the desirability relation of simple games

Mathematical Social Sciences I0 (1985) 155-168 North-Holland

155

T H E D E S I R A B I L I T Y R E L A T I O N O F S I M P L E G A M E S

Ezra EINY Institute o f Mathematics, Hebrew University, Jerusalem, Israel

Communicated by F.W. Roush Received 13 November 1984 Revised 12 March 1985

We prove some properties of simple games with a complete desirability relation, and investigate the relationships between the desirabiity relation of a simple game G and that of some simple games that are derived from G. We also provide an example of a proper simple game that has a complete and acyclic desirability relation but is not a weighted majority game.

Key words: Simple game; desirability relation.

1. Introduction

The desirability relation for coalitions of a simple game (see Definition 2.1) was introduced in Lapidot (1968). It was used by Peleg to develop a theory of coalition formation in simple games (see Peleg, 1980, 1981). In this paper we obtain some properties of simple games with a complete desirability relation, and investigate the relationships between the desirability relation of a simple game G and the desirability relations of some simple games that are derived from G. We also show that there exists a proper simple game that has a complete and acyclic desirability relation but it is not a weighted majority game.

In Section 2 we prove some properties of simple games with a complete desirability relation. In Section 3 we investigate the relationships between the desirability relation of a simple game G and that of the subgame and the reduced game that are derived from G. In Section 4 we investigate the relationship between the desirability relation of a proper simple game and that of its constant sum extension. In Section 5 we investigate the relationship between the desirability relation of a simple game and that of its dual. In Section 6 we provide an example of a proper simple game that has a complete and acyclic desirability relation but it is not a weighted majority game.

2. Complete desirability relations

A simDle game is an ordered pair G = (N, W), where N= { 1,..., n} is the set of players and W is a set of coalitions (i.e. subsets of N) which satisfies

0165-4896/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

156 E. Einy / Desirability relation of simple games

[St W and TDS] ~ T t W (monotonicity).

The members of W are called the winning coalitions. We always assume that G is non-null, i.e. N t W. The characteristic function c of G is the function c" 2 N-* {0, 1 } (here 2 N denotes the set of all subsets of N) which satisfies

[ c ( s ) = 1 ,~s t w].

Let G = (N, W) be a simple game. The following desirability relation for coalitions is derived from G.

Definition 2.1. A coalition S is at least as desirable as a coalition T (with respect to G), written S_> T, if for every B C N such that Bf)(SUT)=O:

B U T t W ~ B U S t W.

If S_~ T, but T>_S does not hold, then we write S> T. If SE Tand TES, then we denote S - T.

The relation -> was introduced in Lapidot (1968). It generalizes the relation of desirability for players (see Definition 9.1 in Maschler and Peleg, 1966). Peleg has developed a theory of coalition formation (in simple games) which is based on the desirability relation (see Peleg, 1980, 1981). Usually, the desirability relation of a simple game is not complete. An important class of simple games that has a complete desirability relation is the class of weighted majority games. A simple game G=(N:W) is a weighted majority game if there exists a quota q > 0 and weights w~ > 0, ..., wn > 0 such that

S t W*~ ~ wi> q. i t s

The (n+ 1)-tuple [q; w 1, ..., wn] is called a representation of G and we write G=[q; wl,...,wn]. Clearly, the desirability relation which is derived from a weighted majority game is complete.

Definition 2.2. Let G = (N, W) be a simple game. G is proper if

Se W = N - S ¢ W. (2.1)

G is strong if

S~ W ~ N - S t W. (2.2)

A proper and strong simple game is called a constant sum simple game.

Theorem 2.3. Let G = (N, W) be a simple game. I f the desirability relation o f G is complete, then G is proper or strong.

Proof. Let -> be the desirability relation of G and assume that E is complete.

E. Einy / Desirability relation o f simple games 157

Assume, on the contrary, that G is neither proper nor strong. Then there exist two coalitions, S and T, such that

Denote

SE W and N - S E W.

T¢ W and N - T¢ W.

S l = S - T , S2= T - S .

B~=SAT , B 2 = N - ( S U T ) .

Then by (2.3) and (2.4), we have

BIUS 1 = S E W and B20Sl = N - T ¢ W.

BIUS2= TOi W and B 2 O S 2 = N - S E W.

Thus, Sl and $2 are incomparable with respect to assumption that _> is complete.

The following lemma will be useful in the sequel.

(2.3)

(2.4)

_>, which contradicts our

Lemma 2.4. Let G = (N, W) be a simple game, and let ~ be the desirability relation o f G. I f >- is complete, then for every S, T C N we have

S> T = S - T > T - S .

Proof. Assume that E is complete and let S, T C N such that S > T. Then there exists B c N - ( S O T ) such that B O S e W and B O T e W. Let B * = B U ( S A T ) . Then B * A ( ( S - T ) O ( T - S ) ) = f l and B * U ( S - T ) = B U S E W, while B * O ( T - S ) = BU T¢ W. Since -> is complete, we have S - T > T - S.

Corollary 2.5. Let G = (N, W) be a simple game, and let >- be the desirability relation o f G. I f > is complete, then for every S, TC N

S> T = , N - T > _ N - S .

Proof. Assume that _> is complete and let S, T C N s u c h that S > T. Let B C S n Tsuch that B U ( N - S) E W. We show that B U ( N - T) E W. Since S > T, Lemma 2.4 implies that S - T > T - S. Let B* = B O ( N - (S U T)). Then B* n ((s - T) O ( T - S)) = 0 a n d B * O ( T - S ) = B U ( N - S ) E W. Since S - T > T - S , we obtain that B U ( N - T)= B* O (S - T) E W. Therefore N - T _> N - S.

. Desirability in the subgame and the reduced game

In this section we investigate the relationships between the desirability relation of


a given simple game and the desirability relations of the subgame and the reduced game that are derived from the game.

Definition 3.1. Let G= (N, W) be a simple game, and let S t W. The subgame of G which is determined by S is the simple game G]S= (S, w N 2 s ) , where 2 s is the set of all subsets of S. The reduced game of G with respect to S is the simple game RG(S)=(S, Ws), where Te Ws~*[TCS and T 1 3 ( N - S ) e W].

Proposition 3.2. Let G=(N, W) be a simple game, and let S ~ W. Let >_, 1>- and 2> be the desirability relations of G, G I S and RG(S), respectively. Then for every S 1 , S 2 C S we have

(a) SI >-SE= SlI> S2. (b) SI >-SE=~ SI2>-S2 .

Proof. (a) Let $1, $2 C S such that $1 >- $2. We show that Sl 1 >- $2. Let B C S - ($1 13 $2)

such that BUS2~2st ' )W. Since $1>_$2, we have B U S l e W . But B13SICS. Therefore BUSI ~2sN W. Hence, $1 l>-s2.

(b) Let S l, $2 C S such that B 13 S 2 ~ W s. Then (B 13 $2) 13 ( N - S) e W. Now, B U ( N - S ) c N - ( S I U S 2 ) and $1>-$2. Therefore B 1 3 ( N - S ) U S l ~ W. Hence, B 13 Sl ~ Ws. Thus, Sl 2_> $2.

Corollary 3.3. Let G = (N, W) be a simple game and let S ~ W. I f the desirability relation of G is complete, then the desirability relations o f G IS and RG(S) are complete.

Proposition 3.4. Let G = (N: W) be a simple game, and let S ~ W. Let >-, l> and 2> be the desirability relations of G, G IS and RG(S), respectively. I f >_ is complete, then for every Sl, $2 C S we have

(a) $11 > $2 =~ S1 > $2" (b) S12~ $2=S1 > S 2.

Proof. (a) Assume that > is complete. Let $1, $2 C S such that S 11 > $2" We show that

Sl > $2. Since S l l> $2, there exists B C S-($1 US2) such that B 13S1 ~ W N 2 s and BUS2¢i Wf)2 s. Thus, B13SIW and BUS2~ W. Since B C N - ( S I U S 2 ) and >- is complete, we have $1 > $2.

(b) Assume that >- is complete and let $1, $2C S such that $12> $2. Then there exists BCS-(S1USE) such that BUSI e Ws and BUS2¢ W s (where RG(S)=(S, Ws)). By the definition of W s, we have ( B U S I ) U ( N - S ) ~ W and (BUS2) U(N-S)¢~ W. Let B * = B U ( N - S ) . Then B * C N - ( S 1 US2) and B'13S l W, while B* 13 $2 ~ W. Since >- is complete, $1 > $2.

E. Einy / Desirability relation of simple games 159

The following example shows that the completeness of -> in Proposition 3.4 is necessary.

Example 3.5. Let N = { 1, .. . , 6}. We define a constant sum simple game G = (N, W) by

w *[ISl>3 or I s l = 3 and ISN{1,2,3}I is oddl

(where here and in the sequel if A is a finite set, then [A[ denotes the number of members of A). Let S = { 1,... , 5} and let _>, 1->, 2> be the desirability relations of G, G IS and RG(S), respectively. Then {4, 5} 1 > { 1, 4} and { 1, 4} 2 > {4, 5}, but { 1, 4} and {4, 5} are incomposable with respect to _>.

4. Desirability in the constant sum extension

In this section we investigate the relationship between the desirability relation of a proper simple game (see Definition 2.2) and the desirability relation of its constant sum extension.

Definition 4.1. Let G = (N, W) be a proper simple game (see Definition 2.2). The constant sum extension of G is the simple game Go=(NU {z}, W0), where z ¢ N , and S ~ W0 if

(a) z ~ S and S ~ W, or (b) z ~ S and N - S ¢ W.

Theorem 4.2. Let G = (N, W) be a proper simple game. Then G has a complete desirability relation i f and only i f its constant sum extension has a complete desirability relation.

Proof. Let G0 = (NO {z}, Wo) be the constant sum extension of G. Let _~ and o> be the desirability relations of G and Go, respectively. If o~, is complete, then since G0IN= G, we obtain by Corollary 3.3 that _> is complete. We now assume that -> is complete and show that o> is complete. Let Sl, S2CNO {z}. We distinguish two possibilities.

4.1. z ¢ S l US2

In this case Sl U S2 C N. Since -> is complete, we have Sl -~ S2 or S2 ~ Sl. Without loss of generality assume that S~ -> S2. We now distinguish two subcases.

4.1.1. S 1 > S2

We show that in this case Sl 0-> 82. Indeed, let B C (N O {z}) - (Sl U S2) such that B U S2 ~ Wo. If z~ B, then since Sl > S2, we have B US1 e W. By the definition of


Go, BUS1 ~ W o. Assume that z eB. Then by the definition N - (B O $2) ~ W. Hence

of Go. We have

( N - ( B U S 1 US2))U (S 1 - S2)=N-(BUS2)~. W. (4.1)

Since $1 > $2, we obtain by Lemma 2.4 that $I - $2 > $2- 51. Therefore by (4.1), we get N - ( B U S 1 ) =(N-(BUS1US2))U(S2-SI)~i W. By the definition of Go, we obtain that B U $1 ~ Wo. Therefore $10> $2.

4.1.2. SI -- S 2

Assume, on the contrary, that in this case $1 and $2 cannot be compared by 0>. Then there exist B 1, B2 C (NO {z}) - (Sl O $2) such that

B 1 O S1 e W o and B 2 O S l ~ Wo, (4.2)

B1US2¢ Wo and B2US2e Wo. (4.3)

Since $1->$2 and zC~Sl US2, we have zeBI NB2. Therefore by (4.2), (4.3) and the definition of Go we obtain:

N - ( B 1 U S l ) g W and N-(B2US1)~ W,

N - (B1U S2) e W and N - (B2 U S2) ¢ W.

(4.4)

(4.5)

Let B ~ = N - ( B I USIUS2) and B ~ = N - ( B 2 U S 1 US2). Then by (4.4) and (4.5):

B?U ( S1- S2) = N - (BI U S2) ~ W and B~U (Sl - S2) = N - (B2 U S2) ¢ W,

(4.6)

B~'U (S 2 - 81) = N - (B 1 US1) ¢ W and B~'U (S 2 - $1) = N - (B2U S1) E W.

(4.7)

Since B ~ U B ~ C N - ( ( S l - $2)O(S2-$1)), (4.6) and (4.7) imply that B~ and B~ are incomparable with respect to ->, which contradicts our assumption that ~" is complete.

4.2. z ~ S 1 US 2

Assume, on the contrary, that in this case S1 and $2 cannot be compared by o>. Then there exist B1,B2C(NU {z})-(S1 US2) such that

B1 US1 ~ Wo and B2USI ~i Wo, (4.8)

B 1 US2~ W o and B2US2 F. W o. (4.9)

Since (BIUB2)th($1US2)=O, (4.8) and (4.9) imply that BI and B2 are incomparable with respect to o_>. Since z~B I UB2, this is impossible by Section 4.1.

Proposition 4.3. Let G=(N, W) be a proper simple game, and let Go =


(NO {z}, We) be the constant sum extension o f G. Let 2- and 02- be the desirability relations of G and G o, respectively. Then for every S, T C N such that SG T=O, we have

(a) $2- T~S°2- T. (b) $2- T-~S°2- T.

Proof. (a) Let S, T C N such that S A T = O and S 2 T . We show that S°2-T. Let BC

(NO {z} - (SO T) such that BU T~ W o. We have to show that B O S e We. If zc~B, then since $2- T, B O S e We. Assume that z~B. Then N - ( B U T ) ~ W. Let B*= N - ( B U S U T ) . Since SAT=O, we have B * U S = N - ( B O T ) ~ W. Now, S2-T. Therefore N - ( B U S ) = B * U T~ W. By the definition of We, B U S e W o. Hence, sO_> T.

(b) Let S, T C N such that St') T=O and S > T. Then by (a), S°2- T. Since S > T, there exists B C N - (S U T) such that B U S E W and B O T~ W. Now B O S O TC N. Therefore by the definition of We, B US e W 0 and B O T~ W0. Since S 02- T, we have S 0 > T.

Example 4.4. Consider the proper weighted majority game G = [6; 4, 3, 2]. Let Go be the constant sum extension of Go. Then it is easy to see that Go = [6; 4, 3, 2, 2]. Let > and 02- be the desirability relation of G and Go, respectively. Then {2} 2- {2, 3}, but {2, 3} o> {2}. Thus, the assumption that S A T = O in Proposition 4.3 is necessary for part (a) of the proposition.

If we assume that the relation 2. in Proposition 4.3 is complete, then part (b) of the proposition remains true even if S N T#: 0 (see part (a) of Proposition 3.4). But if 2- is not complete, the assumption that S N T= O is necessary, as we will see in the following example.

Example 4.5. Let N = { 1, 2, . . . , 5 }. We define a proper simple game G = (N, W) by

S t W~,[[SI<3 or IS[=3 and [sn{1,2 ,3}[ is oddl.

Let Go = (NO {z}, We) be the constant sum extension of G, and let 2- and 02- be the desirability relations of G and Go, respectively. Clearly {4,5} > {1,4}. Now {1,4,z}~W0, while {4, 5, z} ~ Wo. Therefore {1,4} and {4,5} are incomparable with respect to 02-.

The following example shows that the desirability relation of a simple game may be not complete even if its restriction to a one-player coalitions is complete and the game is constant sum.

Example 4.6. Let N = { 1, 2, . . . , 8}. We define a proper simple game G = (N, W) by specifying W n = { { 1, 2, 3,, k}, { 1, 2, 4, 5, 6} [j, k ~ N - { 1, 2, 3} } (where, here and in the sequel, if G = (N, W) is a simple game, then W m denotes the set of minimal


members of W with respect to the set-inclusion relation). Let 2 be the desirability relation of G. Since the desirability relation for one-player coalitions is transitive (see Theorem 9.2 of Maschler and Peleg, 1966), one can easily check that

2 , . . . , _~ {8}. (4.10) {1}Z{2} Let

$1={3},

Then we have

B1 13 S1 ~ W,

$2= {4,5}, BI = {1,2,7,8}, B2= {1,2,6}.

BI US2~ W, B 2 U S 2 e W , B2US1 ¢ W.

Since BIUB2CN-(SIUS2), we obtain that SI and $2 are incomparable with respect to 2 . Thus, 2 is not complete. Let Go = (N U {z}, Wo) be the constant sum extension of G, and let 02 be the desirability relation of Go. Since 2 is not complete, by Theorem 4.2, °2 is not complete. Now (4.10) and part (a) of Proposition 4.3 imply that

{1} 02 {2} °2, ..., °2 {8}. (4.11)

We now show that for every i~N we have {z}°2{i}. Indeed, let i e N and B C N - { i } . If B13{i}e Wo, then BN{1,2}:g0. Therefore N-B~. W. By the definition of Wo, B13 {z} e W0. Therefore {z} 02 {i}. Since the desirability relation for one-player coalitions is transitive,

{z} 0--> {1} 02 {2} °2 , . . . , ° 2 {8}.

The following example shows that Theorem 4.2 does not remain true if instead of the desirability relation for coalitions we consider the desirability relation for players (i.e. the restriction of the desirability relation to one-player coalitions).

Example 4.7. Let N = {1, ..., 7}. We define a proper simple game G= (N, W) by specifying

Win= { { 1,2, i,j}, {1,3,4,5} [i, jeN-{1,2}}.

Let 2 be the desirability relation of G. Then it can be easily checked that { 1} 2 {2}, ..., 2 {7}. Let Go=(N13{z}, Wo) be the constant sum extension of G. We show that {1} and {z} cannot be compared by 02. Indeed, let BI = {2,6, 7} and B2= {2,3}. Then N - B 1 ~ W. Therefore Bl 13{z} ~ W0, while B 1 13{1} e W0. On the other hand, N - B2 ¢ W. Therefore B213 {z} e W0, while B2 13 { 1 } ~ W 0. Thus, the restriction of °2 to one-player coalitions is not complete.

5. Desirability in the dual game

In this section we investigate the relationship between the desirability relation of a simple game and that of its dual.


Let G = (N, W) be a simple game. The dual game of G is the simple game G*= (N, W*), where

S~ W * * ~ N - S ¢ W.

For a detailed discussion of the dual operation the reader is referred to Shapley (1962). We just note that G**= G.

Theorem 5.1. Let G= (N, W) be a simple game. Then the desirability relation o f G is complete i f and only i f the desirability relation o f its dual is complete.

Proof. Let G* = (N, W*) be the dual of G. Let -> and *_> be the desirability relation of G and G*, respectively. Since G**= G, it is sufficient to show that the completeness of -> implies the completeness of *_>. So assume that ~ is complete, and assume, on the contrary, that *_> is not complete. Then there exist S~, S2CN and B I , B 2 C N - (Sl US2) such that

Bl OSi E W* and B2 tJSI ~ W*, (5.1)

B~ O $2 ~ W* and B2 t.) $2 e W*. (5.2)

Now, by (5.1), (5.2) and the definition of G* we have

N - ( B I OSl)¢ W and N - ( B 2 U S I ) ¢ W, (5.3)

N-(B~ tJS2) e W and N - ( B 2 US2)¢ W. (5.4)

Denote

T1 =Sl -$2 , Te=S2-Sl,

B~ = N - (B l US l US2), B~ = N - (B2 U S1 USe).

Then (5.3) and (5.4) imply that

B?t.) Tl = N - (B~ US2)~ W and B~U 7"1 = N - (B2 U S2) ¢ W.

B? U T2 = N - (B~ US1) ¢ W and B~U T2 = N - (B2 tJ S~) e W.

Since B~UB~CN-(TIUT2) , the coalitions Tl and T2 are incomparable with respect to ~ , which contradicts our assumption that -> is complete.

Proposition 5.2. Let G = (N, W) be a simple game. Let >_ and * >_ be the desirability relations o f G and its dual, respectively. Then for every S, T C N such that S N T= 0 we have

(a) S>_ T,~S *>_ T. (b) S> T<*S*> T.

Proof. Let G*= (N, W*) be the dual of G. (a) Let S, T c N such that SN T=~. Since G**= G, it is sufficient to show that


S E T implies S * E T . So assume that S E T and let B C N - ( S O T ) such that B t3 Te W*. Then N - (B O T) ¢ W. Let B* = N - (B 13 S O T). Then since S N T= O, B * O S = N - ( B O T ) ¢ W . Since S E T , we have B*OT¢iW. But B*OT= N - ( B O S ) . Therefore B O S e W*. Hence, S * >_ T.

Co) Let S, TCNsuch that S A T=O. It suffices to show that S > T implies S *> T. So assume that S > T. By (a) we have S E T. Therefore it suffices to show that there exists B C N - ( S O T ) such that B O S e W* and B O T ¢ W*. Now, since S> T, there exists B * C N - ( S O T ) such that B * O S e W and B * O T ¢ W . Therefore N - ( B * t 3 S ) ¢ W * and N - ( B * O T ) e W * . Let B = N - ( B * O S O T ) . Then BC N - (S O T) and since S N T= 0, we obtain that B O S = N - (B* O T) e W* and B O T = N - ( B * O S ) ¢ W*. Therefore S *> T.

Since the order which is determined by the desirability relation for players is preserved by some important solutions of cooperative games (see section 9 of Maschler and Peleg, 1966), the following corollary of Proposition 5.2 might be useful.

Corollary 5.3. Let G = (N, W) be a simple game. Let ;~ and *E be the desirability relations o f G and its dual, respectively. Then for every i, j e iV, i ~j , we have

(a) {i} >_{j} ,~{i}*E{j} . (b) {i} > { j}~,{ i}*> {j}.

Example 5.4. Let G = (N, W) be the simple game that is defined in Example 4.5. Let G* = (N, W*) be the dual of G, and E and *E be the desirability relations of G and G*, respectively. Then {4,5}>{1,5}. But {1,5}eW* and {4,5}~W. Thus {1,5}*>{4,5}. Therefore the assumption that SNT=O in Proposition 5.2 is necessary.

Definition 5.5. Let GI = (N, Wl) and G2 = (N, W2) be two simple games. Let 1E and 2E be the desirability relation of Gl and G2, respectively. Gl and G2 are said to have the same desirability pattern with respect to disjint coalitions if for every S, TC N, such that S N T= fJ we have

S 1> T¢~$2> T.

Proposition 5.2 implies that a simple game and its dual have the same desirability pattern with respect to disjoint coalitions. We show that this is the only possible way that two games with complete desirability relations have the same desirability pattern with respect to disjoint coalitions. First we need the following lemma.

Lemma 5.6. Let Gl = (N, WI) and G2 = (N, W2) be two simple games. I f G and G2 are both proper or strong, and they have the same desirability pattern with respect to disjoint coalitions, then GI = G2.


Proof. Let 1~ and 2~. be the desirability relations of GI and G2, respectively. Assume first that G 1 and G2 are proper. Let S ~ Wl. Since G1 is proper, N - S ~ W 1 . Therefore S I > N - S . Since Gl and (32 have the same desirability pattern with respect to disjoint coalitions, S 2> N - S . Hence, S ~ W l . Thus, WI C W 2. The same argument gives that W2C Wl. Therefore GI = G2. If G 1 and G2 are strong, then the dual games G~ and G~ are proper, and by Proposition 5.2 they have the same desirability pattern with respect to disjoint coalitions. Therefore G~ = G~, and hence

G1 = G2.

Corollary 5.7. Let GI and G2 be two simple games. I f the desirability relations o f GI and G2 are complete, and they have the same desirability pattern with respect to disjoint coalitions, then G1 = G2 or Gl = G~ (here G~ is the dual o f G2).

Proof. Since the desirability relations of G1 and (32 are complete, by Theorem 2.3, G1 is proper or strong and the same holds for G2. If GI and G2 are both proper or both strong, then by Lemma 5.6, GI = G2. Assume that Gl is proper and G2 is strong (the case where GI is strong and G2 is proper is similar). Then G~ is proper. Also by Proposition 5.2, G2 and G~ have the same desirability pattern with respect to disjoint coalitions. Therefore the same holds for Gl and G~. By Lemma 5.6, G1 =GL

Proposition 5.8. Let G = (N, W) be a simple game. let >_ and * ~ be the desirability relations o f G and its dual, respectively. I f ~_ is complete, then for every S, TC N we have

S > T~ S *~- T.

Proof. Let G*= (N, W*) be the dual of G. Let S, T C N such that S > T. We show that S *_> T. Let B C N - ( S U T) such that BU T~ W*. Then N - ( B U T)¢~ W. Now since S > T and -> is complete, Lemma 2.4 implies that S - T > T - S. Let B* = N - (B U S L/T). Then B* U ( S - T) = N - (B U T) ~ W. Since S - T > T - S, we have B * U ( T - S ) ~ W. Thus, N - ( B U S ) ¢ i W. Therefore B U S e W*. Hence, S *-~ T.

Example 5.9. Consider the w e i g h t e d majority game G = [6; 4, 3, 3, 2]. Let G* be the dual of G. Then G* = [7; 4, 3, 3, 2]. Let ~ and *E be the desirability relations of G and G*, respectively. Tnen { 1, 4} > {3, 4} and { 1, 4} - * {3, 4}. Thus, Proposition 5.8 cannot be sharpened.

6. An example

Let G = (N, W) be a simple game, and let ~> be the desirability relation of G. Lapidot (1968) has shown that > may be cyclic. It is also shown by Kim and Roush that > may be cyclic even if ~ is complete. Howeve, if G is a weighted majority

166 E. Einy / DesirabtTity relation of simple games

game, then :~ is complete and > is acyclic. Therefore a natural question which was raised by B. Peleg is: Is it true that each simple game that has a complete and acyclic desirability relation (i.e. the strict relation that is derived from its desirability relation is acyclic) is a weighted majority game? In this section we answer the question negatively.

Let N= { 1,. . . , 9} and let W be the minimal family of subsets of N that has the following properties:

If S c N and [S[>__7, then S~ W. (6.1)

W ~ {{2, 3, 4, 5, 6, 9}, { 1, 2, 3, 7, 8, 9}, {1,4,5,6, 7,8}}. (6.2)

If S~ W, then for each i~S and j 6 N - S (6.3).

j < i ~ ( S - { i } ) U { j } ~ W.

Consider the simple game G = (N, W). Let G* = (N, W*) be the dual of G. We note that since each five-player coalition in G is losing, any coalition that has at least four players is winning in G*.

Let 2. and *2- be the desirability relations of G and G*, respectively.

Claim 6.1. "2- is complete.

Proof . By (6.3), for each i , j~N if i<_j, then {i} 2- {j}. Thus, any two one-player coalitions in G are comparable with respect to 2-. Therefore by Corollary 5.3 any two one-player coalitions in G* are comparable with respect to *2-. Assume now, on the contrary, that *_> is not complete. Then there exist SI,SaCN and BI, B2 C N - ($1 U $2) such that

B1 U $1 e W* and Bl U $2 ~ W*,

BaUSz~ W* and BzUSI~ W*.

Since B1 U $16 W*, we have I B~ O S~ I - 3. Therefore I Bl I > 2 or ] $1[ -> 2. Without loss of generality IB1]>_2. Then since BIOSzeW*, we have [SzI=I. Since B21.JS2E W*, IS2l >_2. Now, BzOS,¢ W*. Therefore ISll= 1. Thus, ISl] =]S2l = 1. Since any two one-player coalitions in G* are comparable with respect to "2-, we have $1 *2- $2 or $2 *2- S1. Without loss of generality, Sl *2- Sz. Then since Bz U $2 W* and Bz C N - ( $ 1 0 Sz), we have Bz O $1 ~ W*, which is a contradiction.

Claim 6.2, * > is acyclic.

Proof . Assume, on the contrary, that there exist coalitions S~, ..., Sm in G* such that

$ 1 " > $ 2 " > ...*> S~*> SI.

Now, 1 ___ ISil-< 3 for each i= 1, . . . , m. Since for each SCN, IS]>_4 implies S e W*,


there exists 1 <i<_m such that [Si[ <3. Since the restriction of *-> to one-player coalitions is transitive, there is 1 < i_< m such that I Si[> 1. We distinguish now two

possibilities. (a) There exist 1 <_i,j<_m such that [Si[ ~ ISjl. In this case there is 1 < k < m such

that ISk] < [Sk+l(mod m)l. But this is impossible because, from the construction of W, if S, T c N and ISl>lTI, then S*>_T.

(b) I Sil = 2 for each i= 1, ..., m. In this case there exist coalitions BI, ...,Bin in G* such that for each l < i < m , BiN(SiUSi+l(modm))=O, B i U S i E W * and BiUSi+I~ W*. Since [Si[ =2 for each l<_i<_m, we have [Bi[ = 1 for each 1 <i<_m. By the completeness of *->, we have Bi+l(modm)*>Bi for each l<_i<m. Thus, B m * > B m_ 1 " > . . . *> B 1 * > Bin, which contradicts the fact that the restriction of *_> to one-player coalitions is transitive. Thus, in each one of the cases we get a contradiction. Therefore * > is acyclic.

Claim 6.3. E is complete and > is cyclic.

Proof. Since *_> is complete by Theorem 5.1, ~ is complete. Now it can be easily checked that (see (6.2))

{2,5,9} >{1,7,9} >{5,6,7} >{2,5,9}.

Therefore > is cyclic.

Corollary 6.4. The game G* has a complete and acycic desirability relation but it is not a weighted majority game.

Proof. Assume, on the contrary, that G* is a weighted majroity game. Then it is well known that G**= G is a weighted majority game (see Shapley, 1962). But this is impossible because > is cyclic.

Remark 6.5. It is easy to see that the game G that is defined above is proper (see Definition 2.2). Therefore it has a constant sum extension (see Definition 4.1). Since

is complete and > is cyclic (see Claim 6.3), by Theorem 4.2 and Proposition 3.4, the desirability relation of the constant sum extension of G is complete and cyclic. Thus there exists a constant sum simple game that its desirability relation is complete and cyclic.

The game G* that is defined above is not proper. We now use G* to construct a proper simple game that has a complete and acyclic desirabiity relation but it is not a weighted majority game.

Let No= {1,..., 10}. We define a simple game G0=(N, W0) by

S e W0~* 10eS and S - {10} ~ W*.

Proposition 6.6. The game Go = (N, W o) is a proper simple game that has acom-

168 E. Einy / Desirability relation o f simple games

plete and acyclic desirabilily relation but it is not a weighted majority game.

Proof. It is clear from the definition of W0 that Go is proper. Let °2 be the desirability relation of Go. We first show that o2 is complete. Let $, T C N o. If 1 0 e S - T, then by the definition of W 0, S 02 T. If 10~SU T, then since "2 is complete we have S *2 T or T "2 S. Without loss of generality, S "2 T. We show that S 0_> T. Indeed, let B C N o - (SU T) such that BU Te W 0. Then lOeB. Therefore ( B - { I O } ) U T e W * . Since S *2T, ( B - { I O } ) U S e W * . Therefore B U S e W o. It remains to check the case where 10 e S N T. In this case we have, by the completeness of *2 , S - {10} *2 T - {10} or T - {10} *_>S- {10}. Without loss of generality, S - { 1 0 } ' 2 T - { 1 0 } . Let B C N o - ( S U T ) such that B U T e W 0. Then B U ( T - { I O } ) e W*. Therefore B U ( S - { I O } ) e W*. Hence, B U S e W o. Thus, S °2 T. We now show that 0> is acyclic. Assume, on the contrary, that there exist coalitions SI, . . . , Sm in Go such that

$10> $20>, ..., 0> Sin0> $1"

m m Since for S, TCN0, 1 0 e S - T implies S 02 T, we have 10e ~i=~ Si or 10~ U~=m si.

m

If 1 0 e ~ _ ~ S i , then (S l - { lO} , . . . , Sm-{ lO}) is a cycle with respect to *>. If m

10~ Ui=~ si, then ($1,..., Sin) is a cycle with respect to *>. Thus, in both of the cases we get a contradiction to the fact that * > is acyclic.

It remains to show that Go is not a weighted majority game. Assume, on the contrary, that there exist a quota q > 0 and weights wl->0, ..., Wlo_>0 such that Go-- {q; wl, ..., Wlo]. Then the definition of W 0 implies that G*= [q-wl0; wl, ..., Wg], which contradicts the fact that G* is not a weighted majority game.

Acknowledgements

I am indebted to Professor B. Peleg for introducing me to the subject and for very helpful discussions concerning this work. I am also very grateful to Professors K.H. Kim and F. Roush for showing me an example of a simple game that has a complete and cyclic desirability relation. That example opened the possibility to construct the example in Section 6.

References

E. Lapidot, Weighted majority games and symmetry groups of games, MSc. Thesis in Hebrew, Tech- nion, Haifa 0968).

M. Maschler and B. Peleg, A characterization, existence proof and dimension bounds for the kernel of game, Pacific Journal of Mathematics 18 0966) 289-328.

B. Peleg, A theory of coalition formation in committees, Journal of Mathematical Economics 7 0980) I15-134.

B. Peleg, Coalition formation in simple games with dominant players, International Journal of Game Theory l0 (1981) 11-33.

L.S. Shapley, Simple games: An outline of the descriptive theory, Behavioral Science 7 0962) 59-66.

the desirability relation of simple games

Documents