ljJ (v, 91) = 1 - ~ = 1 - -;;-~ 1 - - > 1 - - = ljJ (v,9I)mk m m m

- 2 1 1ljJ2(V,9I)= --- = -;;- ~ _1 > ljJ2(V,9I)

mk(k - 1) m n -

the last inequality being (4.5). This shows that 91 is neither y- nor <5-stable.

Case(iii): k ~ 2 and mk > 6. Let T= N\{I} and 91 = {{I}, B1\{1}, N\B.};then m = 3, k = 1,

1 - - 2 - IljJ (v, 91) -3 . 2. (n - k) > 0 -ljJ (V,91)

for all i E N\B1, and

. - 2 2 .ljJ'(v,9I) = 3 .2. (k - 1)

> mk(k - 1) = ljJ'(v,9I)

for all i E B 1\ {I} (recall that mk > 6). This proves that 91 is neither y- nor <5-stable.

Case (iv): k ~ 2 and mk ~ 6. First, let k = 2; then m = 2 or 3. Without loss ofgenerality, B1 = {1,2}. For n ~ 5, all i # 2 are better off in a = [134...nI2], since

- 2 n-2 2 2 1ljJl(V,9I)= 1 - - - > - ~ 1 - - = ljJ(v,9I)2(n - 1) n - 132m

and

. - 2 .ljJ'(v,9I) = 2. 1 '(n - 1) > 0 = ljJ'(v,9I)

L


for i ;el, 2. When n = 4, the strict inequality for cjJ1 does not hold if m = 3; it can beeasily checked that the corresponding coalition structures, namely [121314],[131214] and [141213], are indeed stable in both senses: the two members of B. gettheir highest possible CS-value there.

Second, let k = 3, then m = 2; without loss of generality, B1 = {I, 2, 3}. Forn ~ 6, let T = N\{2,3}, ,jIY)

= [145...nI213] and,j16) = [145...nI23]; for n = 4,5, let T = N\ {1} and ,j

= [1123... n]. In all cases, it can be readily checked that allmembers of T are better off. . Q.E.D.

5. Symmetric Majority Games

. This last section is devoted to the study of n-person symmetric majority games,which are defined by

v(8) ={

I, if ISI~. k,0, otherwise,

where k is a given parameter - the "majority" needed to win. This game is deRoted by(n, k); it is usually assumed that k is an integer larger than nl2 (so that the game willbe super-additive). These games were first studied by Bott [1953], who characterizedall their symmetric solutions.

This class of gameswill provide examples with no stable coalition structures; notonly in the y- and l5-sense, but also in the broader a- and p-sense [cf. HartlKurz]. Wefirst recall these definitions.

Given the game v and its finite carrier N, the two derived normal form games rand!::. both lead to the same characteristic function form games [without sidepayments, cf. Aumann]. For each 8 c N, let Ls = DieS L" and define

011)(8) = {(x/)/eS e IRsI

there is (O"/)/eS e Ls such that for all (O"J)J&N\Se LN\S

cjJ/(v,£f,,) ~ x for all i e 8},0P) (S) = {(xl)leS e IRs1for all (O"J)JeN\Se LN\S there is (O"/)/eSe Ls such that

cjJ/(v,£f,,) ~ Xl for all i e S}.

Definition 5.1: The coalition struction dI is a-stable (resp., p-stable) in the game(v,N) if cjJ(v,£f) belongs to the core of (VIII),N) (resp., (VIP),N»); i.e. if there exists no non-empty TeN and no ye V(II)(T) (resp., VP(T») such that yl > cjJ/(v,dI)for all i e T.

It is easy to see thaUhis definition is equivalent to the following: dI is a-stableif there exists no non-empty coalition TeN and no (a')/eT e IT such that, for all(aJ)jeN\T e LN\T, cjJl(V"j) > cjJ/(v,dI)for all i e T (where,j corresponds to a). dI is p-stable if there exists no non-empty ~oa1ition TeN sl!ch that for a1l18j)jeN\T e LN\Tthere exists (8/)ieT e LT with cjJ/(v,dI)> cjJ/(v,dI)for alf i e T (again, dI corresponds toa). Thus, £f is IX-stableif no T can guarantee an improvement for all its members no

L


matter what the others do, whereas tJI is p-stable if every T can be prevented. by itscomplement from getting better off. Clearly, y-stability and <5-stability each imply P-stability, which in turn implies a-stability.

We return to our analysis of the (n, k) games. It is clear that the minimal winningcoalitions - those of size k - are important in these games. However, the minimalblocking coalitions - those of size p ==n - k + 1, which prevent their complementfrom winning - are no less crucial. Indeed, the symmetric solutions of these games arebased on such coalitions [cf. Bott]. It should be pointed out that the dual game ofv = (n, k) is precisely v* = (n, p) (see the definition preceding Proposition 3.2); hencethe CS-value does not distinguish between the two games. The following lemma will.show the importance of the minimal blocking coalitions in terms of stability; it will be auseful tool in the analysis of these games.

Lemma 5.2: Let v = (n,k) and tJI = {Bl,B2,...,Bm} a coalition structure onN = {1,2,...,n}. IflB11 ~ p, then

14>(v,tJI)(B1) ~

1 + l'

where

. n-IBll1= the Integer part 7) of .p

(5.3)

(5.4)

Moreover, (5.3) becomes an equality if N\B1 is partitioned into 1blocking coalitions(Le.,of size ~ peach).

.

An implication of this lemma is that, if a blocking coalition S forms, the best - interms of the total CS-value - its complement N\S can do is to form as many as possible- 1 - disjoint blocking coalitions. Indeed, this will keep the CS-value of S to itsminimum, hence the CS-value of N\S to its maximum. Note that this does not imply

. that the CS-value of each member of N\S will be maximal- only the total.

Proof: By Corollary 2.4, 4>(v,tJI)(B 1) is the probability of B 1 of being a pivot in arandom order of {Bj}j=I' Since IBd ~ p, this happens whenever the total number ofplayers belonging to those Bj that come after B1 is less than p. We further note that, toobtain a random order of all {Bj}j=1>one can take.a random order of {Bj}j"'l andinsert Bh with equal probability, in anyone of the m available places (before the first,after the first, after the second, ..., after the (m - l)st)... Let ~ be an order on {BJj", 1; we will denote by Qj ==Qj(9I) thej-th element in

that order (thus, (Qj)j=-l is a permutation of (Bj)j",d. For each h = 1,2,... m,define

"-1a" ==

a,,(9I) = L IQm-jl;j=1

7) The integer part of a number x is the largest integer that does not exceed x.-

L


we have at = 0 and am = n - IBtl' Given a subset I of the real line, let

~ == VI(9l) = I{h 11 ~ h ~ m, ah(9l) E I} I.

VI is the number of "visits" of the sequence {ah}h'=t in the set I. With this notation, itfoHows from our previous remarks that

1t/J(v,BI)(Bd =

mE{V[oopj(9l)}, (5.5)

where E denotes expectation over all (m - I)! orders 9l of {Bj}j"lo each with the sameprobability 1/(m - I)!

Let 0 ~ b ~ n -lEt!; we will show that

E(V[Oopj) ~ E(V[b,b+pj); (5.6)

thus, the number of visits in [0, p) is the largest among aH intervals of the same length.Indeed, given an order tJr= (Qt, Q2"'" Qm-t), let q be the smallest integer such thata, ~ b; thus,

,-2 ,-tL IQm-jl < band L IQm-jl.~ b. '

}=t }=t

We define another order 9l' as follows:

9l' = (Qm-t,Qm-2,...,Qm-,+1,Qt,Q2,...,Qm-,)

(we took the last q - 1 elements, whose total number of members first exceeded b, andput them at the beginning, in reverse order). It is easy to check that a,(9l) ~ b impliesthat to each visit in [b, b + p) for 9l there corresponds a visit in [0, p] for 9l' (but theconverse is not necessarily true). Therefore,

£100pj(tJr') ~ £1b, b+ pj(tJr). (5.7)

The correspondence 9l --t tJr' is one-to-one (and onto): To get tJr from tJr' =(Q't, Q2"'" Q:"- d, let r'be such that

,-1 ,LIQjl<b and LIQjl~b,

}=1 }=t

then

9l = (Q~+I,Q~+~,...,Q:"-t,Q~,Q~-1>...,Q'I)'

L


But all orders are equally probable; therefore, (5.6) is obtained by taking expectationin (5.7).

The interval [0, n - IB 11] can be covered by disjoint intervals of length p, asfollows: [0, p), [p,2p), [2p,3p), and so on. The number of such intervals is preciselyI + 1,with I given by (5.4).For each s = 0, 1,2, . . .,

"we have by (5.6)

.

E(l!ro.p) ~ E(l!rSP,(S+l)P))'

therefore

1 I ,-E(l!ro,p)) ~

I + Is~oE(l!rsp,(s+l)JI))'

But the sum on the right-hand side is E(Uro.,,-IB,II)= m (total number of visits isalways m),which proves our assertion by (5.5).

Finally, in the case that £iIconsists of I + 1 blocking coalitions, each one is apivot whenever it is last in the random order. Therefore, all have the same value, namely1/(1+ 1),and (5.3)becomes an equality. Q.E.D.

For a coalition S, we denote byes the vector (1, 1,...,1) e IRs.

Corollary 5.8: Let v = (n,k) and S c: N = {I, 2,...,n} with s =ISI ~ p. Then~es e V(S) if and only if --

1~ ::Ss(1 + 1)'

where V(S) stands for either V(GI)(S)or V(/)(S), and I is the integer part of (n - s)/p.

Proof: The strategy the members of S will use is to form one coalition - S itself,whereas N\S will form I blocking coalitions. The result then follows from Lemma 5.2.

Q.E.D.

We will study five examples of (n,k) games. We will find stable coalitionstructures - both y-and ~- - in the first three examples, and none in the last two. If weconsider IX-and /I-stability instead - only the last example will not have any stablecoalition structures. This proves Proposition 3.6 in Hart/Kurz [1983] (see also thediscussion following Definition 3.7 there).

Example 5.9: n = 5, k = 3, p = 3. The following table summarizes the CS-values of the various coalition structures (again, due to symmetry, we omit permu-tations of those given).

.

It is clear that [1231415], [123145] and their permutations are stable (in all foursenses IX-,/I-, y- and ~-). Moreover, a three player coalition can guarantee 1/3 to each ofits members (Corollary 5,8), therefore all the other coalition structures can be improvedupon by such a coalition (again, in all senses).

L

252 S. Hart. and M. Kurz

,p'(v.£f)

eI i = 1 i=2 1=3 i=4 1=5 stability

[112131415]1 1 1 1 1

5 5 5 5 5

[12131415]1 1 1 1 14 4 6 6 6

[1213415]1 1 1 1 16 6 6 6 3

[1231415]1 1 1

0 0 a, fl. y.~3 3 3

[123145]1 1 1

0 0 a, fl. y.~3 3 3

[123415]1 1 1 1

04 4 4 4

1 1 1 1 1[12345]

5 5 5 5 5

In this example we find the formation of a three player coalition - which is at the, same time minimal winning and minimal blocking (k = p).

Example 5.10: n = 5, k = 4, P = 2.

,p'(v.eI)

eI 1= 1 i=2 i=3 1=4 i"" 5 stability

[112131415]1 1 1 1 15 5 5 5 5

[12131415]1 1 1 1 14 4 6 6 6

[1213415]1 1 1 1

0 a, fl. y.~4 4 4 4

[1231415]2 2 2 1 19 9 9 6 6

[123145]1 1 1 1 16 6 6 4 4

[123415]1 1 1 1

0 a, fl. y.~4 4 4 4

1 1 1 1 1[12345]

5 5 5 5 5

l


A two player coalition can guarantee 1/2. hence 1/4 to each member (by Lemma 5.2 andCorollary 5.8, since p = 2). Therefore, all coalition structures for which the two smallestcoordinates of the CS-value are both less than 1/4 are not stable. This leaves only[1213415] and [123415] (and their permutations) - and they are indeed stable.

This example exhibits the formation of both minimal blocking and minimalwinning coalitions.

Example 5.11: n = 6. k = 4. p = 3.

~

[112131415/6]

[1213/4/516]

[121341516]

[12134156]

[1231415/6]

[12314516]

[1231456]

[123415/6]

[1234156]

[1234 516]

[123456]

;= 1

16

15

16

16

14

29

16

14

14

15

16

,p1(V.~)

;=2 ;=3 ;=4 ;= 5

16

320

16

16

112

112

1

"6

0

0

15

16

;=6 stability

16

15

16

320

16

320

16

3.20

1

6

1

6

1

12

1

6

1

6

0 a., {J.1. ~

1

6

1

6

1

6

1

6

0 a.,{J.1. ~

1

6

1

6

1

4

2

9

1

12

1

12

0

1

6

By Corollary 5.8. a three player coalition can guarantee 1/6 to each member. and a fourplayer coalition. 1/4 to each. This eliminates all coalition structures but [1 2341516]and [1234156]. which are stable (in all four senses).

We obtained here only minimal winning coalitions. and not minimal blockingones. It turns out that the difference between this example and the previous ones is thathere. p does not divide k. In example 5.10, the two "trends" - towards forming

"--

14

29

16

14

16

14

16

14

14

15

14

15

14 -

15

16

16

16

~'(v,a)

a i = 1 i= 2 i = 3 i = 4 i= 5 i= 6 stability

[11213141516]1 1 1 1 1 16 6 6 6 6 6

[1213141516]1 1 3 3 3 35 5 20 20 20 20

[121341516]5 5 5 5 1 1

24 24 24 24 12 12

[12134156]1 1 1 1 1 16 6 6 6 6 6

[123141516]1 1 1 1 1 16 6 6 6 6 6

[12314516]1 1 1 1 1

a.,fJ06 6 6 4 4

[1231456]1 1 1 1 1 16 6 6 6 6 6

[12341516]1

.1 1 1 1 1

6 6 6 6 6 6

[1234156]1 1 1 1 1 18 8 8 8 4 4

[1234516]1 1 1 1 15 5 5

05 5

[123456]1 1 1 1 1 16 6 6 6 6 6


coalitions of size k and coalitions of size p - were consistent, since two coalitions of sizep have together exactly k members. This is not the case here; the two trends arecontradictory, and only coalitions of size k are stable.

Example 5.12: n = 6, k = 5, p = 2.

A coalition of size 2 guarantees 1/6 to each member, and a coalition of size 5 - 1/5 toeach. The only {JI'sthat cannot be improved upon by the worse two or the worse five are[1 2314516] and [1 234516]. The former can be y- and b-improved upon by T = {1,2} with at = a2 = {1, 2}; however, it is both ex-and p-stable. The latter is not stable inany sense: T = N\ {5}, organized as [1213416], guarantees an improvement to all itsmembers.

Thus, there are neither y- nor b-stable coalition structures in this example. Theonly one that is ex-and p-stable consists of two blocking coalitions which together havek members. Since p does not divide k, these blocking coalitions are of unequal size (andone is not minimal).

L

tj>'(v,1I)

11 2 3 4 5 6 7 8 9 0

1 1 1 1 1 1 1 1 1 1[1121314151617181910] - - - - - - - - - -

10 10 10 10 10 10 10 10 10 10

1 1 7 7 7 7 7 7 7 7[121314151617181910] - - - - - - - - - -

9 9 72 72 72 72 72 72. 72 72

3 3 3 3 2 2 2 2 2 2[12134151617181910] - - - - - - - - - -

28 28 28 28 21 21 21 21 21 21

1 1 1 1 1 1 1 1 1 1[1213415617181910] - - - - - - - - - -

10 10 10 10 10 10 10 10 10 10

11 11 11 11 11 11 11 11 2 2[121341561781910] - - - - - - - - - -120 120 120 120 120 120 120 120 15 15

1 1 1 1 1 1 1 1 1 1[12134156178190] - - - - - - - - - -10 10 10 10 10 10 10 10 10 10

1 1 1 5 5 5 5 5 5 5[12314151617181910] - - - - - - - - - -8 8 8 56 56 56 56 56 56 56

8 8 8 3 3 17 17 17 17 17[1231451617181910] - - - - - - - - - -63 63 63 28 28 210 210 210 210 210

23 23 23 11 11 11 11 1 1 1[123145167181910] - - - - - - - - - -180 180 180 120 120 120 120 12 12 12

2 2 2 3 3 3 3 3 3 3[12314516718910] - - - - - - - - - -15 15 15 40 40 40 40 40 40 20

2 2 2 2 2 2 1 1 1 1[123145617181910] - - - - - - - - - -

15 15 15 15 15 15 20 20 20 20

23 23 23 23 23 23 1 1 1 1[12314561781910] - - - - - - - - - -180 180 180 180 180 180 15 15 20 20

5 5 5 5 5 5 1 1 1 1[1231456178190] - - - - - - - - - -36 36 36 36 36 36 24 24 24 24

1 1 1 1 1 1 1 1 1[1231456178910] - - - - - - - - - 0

9 9 9 9 9 9 9 9 9

3 3 3 3 2 2 2 2 2 2[1234151617181910] - - - - - - - - - -

28 28 28 28 21 21 21 21 21 21

13 13 13 13 7 7 1 1 1 1[123415617181910] - - - - - - - - - -120 120 120 120 60 60 12 12 12 12


Example 5.13: n = 10, k = 8, p = 3. The last player will be called 0 (insteadof 10).

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q,'(v,£r)

2 3 4 5 6 7 8 9 0£r

13 13 13 13 11 11 11 11 1 1- - - - - -[123415617 81910] - - - -

120 10 10120 120 120 120 120 120 120

1 1 1 1 1 1 1 1 11- - - - - -[1234156178190] - - - -

12 12 128 8 8 8 12 12 129 9 9 3 3 3 1 1 19

- - - - -[12341567181910] - - - - - 30 30 3080 80 80 80 20 20 20

5 5 5 5 5 5 1 1 15- - - -[1234156718910] - - - - - - 24 24 1248 48 48 48 36 36 36

1 1 1 1 1 1 1 1 11- - - - - -[123415671890] - - - - 9 9 912 12 12 12 9 9 9

1 1 1 10[1234156781910] - - - - - - 0- -

8 88 8 8 8 8 8

1 1 1 1 1 1 1 10- - - - - 0[123415678190] - - -

8 88 8 8 8 8 8

1 1 1 1 1 1 1 1 1 1- - -[123451617181910] - - - -- - - 10 10 10 1010 10 10 10 10 10

1 1 1 1 1 1 1 1 1 1- - - - - -[123451671819\0] - - - - 8 12 12 1210 10 10 10 10 8

1 1 1 1 1 1 1 1 11- - - - -[1234516718910] - - - - 12 12. 12 12 610 10 10 10 10

1 1 1 1 1 1 1 1[1234516781910] - - - - - 0 0- - - 610 10 10 10 10 6 6

1 1 1 1 1 1 1 1- - 0 0[123451678190] - - - - -

610 10 10 10 10 6 6

1 1 1 1 1 1 1 1 1[123451678910] - - - - - - 0- - - 8 810 10 10 10 10 8 8

1 1 1 1 1 1 1 1 11- - - - - - -[12345167890] - - - 10 10 10 1010 10 10 10 10 10

1 1 1 1 1 1 1 1 11- - - - -[12345617181910] - - - - - 10 1010 10 10 10 10 10 10 10

7 7 7 7 7 7 1 1 1 1- -[1234561781910] - - - - -- - - 72 8 8 12 1272 72 72 72 72

1 1 1 1 1 1 1 1 1 1- - - -[123456178190] - - -- - -12 12 12 129 9 9 9 9 9

l

tJI

[123456178910]

[12345617890]

[1234567/8191°]

[12345671891°]

[12345671890]

[12345678191°]

[12345678190]

[123456789/0]

[1234567890]

Stable Coalition Structures

1

12

1

12

3

28

2

21

1

14

1

8

1

8

1

9

1

10

2

1

12

1

12

3

28

2

21

1

14

1

8

1

8

1

9

1

10

3

1

12

1

12

3

28

-2

21

1

14

1

8

1

8

1

9

1

10

4

1

12

1

12

3

28

2

21

1

14

1

8

1

8

1

9

1

10

tP/(v,tJI)

5

1

12

1

12

3

28

2

21

1

14

1

8

1

8

1

9

1

10

6

1

12

1

12

3

28

2

21

1

14

1

8

1

8

1

9

1

10

7

1

6

1

8

3

28

2

21

1

14

1

8

1

8

1

9

1

10

8

1

6

1

8

1

12

1

12

1

6

1

8

-8

1

9

1

10

257

9

1

6

°

°181

12

1

12

1

6

1

8

1

12

1

6

1

6

° °0 0

1

9

1

10

0

1

10

As in the previous example, almost all coalition structures are eliminated by IT I= 3(with 1/9 to each member) or ITI = & (with 1/8 to each member). The only ones thatremain are those with tj>(v,~) = (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 0, 0), namely:[1234156781910], [123415678190], [123456781910] and [12345678190].However, let T = N\{4, 8}; if they organize as [1231567190], their CS-value will be:either 23/180 to each of 1,2,3,5,6, 7 and 1/15 to 9, 0; or 5/36 and 1/24, respectively(depending on whether 4 and 8 are separate or together). In any case, these are bettertnan 1/8 and 0 (respectively), therefore there are no stable coalition structures for thisgame, in any sense. Note that, again, p does not divide k (actually, k ==p - 1(mod p»).

This last example thus proved

Proposition 5.14: There exists a (monotone and super-additive) game v with nostable coalition structures (in all four senses oc-,{J-,y- and <5-).

As discussed in Hart/Kurz [1983], the merit of studying the stability propertiesof coalition structures lies in the fact that when a game has such a structure, the theoryprovides an important prediction of the ultimate organization. However, as we haveseen there are also games that are intrinsically unstable ~ they have no stable coalition

structure. Some alternative concepts - which may provide insights into these cases aswell- are suggested in Hart/Kurz [1983].

L


References

Aumann, RJ.: A Survey of Cooperative Games Without Side Payments. In: Essays in MathematicalEconomics. Ed. by M. Shubik. Princeton 1967,3-27.

Aumann, R.J., and J.H. Dreze: Cooperative Games with Coalition Structures. International Journal of GameTheory 3,1974,217-237.

Bott, R: Symmetric Solutions to Majority Games. Annals of Mathematics Studies 28. Contributions to theTheory of Games, Vol. II, Ed. by H.W. Kuhn and A.W. Tucker. Princeton 1953,319-323.

Harsanyi, J.C.:Rational Behavior and Bargaining in Games and Social Situations. Cambridge 1977.

- : The Shapley Value and the Risk-Dominance Solutions of Two Bargaining Models for Characteristic-Function Games. Working Paper CP-417, Center for Research in Management Sciences. Universityof California, Berkeley 1979 (mimeo).

Hart,S.,and M.Kurz: Endogeneous Formation of Coalitions. Econometrica 51, 1983.Neumann, J. von, and O. Morgenstern: Theory of Games and Economic Behavior. Princeton 1944.Owen, G.: Values of Games with a Priori Unions. In: Essays in Mathematical Economics and Game Theory.

Ed. by R. Henn, and O. Moeschlin. Berlin-Heidelberg-New York: 1977,76-88.Shenoy, P.P.: On Coalition Formation: A Game-Theoretical Approach. International Journal of Game

Theory 8,1979,133-164.

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Documents

236 S.Hart,andM