The Fundamental Theorem of Voting Schemes

Journal of Combinatorial Theory, Series A  2619 journal of combinatorial theory, Series A 73, 120129 (1996) Article No. 0006

The Fundamental Theorem of Voting Schemes D. E. Loeb* LaBRI, Universite de Bordeaux I, 33405 Talence, France Communicated by Gian-Carlo Rota Received December 9, 1993

in memory of rabbi donald starr

19361994

1. Introduction Let V be a set, and call the elements of V voters or players. A subset AV is called a coalition. The compliment V&A of a coalition A is denoted A. A set of coalitions S is a game if all supersets of winning coalitions are winning as well. A coalition AV is said to be blocking (in S) if A  S. The set of all blocking coalitions is denoted S*, and is called the dual of S, since (S*)*=S. A game S is simple if A # S (A wins) implies A  S (A blocks); i.e. SS*. Conversely, a game S is strong if A  S (A blocks) implies A # S (A wins); i.e. S*S. Given a weight w i # N for each voter i # V=[1, 2, ..., n] and a quota q # N, we can define the quota game (w 1 , w 2 , ..., w n ) q =[AV :  i # A w i q]. Democracy over an odd number of voters is a strong simple game. Dem 2n+1 =(11 . . . 1 ) n+1 =[A[1, ..., 2n+1] : |A| >n]. 2n+1

Similarly, dictatorships are also strong simple games. Dict n =(100 .. . 0 ) 1 =[A[1, ..., n] : 1 # A]. n&1

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Strong simple games are also called voting schemes since they provide an effective method by which a group of voters may combine their individual preferences into a group decision for one of two choices. * Author supported by URA CNRS 1304 and NATO CRG 930554. E-mail adress: loeb labri.u-bordeaux.fr; WWW: http:www.labri.u-bordeaux.frtloeb.

120 0097-316596 12.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Shapley and Billera studied composing such games into ``tournaments'' G[H 1 , ..., H n ]. Each of the n players in the game G represents the winning coalition in a game H i . Interpreted as voting schemes, G[H 1 , ..., H n ], is a ``bureaucracy.'' The voters of each H i vote independently as ``committees''. The group decisions of each H i are inputs to the collective choice function G which combines the decisions of the n committees. Games formed from simpler games in this way are said to be decomposable, and all others are said to be prime. [3, 18] Obviously, all games can be factored into prime components. This decomposition is not necessarily unique unless we restrict our attention to full games. A full game is one in which each voter appears in at least one minimal winning coalitions. (Voters which appear in no minimal winning coalitions are called dummies.) The decomposition of full games into prime games is unique (up to rearrangement of parenthesis, see Eq. 1). This is Shapley and Billera's game theoretic analog of the fundamental theorem of arithmetic. Shapley and Billera also studied simple games, and proved the ``fundamental theorem'' for them. Up to associativity (Eq. 1), all simple games have a unique factorization into prime simple-games. Note that a prime simplegame is a simple game which has no non-trivial decompositions into simple games; it may very well have decompositions into non-simple games. Analogously, certain numbers such as 2 have decompositions among the Gaussian integers (i+1)(1&i) although they have none among the integers. Unique decomposition of strong games into prime strong-games holds by a simple duality argument. A[B 1 , ..., B n ]*=A*[B *, 1 ..., B *] n

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where G* is the set of blocking coalitions for G, i.e. coalitions whose compliment is not in G. The main result of this paper will be the proof of the ``Fundamental Theorem'' of strong simple games, namely, all full strong simple games have unique decompositions into prime strong-simple-games. The notion of strong simple games has arisen independently in a number of contexts. They can be interpreted as voting schemes or social choice functions [6], as ipsodual elements of the free distributive lattice [19], as members of the free median set [15, 16, 14], as self-dual anti-chains (or clutters) [13], as maximal intersecting families of sets [5, 12], as (ultra)-filters [4], as non-dominated coteries [20], or as critical tripartite hypergraphs [1, 2]. Games are thought of in reliability theory as semi-coherent structure functions [17, 7]. Linear programmers tend to think of games as boolean functions; threshold functions (or majority games or quota games) in particular are of special interest [12, 8]. Each rediscovery of a theory gives birth to alternate notation and terminology. An attempt has been made here to choose a consistent terminology

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which makes our results as clear as possible. The above references are useful in adapting our results to other fields of interest. In the sequel to this article [14], we will relax the condition that all committees be disjoint.

2. Prime Games Once a certain number of voters have voted, the game can be reduced to the remaining voters by considering what sets of remaining voters together with the voters who have already voted ``yes'' are needed in order to win. More formally, given a game S and a coalition of voters CV, let I C (S) be the function from the set 2 C of subsets of C to the set FD(C) of games on the set C defined as follows: (I C (S))(B)=[AC : A _ B # S]. The notation FD(C) derives from the fact that when ordered by inclusion, these games form the free distributive lattice with 0 and 1 generated by the set C. 0 =<=(0. . . 0) 1 is the all losing game, whereas 1 =2 C =(0. . . 0) 0 is the all winning game. Obviously, S is completely determined by the function I C (S). Note that (I C (S))(B)*=I C (S*)(C &B). Given a game S on n voters, and n games T 1 , ..., T n on disjoint sets of voters V 1 , ..., V n respectively. Then one can define the composition of S with T 1 , ..., T n to be the set of coalitions AV 1 _ } } } _ V n such that [i: A & V i # T i ] # S. In other words, S[T 1 , ..., T n ] is the game in which the voters vote by committee. Each committee votes according to its own rules T i and results are combined via the game S. Let C 2 be the two-element poset [0, 1] with 0<1. n-player games can be associated with weakly increasing functions f S : C n2  C 2 . The composition S[T 1 , ..., T n ] corresponds to the function f S b ( f T1 _ } } } _f Tn ): C k21 + } } } kn  C 2 where k i = |V i |. These compositions obey the following associative law: S[T 1[U 11 , ...], T 2[U 21 , ...], ...]=(S[T 1 , T 2 , ...])[U 11 , ..., U 21 , ...]. That is to say, ( f S b ( f T1 _f T2 _ } } } )) b (( f U11 _f U12 _ } } } )_( f U21 _f U22 _ } } } )_ } } } ) File: 582A 261903 . By:BV . Date:17:01:96 . Time:11:30 LOP8M. V8.0. Page 01:01 Codes: 2927 Signs: 1881 . Length: 45 pic 0 pts, 190 mm

=f S b (( f T1 b ( f U11 _f U12 _ } } } ))_( f T2 b ( f U21 _f U22 _ } } } ))_ } } } ).

(1)

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These compositions involve voting by committees in which each committee is divided into subcommittees, or equivalently voting by committees whose results are passed through ``super-committees'' before being combined. For example, the American presidential voting scheme is the composition of a 52-voter quota game 1 with 52 subgames. A smaller yet non-obvious example is the following: the weighted game (2, 2, 1, 1, 1) 4 is given by the composition Dem 3[Dem 1 , Dem 1 , Dem 3 ]. The following trivial decompositions are true for all games S. S=Dem 1[S]

(one committee),

(2)

(one-person committees).

(3)

and S=[Dem 1 , ..., Dem 1 ]

A strong simple game S is said to be a prime strong-simple-game if all its decompositions into strong simple games are of the form 2 and 3. We will prove the following result concerning prime strong simple games. Theorem 2.1. All full strong simple games admit unique decompositions into prime strong-simple-games. Note however that games with dummies do not in general have unique decompositions. For example, if V=W _ D where all the voters in D are dummies, then S=Dict 2[S|W, T] where S |W is the restriction of S to W (i.e. S|W=[AW : A # S]=S _ 2 W ) and T is any game on D. The important concept in the proof of Theorem 2.1 is the committee. We shall first define and study committees, and then conclude this section with the proof of Theorem 2.1. Consider two coalitions W, CV. Say that W depends on C if W _ C=< and W  S but W _ C # S. In other words, W loses, but could win with the help of C, or equivalently I C (S){0, 1. Let S be a strong simple game. Define a committee of S to be any set of voters CV such that S=T[U 1 , U 2 , ...] where U 1 is a strong simple game on C. By Equation 3, all singletons [v] are committees, and by Equation 2, V itself is a committee. S is a prime strong-simple-game if and only if these are its only committees.

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1 The 52 committees correspond to the 50 states, the District of Columbia, and the United States House of Representatives. Each state (and DC) has a weight given by its number of electoral votes. The House of Representatives votes only in the case of a tie in the electoral college. Thus, the weight of the House can be taken to be 1. The strong simple game adopted by each state is that of democratic vote among the (hopefully odd number of) eligible voters. The actual voting structure of the house is a weak simple game. The 435 congressmen vote in 50 committees, one for each state. If a state is tied, or the 50 states split evenly, there is no provision for breaking the tie.

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The following technical lemma gives other equivalent definitions. Lemma 2.1. Let S be a strong simple game on a set V, and let CV. Then the following are equivalent: 1. C is a committee of S. That is to say, S=T[S C , ...] where S C is a strong simple game on C. 2. I C (S): 2 C  FD(C) takes at most one value other than 0, and 1 (and exactly one other value, namely the game S C if the members of C are not all dummies). 3. I C : 2 C  FD(C ) takes at most two values (and exactly two if the members of C are not all dummies). 4. The set [D # min(S) : D & C{<] of minimal winning coalitions which include some member of C is equal to [A _ B : A # A, B # B] where A and B are antichains in 2 C and 2 C respectively. (S C is the unique filter such that A=min(S C ).) (W) =[AC: W _ A # S]. (i.e., S (W) indicates exactly what 5. Let S C C coalitions AC constitute a win along with W.) Then when they are not (W$) for any two sets W, W$C, and can be written the empty set S (W) C =S C simply S C . Moreover, S C denotes everywhere the same strong simple game on C. Proof. Suppose (1). Then S=T[S C , ...] where S C is a strong simple game on C. (5) now immediately follows. Let B=min([AV : A depends on C]). Then (4) follows. For AC, [I C (S)](A) is clearly equal to 0 when A # S and 1 when A # S. The remaining possibility is that neither A nor A are winning coalitions. In that case, the result depends on the committee C, and [I C (S)](A)=S C . Thus, (2) follows. Observe finally that a committee unites several votes into a single vote, thus a poll of C can only choose between two elements of FD(C ). Moreover, if only one element is in the image of I C (S) then the committee C has no decisive power; i.e. it consists entirely of dummies. Hence, (3). Suppose (2). If I C (S) takes only values 0 and 1, then all the members of C are dummies, and this C is a committee. Otherwise, let S C be the intermediate value taken. S C is a strong simple game on C since S* C =S C . Now, let T=[AC : [I C (S)](A)=1 ] _ [A _ [] : AC, [I C (S)](A)=S C ].

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Then T is a strong simple game on C _ [], and S=T[S C , Dem 1 , Dem 1 , ...]. Hence, (1).

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Suppose (3). Let [I C (S)](<)=:. Then : and ;=:* are the two values taken with :;. Then these two games on C can be combined into the following single strong simple game T on C _ [] where  is a new voter: T=: _ [A _ [] : A # ;]. Let S C =[I C (S)] ( &1)(;). Then S C is a strong simple game on C, and S=T[S C , Dem 1 , Dem 1 , ...]. Hence, (1). denote [I C (S)](W). By hypothesis, Suppose (5). Let W # C. Let S (W) C (W) S C is either 2 C =1, <=0, or a third constant. Hence, (2). K The following lemma permits us to understand subcommittees. Lemma 2.2. Let S be a full strong simple game on V. Let D be a committee of S. Then 1. D inherits a unique voting structure S D from S. 2. If CDV is a committee of S, then C is also a committee of S D and S C =(S D ) C . 3. Conversely, if C is a committee of S D , then C is a committee of S, and S C =(S D ) C . Proof. (1) By the previous lemma, S C is an inherent property of S and C and not related to the particular decomposition of S. (2) Note that [I C (S)](B)=[I C ([I D (S)](B&D))](B _ D). In other words, the effect of a group of voters leaving the room (see above) is equivalent if they leave in one group or in two groups. (3) By hypothesis, S=T 1[S D , ...] and S D =T 2[(S D ) C , ...]. Thus, by associativity (Eq. 1), S=(T 1[T 2 , Dem 1 , Dem 1 , ...])[(S D ) C , ..., ...]. K Lemma 2.3. Let S be full, and let C and D be two of its committees. C and D are then either disjoint, or one contains the other.

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Proof. First, we treat the following special case: Suppose that C _ D=V. Without loss of generality, C # S. Thus, I C (S) takes the values 0 and 1. Thus, the elements of C are dummies. Hence, C=V, and DC. In general, there are two cases. If C  S then the above reasoning applies. Otherwise, C # S. Let AC _ D be a coalition which depends on D. Let Z # min(S D ). If Z & C{<, then (A _ Z)&C depends on C. Thus, Z & C # min(S C ) (and visa versa).

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Thus, if Z # min(S D ), then 1. either Z & C=<, or else 2. ZC & D, and Z # min(S C ). It is impossible that some Z fall in case 1, and other in case 2, since S D is a strong simple game. If 1 is always true, then C & D=<. Whereas, if 2 is always true, then the members of D&C are all dummies. Thus, by hypothesis, D&C=<, and DC. K Lemma 2.4. Let S{Dem 1 be a full strong simple game on V. The set of maximal proper committee of S is a partition of V. Proof. We have remarked that [v] is a committee. Furthermore, by the above lemma, no two maximal committees intersect. K Proof of Theorem 2.1. Let [M 1 , M 2 , ..., M n ] be the maximal committees of S. Then S=T[S M1 , S M2 , ..., S Mn ] with

{

T= A[1, 2, ..., n] : . M i # S i#A

=

is the unique decomposition of S into smaller strong simple games where T is a prime strong-simple-game. By iterating this process, S can be completely and uniquely factored into prime strong-simple-games. K

3. Transitive Games A game S on a set of voters V is called (k& )transitive if it is invariant under a (k&)transitive group of permutations of V. Such games are also called fair or homogeneous. [9] If S is a transitive game on n voters, and T is a transitive game on m voters, then S[T, ..., T ] is a transitive game on nm voters. Thus, the set n

A=[n # N : there exists a transitive strong simple game on n voters] is multiplicatively closed. [9] We also have the following result.

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Proposition 3.1. Let S, T 1 , ..., T n be strong simple games. If S[T 1 , ..., T n ] is transitive, and S is also transitive, then the games T 1 , ..., T n must all be isomorphic transitive strong simple games.

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By Theorem 2.1 and induction, it suffices to consider S prime. Proof. Two voters in the same maximal committees must have the same role. Thus, each T i is transitive. Moreover, since voters in different committees must have the same role, there must be an automorphism mapping any committee onto any other. Thus, the T i are isomorphic and S is transitive. K If we consider k-transitive games, for k2, we find the following result. Proposition 3.2. All 2-transitive strong simple games are prime strong-simple games. Proof. Let S be a non-prime game. Then there exists at least two proper committees, at least one of which contains at least two voters. So let x and y be voters in the same committee, and z be a voter in another committee. (x, y) and (x, z) are not in the same orbit under the action of Aut(S). Thus, S is not 2-transitive. K

4. Quota Games Recall the definition of a quota game. In this section, we will examine quota games in light of our results on the composition of games. Note that if the total weight of a quota game is 2k+1, then the game is strong if the quota is at most k+1, and simple if the quota is at least k+1. Clearly, if U=S[T 1 , ..., T n ] is a quota game, then S, T 1 , ..., T n are all quota games. S is a quota game given weights for each voter equal to the total weight of the corresponding committee in U. Similarly, T i is a quota game given the same weights for its voters as for the same voters in U. The converse is not in general true. For example, S=Dem 3[Dem 3 , Dem 3 , Dem 3 ]

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is transitive. However, S{Dem 9 . (Notice that S is not 2-transitive, or that S has winning coalitions of cardinality four.) In fact, democracies are the only transitive quota games. However, the condition for the converse remain an open question. Under what conditions is the composition of several strong simple games a quota game?

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The order of the composants plays an important role. For example, given S = (2, 1, 1, 1) 3 , S[Dem 1 , Dem 1 , Dem 1 , Dem 3 ] = (4, 2, 2, 1, 1, 1) 6 whereas S[Dem 3 , Dem 1 , Dem 1 , Dem 1 ] is not a quota game. 2 The composition of several quota games will not in general be a quota game. Nevertheless, in the following case, new quota games are found. Let S be a strong simple quota game on a set V. Then Dem 2n+1[ S , Dem 1 , Dem 1 , ..., Dem 1 ] 2n

is a strong simple quota game. The members of V conserve their original weights, and the one-member committees are given identical very large weights. See also the examples in [16, 96]. References 1. C. Benzaken, Critical hypergraphs for the weak chromatic number, J. Combin. Theory Ser. B 29 (1980), 328338. 2. C. Benzaken, Hypergraphes critiques pour le nombre chromatique et conjecture de Lovasz, Ann. Discrete Math. 8 (1980), 91100. 3. L. J. Billera, On the composition and decomposition of clutters, J. Combin. Theory 11 (1971), 234245. 4. G. Choquet, Sur les notions de filtre et de grille, C.R. Acad. Sci. Paris Ser. A 20, No. 223 (1947). 5. P. J. Cameron, P. Frankl, and W. M. Kantor, Intersecting families of finite sets and fixed-point-free-2-elements, European J. Combin. 10 (1989), 149160. 6. G. Th. Guilbaud, Les theories de l'intere^t general et le probleme logique de l'agregation, Econom. Appl. 5 (1992), 4. 7. H. Hannisch, D. J. Hilton, and W. M. Hirsch, Algebraic and combinatorial aspects of coherent structures, Trans. N. Y. Acad. Sci. 2 (1969), 31. 8. J. R. Isbell, On the enumeration of majority games, Math. Tables Other Aids Comput. 13 (1959), 2128. 9. J. R. Isbell, Homogeneous games, Math. Student 25 (1957), 123138. 10. J. R. Isbell, Homogeneous games, II, Proc. Amer. Math. Soc. 11 (1960), 159161. 11. J. R. Isbell, Homogeneous games, III, in ``Advances in Game Theory,'' Annals of Mathematical Studies, Princeton Univ. Press, Princeton, NJ, 1964, pp. 255265. 12. A. D. Korshunov, Families of subsets of a finite set and closed classes of Boolean functions, in ``Extremal Problems for Finite Sets (Visegrad, 1991),'' Bolyai Society Mathematical Studies, Vol. 3, Janos Bolyai Mathematical Society: Budapest, 1994, 375396. 13. D. E. Loeb, Self-dual anti-chains, manuscript in preparation. 2

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Note that when determining the weights of voters in the components S, T 1 , ..., T n from the weights of voters in the compound U=S[T 1 , ..., T n ] further simplifications might be possible even when the simplest possible weights for U are given. In the example above, the weights U=(4, 2, 2, 1, 1, 1) 6 are considered homogeneous since all minimal winning coalitions yield exactly the quota, and all maximal losing coalitions yield exactly one less than the quota. However, the weights for S we derive by adding up the weight of each committee are (4, 2, 2, 3) 6 which can be simplified to the homogeneous (2, 1, 1, 1) 3 .

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14. D. E. Loeb, A new proof of Monjardet's median theorem, J. Comb. Theory Ser. A 72 (1995), 130141. 15. B. Monjardet, Characterisation des elements ipsoduaux du treillis distributif libre, C.R. Acad. Sci. Paris Ser. A 274 (1972), 1215. 16. B. Monjardet, Elements ipsoduaux du treillis distributif libre et familles de Sperner ipsotransversales, J. Combin. Theory Ser. A 19 (1975), 160176. 17. K. G. Ramamurthy, ``Coherent Structures and Simple Games,'' Theory and Decision Library, Series C, Kluwer, Dordrecht, 1990. 18. L. S. Shapley, ``Compound Simple Games III. On Committees,'' RM-5438-PR, The Rand Corporation, October 1967, also, in ``New Methods of Thought and Procedure'' (F. Zwycky and A. Wilson, Eds.), Springer, New York, 1968. 19. Th. Skolem, Uber gewisse ``Verbande'' ober ``Lattices,'' Avh. Norske Vid. Akad. Oslo (1936), 116. 20. H. Garsia-Molina and D. Barbara, How to assign votes in a distributed system, J. Assoc. Comput. Mach. 32 (1985), 841860.