Information, associativity, and choice requirements

JOURNAL

OF ECONOMIC

Information,

THEORY

52, 440-452 (1990)

Associativity,

and Choice

Requirements

MARK R. JOHNSON* Department of Economics, Finance and Legal Studies, The University of Alabama, Tuscaloosa, Alabama 35487 Received May 30, 1988; revised February 26, 1990

The common view of cross-set consistency requirements is to hold them as either ethical or technical requirements. The line of research presented here views consistency requirements as information processing eft?ciency requirements on a choice process. It is shown that path independence axioms impose a hierarchy on the binary system, originally considered by Plott, that is consistent with intuitive notions of information processing efficiency. The hierarchy provides discrimination among the five principal path independence axioms. Implications for information processing in group decision-making mechanisms are noted. Journal of Economic Literature Classification Numbers: 022, 024, 025, 026. 0 1990 Academic Press, Inc.

1. INTRODUCTION

By common experience and general lore we recognize differences in how effectively institutions operate. In particular, there appears to be a wide variation in the amount and type of information required for decision making by different choice mechanisms. There are, in fact, several intuitively appealing folk tales about the information cost of decision making in alternative organizational and political structures. These folk tales claim that highly centralized organizations with narow distributions of power are efficient, while more democatic institutions, generally versions of majority rule, are slow and cumbersome decision-making machines. In the following, decision-making institutions are viewed as choice functions and the information processing efficiency of classes of choice functions is investigated. The link to information processing efficiency of a choice function is obtained through the path independence axioms (consistency requirements) imposed. The view of the consistency requirements is that *I thank Theodore Groves, Walter Heller, Vincent Crawford, and Richard Dean for valuable discussion and suggestions. I also have had the benefit of comments by an Associate Editor of this journal. All errors are my own.

440 0022-0531/90 $3.00 Copyright Q 1990 by Academic Press, Inc. All rights of reproduction in any farm reserved.

INFORMATION,

ASSOCIATIVITY,

AND CHOICE

441

they are not ethics; rather they are a means of deterrn~~~~g ~~forrnat~o~ processing costs of decision-making. The basis for this approach stems ant from Plott’s seminal paper on path independence, in which two arguments to support the property’s reasonab~~nes t, suggested that choice processes might be inherently restricted to ‘“small” sets (e.g., simple-majority rule is defined only for two element sets) and there might be a need to extend choices made on “small” sets to ““larger” lott suggested that path indcpende~ce re~~~~erne~ts mi be the basis of a “computational efficiency” requirement. tured that associativity might be the key mathematical ~~de~ende~ce axioms (Plott [6]). An example presented in Section 3 builds on Plott’s suggestion of extending choices from small sets to larger sets to deveIop a notion of computational efficiency (called information processing efticiency here) that can be order rational-choice processes in an i~t~it~ve~y a~~ea~~~g manner. tion to ordering the choice processes, three co~t~b~tors to the difin the information processing efficiency for the rational-choice case intuition derived from the example then is used to a ocessing in choice processes hich are not required ction 4, it is shown that pa hy on the classes of the binary system of choice f~~ct~o~s ered by Plott. This hierarchy, in which associativity plays trongly associated with the information processing ~ot~~~s from the rational-choice process ex associativity, several other properties important tion among the classes of binary systems are noted. A final result sheds t on Bimits to Plott’s conjecture about the i rlance of asso~at~~ty in pat independence requirements. Section 5 comment on links to information processing eficie making structures. Definitions and notation follow ~rnrne~~ate~~ below in Section 2.

2. DEFINITIONS

AND NOTATION

universal set V is composed of a finite number of distinct alterand 2” is the power set of K Unless otherwise stated, the cardinality of V, denoted 1VI, is t, and the cardinality of 0 E 2’ is n; note that n d t. When necessary to avoid ambiguity, distinct su scripted with an integer in (1, .... 2’); in such cases {ui if i =j. A choice function is a mapping 47 : 2’ --i 2’, su and for all v E 2’- a, C(U) # @ and C(U) E U. A choice function C(* ) is

442

MARK

R. JOHNSON

rational if and only if there exists a relation C(v) = G(v; R), where G(v; R)=

R such that, for every v E 2 “,

(XEV (xRy, Vy~v}.

For a relation R defined on a set v, the transitive closure, R,*, is defined as xR,* y o (3 (zl, .... zk} E v ( xRz, , .... zk RJ.

Given R:, the transitive closure of a relation R, a choice function C( * ) is said to be indirectly rationalized if and only if, for every v E 2’, C(u) = H(u; R,*), where H(u; R,*)=

{x~vjxR;y,

Vy~v}.

Path independence requirements on choice functions follows. Strong Axiom of Preference (SAP): 0)

V-T YE K

(ii)

Vu1v2 E V,

are defined as

xEC((X,Y})~Y~C({x,Y}). V,EV,==-{v,nC(v,)}=

{

go;. 1

Weak Axiom of Revealed Preference (WARP):

vu,,v2-cv,

v,_cv,*{v,nC(v,))= go;. i 1

Independence

of Path (IP):

vu,, VI c v, Acyclic Independence VVl, v,c v, Weak Independence vu,, v2 c v,

C(C(v,) u C(v1)) = C(v, u vz). of Path (IP*): C(C(Vl) u C(h)) 2 au1 u vz). of Path (*IP): C(C(Vl) u C(v,)) E C(Vl u v*h

Each of these axioms represents a different restriction on the way in which a search for the best (or choice) elements of a set can operate. These restrictions affect the efficiency of the choice process. For example, a choice process consistent with IP must meet the following requirement: if the feasible set is subdivided into two sets and a choice set is selected from each of the two subsets, then the choice from the union of the subset choices

INFORMATION,

ASSOCIATIVITY,

AND

CHOICE

3

exactly equals the choice set from the original feasible set. For choice processes consistent with IP, subdivision of the choice process identifies all members of the choice set from the original feasible set. This complete identification occurs even if the full feasible set is never available as a whole. In contrast, a choice process consistent with *IP, applied in the same sequence of divide, choose, and choose again, will result in a choice from the union of subset choices that is guaranteed to be contained wishin the choice set from the original feasible set. That is, in a choice process consistent with *IP, subdivision of the choice process may result in some memers of the choice set not being identified as such. The results iu Section 4 identify structural properties of choice functions that are associated with differences in the processing efficiency of choice processes consistent with different path independence axioms. While the focus of this paper is on the algebraic properties of the strueture of choice functions, intuition is derived from an example using ratio choice functions. For this example, the standard de~n~tions are provid

(1) (2) (3) (4) (5) (6)

completeness: Vx, p E V, (x #v) = x&, yRx, reflexive: Qx E I/* xRx. anti-symmetry: Vx, y E V, xR.y * - (~WX). transitivity: Vx, y, z E V, [(xRy) A (y&J] -j quasitransitivity: Yx, y E K let(xPy) e [x is transitive. acyclicity: Vx, x,, x2, .... X,,:, E k/, (XI%,, - (YPX).

or both.

(x

and -(yRx)]

and P

x,PX2, ...) X,Q)

*

The first two of these requirements are tee roper-ties that are assumed for all relations considered here. The remaining four requirements are different types of consistency requirements. Relations satisfying ( I)? (2>, (3 ), and (4) are called linear orders; (I), (2) and (4) weak orders; (9 ), (a), and (5) quasitransitive relations; and (I), (2), and (6) acyclic relations. For complete, reflexive binary relations, acychcity is the m requirement for directly rationalized choice (see Sen Richter [7-J showed that weaker relations may also rationalize choice functions indirectly through the transitive closure relation, W: . If R is compk~te and reflexive, then Rz is a weak order on any fixed v. Given a rational choice function, choices ra~~~a~ize~ by linear orders satisfy SAP, weak orders satisfy WARP, quasitransitive r&t and acyclic relations satisfy IP*. For indirectly rationalize tions, the transitive closure of a complete reflexive binary relation will satisfy “EP. When necessary, relations (either the base relation or the

444

MARK R.JOHNSON

transitive closure) are represented by an incidence matrix. The incidence matrix for a relation R defined on V is I = [a,], where a,=

1 0

if xiRxj otherwise.

The value of an incidence-matrix entry can be determined by making one binary comparison, i.e., the answer to the question, “Is xj related to xj?” determines ay. It is intuitively appealing to compare the zero-one entries in the incidence matrix with the “bit” from information theory.

3. CHOICE AND INFORMATION

PROCESSING

The classic view of economic choice presumes that a set of alternatives can be scanned and a choice set selected in a single act. This view seems to be derived from the model of individual choice. In that model, considerable effort has been expended deriving conditions under which it is possible to move from choices to an ordering over alternatives to representation by a utility function. Once the utility function is obtained, the consumer choice problem is stated as a straightforward maximization problem. This formulation may oversimplify the problem of finding the best elements of a set. Clearly, in the case of the rational-choice model, given a complete description of R on v, it is easy to find maximal elements of Y. For example, when R is represented by its incidence matrix, the choice set for u is simply those elements that have row sums equal to 1~1. Choice based on knowledge of the complete incidence matrix may be thought of-as a “full-information choice”. While “full-information” is certainly one way to make choices, it is not the only way. In many cases, it is not necessary to know the complete incidence matrix. Indeed, many computer search techniques shorten processing time by not making all the binary comparisons required to construct a complete incidence matrix (see Knuth [ 51) and most agendabased choice processes do not collect all the data. The minimum number of binary comparisons required to correctly complete a search may be called the “bit cost” of the search. Depending on the situation considered, one search process may have a lower bit cost than some other search process. The following example (using a rational choice function) demonstrates the importance of the path independence assumptions in sorting through a set to determine the choice set. EXAMPLE

1. Let the universal set be V= (x,, x2, x3, x.+) and suppose

INFORMATION,

ASSOCIATIVITY,

AND CHOlCE

445

choice is rationaiized by a member of one of the five classes deftned in Section 2. Further, imagine that, through a desire to reduce the number of incidence-matrix entries that must be determined, the set I/ is rummage through in a sequential manner (possibly an agenda), and assume that in some manner the values of the partial incidence matrix representing R on V have been determined as in Table I. Further assume that the search process for the choice from V has proceeded through the point where x1, x2} has been determined. How much more of the qx1~x2A))=I incidence matrix for V needs to be determined in order to know C( V)? The answer depends on the class of binary relation that rationalizes C(* ) on P’. Since C(m) is not single-valued on V, R cannot be a linear order; if a weak order then two values may need to be determined (either aI4 and and a4& if R is a quasitransitive relation then four values a4t5 Or a24 (a t4, a41ya24s and aQ2) of the incidence matrix may need to be determine And so on. Tbe remark below extends the intuition n with choice set of size k.

of the example to any set of size

REMARK 1. Consider v1 cv,~ V, v1 = (xl, .... x,), C(ol)= (xi7 ..~,xk), let u2= (u, u x0>, and let R rationalize C( * ) op2V. The maximum number of entries for the incidence matrix representing R that must be determined ivi order to construct C(v,) is the following:

(i) (ii) (iii)

2, if R is a weak order,

(iv)

II+ k, if R is an acyclic relation, and

1, if R is a linear order, 2k, if R is a quasitransitive relation,

(v) 2k+;(n-k)(n-kfl), ifR is a complete, reflexive relation Ehat rationalizes choice indirectly through the transitive closure Rz. ProoJ

Exercise.

Two points should be noted in the example and remark. First, for the TABLE I Partial Incidence Matrix for Example !

Xl x2

X3 X4

642/52/2-14

XI

x2

X3

x4

1 1 0

1 1 0

I 1 1

1

4.46

MARK

R. JOHNSON

rational-choice case, the number of incidence-matrix entries that must be determined have three natural interpretations: they may be viewed as a “comparative-statics” result for a choice process where the size of the feasible set is perturbed, they may be thought of as the size of the “institutional memory” required for a particular procedure to operate, and finally, they may be considered as the “bit-cost” of preserving independence of path as a set is rummaged through. This last concept of information efficiency seems especially appropriate for choice-based probems. Second, differences in the number of incidence-matrix entries that may have to be determined for each class of relation derive from several sources. Principal contributors are (a) whether a representative element exists, (b) whether the new entries are from the original choice set alone or are comparisons involving previously passed over alternatives required, and (c) whether the required entries involve only the new alternative and original alternatives or also involve pairs of alternatives in the original set. For any relation weaker than a weak order, there is no representative element. For relations weaker than quasitransitive relations, comparisons outside the initial choice set may be required. If the relation is not at least acyclic (e.g., choice is rationalized indirectly, by the transitive closure relation R,*), then comparisons between pairs of previously passed over alternatives may be required. For choice functions that are not required to be rational, it is not possible to rely on incidence-matrix entries for a measure of information processing requirements. As an alternative, it is possible to use searchrelated properties of the structure imposed on a binary system by different path independence axioms to obtain discrimination among choice processes and a hierarchy on information processing. Some of these properties are clearly “order-invariance” properties; one is a “representative” property. A binary system is defined by specification of a domain, a range, and an operation. The system considered here uses 2’ as the domain and range, and adopts Plott’s binary operation defined on the Cartesian product

2vx2v: vv,,v,E2V,

VI . vz = C(C(v,) u C(Q)).

The binary system for V is denoted ( ., 2”). Algebraic properties that are useful in discriminating among choice methods are the following: (1) Closure:V(u,,u,)E2Vx2v,v1.V2E2Y (2) Commutative: V(v,, v2) ~2’x 2’, v1 I v2 = v2. v1 (3) Associative: V(v,, v2, v,)E~~, ul. (vz.v3) = (vl .v2)-v3. In addition to these algebraic properties, it is useful to introduce subalgebraic property that will be termed an associative kernel:

a

INFORMATION,

(4)

ASSOCIATIVITY,

Associativekernel:V(v,,v,,u,)E2”,

AND CHOICE

447

(01.(v2.~j)jn((u,.v2).03)

#0. The associative kernel property is clearly implied y associativity but is clearly weaker; just as clearly, since it is defined at the set level, this property is not a fully algebraic requirement. Closure seems to be a basic property of a choice process (i.e., choice ciative properties from a set must be in the set). The commutative and are separate forms of independence from sequence or o r. As a restriction on choice processes, commutativity is the requirement of invariance between pairs, while associativity requires invariance with respect to choice between pairs of subsets from among three or more subsets. Although commutativity and associativity are both forms of invariance with respect to order, each represents a different level of complexity. Specifically, if ( a92 is a finite binary system, testing for commutativity amounts to verify that the operation table is symmetric, a process that grows with the square of the number of elements in the system. Q~ri~~ation of asso~~at~v~ty~ however, is a process that grows with the cube of the number of elements in the system. Following the terminology of Bruck [Z], a binary system ( ., 2V) satisfies (1) is called a groupoid. A groupoid that satisfies (2) is call abelian groupoid. An abelian groupoid that satisfies (4) will be call abelian groupoid with associative kernel. A binary system that satisfies (11, (2), and (3) is called an abeZian semigroug. Each of these binary systems belongs to a class, thus, a binary system that is an abelian semigroup belongs to the class of abelian semigroups. Finally, it is useful to identify a special subset of the members of a semigroup. DEFINITION. Let T= ( ., 2 “> be a semigroup, and let A c 2 V be a nonempty subset of 2v. If for all t~2’and aEA, a.tEA and t.aEA, t is an ideai. If A is an ideal and if A = 2c(“) - { 0 ), then A is called a chiqf ideal

Algebraically, an ideal is a subset of the members of a semigroup, such that whenever any element of the ideal is operated on by any member of the semigroup the result is always a member of the ideal. For choice functions, the presence of an ideal identifies a type of commonality among alternatives. Given closure, all semigroups have an ideal of the whole; however, not all semigroups have a chief ideal. Note that the four classes of algebraic structures, (i) the class of abelian groupoids, (ii) the class of abelian groupoids with associative kernel, (iii) the class of abelian semigroups, and (iv) the class of abelian sern~~ro~~~

448

MARKR.JOHNSON

with chief ideal, form a nested containment hierarchy: (i) 2 (ii) 2 (iii) 2 (iv). Clearly, an abelian semigroup with chief ideal is the most restrictive structure and an abelian groupoid the weakest.

4. ALGEBRAIC PROPERTIES OF CHOICE FUNCTIONS Plott proved that the binary system ( ., 2”)formed an abelian semigroup if C( *) satisfied IP on V. Theorem 1 identifies a number of structural properties of the binary system ( ., 2’) that are associated with the information processing efficiency of the choice process imposed by different path independence axioms and establishes a range of structures on binary systems relevant to choice problems. Following Theorem 1 are several remarks that refine the theorem’s implications. Theorem 2 addresses Plott’s conjecture about the importance of associativity in path independence requirements. THEOREM 1. Given a set V, C( * ) any choice function on V, and (, ) as dejined above,

(a) ( ., 2’) is an abelian groupoid, (b) ( ., 2’) is an abelian groupoid satisfies IP* on V, (c) ( ‘, 2 ‘) is an abelian semigroup (d) ( ., 2’) is an abelian semigroup WARP on V, and (e) ( -, 2 “) is an abelian semigroup satisfies SAP on V.

with associative kernel if C( * ) if C( * ) satisfies IP on V, with chief ideal if C( *) satisfie with singleton chief ideal if C( * )

ProoJ: Part (c) was proved by Plott [6, Theorem 21. Proofs of parts (a), (b), (d), and (e) follow. For (a), since C( *) is defined VV~E2”, then Vu,, v2 E 2’, v1 . v2 E 2 “, and thus ( ., 2” ) is closed. Commutativity follows immediately by noting that Vul, v,rz2’, u1 .v,=C(C(v,)u C(Q))= Part (b) requires only that v1 . (v2. vj) and c(c(v,)uc(v,))=v,~v,. (vl . vz). v3 have a non-empty intersection. But note, by IP*, that VI ’ (vz. v3) = C(C(v,) u C(C(v,) u C(Q))) 2 C(v* u v.2u vg) and (v,

.v2).

v3

= C(C(C(v,)

u C(Q))

u

C(v3))

2 C(v, u v* u Vj).

Thus, iv1 (v2 .v3)3- n {( V~'v*)~Vg)3C(v1uU~uu3)#~.

INFORMATION,

ASSOCIATIVITY,

AND CHOICE

449

Thus if C( * ) satisfies IP” on v, ( ., 2’ ) has an associative kernel, an is verified. Part (d) follows by noting that a choice function that satisfies WARP also satisfies IP and therefore the semigroup property obtains immediately. It remains only to show that WARP guarantees a chief ideal and that SAP guarantees that the chief ideal is a singleton. Let A = 2c(v) - (@I), and note that for any a,, a2 E A, a, . a2 E A. Next let b E 2” and consider three cases : (1) b n C( V) = b, (2) b n C( V) = as and (3)bnC(V)=C(b)#@, C(b)#b. In case (1) we have a, b~A=>a.beA. In case (29, a. b = a E A. Case (3) gives a. b = {au C(b) Thus WA C(V)1 =+{au C(b)) c {C(W)1 =>a.bEA. chief ideal and (d) is verified. Part (e) follows immedia when SAP is satisfied, C(V) is a singleton. RP, and SA Although the path independence axioms *II?, IP*, II?, form a hierarchy, the theorem shows that t do impose y on the classes of algebraic structures for the binary system ( ., 2”). The hierarchy would be of little value, however, if the results were vacuously true. The following remarks demonstrate that this is not the case. REMARK 2. There exist choice functions C( * ) defined on V so that * is satisfied and ( ., 2’) is an abelian groupoid but ( ., 2’) does not have m associative kernel.

ProoJ

Eetv=(x,,x,,x,3,C((x,})=(x,f,C(V)=&/,

Cl{%9 x2))=

(Xl},

C({Xl> x3))=

(X3>?

c&G,

x3))=

(x2).

Note that C( * ) satisfies *IP and ( ., 2’) is an abelian groupoi However, let vi = {x1, x2), v2= (x1, x3), us== {xi, xj) and note that vl.(%‘%)=

(x1-L

(v1’%).“3=

{x31.

while

Therefore

and there is no associative kernel.

fl

REMARK 3. There exist choice functions C(* ) defined on V such rhat IP* is satisfied and ( ., 2”) is an abelian groupoid with associative ~e~~e~ but such that ( ., 2v) is not associative.

450 ProoJ:

MARK

R. JOHNSON

Let V= {x1,x2, x3>, C((X~})= u1= C(Vl)

=

{Xl>

x2),

v2

=

(x2,

{-%

x2),

C(v2)

=

{xj}, C(V)=

x3), {x2L

{x1},

v3 =

{%X31,

C(v3)

=

{Xlt

x31.

Note that C( * ) satisfies IP* and ( ., 2V) is an abelian groupoid with an associative kernel. Further note that

and

(Vl .v2).

03 =

(Xl>,

so that

but ~1~(~2.~3)Z(~1.~2)~~3~

I

REMARK 4. There exist choice functions C( * ) defined on V such that IP is satisfied and ( .,2 v > is an abelian semigroup but such that ( . ,2 v ) does not have a chief ideal

Pro4 c({x,~

Let V= (x1, x2, x3}, %})=

(-%%},

c((x2~

C((-q))={xi>,C(V)=(x1,x2>, %}I=

ix22

x3},

C({Xl>

x3))=

{Xl).

Note that C( * ) satisfies IP on V and ( ., 2’) is an abelian semigroup. Let A =2c(V)- (@I; then A = {{x1, x2}, {xi}, {x2)}. Define vi = {x2}, v2=(x3), and v3=(x2,x3}, and note V,EA and v~.v~=v~.v~=v~~A. Thus ( ., 2 “> does not have a chief ideal. [ Comment. There exist choice functions C( *) defined on V so that WARP is satisfied and ( ., 2’) is an abelian semigroup with a chief ideal that is not a singleton. These remarks combined with the theorem show that the structural results on ( ., 2’) can be used to provide a reasonably fine cleavage among classes of binary systems. It is evident that associativity plays an important role in providing discrimination. This is consistent with Plott’s conjecture; however, it is not clear that associativity is the defining feature, as the following theorem demonstrates. THEOREM 2. There exist binary systems ( ., 2’) groups and C( * ) does not satisfy IP on V.

that are abelian semi-

INFORMATION,

ASSOCIATIVITY,

451

AND CHOICE

An example suffices. Consider V= (x1, x2, x33 and let C( {Xi >) = cC~v>)={x21~

ProoJ fxjl,

c(Iol)=a

C(hx2l)=C(~

x1,-%))=

@I>?

Note that ( ., 2V) forms an abelian semigroup in Appendix 1). However, note that W(h

x2)) u qx2>

4,

q~2~4=

{x21.

(see the operation

table

= {Xl)

while

Thus

C(C(lXl>x*l)u CC{x2>

x3>,,

f

C((Xl>

x2$

”

{x2,

x3),.

Therefore C( L ) does not satisfy IP on V. The example provided in the proof is not capricious. In this case x1 is a winner in every pairwise comparison and is defeated on the whole set. Examples of this sort, while not desirable, are not uncommon in choice theory (e.g., whenever a Condorcet winner is not chosen on the full set).

5. SUMMARY It is seen that path independence axioms impose a hierarchy on the classes of the binary system ( ., 2v) that is consistent with intuitive notions of information processing efficiency. For each of the path independency axioms considered, a different set of necessary algebraic properties for t class of binary systems is identified. The commutative and associative properties are different forms of in pendence from the sequence of choice. The commutative property is necessarily present in each of the classes of binary systems considered. Some form of associative property (i.e., associativity or an associative kernel) must be present in all of the binary systems considered except those where the choice function satisfies *IP. The importance of associativity in the induced structure on binary systems along with the intuition from the rational-choice example supports Plott’s conjecture about the information processing implications of path independence requirements. Mowever, since not all abelian semigroups satisfy IP, associativity is not a s~fflcient property for path independence In addition to the commutative and associative properties, a chief i

452

R. JOHNSON

MARK

can also help provide discrimination between classes of binary systems. The presence of a chief ideal is guaranteed only when the choice function satisfies WARP or SAP. In both of these cases, the choice process can be rationalized and the relation that does so is transitive. Finally, since dictatorships satisfy WARP, oligarchies satisfy IP, collegial polities and acyclic-majority rules satisfy IP*, while extended-majority rule satisfies only the weaker *IP requirement (see Ferejohn and Grether [3], Blau and Brown [ 11, and Kelly [4] for the necessary results) it seems that, at least for neutral and monotonic processes, the intuitively appealing folk tales about the efficiency of more centralized decision-making structures have some basis.

APPENDIX

1

Operation Table for Counterexample Used in Theorem 2 0

Xl

x2

x3

Xl>%

x1,x3

0 XI

0

Xl

x2

7.3

Xl

Xl

Xl

Xl

Xl

Xl

Xl

x2

X2

Xl

x2

x2

Xl

x3

X3

Xl

X2

x3

Xl

Xl Xl Xl

Xl7

x2

Xl

Xl

XI

Xl

Xl

Xl.

x3

Xl

Xl

Xl

Xl

Xl

x2>

x3

x2

Xl

X2

x2

Xl

Xl Xl Xl

Xl>

x2,

X2

Xl

x2

x2

Xl

x2

x3

x2,

x3

Xl>

x2,

X2 Xl

x2

X2

X2

x2

X2

Xl

Xl

Xl

x1

x2

x2

x2

X2

x3

Xl

REFERENCES 1. J. H. BLAU AND D. J. BROWN, The structure of neutral monotonic social functions, Social Choice and W’erfare 6 (1989), 51-61. 2. R. H. BRUCK, “A Survey of Binary Systems,” Springer-Verlag. Berlin/New York, 1958. 3. J. A. FEREJOHNAND D. ÐER, Weak path independence, J. Econ. Theory 14 (1977), 19-31. 4. J. S. KJXLLY, “Arrow Impossibility Theorems,” Academic Press, New York, 1978. 5. D. E. KNUTH, “The Art of Computer Programming: Volume 3/Sorting and Searching,” Addison-Wesley, Reading, MA, 1973. 6. C. R. PLOTT, Path independence, rationality and social choice, Econometrica 41 (1973), 19751091. 7. M. K. RICHTER, Rational choice, in “Preferences, Utility and Demand,” (John S. Chipman, Leonid Hurwicz, Marcel K. Richter, and Hugo F. Sonnenschein, Eds.), Harcourt Brace Jovanovich, New York, 1971. 8. A. K. SEN, “Collective Choice and Social Welfare,” Holden-Day, San Francisco, 1970.