Unique finite conjoint measurement

Mathematical

Social

Sciences

107

16 (1988) 107-143

North-Holland

UNIQUE

FINITE

CONJOINT

MEASUREMENT

Peter C. FISHBURN* AT&T Bell Laboratories,

Murray Hill, NJ 07974, U.S.A.

Fred S. ROBERTS** Department

of Mathematics.

Communicated Received

by K.H.

NJ 08903. U.S.A.

Kim

3 May 1988

This paper

studies

social sciences,

one of the most important

the additive

Tukey (1964). Lt also considers types of measurement in Cartesian

products

conjoint

types of measurement

measurement

the variant

that has arisen

by Debreu

conjoint

comparisons

of sets. This paper initiates uniqueness

introduced

we call additive

are based on qualitative

all sets are finite. It considers conjoint conjoint

Rutgers University, New Brunswick,

extensive

between

measurement.

multiattribute

a study of their uniqueness

from the

(1960) and Lute and Both

alternatives

for the case in which

up to similar positive affine transformations

for additive

measurement, and uniqueness up to similar proportionality transformations for additive extensive measurement. Both types of uniqueness are related to sets of ‘indifference’

comparisons

that

representation.

correspond

to sets of linearly

After we explicate

necessary

specific aspects of sets of unique solutions and two-factor

additive

conjoint

extensive

independent

and sufficient for two-factor

equations

conditions (two-set)

for the measurement

for uniqueness.

additive

conjoint

we explore measurement

measurement.

1. Introduction

The modern representational theory of measurement (Pfanzagl, 1968; Fishburn, 1970; Krantz et al., 1971; Lute et al., 1988; Suppes et al., 1988; Roberts, 1979; Narens, 1985) was developed with the goal of putting measurement on a firm mathematical foundation. Problems of measurement in the social sciences have led to new types of measurement not encountered in the physical sciences. One of the most important examples of this, the additive conjoint measurement introduced by Debreu (1960) and Lute and Tukey (1964), is the subject of this paper. One of the two fundamental problems of interest in the representational theory of measurement is the uniqueness problem, which asks how unique a scale (or

* The authors comments.

would

like to thank

Garth

Isaak,

** Dr. Roberts would like to thank the National number ET-86-04530 to Rutgers University.

Suh-ryung Science

Kim and Barry Foundation

Tesman

for their helpful

for its support

under

grant

108

P.C. Fishburn, F.S. Roberrs / C.nique finire conjomr measuremenr

representing function) is. This paper studies a type of uniqueness in additive conjoint measurement for finite sets that is identical to a form of uniqueness that arises in infinite-sets additive conjoint measurement. Throughout, K is a positive integer, K2 2, Ak is a nonempty set, uk and Usare real-valued functions on Ak for k = 1, . . . . K, A=A,x -a-xAk;, and z is a binary relation on A. We say that (A, 2) has an additive conjoint representation (Krantz et al., 1971; Roberts, 1979) if there are ux that satisfy

for all (a,, . . . . ah.), (b,, . . . . bK) EA. If, in addition, the uk are unique up to similar positive affine transformations, then we say that (A, 2) has a unique additive conjoint representation. Uniqueness up to similar positive affine transformations means that, given (1), o,, . . . . oK satisfy (1) in place of ul, . . . . uK if and only if there are real numbers a > 0 and PI, . . . ,pK such that nk = (YLiI, + pk for k = 1, . . . , K. Well-known axioms for (A, 2) due to Debreu (1960) and Lute and Tukey (1964) (see Fishburn, 1970, and Krantz et al., 1971, for refinements and proofs) give sufficient conditions for the existence of an additive conjoint representation. They imply that (A, 2) has a unique additive conjoint representation. However, these theories also imply that each nontrivial Ak is infinite, with at least two such factors, and they obtain uniqueness under solvability and continuity/Archimedean axioms that are not entirely necessary for (1). Scott (1964), Fishburn (1970) and Roberts (1979) present axioms for (A, 2) that are necessary as well as sufficient for (A, 2) to have an additive conjoint representation when every Ak is finite. In this case the 11~are only unique up to a system of inequalities that adheres to (1). Finite-sets uniqueness up to similar positive affine transformations can be obtained only if additional conditions are imposed. Moreover, because continuity/Archimedean axioms have no bearing on the purely finite formulation, we must approach uniqueness from another direction. An appropriate direction, which can be viewed as a generalization of typical solvability conditions (Krantz et al., 1971; Roberts, 1979) and is transparent once the structure of the problem is understood, involves systems of linear equations of the form

C

EjZj

=

0

(2)

in which z,,~~,... are positive-valued variables and the Ej are coefficients in (0, 1, - 11. These equations arise from - comparisons in (I), where - , usually referred to as an indifference relation or qualitative equality, is the symmetric part of 2, i.e., a-b if azb and bza. For example, if A ={al,b,) x {at,&} and (1) holds with u,(a,) > u,(bl) and uz(at) > u2(bz), then we have a unique additive conjoint representation if and only if u,(aJ - u,(b,) = uz(az) - Q(&), or Zl -zz = 0, when z, = rr,(a,) - u,(&) and z2 = u2(a,) - ~~(6~). This equation can also be written

P.C.

Fishburn.

F.S. Roberts

I Lnique finite con~ornl meusuremenl

109

as ~,(a,) + u?(b?) = u,(b,) + ul(u3). which corresponds to (a,, b,) - (b,, a:) in (1). The next section describes in detail how vve get from (1) to (2). It then states a condition on - that is necessary and sufficient for uniqueness up to similar positive affine transformations with finite Ak. This basic condition involves the notion of linearly independent equations like (2). Section 2 also discusses a variant of additive conjoint measurement that is suggested by previous work on unique subjective probabilities with finite sets (Fishburn and Odlyzko, 1986; Fishburn and Roberts, to appear, a; Fishburn, Roberts and Marcus-Roberts, 1988). This variant, which figures in subsequent developments in the paper, allows comparisons that go beyond those used in conjoint measurement. In the two-factor (K= 2) case it applies, for example, to a situation in which a comparison is allowed between any nonempty subset of the first factor and any nonempty subset of the second factor, but, like conjoint measurement, it prohibits a direct comparison between subsets of the same factor. We refer to the general case of this type of measurement as additive conjoint extensive measurement, and will discuss the additive conjoint extensive representation and its uniqueness properties in the next section. Other combinations of the conjoint and extensive themes are discussed in Chapter 10 of Krantz et al. (1971). The linear solution theory for uniqueness that is developed in $2 gives rise to a host of interesting and often difficult problems in algebra, combinatorics and number theory. Several of these problems that involve either additive conjoint measurement or additive conjoint extensive measurement with two factors are described in $3. Presently-known results are summarized there, and later sections deal mainly with proofs of those results. Our focus on only tvvo factors reflects both the fact that this case alone presents a large number of intriguing mathematical questions, and the need for additional research on three or more factors. The paper concludes with a brief summary, including open problems and remarks on further research. We also mention that the present work is part of a larger research project on uniqueness in the theory of measurement for finite sets. Elsewhere we consider uniqueness for representations of subjective probability (Fishburn and Odlyzko, 1986; Fishburn and Roberts, to appear, a; Fishburn, Roberts and Marcus-Roberts, 1988; see also Van Lier, 1988), for representations of finite algebraic difference structures (Fishburn, Marcus-Roberts and Roberts, 1988; Fishburn, Odlyzko and Roberts, to appear), and for extensive measurement (Fishburn and Roberts, to appear, b). An overview of this work is provided by Fishburn and Roberts (to appear, b). In addition, Fishburn (1981) examines uniqueness in two-factor additive conjoint measurement when A, is a finite set and A1 is a real interval with uz increasing on Al.

2. Finite uniqueness

for conjoint

measurement

We say that 2 on A =A, x.** x.4,

is a lceak order if it is transitive (a~ b and

110

P.C. Fishburn. F.S. Roberts / Unique finite conjoint measurement

b 2 c= a r c) and complete (for all a, b E A, a L b or b L a). Also, (a,, . . . , ue) is sometimes written as [Us.,~~1, where utk) is the ordered list of all components of (a ,, . . . ,uK) except ak. The following necessary axioms for the additive conjoint representation will help to explain our approach to uniqueness. Al (order). 2 on A is a weak order. A2 (independence). For all k E ( 1, . . . , K}, and u/I (a,, . . . , uK) and (b,, . . . , bK) in A,

bk9Q(k)1 2 hr Q(k)1 * h, &,I 2 hr b(k)]. The axiomatizations for additive conjoint measurement cited earlier require strengthenings of A2, but A2 by itself is sufficient for our present purposes. It is easily seen that Al and A2 imply that for each k there is a weak order zk on Ak such that, for all uk,bk~Ak and all c~A~x...xA~_,xA~+,x...xA~, uk 5 kbk - [uk, c] L [b,, c]. Given the (Ak, zk) as thus induced from (A, r), let -k denote the symmetric part of Tk on Ak. Then -k on Ak is an equivalence relation (reflexive, symmetric and transitive). The next step is to divide out the symmetric part -k of z k and work with the linearly ordered equivalence classes of Ak thus obtained. For simplicity, we skip the full details of this routine step and assume directly that the asymmetric part >& of z/, is a linear order. (If one prefers, the Ukjthat follow can be viewed as equivalence classes in Ak rather than single elements of Ak.) For convenience, we shall work with the dual c k of > k (XC k~ if y >k~), assume that Ak is finite with elements ukl, . . . . ok&+,, and, supposing that (1) holds with Ck a linear order on Ak, take this order as akl

The additive conjoint representation Ukbk,)

< Ukbk2)

<

then gives

*‘- < Uk(ak,nk+

1).

for k= 1, .._,K. Since nk vanishes from (1) if IAkl = 1, we assume henceforth generality that nk L 1 for each k. Let dkj = uk(ak,j + I ) - uk(akj)

with no real loss Of (3)

nk and k= 1, ..,, K. Hence each dkj is a positive difference of adjacent forj=l,..., uk values. Also let d= (d,,,...,d,,,;dz,,

. . . . d,,z;...;d,,,...,d,,,),

(4)

so that d is a positive real vector with C n, components that can be used to characterize a complete solution for (1). In particular, if d as thus obtained from (1) is known,andifukonAkaredefinedby,fork=l ,..., Kandj=l,..., nk+1,

P.C. Fishburn. F.S. Roberts / Unique finiie conjoint measurernenf

III

J-1 uk(akj)

=

a C dkr+ Pk i= I

with (Y> 0 and pi, . . . . pK any real numbers, then the uk satisfy (1) in place of the uk. If, given (l), (A, 2) does not have a unique additive conjoint representation, then, given d as derived from one {uk} solution for (l), there will be {u/i} that also satisfy (1) but cannot be obtained from d as in the preceding paragraph. On the other hand, if (A, 2) has a unique additive conjoint representation, then all possible {uk} solutions for (1) are precisely the uk SetS obtained from (5) for all possible values of a > 0 and p, through PK. Since the pk vanish when we take differences of uk or uk values for any fixed k, it follows that, given (l), (A, L) has a unique additive conjoint representation if and only if d is unique up to proportionality transformations. This means that if d=(d ,,,..., dKnK)characterizes one solution to (1) by positive uk differences, then e=(e,,, . . . . eKnk-)also characterizes a solution to (1) in a similar way if and only if there is an (Y> 0 such that ekj = “dkj

for

k = 1, . . . ,K

and

j = 1, . . . , nk.

We have thus shown that uniqueness up to similar positive affine transformations in our original formulation translates into uniqueness up to proportionality transformations for a real vector of the form given by (4). We say henceforth that d is unique if it is unique up to proportionality transformations. To explain how unique d’s arise from - comparisons in our original formulation, we first state a general lemma and then apply it to the conjoint structure. We use the statement in Fishburn and Roberts (to appear, b), and refer readers to Fishburn, Marcus-Roberts and Roberts (1988) for the essentials of its proof. Recall that the equations in a set of linear equations are linearly independent if none of them can be derived from the others under the operations of addition and multiplication by real numbers. Lemma 1. SupposeNz ties of the form

1 andd=(z,,...,

z~) is the solution to a set S, of inequali-

N

and a set S2 of equations of the form i

&ijZj=

0

j=l

with all E;j coefficients in { 1,0, - 1 }, and suppose that the inequalities in S, imply that some Zj has a fixed nonzero sign. Then d is unique up to proportionality transformations if and only if S, contains a subset of N- 1 linearly independent equations.

P.C.

112

Fishbum,

F.S. Robercs

/ Unique

finite

conjoint

measurement

Lemma 1 avoids the question of conditions that imply the existence of a solution to S, and S2 so as to focus on uniqueness. In the particular case of additive conjoint measurement for finite sets, the existence question is answered by Scott’s (1964) axioms, or by those in Fishburn (1970) or Roberts (1979). To apply Lemma 1 to the additive conjoint case, we take (z,, . . . . z,v) = (d,,, . . . . d,,,; . . . . dK,, . . . . dKnh) with N= c nk. We refer to subsequence (dklr. . ..dknr) here or in (4) as block k. With > the asymmetric part of r on A, the ith > comparison of the form (al,u2, . . . . 0~) > (b,, 62, . . . , bK) yields an inequality ,g, [u&k) -

%(bk)l

> 0

via (1). which translates into ,i,

j$

&i.kjdkj

(6)

> o

under the correspondence between d and the uk. Similarly, the ith qualitative equality comparison (a,, . . . , aK) - (b,, . . . , bK) yields an equation kt,

b&k)

- Uk(bk)l

= 0

via (1), which translates into

kt,j$, kjdkj= 0.

(7)

Ei,

Each inequality (6) is an instance of an inequality C EuZj> 0 for St in Lemma 1, and each equation (7) is an instance of an equation C EcZj=O for S2 in Lemma 1. All E coefficients in (6) and (7) are in { LO, - 1) by the correspondence between d and the uk. For example, Uk(akr)

-

ukbkr)

- uk@ks)

uk(akr)

-

The distinguishing

uk(aks)

uk(akd

= dks + dk, s + I +

*-* +

d+

,

ifs=r

= 0

= - dkr - dk,r+

if s
I-

.-.

- dk,s-l

if r
features for (6) and (7) in the additive conjoint case are

For fixed i, all &i,kjin block k are either in (0, 1) or in (0, - l}, P2. For fixed i, all nonzero &i,kjin block k are contiguous, or form an interval of consecutive j values. Pl.

In addition, since (1) holds, since 0, the equations like (7) in Sz also require P3. For fixed i, if some &i,kj#O, then both + 1 and - 1 appear among the &i,kj values.

P.C.

Fishburn.

F.S. Roberts

/ Unique finite

conjoint

measurement

113

Any equation like (7) that adheres to Pl, P2 and P3 is a candidate for S, in the additive conjoint setting, On the other hand, not all sets of C nk - 1 linearly independent equations subject to the preceding restrictions correspond to an additive conjoint representation. This is true, in particular, if their solution forces some dkj to be 0, or if every nontrivial solution has dkj of opposite signs. However, if they have a solution in which all dkj are nonzero and have the same sign, then they must characterize some additive conjoint representation. To see this, take dll > 0 as the only inequality in S,. Then, by Lemma 1. d is unique (with all dkj > 0). We can then specify Ak = {ski, . . . , ok.nr + ,} for each k, define uk on Ak by j- 1 uk(akj)=

c

i=I

dk; for

j=

l,....nk+l,

and finally define r on A by (a 1,“. ,a,)r(b,,...,

bK)

if

1

Ukbk)

1

c

Uk@k)-

Then (A, 2) has an additive conjoint representation by definition, and its uk are unique up to similar positive affine transformations by the uniqueness of d. We thus arrive at the following implication of Lemma 1. Corollary 1. If (1) has a solution {uk} for a given (A, 2 ), and d is defined as indicated through (3) and (4) with nk L 1 for all k, then d is unique if and only if on A gives rise to 1 nk - 1 linearly independent equations like (7) in which all E coefficients are in (LO, - l} and satisfy Pl, P2 and P3. Moreover, if d is an arbitrary vector like (4), unspecified except for block sizes, and if this vector is then required to satisfy C nk - 1 linearly independent equations like (7) in which all coefficients are in { l,O, - 1) and satisfy Pl, P2 and P3, then there exists an (A, 2) that has a unique additive conjoint representation whose uk differences correspond to d as in (3) if, and only if, the C nk- 1 equations have a strictly positive solution.

It should also be noted that since the coefficients used in Lemma 1, or in (6) and (7), are rational, every positive unique d can be expressed using only integers for all dkj. To avoid duplication of d’s that are related by a proportionality transformation, we usually express them in smallest-integer format: If d = (d, ,, . . . , dKnK)is in smallest-integer format, then every dkj is a positive integer and the only integer (Y> 0 that divides all dkj is a = 1. We illustrate the preceding developments with two examples. Example 1. Suppose K=4, Ak= {ak, bk} for each k, and (A, 2) has an additive conjoint representation in which

dk, = #k(ak) - uk(bk) > 0

for

k = 1,2,3,4.

Thus each block of d= (d,,; d2,; d3,; d4J has one term. By Corollary

1, d is unique

114

P.C.

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F.S. Roberts

/ Unique finite

conjoint

measurement

if and only there are 1 nk - 1 = 3 linearly independent equations in the components of d that arise from - comparisons of the type allowed by Pl-P3. Three such equations and their - comparisons are dll-t-d31 d2, + 4,

=&I

a1b2w.t

= 4,

a,a2a3b, - a,b2b3a4

-

b2bd

a,a2b3b, - b,bza3al.

d,, +dz, = d3, +d,,

The unique positive solution in smallest-integer

format is d = (2;3; 1;4).

Example 2. Suppose K= 2 and the block sizes are 4 and 3. For convenience, d as d = (dl.d2,d3,d~;el,ez,e3),

write

dj = d,j and ej = dzj in prior notation. With respect to the latter part of Corollary 1, consider the following 6 = 4 + 3 - 1 linear equations:

SO

d, = e2 d4 = e2 d2 + d3 = e2 d, + d2 = e3 d3 + d4 = e3 d,+dz+d3+d4 These adhere to Pl-P3, set is

= el. they are linearly independent,

and their complete solution

Since there is a strictly positive solution (in smallest-integer format when a= l), Corollary 1 tells us that d= (2,1,1,2; 6,2,3) is the unique d for a two-factor unique additive conjoint representation. The simplest A in this instance has IA, / = 5 and IA21= 4 with Mart),

. . . . ut(ars)) = (0,2,3,4,6)

@2@2,),

. . . , u2@2J)

= to,6

&11).

Here the successive positive differences of adjacent u, values are the dj, and the successive positive differences of adjacent u2 values are the ej. The appropriate definition of r on A is that obtained from (1). We now consider the type of measurement referred to in the introduction as additive conjoint extensive measurement. To illustrate this concept for the two-factor case, suppose that A, is a finite set of red pebbles and A2 is a finite set of blue pebbles. We are interested in a comparative assessment of the weights of all pebbles in A, U A2. We are provided with a two-pan scale that tells which of two disjoint sub-

P.C. Fishburn. F.S. Roberfs i Unique finite conjoint meosuremenr

115

sets of pebbles is heavier, or whether their weights are equal, but nothing more. In addition, pebbles of the same color are not permitted in both pans, so every allowable and nontrivial comparison is between a red subset and a blue subset. (For K 2 3 we use K colors, the same two-pan scale, and the restriction that pebbles of the same color are never allowed in both pans.) We would like to determine the exact weight of every pebble relative to the others. To formalize this, with K 12 and Ak a nonempty finite set as before, let J={(B,

,..., B,):BkcAk

for

k= l,..., K},

the family of all collections of subsets of the factors. We denote by zc, a binarycomparison relation which applies to d under the restriction (B ,, . . ..BK) r,(B;,

. . . . Bk)=Bk=O

or

Bi=@

forallk.

In addition (B,, . . . . BK) -o(B;, . . . . Bk) means that ((Bk)) z O(W)) and WW 2 OWkN. We say that (d, zo) has an additive conjoint extensive representation if there are strictly positive real-valued functions wk on Ak for k= 1, . . . , K such that, for all (B ,r . ...&), (B,‘, . . . . Bi) E SL’for which Bk = 0 or B/ = 0 for each k, K (B I,....BK)Z~(&’

,...

,B;)o

; k=I

c wkbk) aieBI-

2

C k=l

C are&

(8)

WkMkh

If, in addition, the wk are unique up to similar proportionality tranSfOrmatiOnS for all k, with U> 0), then we say that (Jd, 2,) has a unique additive conjoint extensive representation. In the present setting we take nk= /Akl, enumerate Ak as {aktr . . . , aknr}, and define dkj directly from wk by (wk+awk

(9)

dkj = wdakj)

when (8) holds. As in (4) we write d = (d, 1, . . . , d,,,; . . .; dKI, . . . , dK,,& and refer to nk as the size of block k. To apply Lemma 1, we again suppress the existence question to focus on uniqueness. As before, we say that d is unique if it is unique up to proportionality transformations. Let Ck and CL denote subsets of { 1, . . . , nk} subject to Ck = 0 or Ci = 0 for all k. Then, using dkj in place of in (8), each strict inequality for (8) translates into wk(akj)

k=l

jsCkUC*

Ei,kjdkj> 0,

and each equation for (8), corresponding &i,kjdkj = 0. k=l

jeC,UCl

(10) to an -o comparison,

translates into

(11)

It is easily checked that Pl must hold for (lo), and Pl and P3 must hold for (1 I), but P2 is no longer relevant since we are working with arbitrary subsets of factors.

116

P.C. Fishburn. F.S. Roberts / Unique finite conjoint measurement

Consequently, the only substantial difference between our formulation for unique d in the additive conjoint and the additive conjoint extensive cases is the removal of P2 from the former. The following companion of Corollary 1 summarizes the uniqueness situation for finite additive conjoint extensive measurement. Corollary 2. If (8) has a strictly positive solution { wk} for a given (4 zO), and d is defined in (4) and (9). then d is unique if and onl_v if -a on sl gives rise to C nk - 1 linearly independent equations like (11) in which a/i E coefficients are in { l,O, - 1) and satisfy Pl and P3. Moreover, ifd is arbitrary, unspecified except for block sizes, and if d is then required to satisfy 1 nk - 1 linearly independent equations like (11) in which all coefficients are in ( 1,0, - 1) and satisfy Pl and P3, then there exists an (J, z,,) that has a unique additive conjoint extensive representation whose wk correspond to d as in (9) if, and only if, the c nk - 1 equations have a strictly positive solution. In the present case, if d = (d, I, . . . , d,,,; . . .; dk,, . . . , dknK) is a unique solution for some additive conjoint extensive representation, then so is every d’ obtained from d by permuting the entries within each block. This is not true, because of P2, for the additive conjoint setting of Corollary 1. Example 3. As in Example 2 with block sizes of 4 and 3, write d as (d,,dz,d,,d4; e,,ez,e3). With the removal of P2 we consider 6 linear equations that satisfy Pl and P3: d, + d2 = e2 d, + ds = e3 d,+d3=ei+e2 d2 + d3 = el -t e3 d,+d., = e2+e3 d4 = el + e3. It is not hard to check that these are linearly independent and that their unique positive solution in smallest-integer format is d = (2,3,4,7; 1,5,6). We cannot begin to get close to this solution if P2 is imposed since only the two equations d, + d2 = e2 and d3 + d4 = e2 + e3 satisfy its contiguity requirement.

3. Two-factor unique measurement To describe our results and several open problems for the two-factor write d as in Examples 2 and 3, d=(d, ,..., d,,,;e, ,..., e,),

case, we

P.C.

Fishburn,

F.S. Roberrs

j’ L’nique finite

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measuretnen~

117

and assume henceforth that m 2 n so that block 1 has at least as many terms as block 2. We also write unique positive d in smallest-integer format. Recall from Corollaries 1 and 2 that we get a unique d if and only if there is a strictly positive solution to M + n - 1 linearly independent equations of the form (12) Cd,= jFJejv iPl where every I and J must be nonempty intervals of integers in { 1, ..,,m} and (A..., n} respectively for the additive conjoint case, and I and J can be any nonempty subsets of { 1, . . . , m} and { 1, . . . , n> respectively for the additive conjoint extensive case. Given m 1 n 1 1, let C(m, n) be the set of all positive unique d in smallest-integer format for the additive conjoint case, and let E(m, n) be the set of all positive unique d in smallest-integer format for the additive conjoint extensive case. Because the restrictions on I and J in (12) are more demanding for the additive conjoint case, C(m, n) G E(m, n). Clearly, C(l, l)=E(l, 1)= ((1; l)} and C(2,l) =E(2, l)= {(l,l; I)}. Moreover, C(2,2) = E(2,2) = {(Ll; 1, l), (Al; 1,2), (Ll;2,

I),

(192; 1, l), (231; 1,1)}. But C(m,n)C E(m,n) for all other (m, n). For example, E(3,l) via d, + dz = el

contains (&I, 1;2)

d, + d3 = el d,+d,

= e,,

while (1, 1,l; 2) is not in C(3,l) ditive conjoint case.

because d, + d3 =e, is not admissible for the ad-

3. I. Conjoint results Our first theorem notes that C(m, 1) has only one unique d. Theorem 1. C(m, l)= ((1.1, . . . . 1; l)}. The simple proof of this in the next section shows that there is exactly one way to get m linearly independent equations like (12) when J= {l} and each lis an interval of integers in ( 1, . . . , m}. In contrast, we note shortly that, when the interval restriction is removed, IE(m, l)/ grows rapidly in m. C(m, 2) is substantially richer than C(m, 1). For example, IC(2,2)1= 5 (see above), IC(3,2)/ = 19, (C(4,2)1 =79and IC(S,Z)l 1335. Table 1 lists all unique din C(m,2), up to complete reversals of blocks (e.g., replace d,, . . . . d,,, by d,,,, . . . . d,), for m = 3,4, and notes some of the unique d for m = 5. We do not presently have a good estimate of IC(m, 2)( for large m, but do have some results on the structure of C(m,2).

P.C. Fishburn, F.S. Roberts I Unique finite conjoint measurement

118

Table

1

All C(m.2) solutions solutions for m = 5.

m=3

block

m =J (cont.)

reversals

for m =3,4;

some

Some for m=S

111111

1111

12

12211 i 13

l12lll

1112

12

12212 ~ 13

1211 II

1113

12

12213

122 1 I1

1121

I2

212 111

112’

I2

Ill

/ I2

1123

12

1121 I2

1131

12

11112 I 14

113 112

1132

12

Ill

113

‘3 12214 I 13 11111 i 14

1133

12

11113 I 14 11114 I 14

2112

I2

Ill15

2113

I2

&

3113

12

1111~11

1111

I3

I I4

12121 I 14

I 23

lll2l

11

1112

13

21121

1121

II

1221

13

1122

11

1113

13

21122 I 23 21123 I 23

1212

11

1114

13

21125 ~23

1221

11

1111

14

2112

II

1111

23

1222

II

2112

23

2122

11

Ill11

I 15

2. Zf (d,, . . . . d,,,; q, e,) E C(m, 2), then e, and e2 are relatively prime, i.e.,

Theorem (q,e2)=

up to

1.

Theorem

3. Zf (e,, e2) = 1 and m = el + e2 - 1, then (1, 1, . . . , 1; et, e2) E C(m, 2).

Proofs are given in the next section. Theorem 2 implies that, when el =e2, all solutions in C(m,2) have the form (4, . . . . d,,,; 1,l) with dieeither 1 or 2 for every i. Since it is easily seen that at least one dj= 1 (else we get only m linearly independent equations, which is one too few for uniqueness), the number of d in C(m, 2) that have et = e2 is 2”’ - 1. Theorem 3 shows that, given relatively prime et and e2, there is a unique d= (d,, 1.. , d,,,; el, e2) in C(m, 2) for which m I: et + e2 - 1. We conjecture that m can be no smaller than el + e2 - 1. Conjecture

1. [(e,, e2) = 1, (d,, . . . , d,,,; el, e2) E C(m, 2)] = m 1 e, + e2 - 1.

We have verified this for m I 5 by showing that if its hypotheses hold and m is

P.C. Fishburn.

F.S. Roberts

/ Unique finire conjoint

119

measuremenl

given with m I 5, then e, + el 5 m + 1. Its status for M L 6 remains open. Very little is presently known about C(m, n) for m L n L 3. 3.2. Extensive results Our results for E(m, n) concern only the simplest case in which n = 1. Values of lE(m, l)! for small m are :E(2, l)/ = 1, lE(3, l)l =2, lE(4, 1)~= 11, ;E(5,1)1 = 169 and lE(6, l)j =6639. Table 2 lists all unique d in E(m, l), up to permutations in block 1, for m from 2 through 6. It is apparent from this that jE(m, 1): grows rapidly in m, and that E(m, 1) for large m has a great variety of solutions. Facets of this variety are revealed by the next two theorems. For the first, let K,, denote the maximum number of different d, values in any (d,, . . . , d,; el) E E(m, 1). Table 2 shows that Table

2

All E(m, I) solutions (et) is maximum.

m=3

0

1. The 0 sequences

m=6

are those whose final term

by the number

111222

5

112223

111122

6

112233

;5 j5

111335

11223 !4

111222

6

111123 16

111335

4

11111

l

11122; 5

111222

7

11122316

0

11223

5

111113

3

111233 ,6

l

11123

5

111133

3

112223

111113

4

122233

111133

4

111223

m=l I

d, valrres. m=6

m=6

111/z

llll/

of distinct

11113’4

11122

11 11

in block

for m = 6 are grouped

m=5

m=2 0

up to permutations

The sequences

m=6

111111

I

111113

5

112223

111212

111111 !2

111133

6

112233

0

111113

111111

3

111133

7

122233

0

1112~3

111111 j4

111114

4

11123318

5

j6 /8

(Jd;) 112234

~6 ~6 ~7 17 /7 j7

1111/z

111111

11222517

~4

11123415

~5 16 112334 j 6 112234 112234

11123417 112234

17

122334 17

j8

112334

j8

111114

5

112233

(2d,) 111112,2

222233

8

11112414

111122’2

111123

3

llll2l

111222,2

111223

3

11112416

19 /5 112235 / 5

11122

111112,3

111233

3

11122416

112335 17

11111

111122,3

111123

4

11122417

112335

11112

Ill222

3

111223

4

31113414

1112213

111112

4

112223

4

11133416

(5d,) 112345

1111313

111122’4

112233

4

11113417

1112313

11122214

111123

5

111334;7

0

j7 /8 112345 / 9

11111)4

111112

111223

5

111225 I5

l

122345 19

1111214

111122;5

111233

5

111225

m=S 11111 11111

I /

11122414

(3di)

5

122334 18 l

/5

111224

17

112234

111235

)8

122315

120

P.C.

Fishburn.

F.S.

(ki,K,,K,,&‘i,K,)

Roberrs

/ Unique finire

conjoint

measurement

= (1,1,2,3,5).

For example, K6 L 5 since d = (1, 1,2,3,4,5; 7) is in E(6, l), and K6 < 6 since no d in E(6,l) has different d, for all six terms in block 1. For m 2 7 we can always obtain di # d/ whenever i # j. 4. K,,, = m for all m L 7.

Theorem

Our second theorem for finite unique two-factor conjoint extensive measurement with n = 1 concerns the largest possible value of et for solutions in E(m, 1) expressed in smallest-integer format. Let e(m)=

max e,. deE(rn,I)

Table 2 reveals that er2) through e(@are 1,2,3,5 and 9. The d with e, = e”“’ are identified by 0. To get an idea of the magnitude of er”‘) for larger m, we first define a denumerable sequence (ct. cz. . . . ) by an unusual odd/even recurrence as follows: c, = Cl = C2j+,

c3 = c, = cg =

=C21_1+C2i_2

1,

for all ir3, for all i23.

C2i=C2,+I-C,-l

The first 16 terms of this sequence after c, are cg,...,

C‘,

= 1,2,2,3,4,5,8,9,16,

17,31,33,62,64,123,

126.

Each of the first five pairs of terms in this part of the sequence have the form (2j, 2j+ 1). Thereafter, the first member of each pair is somewhat less than a power of 2. Although we developed (c,,c2, . . . ) to obtain a lower bound on e@), it is of some interest in its own right. Two aspects of the sequence are noted in the following lemmas. Lemma

2.

C2i=

Cy&‘Cj

for al/ i2 3.

Lemma 3. There is a constanf czi+ ,/(co2i-3)-+ 1 as i- 03.

co, with 0.9479939

< co < 0.9479940,

such

that

Our lower bound for e”“’ is given by Theorem

5. ecm’2 c2,,, + , for all m 2 1.

Table 2 shows that equality holds in the theorem for m I 6. Moreover, we have not been able to obtain a value of e”“’ larger than czm+, for any m, and are fairly confident that e(‘)= cl5 = 17 and e@)= cl7 = 33. In any event, Lemma 3 and Theorem

P.C. Fishburn, F.S. Roberrs / Unique finire conjoinr measurement

I21

5 show that e’“’ is nearly as large as 2m-3, if not larger than Zmd3, for all m. Proofs of the two preceding theorems and lemmas appear in $5. 3.3. Biregular results The theorems stated thus far involve only the smallest values of n, the block size of the second factor. We now present results that apply to all n, but which deal only with special subsets of C(m, n) and E(m, n). These subsets are based on a notion of a biregular sequence (cf. regular sequence, Fishburn and Odlyzko, 1986). A two-block sequence d = (d,, . . . , d,,; e,, . . . , e,) is biregrrlar if, beginning vvith one 1 in each block, its other m + n - 2 terms can be specified one at a time in the following way. Each new term specified is adjacent to the terms already specified for its block (including the beginning 1) and is equal to a nonempty sum of terms already specified for the other block. If the latter sum must be composed of successively adjacent terms, we say that the biregular sequence is interval-restricted. For example (2,3,1,1,7; 6,1,2,14) is an interval-restricted biregular sequence for (m,n) =(5,4) since it can be built up as follows:

1I 1

begin

1 1 11

l=l

11

2=1+1

j 12

311112

3=1+2

2311112

2=2

23111612

6=2+3+1

231171612 23117161214

7=6+1 14=2+3+1+1+7.

If the last term added is 10 (= 3 + 7 = 2 + 1 + 7) instead of 14, we get a biregular sequence that is not interval-restricted. Let BC(m, n) be the set of al! interval-restricted biregular sequences with M terms in block 1 and n terms in block 2, and let BE(m, n) be the corresponding set without the interval restriction. Because the equations for the construction of such sequences adhere to (12), including d,= ej for the beginning l’s, we have BC(m, n) G C(m n), Clearly, BC(m,l)=BE(m,l)={(l,...,l;

BC(m, n) c BE(m, n) C EM, n). l)}. Moreover:

Lemma 4. BC(m, 2) = BE(m, 2) for al/ m L 2.

The values of jBC(m,2)1 for m from 2 through 8 are 5,19,69,243,841,2859 and 9573, and /BC(m,2)1 behaves roughly as 3”’ as m gets large. The next theorem, which is proved in $6, gives order-of-magnitude values for jBC(m,n)l and 1BE(m, n)i. In the theorem, o(l) denotes a function of m that approaches 0 as m gets large.

122

P.C.

Theorem

Fishburn,

F.S. Roberrs

/ Unique finire

conjoint

measurement

6. For each fixed n 2 1,

c) n+

/BC(m,n)/

=

IBE(m,n)l

= (2”-

1 NI *o(l))

2

for all mm,

I)m(‘+O(‘)) for all mzn.

An easy corollary of the theorem is that for each n 2 3, IBC(m,n)I/jBE(m,n)l +O as M+Oo. 3.4.

Maxima in biregular sequences

Our final results focus on the maximum value of the largest term in all biregular sequences for (m,n), defined formally as

i4m 4 =

max

deBE(m,n)

[max {d,, . . . , d,,,, e,, . . . , e,}].

Since the construction of a biregular sequence that has the largest possible term will always take the sum of all terms specified thus far for the other block when a new term is added, BE(m, n) in the definition of ,~(m, n) can be replaced by BC(m, n) without loss of generality. The determination of p reduces to the specification of an optimal block-selection strategy for adding new terms. When m = n, we can do no better than to alternate between blocks: 11 1, 11 j 1, 11 112, 311 112, 311 1125, 8311 1125,

and so forth. The numbers that arise from this alternating strategy are the Fibonacci numbers 1, 1,2,3,5,8,13, . . . , which arise from f, =f2 = 1 and the recurrence fj= h_,+J;_, for every j23. Theorem

7. For each n 2 1, p(n, n) =fi,,- I and p(n + 1, n) =f&,.

The next theorem gives precise values of p(m,n) for small values of n. Theorem

8. For all m r n with n E { 1,2,3,4}, ,dm. 1) = 1 p(m, 2) = m

m(m + 4)/4 p(my 3, = { [m(m + 4) - 1]/4

for even m for odd m

t3+4t2+3t Am, 4) = t3+5t2+6t+1 { t3+6t2+10t+4

if m=3t ifm=3t+l if m = 3?+2.

General bounds on p for larger values of n are noted in our final theorem.

P.C.

Fishburn.

F.S. Roberrs

9. For all mznz4,

Theorem

,I Unrque

conjoinr

if m=(n-l)t+b

@+2)b(t+ l)“-‘-+@?,n)5 Corollary

finire

with Osbsn-2, b+Z(n-2)

t+

n-1

3. For each fixed n I 2, p(m, n)/[m/(n

The corollary

follows

directly

123

measurement

then

1. n-1

- l)]“- ’ - 1 as m -+CQ.

from Theorems

8 and 9. The proofs

of Theorems

7-9 are given in $7.

4. Proofs Theorem

for C(m,l)

and C(m,Z)

1. C(m, 1) contains only (1, . . . . 1; 1).

Proof. Clearly C( 1,1) = {( 1; l)]. Proceeding by induction, suppose m L 2 and the theorem is true at m - 1. Assume that d = (d,, . . . , d,,; e,) is in C(nr, 1). Then d, must be in one of our m linearly independent equations, say

d, + e.0 +d, = e,, and it can only be in one such equation since all d,>O, the d, in an equation are the other nz - 1 linearly incontiguous, and every right hand side is el. Therefore, dependent equations do not involve d, and, by Lemma 1 or Corollary 1, they must hypothesis, (d,, . . . , d,,,; e,) = yield a (d,, . . . , d,,; e,) in C(m - 1, 1). By the induction (1, ***,1; 1). Since every di > 0, the only possibility for the equation involving d, is

d,=el.

Therefore

Theorem

(d, ,..., d,,,;e,)=(l,...,

1; 1).

2. (d,, . . . . d,,,; e,, e2) E C(m, 2) = (e,, ez) = 1.

Proof. Given the hypothesis of the theorem, that (e,,e,)=g> 1. Then el =pg and e2 =qg, Write

0

out a sequence

of C dj l’s in m groups

1 ... 1 --

1 .. . 1

d,

dz

...

suppose, contrary to its conclusion, where p and 4 are positive integers. as

1 ... 1 (13) &

Now change every gth term in (13) from 1 to 2, beginning with the first term, and call the resulting sequence (*). Define d; to be the sum of terms in the corresponding ith group of (*). Let

p’=e,+p

and

q’=e?+q.

Claim. (d,‘, . . . , dh;p’, q’) satisfies the same m + n - 1 =m + 1 linearly independent as (d,, . . . , d,,,; e,, eJ when d; is replaced by di: e, by p’, and e2 by 4’.

equations

I24

P.C.

The equations

Fishburn.

F.S. Roberrs

/ Unique finire

conjoint

meuswemenr

for the original d have the form

,z,dI = r

with

t-E {e,,e?,e, +ez}.

Each I here is a nonempty interval of integers. If C ,d, = e, =pg, then there are pg l’s in (13) for I, and hence there are 2’s in the corresponding places in (*). Thus

C d,‘= iF,di+p = e, +p = p’. icl

Similar results can be obtained when TE {e?, el + ez}, and the claim follows. The claim and Lemma 1 imply that (6, . . . . d,k;p’,q’)

= cc(dr,...,d,,,;e,,e&

for some cr>O. In fact, (p’=ae,)= (e, +p=ae,)= [ p(g+l)=apg]=a-1=1/g. With Ai the number of l’s changed to 2’s in group i iln going from (13) to (*), we have di + A, = d,‘= adi, hence Ai = (a - l)d, = di/g. But then each di is divisible by g, and therefore (d,, . . . , d,,,; e,, e?) is not in smallestinteger format, contrary to hypothesis. Hence e, and ez must be relatively prime. 0 The following is a convenient

rewording of Theorem 3.

Theorem 3. If p and q are relatively prime positive integers, then (1, . . . , 1; p, q) with

(p+q-

1) l’s, is in C(p+q-

1,2).

Proof. Consider the following equations for a proposed unique d= (d,, . . ..dptq_.; el, e2), given (P, 4) = 1: a+p- I C i=o

Q+P

di=e,

and

.J+,

a+q-I C i=a

di = el

for

Ocalq-I;

for

Ocalp-

a+q

di = e2 and

i=F+, di=e2

1.

The first pair of equations yields

4 = 4-rp, Ocalq-1;

(14)

the second pair gives d, = docq,

Ocalp-

1;

(15)

and these two together imply that d, = do-pr

pcaSp+q-I

(16)

4, = domq.

qcalp+q-I.

(17)

P.C. Fishburn. F.S. Roberts / Unique finife conjoint measurement

125

Weshowthat(14)through(17)implythatd,=d,foralla~{1.2,...,p~q-1),and that e, =pdr and ez =qdr. Hence, in smallest-integer format, (dl,...,dp+g_l; e,,ez)=(l,..., l;P,q)EC(P+41,2). To verify dP =d, for all a, we study sequences of integers from {1,2,..., p + q - 1). Given p and q, we consider two types of sequences. The first is a (p, q) forward sequence: start with p, add p repeatedly (possibly never) until it is impossible to continue without leaving the set { 1,2, . . ..p+ q- l}, then subtract q once, then add p repeatedly until it is impossible to increase further, then subtract q once, . . . , until it is impossible to continue. For example, a (5,13) forward sequence is 5,10,15,2(=15-13),7,12,17,4(=17-13),9,14, l(= 14- 13),&l&16,3(=

16- 13),8, 13.

A (q,p) backwardsequence starts with q and does the opposite of a (p, q) forward sequence: start with q, subtract p repeatedly as long as we remain in { 1,2, . . ..p + q - l}, add q once, subtract p repeatedly, . . . , until it is impossible to continue. Clearly, a (p, q) forward sequence, if it ends, must end in q, and a (q,p) backward sequence, if it ends, must end in p. Note that a forward sequence uses moves from X, to x,, , defined by (14) and (17), while a backward sequence uses moves from xi to xi+, defined by (15) and (16). It follows that if a (p,q) forward or a (q,p) backward sequence is (s,,.y2, . . ..x~). then (14) through (17) imply that d,=d, for all a,b~ {xt, . . ..x.~>. Lemma 5. If p and q are relatively prime positive integers, then every integer in {1,2,..., p + q - 1) appears in a (p, q) forward sequence and in a (q, p) backward se-

quence. It follows from the lemma by considering a (p, q) forward sequence that dP = d, for all aE{1,2,..., p+ q- l}, thus proving Theorem 3. We prove Lemma 5 by induction on p+ q. If p= q = 1, then the forward and backward sequences each consist of the single term xl = 1, and this covers all of {l,...,p+q-l}=(l). Given (p,q) = 1 with p+q> 2, assume the lemma holds for all (p’,q’) = 1 for whichp’+q’ 1, for otherwisex=(l,2 ,..., p + q - 1) and the conclusion of Lemma 5 for a forward sequence follows trivially. Let q= kp+r,

kr0

and

Olr
Note in fact that 0 1, (p,q)# 1. Observe also that (p, r) = (q,p) = 1. By the preceding definition of k and r, we have p+q= (k+ l)p+ r. Thus, the (p,q) forward sequence x goes from p to 2p to 3p to ... to (k+ l)p=p+ q -r and then down to (p + q-r) - q =p - r. Since p > r, the sequence goes down at least one

126

P.C. Fishburn. F.S. Roberts / Unique finite conjoint measurement

time. Let y=(y,,y2 ,..., y,vl) be the subsequence of s, values that arise from each subtraction of q. Thus, y, =p-r. Let p=k’r+s, k’?O and OIsp, i.e., once f = (k’ + l), then after we reach yr_ , =p - (I - l)r, the sequence x goes next to p-(tl)r+p, p-(tl)r+2p,...,p+q_tr+p, and then down to y,=(p+q-tr+p)-q=p-(tl)r+(p-r), so y,=y,_, +(p-r). In this case, y,_, =p- k’r=s. To summarize, the sequence y of down terms in x that begin the +p runs starts with p - r, goes down by r until it cannot decrease further, then goes up by (p-r). It is straightforward to show that this pattern continues. It follows that y is a (p-r,r) backward sequence. If p’=r and q’ =p-r, then p’ +q’=p
5. Proofs

for E&2,1)

The following result implies Theorem 4, that K, = m for all m 17. Lemma 6. (1,2,3 ,..., Proof.

m-l,m;m+3)~E(m,l)forallmr7.

Seven equations for (12) in the E(7,l)

context are

d, + dz + d, = e, = d3 = d, + dz d3 + d, = el I d, + d3 + d6 = el =d,=d,+d, d4+d6 = e, I d, + d4 + ds = el g d, +d4 = dz+dj dl+dj+dj = e, I d,+dz+d,+d4=e,.

P.C. Fishburn. F.S. Roberrs / Unique finire conjoint measurernenr

127

The second and third implications on the right give d2 =2d,. Then d, =d, + dz = 3d,, dq =d, +d, =4d,, and, by the seventh equation, e, = lOd,. The first, third and fifth equations then yield d, = 5d,, ds = 6d, and d, = 7d,. Thus a unique solution is (d,,2d,, . . . . 7d,; lOd,), which in smallest-integer format is (1,2, . . . ,7; 10). For mz8 we begin with d, + d2 + d,,, = e,

= d3 = d, + dz

d3 + d, = e, d,+d3+d,,,_, d,+d,,_,

= e, =d.,=d,+d,=2d,+dz =e, I-

d,+d,+d,,_, d2+d3+d,,,-2 ds+d,,,-2

= e, = e, = e, >

d5 = d, + d, = 3d, f d2 = d5 = dz + d3 = d, +-2d,

= dz = 2d,.

These give (d,, . . . , d,) = (1,2,3,4,5) after we set d, = 1 for definiteness. Then, for all k such that 6 I kc (m + 3)/2, the pairs of equations d,+d~-,+d,,,+J_k=el dk+d,,,+j_x- = e,

successively yield di; =k. e, =m+3 from

Thus dk =k for all kc

(m +

4 + d,,z + 4, + 7~2 = e,

if m is even

d3+d~,,,_,,,z+d~,,+,~,2

if m is odd.

= et

3)/2. We then find that

gives d,,,=(m+3)-3=m, d,_,=m-l,..., Then dk+d,,,+3_k=e, for k=3,... dcm+4j,2= (m + 4)/2 if m is even or dcm+s,,2 = (m + 5)/2 if m is odd. Finally, if m is odd, we need one more equation to obtain dcm+ 3,,2, and d, + d,,, + ,)/?+ d,m+ 3j,2= e, will do. It is easily checked that exactly m equations have been used and, as noted, their unique solution is (42, . . . , m; m + 3). 0 We now turn to the proofs of Lemmas 2 and 3 and Theorem 5, recalling that (c,, c2, -**) is specified by c,=c2= ... =cs=l, c2i+l

= c2i-l

C2i = C2i,

for all ir 3,

+c2i-2 1 -Ci_

1

for all iZ3.

=C2i_*+C2i_2_Ci_1

Lemma 2. czi= Cfi=;.‘cj for all il3. Proof. This is easily checked for the first few il3. we want c2(i+I)=ci+I+ci+2

+

**’

+C2(i+1)-3*

Assume it for i. Then for i+ 1 (18)

P.C.

128

Fishbum,

F.S.

/ L’nique finiie

Roberrs

conjoinr

measurement

By definition, c?, _ z = c,, _ , + czi - c,. By the induction hypothesis, c, + , + .a= + cl, _ 3= 3 (18) holds if cl;+, = cz,_, + C2i-2, which is true by definition.

czi - c,. Therefore

We note here other identities that follow from the definitions. Parentheses indicate that the index on c increases by 2 from each term to the next. Proofs are left to the reader. i odd, il5: C2,+1 = (Ci+2+Cj+J+ **. +C2,-,) (19) i even,

ir4:

C2;,l

=C;+1+(Ci+Ci+2+

*.* +C2i-2)

(20)

i odd,

irj:

C2i+]

=Cj+(C,_,+Ci+*+

‘**

(21)

i even,

ir 6:

C2i=(Cj_*+CI+

i odd,

ir5:

C?i=Ci+(Ci+[+C,+J+

**.

+C2i_*)

+C2i_2) *‘* +C21_*)*

(2.3

(23)

Our next lemma implies Lemma 3. Define yi by cli+ r = yi2’-’ for all ir 3. Then y3, y4, a*- is strictly decreasing and has a limit co which satisfies 0.9419939 < co < 0.9479940. Lemma 3A.

Proof.

The definition C2i+3

of (cr,q,

e-e) implies directly that for all ir 3.

=ZC2i,*-Ci-I

Therefore Yi - Yi r

c2i+l 1 = T-F= 21-3

c2i+3

ci-

y>o. 2’-_

2i-2

We have from the definitions

I

preceding Lemma 2 that, for iz 10,

Y2i-Y2i+2=(Y2i-Y2i+I)+(Y2i+I-Y2i+2) c2i- I _.-+2!22i-2

=-+

<

c2i22i-2

I

22i-

I

c2i+ I -Ci-1 22i- I

c2(i - 1) + 1 + c2i + 22i-2

-22i-

1 1

< 2i-4 I 2i-3 _ 22i-2

22i-I

,l 2

r-l’

The second < follows from cl9 = 26 and c2; < 2i-3, Cli+, < 2’-’ for i L 10. Therefore co = lim yi exists and

1


1

2r+2

1 *-- =--I. 2’

P.C. Fishburn. F.S. Roberrs /’ Unique

finite

con~omr

129

measuremenl

It follows that, for large i, 1 Yz;-2’<

co < Ylr.

or 1 c-li+ I 2% 3 --2’ccoc-

c;+ I p-3.

At i = 26, we compute 0.94799397 < co < 0.94799400.

3

We conclude this section by proving an extension of Theorem 5. Let Cl,

=

c2,

= (C,,,C,.,r...,CZn,-,;C2m+l)

C3,, =

(C,,I,C,,+1,...rC?,,-,;CZm)

(~,,,,~,~+l~~~~r~2m-I~C2,~;C2m+3)

for all m 1 1. For example, Cl5 = (1,1,2,2,3;4) C2s = (1, 1,2,2,3;

5)

c35 = (1, 1,2,2,3,4;

9)

and Cl,=

(1,2,2,3,4,5;8)

c2, = (1,2,2,3,4,5;

9)

C3,=(1,2,2,3,4,5,8;17). We shall prove that Cl, and C2, are in E(m, l), and C3, is in E(m + 1,l). Theorem 5 then follows from C2, lE(m, 1) since this shows that some sequence in E(m, 1) has el =c2,,,+,. Theorem

5A. For any m L 1, Cl,,,, C2, E E(m, 1) and C3, E E(m + 1,l).

Proof. It is not hard to verify the theorem for small m, say through m = 6, and we assume that it holds for these m. Proceeding by induction, let I(m) denote the hypothesis that Theorem 5A has been established for Cli, C2i and C3i for all i < m. Separate proofs for Lemmas 7A through 7E stated shortly will be used to verify that, for all m > 6,

[I(m), m even] = C2, E E(m, 1) [l(m), m odd] = C2, E E(m, 1) [I(m), m even] = Cl,,, E E(m, 1) [I(m), m odd] = Cl, E E(m, 1). We also show that C2, E E(m, 1) = C3, E E(m + 1,l). Then Theorem by induction.

5A follows

/ tiniqlre finite conjoinr measurernenr

130

P.C.

Lemma 7A.

CZ,, E E(tn, 1) = C3, E E(m A 1, 1).

Fishbum.

F.S. Roberrs

Assume that (c,,c,_, ,..., c~~-~;c~~~+,) is in E(m,l). For C~,=(C,,C,+ ,,..., czmc3), it follows that the m linearly independent equations that yield c2m- 19C?A C2,, E E(m, 1) also imply the partial solution (c,,c,“*!,...,c1,,,-IrX;c,,,_,+.Y),

(24)

where x, which is added to both sides of those m equations, denotes an indeterminate whose value is to be specified by one new equation. The new equation, giving a total of m + 1 for C3,,,, is c,, + c,, + I + a*. $.Gm_, = Czm+, fl, which of course is in the 1 d,= e, form. Thus x = (c,+

.

=

(c,, +

=

C2m

...

+~2,-3~+~~2,~-2+~2nl-l-~2m+1~

.** +

c2,,I- 3)

. .

by the defmmon of (c,,c?, . . . ) and Lemma 2. Hence c2m+3 = c?,, + , + ~2,” implies that (24) is C3,, and therefore C3, is in E(m + 1,l). Lemma 7B. [Z(m), m even] =) C2, E E(m, 1).

We have by I(m) that C3,,2_I C3m,2-r =

is in E(mL?, l), where

(~,,~2-1,~,~2~...~~,-2;~,+1).

The m/2 - 1 linearly independent equations that establish this also yield the following partial solution at m, in which x m,x mf 2, . . . , xzm _ 2 are indeterminates: m/2-l ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

C j=O

( particular, replace it by

if c, + ‘-. + cb = c, + 1 is one of the m/2-

In

m/2-1 jFo

xm-2j+cm+l

1 original equations,

* >

we

m/2-1 Ym+2j=

jgo

xm+2j+cm+19

is not part of the original equation, and y,,,+lj= if this c term appears in the original equation. Since all x’s are on xm+2j+cm/2+j-I both sides of the replacement equation, it is equivalent to the original. We adjoin the following m/2 equations to the Let el= C~~~-1X~+2j+C~+1m preceding m/2 - 1 to get m - 1 altogether: where

Ym+2j=Xm+2j

ifc,/2+j-]

x~~_~+(x~~-~+c,-z) X2m_4+(X2m-~+C~-3)+X2m-2=el

=

9

P.C.

Fishburn.

F.S. Roberls

X ,~+2+(,Y,,~+C,,,

/ L’nique finite

?)+.y,+j+&,+fj+

are cancelled,

in reverse order.

e, +.v~~~-~ =el.

..-

The last reduces

to .u, + c,,,,~ _ I = c,,,_ 1 when

X’S

and therefore &I = Cnt+l - c,n),_

The next to last equation

1 =

c,.

gives x, + z + c,/2 = X,,, +

X ml+2 = cm+1 +c,,,--m,2

The general

131

measurement

*.. +-~;,,,-2 =

x,,+(x,,+c,,,~-,)+x,,+~+x,~_~+

These are solved

conjoin!

C,,, + I = C, + C,, _ I, so

= c,+z.

case is +(Cm+Cm+2+

*** +cm+2j-2)-cm/2+j-I

=Cm+I

+(Cm+Cm+2+

"*

=Cm+l

+(Cm+Cm+2+

'** +Cm+2j-4)-(Cr~~+2j-3+Cm~2j-J)+cm+2j

xm+2j=cm+l

+cm+2j-2)-(cm+2j-I+cm~2j-2-cm+2j)

=c m+2j9 where we use equations C m/2+;-1 cm+2j-l

of (cl, ~2, . . . ) such as

from the definition

and

=cm+2j-I+cm+2j-2-cm+2j =Cm+2j-3

With all x’s determined

+cm+2j-4*

as claimed

for C2,,

we note that

Xm+2j+cm/2+j-I=Cm+2j+cm/2+j-I=cm+2j+I

and,

by (20), that el=

Hence

CXm+2j+Cm+I

(x,,,,x,,,+c,,z-t,...,

Lemma 7C.

=C,+I+(C~+C~+Z+

"*

+C2~71-2)=C2rn+I*

x2m_2,x2m-2+c,-2;e~)=C2,.

[I(m), m odd] = C2, E E(m, 1).

Since the proof of this lemma and its two successors are similar to the proof of Lemma 7B, we present only outlines. For Lemma 7C with m odd, we begin with Cl,,_ I),2 E E((m - 1)/2,1) and use this and its (m - 3)/2 linearly independent equations to get

(m -

. . . . X2m_2,X2m_2+C,-2;X,+

c

3)/2

j=O

*

xm+I+2j+cm-I >

132

P.C. Fishbum.

F.S. Roberrs / tinique finite

conjoint

measurement

We then adjoin (m - I)/2 new equations (for a total of m - 2 thus far):

X mc3 +(x,+3+c~,+1~~2)+-~m+5+xrn+7+

...

x,~+~+~~,+~+c~,-I~,z)+x,,+~+x,+~+

a.* +h-2=e19

+-~z~-z=~~

When these are solved in re-

where ~,=x,,,+(x,+~+x,+~+ .** +x~~-~)+c,,,_~. verse order and we let t =x,,, - c,, we get X m+I+~j=Zjt+C,+I-~j9

j=O,l,...,(m-3)/2,

and el = 2(m-‘)“t+c2m+I. The last of these uses (21). Our final equation is (%?I+1+c~,,_~),~)+(x,,~+~~,+~),~)+

... +(x~~-~+c,-~)

= e19

which, with cancellation of x’s and the use of Lemma 2, yields x, =c, and hence t = 0. The sequence displayed at the beginning of this proof is therefore C2,,. Lemma

7D. [I(m),

m even] =

Cl,

E E(m, 1).

In this case we begin with Clm,2_I and add one more equation di= et) with a new d, to get (c,,z-,,c,,29***, in

The

E(m/2,1).

Xm+Xm+,+

...

+x2,,,_,

(set the final

cm-3rcm-2;cm-2)

m/2- 1 equations to both sides,

used

for

this

give,

after

adding

~Xm,X,+c,,2-l~X,+2rXm+2+cm~2~...~X2m-2rX2m-2+cm-2~~l~

with e, = C&2)‘2x m+zj+Cm_2. Let equations are

x,+2~+1=Xm+2~+Crn/2+j-1.

Then our other

m/2

-*- +x2m-5

x,+x,+I+ x,+x,+1+

+(x,+~+x,+~+

(x,+3+x,+5+X,+7+

= el

*** +X2m-5+(X2m-1)=~1

Xm + *** +X2m-7

xm+&+l

+X2,,,-4+$,,-3

+ (X2m-3

+X2,_

1) = e,

--- +x~~-J

= el

e.0 +X2m-l)

= el.

The final equation gives c,,,/~+ c,/~+ l + -.. ma 2,

+c,_~+c,_~=x,,,+c,,,_~,

SO,

by Lem-

P.C. Fishburn. X,

= c,2+

F.S. Roberts / Unique finite conjoint measurernenr

...

+c,_3

= c,.

Adjacent differences

Also, ~,+,=x,+c,:~-~=c,,,+c,,~_~=~,,,+~. ing equations yield X,+3

= Xm+I+Xm

X m+5

=Xmt3

X2??-I

133

of the preced-

+-Y,+2

=X2m-3+XZ,n-4,

and it follows routinely that xi = cj for j = m, . . . ,2m - 1. Finally, e, = c?,, by (22). Lemma

7E. [I(m), m odd] =$C 1, E E(m, 1).

For this final part of our induction c,,,_~;

C2(,-1)/2=(C(,-1)/2,C(m+I)/zr.~.,

proof of Theorem 5A, we begin with E((m - 1)/2, l), and use its (m - 3)/2

cm) E

equations plus another equation [x,+(x,+~

+x,+~+

-.a +X2,-2)

= ell

to get (

C,,X,+1,Xm+1+C(m-I)/2~Xm+3~~m+3+~(m+I)/2~ (m - 3)/2 .~~,X2~-2~XZm-Z+Cm-2~

jso

xm+I+2j+crn

>

*

Our other (m - 1)/2 equations are C,+X,,+, C/n+Xm+l

+X,+2+ +

...

c,+(x,+~+x,+~+

*** +XZ~-~

+x~~_.~+(x~~._~)

..a

= el = e,

+xZm-d = el,

where Xm+2j=Xm+2j-l +c(tn-3)/2+j- The final equation reduces to cc,,,+1)/t+ -se + by Lemma 2. Adjacent differences of the preceding c,,,_~=x,,,+~, hencex,+t=~,+~ equations, as in the proof of Lemma 7D, then give Xj= Cj for j = m, . . . ,2m - 1. The desired value of c2,,, for e, is obtained by means of (23). 0

6. Biregular counts Recall that biregular sequences in BC(m, n) are constructed away from an initial 1 in each block, and the value of each new term added is a sum of consecutive terms already specified for the other block. The restriction to consecutive terms is removed for BE(m, n).

134

P.C. Fishburn. F.S. Roberrs / Unique finire conjoint measurement

Lemma

4. BC(m, 2) = BE(m, 2) for all m 12.

Proof.

In either case we begin with ... 1 .a. 1 lx or with ... 10.. /xl,

with x to be specified. By the rules of biregular construction, so long as x remains unspecified we can add only l’s to block 1 that are adjacent to l’s already there. Eventually, x is specified as a sum of one or more of those l’s, and we can always presume that consecutive l’s are used for this. Thereafter, in both the BC and BE cases, each new term added to block 1 is either 1 or x or 1 +x. Since 1 and x are adjacent in block C: 2, it follows that BC(m.2) = BE(m,2). Theorem

6A.

1BC(m, n) 1= (” 1 I)“‘(’+O(‘))for all rn 1 n for each fixed n 1 1.

Proof.

With n fixed and m much larger than n, we begin a construction quence in BC(m,n) as follows:

of a se-

. . . 2n-‘2n-3 . . . 4211 1 1248 . . . 2*-I, with the first m-n terms in block 1 yet to be specified. Since block 2 has successive powers of 2, each of its n + (n - 1) + es* + 1 = (“2 ‘) nonempty intervals of terms has a different sum. Therefore, each unspecified term in block 1 can be assigned any one of (” 1’) distinct values, so

On the other hand, there are at most n!rn(T) different sequences that can appear in block 2. For example, if we begin block 2 with el = 1, then ez I m, e3 5 m2, . . . , e,sm”-‘. (These can either be proved directly, or one can use the results of the next section which show that for i L 3, ej is never much larger than about [m/(jl)]‘- ‘.) Hence, considering permutations in block 2, its total number of possible sequences is no greater than n!(m)(m2) . . . (m”-‘) = n!m(l). Since there are (“$ ‘) nonempty lec(m,n)/

5

n+l 2

intervals of terms in block 2, m n!m(2).

( >

The logarithm of the right side here is

-lokf~‘)+(J logm+log(n!)=m[l+o(l)]log and therefore

n+l 2 (

, >

P.C. Fishburn, F.S. Roberts / Unique finite conjoint meusurement

Theorem

6B.

!BE(m,n)~

=(2”-

135

l)‘“(‘to’l)’ for all m 1 n for each fixed n 2 1.

Given the beginning construction of the preceding proof, there are 2”- 1 different sums of the pokvers of 2 in block 2, namely 1,2,3, . . . (2” - 1, and therefore each of the first m -n terms in block 1 can be assigned any one of 2” - 1 values. Hence Proof.

lBE(m,n)l 2 (2”- l)‘“-n = (2”- l)“r(!rO(‘)). As in the preceding proof, there are at most n!rn(!) different sequences that can appear in block 2. For each of these the number of possible sequences in block 1 for BE cannot exceed (2” - 1)“‘. Hence lBE(m,n)/

7. Extreme

5

(2”-

l)“‘n!m(!)

= (2”-

l)‘n(l+o(‘)).

3

values for biregular sequences

Recall that p(m,n) is the largest possible term in all non-interval restricted biregular sequences that have m terms in block 1 and n terms in block 2. With no loss of generality, we assume that biregular sequences used to determine iu are constructed inside-out away from d,,, = e, = 1, and that each new term added to a block is the sum of all terms already specified for the other block. An example for (m, n) = (5,3) is 8831

11

125

with maximum term value 8. However, we can do better since ,~(5,3) = 11, attained by 1133111128

or

44111I1311.

Each such biregular sequence can be reformulated as a sequence S(m, n) of mL (left) and nR (right) terms that begins LR or RL for the initial l’s and tells whether the next term added is on the left or right hand side, along with a function W: { 1,2, . . . , m + n} * Z + defined as follows: w, = w2= 1; for all k=1,2 ,..., m+n-l,m+n, rk= z{Wi:ilk,

(ttXTTli)=R}

fl(= x{w;:isk,

(termi)=L},

and for all k r 2, Wk+’ =

rk fk

if (term k+ 1) = L if (term k+ 1) = R.

136

P.C. Fishburn, F.S. Roberts / tinique jlnire conjoint measurement

The first and third sequences for (5,3) in the preceding paragraph, translate into

beginning with

RL for definiteness,

S(5,3)=R w=l

L L R L R L L

1123588

and S(5,3)=R

rv=l

L L L R L L R

11

1344

11.

It is readily seen that every w sequence is nondecreasing and that for kz 3, w,, , > wk if and only if (term k+ 1) #(term k). Moreover, ,u(m, n) is the maximum value of w,, _ ,1 attainable by an S(m,n) sequence. Before we focus on specific (m, n) pairs, we consider N-term sequences of L’s and R’s that place no restrictions on the relative numbers of L’s and R’s apart from the beginning LR or RL. Let S,,- denote such a sequence and define w for S.v as before: Wr=Wz=Land,fOrkz2, wk_,=fkif(termk+l)=L, wk+t=/kif(termk+l)=R. We refer to S.Vas alternating if (term k+ 1) #(term k) for k= 1, . . . ,h’- 1. We say that a part of S*vis dualized if every R(L) in that part is changed to L(R). Also let P,~ = max w.~. s\ Lemma

8. For all Nr

2, ~(.~=fi~__ ,, where fi is the jth Fibonacci

number.

Each

alternating S,v has w,~=.u.~.

We begin with a few facts about Fibonacci numbers and alternating quences. First, elementary induction proofs give the well-known identities Proof.

fi +fZ+fJ+f6+ fO+fi

se-

*** +fij=fij+1*

+f3 +f5 + *** +f2j-

I =fij*

where fO =O. Suppose a denumerable sequence, referred to as type S,, has I-=T~ and I= /k through term k L 2 and then continues by alternation. If the pattern from term k+ 1 on is LRLRLR m-ethen u’k+ , = r wk_? =r+l wk,3 = ?!r+t wk+4=3r+2f wk-5 = 5r+3! wx_+a= 8r+ 51

and so forth, where the coefficients of r are the Fibonacci numbers f,. f2, f3, . . . and those of I are fo, f,, f2, f3, . . . . The coefficients are deduced from the preceding Fibonacci identities and in general satisfy

P.C.

Fishburn,

F.S. Roberts

wkT, =f,r+f,_,l,

/ Unique finite conjoin! meuwremenl

13:

j = 1,2 ,... .

Similarly, if the pattern from term k+ 1 for S, is RLRLRL ..+ , then Wk+j=f,_lr+fJI,

j=

1,2 ,... .

Suppose now that S,\ is a sequence in {L,R}“’ that begins with RL or LR. With no loss of generality, we assume that S,v ends with R. Suppose S.\ is not alternating, and let k be the smallest index (2 I k I N- 1) such that S,V is alternating from term k + 1 through term N. Let rk = r and lk = 1. If we have LRLR ... LR from term k+ 1 through term N then, by the preceding paragraph,

Moreover, in this case (term k) = L, and it follows from the nondecreasing property of w and its recursive definition that 1~ r. Therefore, if the part of S.Yfrom term 1 through term k is dualized, then, since I and r are interchanged by the dualization, we get

for the partially dualized S,v, and SinCe f 2 r and j& k > f+k_I, we conclude that wj, L w,~. On the other hand, if S,v has RLRL a.0RLR from term k+ 1 through term N, then w.~=J~_~-_ lr +j&k/r r I I since (term k) = R, dualization of the part of SN from term 1 through term k gives w,;.=&_~_ t/+Jv_k~, and again k*,\lZ by\-. Hence, by dualizing the part of S, through term k we increase, or at least do not decrease, wN. At the same time, the dualization changes term k so that S.Y thus modified is alternating at least from term k through term N. It then follows from a series of dualizations with smaller and smaller k that for every S,V that is not alternating there is an alternating SN whose w.\ is at least as large as lt*.Vfor the nonalternating sequence. Hence alternating SN maximize wN, and for such an S., Pciv= w, Theorem

=_tv-z(l)+fN-3(1)

=h-1.

q

7. p(n, n) = fin _ , and J.&I + 1, n) = f2,.

This is a corollary of Lemma 8. With N=2n, the lemma says that we can never exceed Lv_, = f2” _ , for the largest term no matter how many L’s and R’s are used from terms 3 through N, and since an alternating sequence is possible when m = n we actually get &z, n) =r(12n= fin _ , . Similarly, an alternating sequence is possible when m = n + 1, so p(n + 1,n) =p2,,+, =f2,. •1 Proof.

Theorem

8. ,u(m, 1) = 1 for all m, p(m, 2) = m for all m L 2,

P(m’3)=

m(m + 4)/4 [m(m+4)-1114

for even m 2 3 for odd mr3.

138

P.C.

Fishbum,

Mm, 4) =

F.S. Roberrs

t3+4tz+3t t3+5tZ+6t+ t3+6tZ+

/ Unique finite

1 lOt+4

conjoint

measurement

if m = 3t24 if m=3t+lz4 if m = 3t+2?4.

Proof. Since ,~(m, 1) = 1 is obvious, assume henceforth that n L 2. Given m z n 2 2, we remark that there is always an S(m, n) which achieves p(m, n) that begins and ends with R. This follows from two easily-proved facts about sequences and ,u(m, n)

for m z n L 2: two sequences that are identical up to the last two terms, which are RL in one case and LR in the other, have the same w,,,; some sequence that attains ,u(m,n) ends in LR or RL. The proof of the latter fact uses the subsidiary result that p(m, n) strictly increases in m and n. Further details are left to the reader. It follows that RLL ..+ LR is optimal for n = 2, hence that ,u(m, 2) = m. For n=3, an optimal S(m.3) has form RL ... LRL a.. LR, and with k and m - kL’s in the two parts we get W m+3 =

k+(k+l)(m-k)

= -k’+km+m.

It is easily checked that this is maximized at k = m/2 for even m and at k E {(m - 1)/2, (m + 1)/2} for odd m. Insertion of these k’s into w, +3 gives p(m, 3) as in Theorem 8. An optimal S(m,4) has form k, +k,+k,

RL...LRL...LRL...LR, --kl

k2

= m.

k3

Lemma 8 tells us that RLRLRLRL is optimal at m = 4, and LRLRLRLRL is optimal at m = 5, but by previous remarks these have the same w,,,,~ values as RLRLRLLR and RLLRLRLLR respectively, which fit the preceding form. These smallest-m cases have ki > 0 for all i, and that is true also for larger m. It is straightforward to calculate for n = 4 that Wm +J =

k, + k,(k, + 1) + k3(k,k2 + 2k, + k, + 1) = z(k,, k,, k,).

Since z is symmetric in k, and k,, i.e., z(k,, k2, k,)=z(k,, for kz to get

kz, k,),

we substitute

m-k,-k3

z=m+m(k,k3+k,+k3)-[k:+k:+kfk,+k,k$.

(25)

Suppose kl + k3 =s with s fixed for the moment. Then z = m + (k,k, + kl + k3)(m - k, -k,)

+ 2k,k3

= m+s(m-s)+k,k,(m+2-s) = m+s(m-s)+k,(s-k,)(m+2-s).

Differentiation oft with respect to k, shows that z is maximized at k, =s/2. Hence, ifs is even then kl = k, =s/2 maximizes z. Ifs is odd then (since we have a parabola with maximum at s/2) we conclude that {k,, k3} = {(s - 1)/2, (s + 1)/2} maximizes z.

P.C.

Fishburn.

F.S. Roberrs

i Unique fiinrre conjoint

139

measurement

In any event, ,~(m,4) is realized at an S(m,4) that has lk, - k,l I 1. W’hen we set k3 = k, in (2.9, differentiation shows that ; is maximized at k; = [m - 2 + i-]/6, and in this case m/3 + l/6 < k;
when we set k, = k, + 1 in

k;‘= [m-5+im]/6, and here m/3 - l/3
m/3 - l/6. Taking account of the fact that k, and k, must that optimal k, and k, values lie in { Lm/3l, r&31}.

Moreover, by substitution of m - kl -k, for ceding (29, it is easily seen that, at optimality, k,, that kz I kJ. For example, if we begin with values, w, +J will be increased. It follows, with ties, that optimal (k,, k:, k,) triples are

k3 in the expression for LV,,,~~prek, 5 k, and, by symmetry of k, and k, < k2 and then interchange these a few checks on adjacent

U, t, t)

when

m = 3t

(t,t,t+ 1) or (t+ l,t,t)

when

m = 3t+ 1

(f + 1, f, t+ 1)

when

m = 3t-t-2.

When the indicated values of k, and k3 are substituted II ues of p(m,4) shown in Theorem 8. Theorem

into (29,

possibili-

we obtain

the val-

9. For all m 2 n 2 4, if m = (n - 1)t + b with 0 I b I n - 2, then

Proof. The natural generalization of the procedure for n 2 4 that some S(m, n) of the form

R(k, L’s)R(kzL’s)R will maximize

in the preceding

... R(k,_, L’s)R

w, +n. For such a sequence,

let

ai = w value for the ith R bi = w value for each L in the ith run of L’s Bj = kibi = sum of the w values in the ith run of L’s. For the first few i,

al = 1

a2 = k,

6, = 1

bz = k, + 1

B, = k,

Bz = kz(k, + 1)

proof

shows

I40

P.C.

Fishburn,

, L:nique finite conjoinr measuremen!

F.S. Roberrs

and a,=k,+kl(k,+l)<(k,+l)(k,+l) bJ = 1 + k, + [k, + k,(k, + l)] = (k, + l)(kJ + 1) + k, < (k, + l)(k, + 2) B,=k,[(k,+l)(k,+l)+k,]
j+l

bj+l = C a;=aj+I+bj. i=l

(27)

We claim that for j L 3 j-2

C7j<(kr+l)(kj_t+l)n

i=z

(ki+2)

j-l

bj< (k, + 1) fl (ki+2). i=2

This is true for j= 3 by preceding calculations. We show it is true for j+ 1, given its truth for j L 3. Under the induction hypothesis we have

=(k,+l)

j~~(k’+2) [i=2

’

]

[(kj_,+2)-l+kj(kj_,+2)]

j-l

bj+,=aj+l+bj<(kl+l)

(kj+l)+(k,+l)

n (ki+2) i=2

= (/Cl + 1) h (ki+ 2). i=2

Therefore W l?l+n

=a,<(k,+l)(k2+2)(k3+2)...(k,-,+2)(k,-,+l).

The right side of this inequality is maximized (without regard to integers for the ki) when each term in parentheses is equal, i.e., when each term is [ Cki+2(n -2)]/ (n - 1) = [m + 2(n - 2)]/(n - 1). Therefore

P.C. Fishbum,

Am, n) <

F.S. Roberts / Unique finire conjoinr meusurernenl

m+Z(n-2)

n-l

n-l

=

I

t+

b+Z(n-2)

n-1

[

141

n-1

1

when m = (n - 1)t + b, 0 I b 5 n - 2, which verifies the upper bound in Theorem 9. To establish the lower bound, we take kj E {t, t + I} for i = 1, . . . , n - 1. There are b terms ki = t + 1 and (n - 1 - 6) terms ki = t. Although it appears optimal to place the larger ki symmetrically at the ends of S(m, n), for computational purposes here we put them at the beginning: k(=t+l

for

isb

ki = t

for

i>b.

With n L 4, we get aj = t(t + l)(t + 3) L (t + 1)3 if b = 0 a,=(t+l)(t2+3t+1)+t(t+2)r(t+l)‘(t+2) a4 = (t+ l)*(t+3)+t(t+2)

aj = (t+ l)(t+2)(r+4)

if b= 1

2 (t+ l)(t+2)’

I (r+2)3

if b = 2

if bz’3.

Using (26) and (27) for j L 4, we have aji

I =aj+Bj=

(t+2)aj+(t+l)(at+ i (t+l)Uj+t(Ol+

-a* +a,_,) “‘+Uj_*)

ifjlb if j > b.

Hence Uj+1> (t + 2)Qj if j 5 6, and aj+ 1> (t + l)aj if j > 6. These inequalities and the preceding inequalities for Q yield the lower bound of the theorem since ~(rn, n) 1 a,. 0

8. Discussion

This paper has two purposes. The first is to explain necessary and sufficient conditions for types of uniqueness in finite representational measurement structures that are similar to familiar types of uniqueness for infinite structures. This is done for uniqueness up to similar positive affine transformations for additive conjoint measurement, and for uniqueness up to similar proportionality transformations for additive conjoint extensive measurement. Both types are covered by the linear-solution-theory uniqueness result of Lemma 1 as subsequently described in Corollaries 1 and 2. The second purpose of the paper is to explore specific aspects of sets of unique solutions for the two-factor case. Our results for this are summarized in $3, with proofs in later sections. We first considered properties of solution sets when one of the two factors has only a few elements. We then analyzed interesting subsets of two-factor unique solution sets that are based on the notion of biregular sequences. Since no detailed analysis has been made for solution sets for three or more fac-

P.C. Fishburn, F.S. Roberts J LSlrque finite conjoinr measurement

112

tars, this area remains wide open. In addition, many interesting questions remain unanswered for the two-factor case. Primary among these are Conjecture 1 and the cardinality of the solution set C(m.2) for additive conjoint measurement. A more ambitious task would be to develop related results for C(m, n) when n ~3. On the additive conjoint extensive side we have dealt only with the solution set E(m, 1) where the second factor has only one element. We suspect that the extremal inequality for e(‘“)in Theorem 5 may actually be an equality, and we know very little about IE(m, 1)1. E(m,n) for nr2 remains to be explored. With regard to biregular sequences, we would challenge readers to obtain more precise estimates of ,BC(m,n)/ and ]BE(m,n)/ than those given in Theorem 6. Another topic that awaits further development, and has been looked at for other types of measurement (Fishburn and Roberts, to appear, b), is the determination of the lengths of the shortest biregular sequences which contain particular terms or sets of terms.

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