Binary games in constitutional form and collective choice

Mathematical

Social Sciences 16 (1988)

189

189-201

North-Holland

BINARY GAMES IN CONSTITUTIONAL AND COLLECTIVE CHOICE Nicolas G. ANDJIGA Ecole Norrnale SupPrieure.

Communicated

FORM

and Joel MOULEN BP. 47 Yaounde,

Cumerorrn

by R. Selten

Received 2 July 1987 Revised 29 February

1988

In this paper, we define the notion of binary game in constitutional define a core and give a necessary and sufficient We define a representation and characterize

For this game, we form

choice rules which are representable.

the notion of c-social decision function and characterize,

on stability of binary constitutional

c-social decision

form.

for a game to be stable.

of a collective choice rule by a binary game in constitutional

those collective

We finally introduce of our theorem

condition

as an application

games, the collective choice rules which are

functions.

Our representation vious improvement

of a collective choice rule by a binary game in constitutional of the classical representation

Key words: Social choice; aggregation

form is an ob-

by a simple game.

of preferences;

game in constitutional

form.

1. Introduction

The first problem, studied in this paper, is the representation of a collective choice rule by a ‘concrete mechanism’. This is the case of the simple majority vote which can be considered as a concrete model of the simple majority rule. A collective choice rule is an application which assigns a collective relation to each profile of individual preferences. A concrete mechanism of such an application can be used to effectively construct the collective relation associated with a given profile of individual preferences. This problem of representation of collective choice rule (or social collective function, or social collective correspondence) is classical and has been studied explicitly or implicitly by several authors like Peleg (1978) and Arrow (1963). In the search for a representation of a collective choice rule, the notion of decisive coalition for an alternative x against another alternative y is often used. This notion has been used by Arrow in the proof of his famous possibility theorem. To state it formally, let A be the set of alternatives; N the set of players or elec016%4896/88/$3.50

0

1988, Elsevier Science Publishers

B.V.

(North-Holland)

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tors; R’, Jo N, the preferences on A of the player j; RJ is a binary relation on A with constraints of rationality (Definition 2.2.1); R’“= (R’!, E,Vis the profile of individual preferences; and o is the collective choice rule. A subset S of N is a decisive coalition for (x, y) with respect to o if for any profile R‘V, xRjy Vj E S implies xo(R”)y. The set of decisive coalitions for (s,y) with respect to o is denoted by 4(x, Y). Classically the collective choice rule o is represented by a simple game (N, WJ on A where w0 is the intersection of all D,(x,y) for x#y. But Arrow noticed that a coalition can be decisive for x against y without being decisive for y against X. A coalition which is decisive for all but one couple of elements of A will not be in wO, in spite of the fact that it possesses a great part of the power of decision of o. So we can consider that the simple game(N, wO) on A is an approximation of a more precise representation which has to be defined. We define by: (1) go,= {(x,Y)E~*~~,(x,Y)#0}, (2) X0= {SE2%(X,Y)E ZO, SED,(X,Y)}, (3) 6,, the multiapplication from X,, to y(, defined by 6,(x,y) =D,Jx,y). G, = (N, XO,A, Zo, a,), when it exists, could be our representation of Q. G,, the decisive game of cr. is a binary game in constitutional form (Definition 2.1.1). Let us now specify the definition of a representation of o. We associate with the game G, the dominance d, of 0 as follows: to each profile RN, d, assigns the collective preference d,(RN) defined by: V(X,~)EA*, xd,(R”)y if and only if there exists a coalition S in 6,(x,y) such that xR’y for all i in S. (See Moulen, 1977.) Obviously, d, is a collective choice rule and G, can be considered as a concrete mechanism of formation of d,(R”). The mechanism G, is a representation of o if and only if d, = r.7and so we will say that o is representable. It is obvious that, even if o is not representable, the mechanism G, is a better approach to 0 than the simple game (N, w,J and so we think that G, improves the classical representation of 0. In this paper, representable collective choice rules are characterized (Theorem 3.2.7) by natural and classical properties. The second focus of this study is related to collective choice. A collective choice rule o assigns a collective relation a(RN) on A to each profile R”. As Sen (1970), to (A, o(RN)) we will associate a collective choice set which is a stable subset of A with respect to &RN). The classical collective choice sets for (A,a(RN)) are: G(a, R”) the set of greatest elements of A for a(RN), and M(a, R”) the set of maximal elements of A for a(R”). We introduce the core of A with respect to @RN), denoted by 8(0, RN), as follows: ‘?3(a,RN) = {XE A/3y E A, ya(RN)x) . We define two notions of social decision functions: o is an M-social decision function, if M(o, R”) is non-empty for each profile RN, and Q is a c-social decision function, if 8(o,R”) is non-empty for each profile RN. SEN characterizes the collective choice rules which are M-social decision func-

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tions. But his characterization is based on properties of o(R’~) and does not explicitly give the corresponding properties on the structure of CJ. In our study, we characterize the representable collective choice rules which are c-social decision functions (Theorem 4.2.2). Our characterization uses the Theorem 2.2.7 which gives a necessary and sufficient condition of stability for binary games in constitutional form. The proof of the theorem of stability of binary games in constitutional form is technically based on the Keiding’s theorem for the stability of effectivity functions (1985). The main results of this paper are the following: (1) The characterization of stable binary games in constitutional form (Theorem 2.2.7). (2) The characterization of representable c-social decision functions (Theorem 4.2.3). This characterization explicitly uses the structure of decisive coalitions of 0 which is the structure of power of 6. If in real-life, committees are usually formalized as simple games (e.g., parliaments or town councils), this is only due to the fact that, by ethical considerations, for example the equality between candidates, it seems natural to admit the principle of neutrality in such committees (see Proposition 3.2.12). But in real-life, there are committees where such hypotheses are not true. In some international organizations, the set N of countries is partitionned in k classes or regions (for example N, is the set of the african countries of the organization, N2 the set of european countries...). Suppose that, according to its constitution, the organization needs two vice-presidents, one from Africa and the other from Europe. Let A, (respectively A,) be the set of african (respectively european) candidates. Usually the selected candidate from a region is automatically adopted by the general Assembly of the Organisation. The selection of the african and the european candidates can be formalized as simple games: (N,, ?.)&,,A,) and (N2, ‘W2,A2). Such an election, for the organization as a whole, can be formalized as an a priori given BGCF. This type of situations usually also appears in academic councils.

2. Binary games in constitutional form 2.1. Definitions Definition 2.1.1. G is a binary game in constitutional form denoted (BGCF), if G=(N,X,A, E,6) where: (1) N, the set of players, is a non-empty set (2) X, the set of coalitions, is a non-empty set of non-empty subsets of N (3) A, the set of alternatives, is a non-empty set (4) .X, the domain of G, is a non-empty set of a couple of distinct elements of A (5) 6, the constitution of G, is a multiapplication from &to X.

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Remark 2.1.2. (1) Binary games in constitutional form are particular cases of games in constitutional form. In general, a game G is in constitutional form if G= (N,JL, A, 2;s) where: &is a non-empty set of couple of disjoint and non-empty subsets of A. (2) This paper is restricted to the case of binary games in constitutional form with A finite. For the general study of games in constitutional form, (see Moulen, 1985; Andjiga, 1987; and Andjiga and Moulen, 1987). (3) Without lost of generality, we will suppose that: (i) V(x,,xz)E X, 6(xl,xz)f0 (ii) VSrz S, 3(x1,x& E X/SEa(x,,xz). Notation Let M be a non-empty set and K a subset of M. (i) P(M) is the set of all subsets of M.

(ii) 2”=P(M)\{0) (iii) M* = M x M (iv) If M is a finite set, IMI is the number of elements of M. (v) I? is the complementary of K in M. Let G = (N, S,A, K, 6) be a (BGCF). We define D(G) by: D(G) = {x2EA/ZLY~E A, (x*,x2)EEt-). Definition 2.1.3. (1) D(G) is the domain of negotiation of G (2) G is a frank binary game if D(G) = A. Definition

2.1.4.

(1) 2x=

{(~~,~2kA*,x,

(2)

V(X,rX*)E

Notation.

Let G =(N,X,A,

E,6)

be a (BGCF). G is a simple game if:

+x2)

%~(x,,x*)=~

If G = (N, S, A, LX,6) is a simple game, we simply denote G = (N, X, A).

Remark 2.1.5. The notion of a simple binary game in constitutional form is equivalent to the classical notion of a simple game on A (see for example Peleg, 1984).

2.2.1. Core of a binary game in constitutional form Definition 2.2.1 (1) A linear order on A is a complete, antisymmetric, and transitive binary relation on A. (2) R is an acyclic binary relation on A, if: for every finite subset {x,, . . . ,xk) of

A, [X;Rxi+IVi=l,...,

k - l] + not (xkRxI).

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Notation

(a) L is the set of linear orders on A. (b) L” is the set of all functions from N to L. (c) If R”E L.‘, SET” and (X,,X~)EA’, XI fx? x,RSx2++(Vi~S,x,Rix2). (d) If R, and R2 are binary relations on A R, cRz++(Vx,,x2~A,x,R,x2-+x,R.u,). Definition 2.2.2. Let G=(N,&A, x,6) be a (BGCF), RNe L.l’, SE~.~, and (X,,XJ E K (a) x, dominates x2 via S with respect to G and RN, written x,do(R.‘,S)xZ, if: SE~(X,,X~) and x,Rsx2. (b) x, dominates x2 with respect to G and RN, written x,dG(RN)x2, if there exists a coalition S such that x,do(RN,S)x2. (c) The core of G with respect to R N, denoted by f%‘(G,RN), is the set of all undominated alternatives on A with respect to G and RN. (d) G is stable if ‘&‘(G,RN)#OVRN~LN. (e) The dominance of G, denoted by do, is the application which associates to each R.’ in L’”, the dominance relation d&RN) on A.

Let us now characterize Definition

stable binary games in constitutional

form.

2.2.3.

Let G = (N,r(, A, .F, 6) be a (BGCF). A cycle in G is a family of elements of Z x % such that: (C,) SiE6(Xi,yi)Vi=l,...,no. (C,) {Y,,YZ~Y,> =A. (C,) V{i,, . . . ,i,) C { 1, . . . . no}. If Xi,+,=yi,Vj=l,***, r - 1 then n,=, &, = 0 or Xi, f yi,.

((X;,Yi)rSi)i=l,...,n,

Definition

2.2.4.

Let G = (P/,x, A, E, 6) be a (BGCF). G is acyclic if there is no cy-

cle in G. Theorem

2.2.5. Let G= (iV,r(k, A, .%‘,a) be a (BGCF). If G is acyclic then G is

stable. Proof. Suppose that G is not stable. Then ZIR~E LN such that B(G, Rf) = 0.Thus of elements of Zx Z such that: (1) there exists a family ((Xi,_Yi),Si)i= I,...,,,, SiE6(xi,yi)Vi=l,..., no, (2) {y, ,..., y,,}=A, and (3) xiR&iVi=l, . . . . no. Let us prove that ((xi,Yi),Si)i=~,....n~ is a cycle in G. Suppose on the contrary, then there exists {i,, . . . . ir} c {l,..., no} such that: xi,+,=yi,Vj= 1, . . . . r- 1; xi,=yi, and n~=,$,#0. Let &-,En&,S,, since xi,Riy6Vj=1 ,..., r, then xi,R$xi,_,Vj=l ,..., r-l

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and x~,@x, which contradicts the fact that R$‘, a linear order on A, is acyclic on A. Hence ((x,,Y,)~S~),=~,_._.,,” is a cycle in G and G is not acyclic. Let G = (iv, .X, A, 9; 6) be a binary game in constitutional form. If G is stable then G is acyclic.

Theorem

2.2.6.

Suppose that G has a cycle ((.u,,yi),S,);, ,,,,.,n,j. We want to prove that there exists R,\ in Ljv such that E(G, Rz) = 0. It is sufficient to prove that there exists Rr in L.’ such that x,R$yjVi= 1, . . . , no. It is then sufficient to find for each player k, a linear order Rt on A, compatible with the constraints of k which are xiR,$yj for every i such that k E Si. Let k be a player, we consider Dk = {i/l I is no,kE S;} and Mk = U;,~,C(X~) U { yi)). If D, = 0, every linear order on A satisfies the constraints of k. If Dk #0, let Dk={ilr..., i,} . We have t-5 no. Let us define on Dk the binary relation pk by: Proof.

Vh, j EDk,hPkjo_Y,,=xj.

The relation Pk is irreflexive and acyclic on Dk. Pk is irreflexive since V’i= 1, . . . , no x,# yi. Suppose that Pk is not acyclic on Dk, then there exists {jr, . . . . j,} a subset of Dk such that: j,Pkj,+,

VI= l,..., t- 1 and

j,pkj,.

Thisisequivalent tOyj,=Xj,_,VI=l,...,t-1 andy,=X,. But VI=l,...,t,kESj,, SO ni=,S,#t and this contradicts the fact that ((xi, y,)rSi)i=r,,.,,,, is a cycle in G. We then have Pk acyclic on the finite set Dk and thus Dk has a maximal element i;” for Pk. The i,$ maximal element is equivalent to Vh E Dk, yh Zx,?. Define the families 07

and iz for m = 2, . . . , r by:

Dr=Dk-{iT,j=l,...,m-1) a maximal element of

and Dr

for

i$

pk.

Define the families A$ and B,$ for m = 1,. . . , r by: A,* =

{xi;}, m-l

A;={x~$}-,~,

{x~}Vm=2

,...,

r,

B;” = { yi;} - ij A,+, i=l

B:= {ui;> -

[(~,A~)“(MB~)]Vm=2,...,f.

There exists at least one linear order Rk on Mk which satisfies:

N.G. Andjiga,

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RkA;RkB;Rk...

195

RkB;.

Let us prove that Rk satisfies the constraints of k. Suppose on the contrary, then there exists i&, n?o~ { 1, . . . ,I-} such that yizORkx,._. Therefore there exists , . . . ,r} such that m,
2.2.7.

Let G = (N,X, A, 2X”, 6) be a (BGCF). G is stable if and only if G

is acyclic. (1) This characterization of stable binary games in constitutional form is generalized on general games in constitutional form (see Andjiga and Moulen, 1987). (2) A similar characterization of stable effectivity functions, which can be considered as particular games in constitutional form, is due to Keiding (1985). Let us now characterize stable simple binary games in constitutional form. Let G=(N,S,A, .%,a) be a (BGCF). We denote: V(G)=nsaXS. Remark

Definition

2.2.8.

2.2.9

(a) G is a weak game if V(G) #0. (b) V(G) is the set of vetoes of G. If G is not weak. We denote by: (a) CG={OCX/ns,BS=O}. (b) v(G)=min{/~/, 0~ Co) Definition

2.2.10.

v(G) is the Nakamura’s

Theorem 2.2.11. Let G =(N,X,A, or if IA 1< v(G) then G is stable.

number of G.

.Z,6) be a (BGCF). Zf G is non frank or weak

Suppose that G has a cycle ((Xi,yi),Si)i=l,...,no and let US prove that G is frank, not weak, and IAl 2 v(G). (1) SiE6(Xi,Yi)Vi=l,...,no and {yI,...,yn,}=A thus G is frank. (2) G is frank and A is finite so there exists {ii, . . . , i,} G { 1, . . . , no} such that is a cycle in G xi,+, =yi,vj= l,..., r- 1 and Xi, =yi,e But ((Xi,yi),Si)i=l,..,,., thus n~=i.S, =0 and SO G is not weak. (3) rlnoandne=IAl sor~lA[. Since nj=tSi,=0, v(G)lrand lAl~v(G). Proof.

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Remark 2.2.12. The notions of weak (BGCF) and Nakamura’s number of a (BGCF) are generalizations of the corresponding notions defined on simple games by Nakamura (1979). Since each simple binary game in constitutional form is a frank binary game, the previous theorem gives a generalization of the sufficient conditions of stability of a simple game due to Nakamura (1979).

Let G = (N, 3, A) be a simple binary game in constitutional form. If G is stable then G is weak or IA 1< v(G).

Theorem 2.2.13.

Proof.

(see Nakamura,

1979).

We obtain the following characterization Theorems 2.2.11 and 2.2.13.

of stable simple binary games by

2.2.14. Let G = (N, X, A) be a simple binary game in constitutional form. G is stable if and only if G is weak or IAl < v(G).

Theorem

Remark 2.2.15.

The previous theorem is Nakamura’s

3. Binary games in constitutional

form and collective

theorem on a simple game.

choice rules

3.1. Decisive game of a collective choice rule Definition

1.1.

A collective choice rule, denoted (ccR), is an application

CJwhich

associates with each RN in LN, a binary relation a(R.‘) on A. Remark 3.1.2.

Let G be a (BGCF), the dominance of G is a (ccR).

Let 0 be a (ccR), (x1,x& EA’, xl #x2, and SE 2”. S is decisive for (x1,x2) with respect to B if VRNe LN, xlRSx2+xlo(R~‘)x~. Definition

3.1.3.

We denote by D,(x, , x2) the set of all decisive coalitions for (x1, x2) with respect to 0. The notion of a decisive coalition for (.v,,x~) with respect to CFis classical, see for example Arrow (1963) and Sen (1970). Remark 3.1.4.

Let Q be a (ccR), (x1,x2) EA’, x1 #x2, and S, TEE”. (i) If SED,(X,,X~) and SC T then TED,(x,,x~). (ii) D,(x! ,x2) # 0 * NE D,(x, ,x2).

Remark 3.15

Definition

3.1.6.

Let cr be a (ccR):

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(a) cr is locally Paretian if there exists (x,,x~)EA’,

xl fx:,

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197

such that:

VR’~EL.‘, (~,R’.Y,,V~EN)-‘X,~(R~).Y?. (b) o is paretian if: V(X,,~~)E,~‘, VRSYe L.‘, (x,RixlViEN)-+x,o(R.‘)x?. Remark 3.1.7. (1) If o is Paretian

then cr is locally paretian. (2) The notion of Paretian (ccR) is classical and expresses the sovereignty of the society N. (See Sen (1970) and Peleg (1984.) It is a natural property of a (ccR) and so is the locally Paretian condition which is less restrictive. If cr is a locally Paretian (ccR), we define by: (i) ~~=((x,.x2)EA2/D,(x,,x2)#0}. (ii) X0= (SE2’v/ZI(x,,x2)E X0, SED,(x,,xz)}. (iii) 6, the multiapplication from Z,, to rC, defined by:

V(x, ,x2)

E EC79

47(x,

9x2)

= &(x1

9*V?).

3.123. Let o be a locally paretian (ccR). The decisive game of o is the (BGCF) G, = (N, ,XI,, A, 9YD,6,).

Definition

We now give some properties of the decisive game of a (ccR). Let S,,- be the set of all permutations of N. Definition

3.1.9.

0 is an anonymous

(ccR) if: VR’“E L“, V’~ES,~, o(RN) =

a(R ‘(“‘)). 3.1.10. Let rs be a locally Paretian and anonymous (ccR) and G,= (N,3Y,, A, ,5Y0,6,) the decisive game of 0. (i) V(X,,X*)E X,,, VS, TEE”, SE~~(X,,X~) and jS( = 1TJ 4 TEG,(x,,x~). (ii) X,, = (N} or G, is not weak.

Proposition

Proof. (i) Suppose that SE&,(X,,X~) and ISI = ITI. Let RN~LN such that x,Rrx2, we want to prove that x,o(RN)x2. Since IS/ = j Tl, there exists rr E S., such that S=n(T). Let Rr=R n(N). We have x,RTxz, so x,Rfx2, but S~d,(x,,x~), therefore x,o(Rr)x2. o is anonymous and Rfl=R’(“) thus a(Ry)=a(R”) and thus x,o(RN)x2. (ii) Suppose that o is anonymous and X,, # {N}, then there exists S; E sl,, where Si=N- {i}. We have r)i,NSi=0 thus G, is not weak. Definition

3.1.11.

[ViEN,xRhy++uRiu]

0

is a neutral (ccR) if: Vx,y,u, UEA, VR~,R.“E LN; if then [xo(R,j’)youa(R~)o and ya(Rt)x-va(R:‘)3].

This definition of neutrality is due to Sen (1970) and implies the independence

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of irrelevant alternatives.

choice

W’e have this obvious proposition:

Let o be a locally Paretian (ccR) and G, the decisive game of a. If a is neutral then C, is a simple binary game. Proposition

3.1.12.

Remark 3.1.13. Using the notations following Definition 2.1.4, if a is a neutral and locally Paretian (ccR), the decisive game G,, of a, is denoted G, = (N,X,,, A).

3.2. d-representation

of collective choice rules

We have this obvious proposition. Let a be a locally Paretian (ccR), G, the decisive game of a and d, the dominance of G,. We have: d, C a. Proposition

3.2.1.

Definition 3.2.2. Let a be a local Paretian (ccR), G, the decisive game of a, and d, the dominance of a. (a) a is d-representable if d, = a (b) if a is d-representable, G, is the d-representation of a. Remark 3.2.3. A (ccR) is a procedure which assigns to each element RN of LN a binary relation a(RN) on A. We can essentially define a in two ways: (i) for each element RN of LN we explicitly assign the corresponding a(RN). (ii) We give a mechanism of formation of a(RN) using only the action of coalitions on a couple of distinct elements of A. The first way cannot be used if A or N are variable sets or if A and N are fixed but one of them has a large number of elements. In practice, this method is an unrealistic one with which to define a (ccR) and so classically (ccR) are defined by using the second method. If a is locally Paretian, d,, c a and then G, partially explains the mechanism of formation of a. If a is d-representable, this mechanism is entirely described by the decisive game of a. An experiment can be performed on the basis of G, to obtain a(RN) for a given RN in LN. Thus, G, is a concrete (practical) mode1 of a for any d-representable a. * Let us characterize d-representable (ccR). 3.2.4. Let a be a (ccR). (1) a verifies (P,) if: VR”E LN, V(x,,xJ Definition

EA’, x, #x2

[ViE N, not (x2RixI)] + not (x2a(RN)xl). (2) o verifies (P2) if: VR!, RYE LN, V(xl ,x2>E A’, xl #x2 [x,a(Rr)x2

and (x,R~~-*x,R~x~ Vie N)] +x,a(Rr)x2.

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199

Remark 3.2.5. The previous notions, used by Bloomfield (1976) are natural properties of a (ccR). (i) The property (P,), which expresses the collective sovereignty, is at least as natural as the Paretian condition. If a(R’“) is complete for each R” in L”, o is Paretian if 0 verifies (P,). (ii) (P?) is a property of positive responsiveness. (See Sen, 1970.)

We have this obvious proposition. Proposition

3.2.6.

Let G be a (BGCF) and do the dominance of G. do verifies (PI)

and (PI). Let a be a locally Paretian (ccR). r~ is d-represenrable if and only if o verifies (P,) and (P2).

Theorem

3.2.7.

Proof (a) if 0 is d-representable,

d, = Q, and by Proposition

3.2.6, (3 verifies (P,) and

(P2).

(b) Conversely

suppose that u verifies (P,) and (P2), and furthermore, let xl #x1 and R12;in LN such that xla(RN)x2. We want to prove that

(xI,xZ)~A2, x,do(RN)x2. xlc7(RN)x2 and 0 verifies (PI) so K= {iEN/x,R’x2} #0. Suppose that Kg 6,(x1,x2), then there exists R$ in LN such that x,Rtx2 and not (x1~(R:)x2). Let us define an element Rr of LN by: RF= Rt and Rf = R”. By definition of RF, we have [ViEN, xlR’x2++x,R;x2], but s,cr(R.l’)x2 and D verifies (P2) thus x,a(RiW)x2. Let us now verify that [Vi E N, xlRix2 -+x,R,$z2]. Suppose on the contrary, then there exists iOEN such that x,Rpx, and x2R$xl. But since x2R,$x,, iOE i?. Rf = RK then RF = Rio and by definition of K, x2Rpxl and this contradicts the hypothesis x 1R”‘x I 2. We have proved that [ViE N, x,Rix2 *x,R&Y~], but x,a(RjY)~~ and CIverifies (P2) so x,o(R~)x2 and this contradicts the hypothesis not (x,o(R~)xz). Therefore KE~,(x~,x~) and by definition of K, we have xlRKx2, so x,d,(R.‘)x2. We have proved that 0 C d, and by Proposition 3.2.1, d, C or thus 0 = d,., and G is d-representable. Remark 3.2.8. The previous theorem gives the characterization of a d-representable (ccR). Those necessary and sufficient conditions are natural and classically used. This can partially explain the fact that classical (ccR) are usually defined by giving their decisive game and thus are d-representable. But, for example, the rank order method of voting, which violates (P&, is not d-representable in spite of the fact that it has a decisive

game.

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4. Choice sets and collective 4.1.

choice

rules

Choice sets of a (ccR)

Definition 4.1.1. Let 0 be a (ccR) and R” an element of f..V. (1) An element x of A is ma,uimaf for o with respect to R” if there is no element y of A such that:

[ya(R”)x

and not

(xo(RN)y)].

(2) The maximal set of o with respect to R’V, denoted by M(o, R.‘), is the set of all maximal elements for TVwith respect to RN. (3) o is an M-social decision function if VRN~LN, M(o,R“)#O. Let CJbe a (ccR) and RN an element of L’“. (1) An element of x of A is undominated for r-7with respect to R.’ if there is no element y, of A, such that: yo(R”)x. (2) The core of CJwith respect to R N, denoted by B(a,R”), is the set of all undominated elements for 0 with respect to RN. (3) 0 is a c-social decision function if: VR”E LN, %‘(a, R”) #0. Definition

4.12.

If a (ccR) 0 is used to look for a choice subset of A for a given RN in LN, among the stable sets, we can consider as choice sets 8(0, RN) and M(a, RN) if they are non-empty. Intuitively B(a, R”) is preferable to M(o, RN), but it is more often empty than M(a, RN). M(a, RN) is the stable set used to define the classical notion of the social decision function (see Sen, 1970). It is then interesting to characterize (ccR) which are c-social decision functions or M-social decision functions. Remark 4.1.3.

4.2.

c-social decision functions

and M-social decision functions

Let us give some obvious properties of the previous stable sets. Proposition 4.2.1. Let o be a locally Paretian (ccR) and G, the decisive game of o. (1) VR*‘E L”, %‘(a, RN) C M(o, RN) and 8(0, RN) C B(G,, RI\“). (2) If o is d-representable then

VRNc LN ‘6?(a,R”) = b(G,, R”). By the previous proposition, Theorems 2.2.7 and 2.2.14, and by Propositions 3.1.12 and 3.1.14, we have the two following theorems: Theorem 4.2.2. Let o be a locally Paretian (ccR) and G, the decisive game of 6. (1) If Q is a c-social decisive function then G, is acyclic. (2) If o is a neutral c-social decision function then G, is weak or IA\ < v(G,).

201

/V. G. Andjiga. J. ,kforrlen / Binary games and collecrive choice

(3) If rs is a neutral and anonymous IA I < v(G).

c-social decision function

then .Iy, = {‘V} or

Theorem 4.2.3. Let u be a d-representable (ccR) G, the decisive game of a. (I) a is a c-social decision function if and only if G, is acyclic. (2) If CI is neutral, o is a c-social decision function if and on/y if G, is weak or IA 1< v(G,). (3) If o is neutral and anonymous, o is a c-social decision function if and only if Xa= {Nj or ]A( c v(G,). Remark 4.2.4.

(a) If o is d-representable the previous necessary and sufficient conditions for a (ccR) to be a c-social decision function are sufficient conditions for a (ccR) to be a M-social decision function. (b) The previous conditions on (ccR) use the structure of G,. Those properties can therefore be considered as internal properties of cr and are not similar to those which use properties on o(RN) as in the characterization of M-social decision function by Sen (1970). Acknowledgements

The authors are grateful to R. Selten and two anonymous suggestions and comments.

referees for valuable

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