On Fishburn’s questions about finite two-dimensional additive measurement, II

Journal of Mathematical Psychology 82 (2018) 1–11

Contents lists available at ScienceDirect

Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp

On Fishburn’s questions about finite two-dimensional additive measurement, II Che Tat Ng Pure Mathematics, University of Waterloo, 200 University Ave West, Waterloo, ON, Canada N2L 3G1

article

a b s t r a c t

info

Article history: Received 13 October 2017 Available online 23 November 2017

Cancellation conditions play a central role in the representational theory of measurement for a weak order ≾ on a finite two-dimensional Cartesian product set X = X1 × X2 . The order has an additive real-valued representation if and only if it satisfies a sequence of cancellation conditions C (2), C (3), . . .. Given fixed cardinalities m and n for X1 and X2 , there is a largest K , denoted by f (m, n), such that some ≾ on X satisfies C (2) to C (K − 1) but violates C (K ). Fishburn shows that f (3, 3) = 3, f (3, 4) = f (4, 4) = 4. He gives lower and upper bounds for f (m, n), including 4 ≤ f (3, 5) ≤ 7, and asks for the exact values of f (m, n) for some small (m, n) such as (3, 5), (4, 5) and (5, 5). In an earlier article, we obtain f (3, 5) = 4. Here, we report that f (4, 5) = 6. The finding is accompanied by a family of cancellation violating sequences adequate for the detection of all non-additive weak orders for (m, n) = (4, 4) and (4, 5). © 2017 Elsevier Inc. All rights reserved.

Keywords: Preference order Weak order Cartesian product structure Additive utility Additive measurement Conjoint measurement Conjoint commutativity Cancellation condition Functional equation

1. Introduction This section and the next are mostly taken from the earlier article, Ng (2016a). The essential definitions and the setup are repeated so that the present article is self-contained. A binary relation ≾ on the Cartesian product X = X1 × X2 is additively representable (or additive) if there exist real-valued functions ui on Xi (i = 1, 2) such that, for all x = (x1 , x2 ) and y = (y1 , y2 ) in X, x ≾ y iff

u1 (x1 ) + u2 (x2 ) ≤ u1 (y1 ) + u2 (y2 ).

(1)

A utility function representing ≾ is any function u : X → R such that x ≾ y iff

u(x) ≤ u(y),

∀x, y ∈ X .

It is easy to observe that two functions u and u˜ represent a common ≾ if, and only if, one is an order preserving function of the other, say u = Φ (u˜ ) where Φ is strictly increasing. A (utility) function u is additively representable (or additive) if the binary relation ≾ it represents is additive. By the previous observation, we see that u is additive if, and only if, for some strictly increasing function Φ : R → R and real-valued functions ui , it admits the decomposition u(x1 , x2 ) = Φ (u1 (x1 ) + u2 (x2 )),

∀x1 ∈ X1 , x2 ∈ X2 .

E-mail address: [email protected]. https://doi.org/10.1016/j.jmp.2017.10.003 0022-2496/© 2017 Elsevier Inc. All rights reserved.

In the study of functional equations, we normally ask for the general solution (u1 , u2 , Φ ), given u. In this paper, we are interested in the necessary and sufficient conditions on u under which the existence of such a decomposition is assured. A natural ordering condition implied by additive representability, (1), is that ≾ is a weak order - that is, it is transitive and complete. We assume henceforth that ≾ is a weak order. On a finite set, being a weak order is equivalent to the presence of a utility representation. A weak order with the property x ∼ y (x ≾ y and y ≾ x) iff x = y is called a linear order. We shall cover both linear and non-linear orders. Additive relations necessarily satisfy cancellation conditions described below. We use the notation x = (x1 , x2 ), y = (y1 , y2 ) for (typical) objects in X = X1 × X2 ; and (x1 , y1 ), (x2 , y2 ), . . . , (xk , yk ) for (typical) objects in X × X . In the definition below, xi = (xi1 , xi2 ), yi = (yi1 , yi2 ), say, and (xi , yi ) = ((xi1 , xi2 ), (yi1 , yi2 )). Definition 1 (Fishburn, 2001). Given X = X1 × X2 , integer K ≥ 2, sequence of distinct pairs (x1 , y1 ), (x2 , y2 ), . . . , (xK , yK ) ∈ X × X , and positive integers a1 , a2 , . . . , aK , the expression

∑

ak (xk , yk ) ∈ CK

k

means that for the two coordinates i = 1, 2, the sequence a1 x1i [ i.e. x1i repeated a1 times], a2 x2i , . . . , aK xKi is a permutation of the sequence a1 y1i , a2 y2i∑ , . . . , aK yKi in the coordinate set Xi . We shall say that the sequence k ak (xk , yk ) is balanced.

2

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

k k k k We call k ak (y , x ) the co-sequence of k ak (x , y ) and it is clear that a sequence is balanced if and only if its co-sequence is balanced. The K th order∑ cancellation condition on ≾ is k k 1 C (K ): for all ≾ y1 , x2 ≾ k ak (x , y ) ∈ CK it is false that x y2 , . . . , xK ≾ yK and xk ≺ yk for at least one k. The condition C (K ) can be rephrased as follows: for all balanced sequences with K distinct pairs, the comparisons x1 ≾ y1 , x2 ≾ y2 , . . . , xK ≾ yK can only hold under indifferences x1 ∼ y1 , x2 ∼ y2 , . . . , xK ∼ yK .

∑

∑

∑

In Slinko (2009), k ak is called the cardinality of the sequence, K is called its width. Sequences we are dealing with so far have all multiplicities ai = 1. Example 2. Let X1 = {1, 2, 3, 4}, X2 = {a, b, c }. The sequence (x1 , y1 ), . . . ,(x5 , y5 ) defined by k 1 2 3 4 5

xk (1, a) (2, b) (3, c) (4, a) (1, b)

yk (4, b) (3, a) (1, a) (1, b) (2, c)

is balanced because in the first component, 12341 is a permutation of 43112; and in the second component, abcab is a permutation of baabc. If ≾ is additive, (1), then the comparisons x1 ≾ y1 , x2 ≾ y2 , . . . , x5 ≾ y5 translate into k 1 2 3 4 5

xk ≾ yk u1 (1) + u2 (a) ≤ u1 (4) + u2 (b) u1 (2) + u2 (b) ≤ u1 (3) + u2 (a) . u1 (3) + u2 (c) ≤ u1 (1) + u2 (a) u1 (4) + u2 (a) ≤ u1 (1) + u2 (b) u1 (1) + u2 (b) ≤ u1 (2) + u2 (c)

The sum of the ten terms at the lower end equals that of the sum of the ten terms at the higher end, as we see term-for-term matching (cancellation). This implies that all five inequalities are in fact equalities. That is, all five comparisons are in fact indifferences xk ∼ yk . An additive order satisfies C (K ) for all K . The converse is also true (Krantz, Luce, Suppes, & Tversky, 1971, p. 431; Scott, 1964) that a non-additive order violates C (K ) for some K . An important implication of C (2), which is known by various names, including coordinate independence, that we shall be using as a preamble, is that it induces well-defined weak orders ≾i on the coordinate sets Xi by xi ≾i yi if x ≾ y whenever xj = yj for all j ̸ = i (Krantz et al., 1971, §6.1.4). The associated indifference relations xi ∼i yi , defined by xi ≾i yi and yi ≾i xi , are equivalence relations on Xi . The quotient spaces Xi∗ = Xi /∼i are then linearly ordered. The weak order is then seen as a weak order on the product X1∗ × X2∗ which may have smaller sizes. With C (2) imposed we henceforth assume that the coordinate sets Xi are linearly ordered under the induced orders. Definition 3 (Fishburn, 2001). Let X = X1 × X2 be of finite size m by n ( |X1 | = m and |X2 | = n), where m, n ≥ 2. The unique K ≥ 2 such that (i) every weak order on X that satisfies C (2), C (3), . . . , C (K ) is additive and (ii) some weak order on X that satisfies all cancellation conditions (if any) prior to C (K ) is non-additive will be denoted by f (m, n). It is reported in Krantz et al. (1971) that f (2, n) = 2 (pp. 427–428; §9 Exercises, 2). In Fishburn (2001) it is shown that f (3, 3) = 3, f (3, 4) = f (4, 4) = 4 and 4 ≤ f (3, 5) ≤ 7. A call for the exact values of f (3, 5), f (4, 5) and f (5, 5) is made. His paper

contains many other nice results and open problems. In the earlier article, Ng (2016a), f (3, 5) = 4 is obtained. Here, we report that f (4, 5) = 6. Definition 4. A cancellation sequence for a weak order ≾ ∑K violating k k a (x , y ) such that x1 ≾ y1 , x2 ≾ is a balanced sequence k=1 k 2 K K k y , . . . , x ≾ y are valid (true) and x ≺ yk for at least one k. We say that the sequence detects (the non-additivity of ) ≾. A balanced sequence is a cancellation violating sequence for the product X if it, or its co-sequence, detects some (non-additive) weak order on X . It is pointed out in Fishburn (1997) that, for given (m, n), a longstanding open problem is to determine the simplest subset of cancellation conditions that is violated by every order that is not additively representable. With that in mind we shall determine a simplest family, F (m, n), of cancellation violating sequences adequate for the detection of all non-additive weak orders on a product of size (m, n) = (4, 4) and (4, 5). 2. The setting, the canonical representation and the diagrams for C (K ) violations As mentioned in the introduction, we have assumed that our orders are weak orders satisfying C (2) and that on the coordinate sets the induced orders are linear. For a 3 by 5 product, we can map X1 to {1, 2, 3} and X2 to {1, 2, 3, 4, 5} in such a way that the induced linear orders on Xi coincide with the natural order of the integers. In this sense we identify X1 with {1, 2, 3} and X2 with {1, 2, 3, 4, 5}. The weak order ≾ is then representable by a utility function u : {1, 2, 3}×{1, 2, 3, 4, 5} → R. Such u is not unique. It is strictly increasing in its two variables. We shall make it unique by specifying that the range is a set of consecutive integers 1,2,3,. . . , starting from 1. Evidently u(1, 1) = 1 is the smallest and u(3, 5) is the largest. If the order is linear then there will be 15 distinct values beginning with u(1, 1) = 1 and ending with u(3, 5) = 15. Otherwise there will be repeated values and u(3, 5) will be less than 15. We shall call such unique u the canonical representation of the weak order. We shall present u, or the weak order it represents, as a 5 by 3 array u(1, 5) ⎢u(1, 4) ⎢ A = ⎢u(1, 3) ⎣u(1, 2) u(1, 1)

⎡

u(2, 5) u(2, 4) u(2, 3) u(2, 2) u(2, 1)

u(3, 5) u(3, 4)⎥ ⎥ u(3, 3)⎥ . u(3, 2)⎦ u(3, 1)

⎤

The entries of A should be strictly increasing along the rows towards the right and along the columns upwards. For canonical u, we call A a canonical array. The lower left corner entry of such A, A[5, 1], in linear algebra indexing, is always equal to 1 by our specification. This notion of canonical representation naturally extends to products of different sizes. On an m by n product, the arrays used are n by m. Any array with strictly increasing rows and columns, not necessarily canonical, still represents a unique weak order that fits our setting. In particular, the subarrays of A represent the restrictions of the order to sub-products of X . Here is a 3 by 3 example, originally given by Fishburn, mentioned in Slinko (2009) and used in the early paper. We are repeating it once more to bring out the canonical array representation and the terminologies. Example 5. Let X = {1, 2, 3} × {a, b, c }. The linear order 1a ≺ 1b ≺ 2a ≺ 2b ≺ 3a ≺ 1c ≺ 2c ≺ 3b ≺ 3c satisfies C (2) but fails C (3). The linear orders induced on the coordinate sets are 1≺1 2≺1 3 and a≺2 b≺2 c. The coordinate sets are therefore identified with the natural integers by 1 ↦ → 1, 2 ↦ →

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

2, 3 ↦ → 3 and a ↦ → 1, b ↦ → 2, c ↦ → 3. The linear order is represented canonically by

[ A :=

6 2 1

7 4 3

9 8 . 5

]

It fails C (3) because it is detected by the sequence ((1, 2), (2, 1)), ((3, 1), (1, 3)), ((2, 3), (3, 2)). The pictorial presentation of the array and the sequence is conveyed in the diagram where the edges mark the pairs being compared.

3

In Ng (2016a), along with Ng (2016b), we have obtained B for the dimensions 3 × 3, 3 × 4, 3 × 5. The sequences are shown in Appendix A. By transposition, in other words, interchanging the coordinate spaces X1 and X2 , we see that the results for 4 × 3 products are in complete parallel with those for 3 × 4 products. So, by transposing Figure 2 and Figure 3 in Appendix A we get immediately a corresponding set B for dimension 4 × 3, as posted in Appendix B. Going forward we shall report on the results for the remaining dimensions 4 × 4 and 4 × 5. 4. Supporting facilities and supplementary material Maple worksheets are prepared and used by the author to generate the data. The worksheets and the supporting data are made available in electronic forms through the portal reference Ng (2017). 5. On 4 by 4 products

Figure 1 carries two balanced sequences: the sequence ((1, 2), (2, 1)), ((3, 1), (1, 3)), ((2, 3), (3, 2)) and its co-sequence ((2, 1),(1, 2)), ((1, 3),(3, 1)), ((3, 2),(2, 3)). The cancellation condition it represents is the equivalence {(1, 2) ≾ (2, 1), (3, 1) ≾ (1, 3), (2, 3) ≾ (3, 2)} iff {(2, 1) ≾ (1, 2), (1, 3) ≾ (3, 1), (3, 2) ≾ (2, 3)}. The ⊕ and ⊖ signs mark the opposite sides in the pairing without specifying which sign is for the left. We say that Figure 1 detects (the non-additivity of) A when a sequence (carried) in Figure 1 detects A.

3. The underlying strategy Definition 6. We call an array critical-to-inspect if it is non-additive and all its (maximal) proper subarrays are additive. Note: If an array is additive then all its subarrays are additive. Because of that, in the above definition, it makes no material difference whether we take up all proper subarrays or just the maximal ones. In computer programming it is more efficient to use just the maximal ones. In its interpretation it is better to take all. Let X be of a given dimension m × n (m ≥ 3, n ≥ 3) and we wish to find a family, F (m, n), of cancellation violating sequences adequate for the detection of all non-additive n × m arrays. Our strategy is to find a family B(m, n) of cancellation violating sequences adequate for the detection of all n × m critical-to-inspect arrays. Under our current setting and using the reported fact that f (2, n) = f (m, 2) = 2, it follows that all proper subarrays of our 3 × 3 arrays are additive. So all 3 × 3 non-additive arrays are critical-to-inspect, and we may take as a convention that F (2, n) = F (m, 2) = ∅. Suppose a priori that we have already obtained a base family, F (m − 1, n) ∪ F (m, n − 1), which is adequate for the detection of all non-additive arrays of dimensions lower than n×m. We can then take F (m, n) = F (m − 1, n) ∪ F (m, n − 1) ∪ B(m, n). The strategy is recursive in nature, leading to the fact that we can take F (m, n) = ∪3≤i≤m,3≤j≤n B(i, j)

at the end. The main purpose of this paper is to report on F for dimension 4 × 5. The above strategy requires the finding of a B for each of the following dimensions: 3 × 3, 3 × 4, 3 × 5, 4 × 3, 4 × 4 and 4 × 5.

5.1. Conclusion The following figures detect all critical-to-inspect 4 × 4 arrays and constitute B(4, 4).

4

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

5.3. Linear data When we apply the figure detection worksheet(s) to the file ‘‘Linearcriticaltoinspect.txt’’ which holds the (216) critical-toinspect linear arrays, we see that each member is detected by exactly one listed figure, and that each listed figure detects at least one member. A small sample is drawn from the 216-member list for display. Some are illustrated by diagrams with the figures that detect them super-imposed.

In Fishburn (2001) four of the above six types are illustrated. Our data goes well with his result f (4, 4) = 4. 5.2. The steps leading to the conclusion Using worksheet ‘‘generating4by4.mw’’, we get (24024) linear canonical arrays which are saved to file ‘‘4by4Linear.txt’’. Applying worksheet ‘‘findcriticaltoinpectlist4by4.mw’’ to the linear canonical arrays, we obtain (216) critical-to-inspect linear arrays which are saved to file ‘‘Linearcriticaltoinspect.txt’’. Inspecting every (216) critical-to-inspect linear arrays we arrive at the above family of cancellation violating sequences. The family is then full enough to detect all linear critical-to-inspect arrays. Worksheets are then written to perform the detection of the arrays by the figures. There are several versions, tailored to the manner we wish to keep records. ‘‘4by4FigureDetection.mw’’ records the actual arrays detected by the figures. ‘‘4by4Figuredetection IndexOnly.mw’’ records only the index of the arrays being detected in a list. The generation of canonical arrays and the extraction of criticalto-inspect arrays are then extended to the non-linear cases. All resulting data are recorded in text files, and are also viewable by readers without the Maple software. The figure detection worksheet(s) is applied to see if the figures reported are adequate to detect all non-linear critical-to-inspect arrays. That turns out to be the case and hence we draw our conclusion. A note on file keeping. An array may be detected by a figure, say, Figure 1A, in two ways. The first is by the sequence where ⊖ is on the left side in the comparison. Such arrays are kept in files with the key ‘‘Figure1a’’ in its filename. The second is that ⊖ is on the right side in the comparison. Such arrays are recorded in files with ‘‘Figure1are’’ appearing in its filename ( ‘‘re’’ designates a reversal of direction in the comparison).

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

5

5.4. Non-linear data We choose a small sample to display. There are four on the ‘‘D2D3D4D5D7criticaltoinspect.txt’’ list (as suggested by the filename, it contains the critical-to-inspect arrays having a double 2, a double 3, a double 4, a double 5 and a double 7):

⎡

4 ⎢3 ⎣2 1

5 4 3 2

8 7 6 5

⎡

9 6 3 2

10 7 4 3

7 ⎢5 ⎣2 1

⎤ ⎡

11 4 10⎥ ⎢3 , 9 ⎦ ⎣2 7 1

⎤ ⎡

11 7 8 ⎥ ⎢5 , 5 ⎦ ⎣2 4 1

5 4 3 2 8 6 3 2

9 7 6 5 10 7 4 3

⎤

11 10⎥ , 8⎦ 7

⎤

11 9⎥ . 5⎦ 4

The first two are detected by Figure 1A and the last two are detected by Figure 1B. There are four on the ‘‘D4D7D8D9criticaltoin spect.txt’’ list:

⎡

8 ⎢5 ⎣3 1

9 7 4 2

11 9 7 4

12 8 10⎥ ⎢6 , 8 ⎦ ⎣3 6 1

⎤ ⎡

9 7 4 2

11 9 7 4

12 10⎥ , 8⎦ 5

⎡

8 7 4 3

10 9 7 5

12 5 11⎥ ⎢4 , 9 ⎦ ⎣2 8 1

⎤ ⎡

8 7 4 3

10 9 7 6

12 11⎥ . 9⎦ 8

6 ⎢4 ⎣2 1

⎤

⎤

The first two are detected by Figure 2A and the last two are detected by Figure 2B. There are six on the ‘‘detectedFigure4a.txt’’ list:

⎡

7 ⎢3 ⎣2 1

12 9 6 4

⎡

9 7 5 4

6 ⎢3 ⎣2 1

13 11 8 5 13 12 10 8

⎤ ⎡

16 7 15⎥ ⎢3 , 14⎦ ⎣2 10 1

⎤ ⎡

16 6 15⎥ ⎢3 , 14⎦ ⎣2 11 1

12 8 6 4 10 8 5 4

⎤ ⎡

13 11 9 5

16 6 15⎥ ⎢3 , 14⎦ ⎣2 10 1

⎤ ⎡

13 12 9 7

16 6 15⎥ ⎢3 , 14⎦ ⎣2 11 1

10 9 5 4 8 7 5 4

13 12 8 7 13 12 10 9

⎤

16 15⎥ , 14⎦ 11

⎤

16 15⎥ . 14⎦ 11

There are six on the ‘‘detectedFigure4are.txt’’ list and the total number detected by Figure 4A is twelve. There are four on the ‘‘detectedFigure5.txt’’ list:

⎡

9 ⎢5 ⎣3 1

13 7 6 2

15 11 10 4

16 9 14⎥ ⎢6 , 12⎦ ⎣3 8 1

⎤ ⎡

12 7 4 2

15 13 10 5

16 14⎥ , 11⎦ 8

⎡

13 9 6 3

14 11 8 4

16 7 15⎥ ⎢6 , 12⎦ ⎣2 10 1

⎤ ⎡

12 9 4 3

14 13 8 5

16 15⎥ . 11⎦ 10

7 ⎢5 ⎣2 1

⎤

⎤

There are four on the ‘‘detectedFigure5re.txt’’ list:

⎡

10 ⎢5 ⎣3 1

⎡

8 ⎢5 ⎣2 1

11 8 4 2 11 10 4 3

15 13 9 6 14 13 7 6

⎤ ⎡

16 10 14⎥ ⎢ 4 , 12⎦ ⎣ 3 7 1

⎤ ⎡

16 8 15⎥ ⎢4 , 12⎦ ⎣2 9 1

12 8 6 2 12 10 6 3

15 11 9 5 14 11 7 5

⎤

16 14⎥ , 13⎦ 7

⎤

16 15⎥ . 13⎦ 9

The total number detected by Figure 5 is eight.

6. On 4 by 5 products 6.1. Conclusion The following figures detect all critical-to-inspect 5 × 4 arrays and constitute B(4, 5).

6

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

⎡

8 ⎢5 ⎢ ⎢4 ⎣3 1

9 7 5 4 2

15 12 11 10 6

⎤ ⎡

17 8 16⎥ ⎢5 ⎥ ⎢ 14⎥ , ⎢4 13⎦ ⎣3 11 1

10 6 5 4 2

⎤

14 12 11 9 7

17 16⎥ ⎥ 15⎥ . 13⎦ 11

Figure 2A detects all. ‘‘D3D6D10D13criticaltoinspect.txt’’ holds two arrays:

⎡

6 ⎢4 ⎢ ⎢3 ⎣2 1

6.2. The steps leading to the conclusion The steps are similar to those for Section 5.2. Worksheets ‘‘generating5by4.mw’’, ‘‘findcriticaltoinpectlist5by4.mw’’ and ‘‘5by4FigureDetection.mw’’ are written to handle the calculations and detections. The record keeping for detections by the figures is different in that we no longer separately record the arrays detected by each figure into two files (vs. the record keeping note in Section 5.2).

There are (1662804) linear canonical arrays (see Appendix C concerning memory handling). Out of that, there are (28) criticalto-inspect arrays, as posted in Appendix D. 6.4. Nonlinear data A small sample is chosen for display. We intend to include all figures and use arrays with different levels of non-linearity. ‘‘D3D6D15criticaltoinspect.txt’’ holds ten arrays. Figure 1A detects six:

⎤

⎡

8 ⎢5 ⎢ ⎢3 ⎣2 1

15 12 9 6 3

16 14 10 7 4

17 8 15⎥ ⎢4 ⎥ ⎢ 13⎥ , ⎢3 11⎦ ⎣2 1 6

⎤ ⎡

15 10 7 6 3

16 13 12 9 5

8 17 15⎥ ⎢4 ⎥ ⎢ 14⎥ , ⎢3 11⎦ ⎣2 1 6

⎤ ⎡

15 9 7 6 3

16 13 12 10 5

17 15⎥ ⎥ 14⎥ , 11⎦ 6

⎡

15 9 8 6 3

16 14 11 10 5

7 17 15⎥ ⎢4 ⎥ ⎢ 13⎥ , ⎢3 12⎦ ⎣2 1 6

⎤ ⎡

15 10 8 6 3

16 14 11 9 5

9 17 15⎥ ⎢5 ⎥ ⎢ 13⎥ , ⎢3 12⎦ ⎣2 6 1

⎤ ⎡

15 12 8 6 3

16 13 11 7 4

17 15⎥ ⎥ 14⎥ . 10⎦ 6

Figure 1B detects four:

⎡

⎤ ⎡

⎤

12 ⎢6 ⎢ ⎢4 ⎣3 1

13 8 6 5 2

15 11 10 7 3

17 11 16⎥ ⎢ 6 ⎥ ⎢ 15⎥ , ⎢ 5 14⎦ ⎣ 3 9 1

13 8 6 4 2

15 12 9 7 3

17 16⎥ ⎥ 15⎥ , 14⎦ 10

⎡

13 7 6 5 2

15 11 10 8 3

17 11 16⎥ ⎢ 6 ⎥ ⎢ 15⎥ , ⎢ 5 14⎦ ⎣ 3 9 1

⎤ ⎡

13 7 6 4 2

15 12 9 8 3

17 16⎥ ⎥ 15⎥ . 14⎦ 10

12 ⎢6 ⎢ ⎢4 ⎣3 1

6

⎢5 ⎢ ⎢4 ⎣2 1

9 8 5 4 3

15 13 11 10 7

⎤ ⎡

16 6 14⎥ ⎢4 ⎥ ⎢ 13⎥ , ⎢3 11⎦ ⎣2 9 1

10 7 6 5 3

15 13 11 10 8

⎤

16 14⎥ ⎥ 13⎥ . 12⎦ 9

Figure 2B detects both. ‘‘D4D6D8D9criticaltoinspect.txt’’ holds two arrays:

⎡

9 ⎢7 ⎢ ⎢4 ⎣2 1

13 11 8 5 3

15 12 9 6 4

⎤ ⎡

16 9 14⎥ ⎢7 ⎥ ⎢ 10⎥ , ⎢4 8 ⎦ ⎣2 6 1

13 10 8 5 3

14 12 9 6 4

⎤

16 15⎥ ⎥ 11⎥ . 8⎦ 6

⎤ ⎡

17 6 16⎥ ⎢5 ⎥ ⎢ 14⎥ , ⎢4 12⎦ ⎣2 11 1

10 7 5 4 3

14 13 11 9 8

⎤

⎡

10 ⎢9 ⎢ ⎢5 ⎣2 1

11 10 7 3 2

13 12 9 5 4

⎤ ⎡

10 15 14⎥ ⎢ 9 ⎥ ⎢ 11⎥ , ⎢ 5 8⎦ ⎣2 1 6

11 10 6 3 2

13 12 9 5 4

⎤

15 14⎥ ⎥ 11⎥ . 8⎦ 7

Figure 3B detects both. Remark 7. The number of non-linear canonical arrays we need to generate and work with is quite large. To reduce the workload we exploit a role played by the reverse order (also known as the dual order). An order is additive if and only if its reverse order is additive. In that sense additivity is dual-invariant. The family of figures we obtained is also dual-invariant. For example, for 5 × 4 arrays, those carrying a double 2, a double 5, a double 9, a double 10 and a double 11 (in a file with the key D2D5D9D10D11 in its name) have duals carrying a double 5, a double 6, a double 7, a double 11 and a double 14. After handling the D2D5D9D10D11 files, we can skip the D5D6D7D11D14 files because the two classes are in dual. The fact that D2D5D9D10D11 has two critical arrays detected by Figure 3B implies that D5D6D7D11D14 has two critical arrays detected by Figure 3A. In that sense Figure 3B is the dual of Figure 3A. This small observation cuts the workload nearly by half. As a check we process D5D6D7D11D14 and obtain two critical arrays:

⎡

10 ⎢8 ⎢ ⎢5 ⎣2 1

12 11 7 4 3

14 13 9 6 5

⎤ ⎡

15 9 14⎥ ⎢8 ⎥ ⎢ 11⎥ , ⎢5 7 ⎦ ⎣2 6 1

12 11 7 4 3

14 13 10 6 5

⎤

15 14⎥ ⎥ 11⎥ . 7⎦ 6

⎤

‘‘D4D5D11criticaltoinspect.txt’’ holds four arrays:

⎡

15 13 12 10 7

Figure 3A detects both. ‘‘D2D5D9D10D11criticaltoinspect.txt’’ holds two arrays:

6.3. Linear data

7 ⎢4 ⎢ ⎢3 ⎣2 1

10 8 6 5 3

They are indeed the duals of the former reported pair, and are detected by Figure 3A. Worksheet ‘‘findDual5by4.mw’’ is written to find the dual of an array.

7. How ‘simple’ are the sequences given in Sections 5.1 and 6.1?

⎤

17 16⎥ ⎥ 15⎥ , 12⎦ 11

An aspect in the longstanding open problem pointed out in Fishburn (1997) is the desire of ‘the simplest’ subset of cancellation conditions. Those given in Sections 5.1 and 6.1 are in fact the simplest possible in a certain sense.

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

Definition 8. Let k ak (xk , yk ) be a cancellation violating sequence which detects ≾ (or an array that represents ≾). A particular pair {xk , yk } is said to be basic for (the detection of) ≾ if every cancellation violating sequence which detects ≾ involves {xk , yk }. ∑ k k k ak (x , y ) (or the figure that carries it) is said to be basic for ≾ if k every {x , yk } is basic for ≾.

∑

Proposition 9. Let A be a critical-to-inspect 4 × 4, or 5 × 4, linear canonical array. Every figure on the list B(4, 4), or B(4, 5), which detects A is basic for A. Proof. It is not difficult to observe that all the sequences in the listed figures are irreducible and in unitary balance (using Lemma 3.2 of Fishburn, 2001). For the case of a 4 by 4 A, say

⎡

9 ⎢5 A=⎣ 3 1

13 7 6 2

15 11 10 4

⎤

16 14⎥ . 12⎦ 8

It is detected by Figure 5. Consider the induced array

⎡

9 4 ⎢ A˜ := ⎣ 3 1

12 7 6 2

15 11 9 4

⎤

16 14⎥ 12⎦ 7

which is obtained by ‘equalizing’ the values at the points being compared under Figure 5. We can confirm that (a) A˜ is additive ˜ and (b) the order ≺ defined by A, and the order ≺0 defined by A, are tied by the relation described in Fishburn’s Theorem 3.1 (see Appendix E). The confirmation of (a) is easy using the worksheet, and (b) is primarily based on the observation that the values of A at the pairs being compared are ‘adjacently-valued’ (e.g. 4 and 5; 7 and 8; 9 and 10; 12 and 13). So, by Fishburn’s theorem we arrive at the statement: every violation of cancellation for ≺, or equivalently, for A, must involve the four pairs presented by Figure 5. In other words, the sequence in Figure 5 which detects A is basic for A. Similarly, without exception, all (216) critical-to-inspect 4 by 4 linear canonical arrays pass the above scrutiny. For the case of a 5 by 4 critical-to-inspect linear canonical array, say

⎡

10 ⎢5 ⎢ A := ⎢ 3 ⎣2 1

17 12 9 8 4

19 15 14 11 6

⎤

20 18⎥ ⎥ 16⎥ . 13⎦ 7

It is detected by Figure 1A. Consider the induced

⎡

9 ⎢5 ⎢ A˜ := ⎢3 ⎣2 1

17 12 9 7 3

19 15 13 11 6

⎤

20 17⎥ ⎥ 15⎥ . 13⎦ 7

We are able to confirm that (a) A˜ is additive and (b) the order ≺ ˜ satisfy the relation defined by A, and the order ≺0 defined by A, described in Fishburn’s Theorem 3.1. It follows from the theorem that the sequence in Figure 1A which detects A is basic for A. Without exception, all (28) critical-to-inspect linear 5 by 4 canonical arrays pass the scrutiny. □ Proposition 9 implies in particular that for each critical-toinspect 4 × 4, or 5 × 4, linear canonical array A, the figure from the list which detects it is unique. It also implies that the cancellation conditions represented by the figures constitute an independent family. F (4, 5) reported is the simplest possible as each Figure is basic for at least one array it detects.

7

Corollary 10. f (4, 5) = 6. Proof. The maximum widths of the sequences in F (4, 5) is six. Thus f (4, 5) ≤ 6. At least one width six sequence in B(4, 5) is basic for at least one array. So f (4, 5) ≥ 6. □ Fishburn’s Theorem 3.1 is stated for linear arrays. How about the detection of non-linear arrays? We shall handle that question with the following theorem which is applicable to both linear and non-linear arrays. Theorem 11. Let ≾ and ≾⋆ be weak orders on a finite product X = X1 × X2 . Suppose that ≾ is non-additive, ≾⋆ is additive, and the two weak orders differ in just one distinct pair x0 and y0 in the sense that x ≾ y if and only if x≾⋆ y for all distinct x and y, {x, y} ̸ = {x0 , y0 }. Then {x0 , y0 } is basic in the detection of ≾.

∑K

k k Proof. Let k=1 ak (x , y ) be any cancellation violating sequence which detects ≾. If ∑ it were true, that {x0 , y0 } ̸ = {xk , yk } for all K k k k = 1, . . . , K , then k=1 ak (x , y ) is also a cancellation violating ⋆ sequence which detects ≾ because there is no distinction between xk ≾ yk and xk ≾⋆ yk , and there is no distinction between xk ≺ yk and xk ≺⋆ yk , for all k = 1, . . . , K . This contradicts the additivity of ≾⋆ . □

Example 12. Consider the 4 × 4

⎡

4 ⎢3 A := ⎣ 2 1

5 4 3 2

⎤

8 7 6 5

11 10⎥ 9⎦ 7

which is detected by the sequence SA : ((1, 2), (2, 1)), ((2, 3), (1, 4)), ((3, 4), (4, 2)), ((4, 1), (3, 3)) of Figure 1A in Section 5.1. We shall show that the sequence is basic for A. By checking out that the four arrays

⎡

4 ⎢3 A1 := ⎣ 2.5 1

⎡

4 ⎢3 A3 := ⎣ 2 1

5 4 3 2 5 4 3 2

⎡

⎤

4 11 10⎥ ⎢3 , A2 := ⎣ 2 9⎦ 1 7

8 7 6 5

⎡

⎤

4 11 10⎥ ⎢3 , A4 := ⎣ 2 8⎦ 1 7

8 7 6 5

5 4.5 3 2 5 4 3 2

8 7 6 5

8 7 6 5

⎤

11 10⎥ , 9⎦ 7

⎤

11 10 ⎥ 9 ⎦ 7.5

are additive, and that their induced orders differ to that of A only at the corresponding pairs of SA , we conclude that the sequence is basic for A according to Theorem 11. Next we consider the non-linear 5 × 4

⎡

8 ⎢5 ⎢ B := ⎢3 ⎣2 1

15 12 9 6 3

16 14 10 7 4

⎤

17 15⎥ ⎥ 13⎥ . 11⎦ 6

It is the first member listed in the ‘‘D3D6D15criticaltoinspect.txt’’ file, and is detected by the sequence SB := ((2, 1), (1, 3)), ((1, 5), (2, 3)), ((4, 4), (2, 5)), ((4, 3), (3, 4)), ((3, 3), (4, 2)), ((2, 2), (4, 1)) of Figure 1A in Section 6.1. We shall show that the sequence is basic for B. Corresponding to the above six terms of SB , we consider the following six arrays:

⎡

8 ⎢5 ⎢ B1 := ⎢3 ⎣2 1

15 12 9 6 3.5

16 14 10 7 4

⎤

⎡

17 9 ⎢5 15⎥ ⎥ ⎢ 13⎥ , B2 := ⎢3 ⎦ ⎣2 11 6 1

15 12 8 6 3

16 14 10 7 4

⎤

17 15⎥ ⎥ 13⎥ , 11⎦ 6

8

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

⎡

15 12 9 6 3

16 14 10 7 4

17 8 ⎢5 15.5⎥ ⎥ ⎢ 13 ⎥ , B4 := ⎢3 ⎦ ⎣2 11 6 1

⎡

15 12 9 6 3

16 14 11 7 4

17 8 ⎢5 15⎥ ⎥ ⎢ 13⎥ , B6 := ⎢3 ⎦ ⎣2 10 6 1

8 ⎢5 ⎢ B3 := ⎢3 ⎣2 1 8 ⎢5 ⎢ B5 := ⎢3 ⎣2 1

⎤

⎡

⎤

⎡

15 12 9 6 3

16 13 10 7 4

15 12 9 6.5 3

16 14 10 7 4

⎤

17 15⎥ ⎥ 14⎥ , 11⎦ 6

⎤

17 15⎥ ⎥ 13⎥ . 11⎦ 6

Let ≾ denote the weak order defined by B, and ≾⋆i denote the weak ordered defined by Bi (i = 1, 2, . . . , 6). It is straight forward to observe that, for each i, ≾ and ≾⋆i differ only at one distinct pair, and is the pair associated with the ith term of SB . We can also confirm that all six Bi are additive (using our worksheets). Hence, by Theorem 11, every pair in SB is basic for B, i.e. the sequence is basic for B. Each non-linear example shown in Sections 5.4 and 6.4 is so checked and we are lead to the same conclusion: the sequence in the figure that detects it is basic. Remark 13. In contrast, it is reported in Li and Ng (2016) that 12

⎡

⎢11 ⎢ ⎢9 C := ⎢ ⎢4 ⎢ ⎣2 1

17

18

⎤

15

16⎥

13

14⎥ ⎥

7 5 3

⎥

10⎥ ⎥ 8⎦ 6

is a 6 by 3 critical-to-inspect linear array which is detected by two distinct irreducible unitary balance sequences having the same width:

can be confirmed. For i = 1, say, the weak order ≾ given by C and the weak order ≾⋆1 by C1 differ only in the pair {(2, 1), (1, 3)}. Theorem 11 is applicable, implying that {(2, 1), (1, 3)} is basic for C . Similarly, for i = 2, 3, 4, we get that {(3, 1), (2, 2)}, {(2, 4), (1, 6)}, {(2, 6), (3, 5)}, respectively, are basic for C . Following the actual ordering by C , this translates into the more exact statement that the four ordered pairs ((2, 1), (1, 3)), ((2, 2), (3, 1)), ((1, 6), (2, 4)), ((3, 5), (2, 6)) must be included in any cancellation violating sequence for C . The fact that both Figure 6 and Figure 8 detect C is consistent with that statement, and their presence implies also that the four pairs are the only basic ones for C . 8. A connection to conjoint commutativity Conjoint commutativity has its roots in Falmagne (1976) and is brought up in Luce and Steingrimsson (2011). The definition below is inspired by Falmagne’s original paper, and is an adapted version. Definition 14. Let M be the function defined by Mxy (a) = b if (a, x) ∼ (b, y). Conjoint commutativity refers to the equality Mz w (Mxy (a)) = Mxy (Mz w (a)), holding for a ∈ X1 , x, y, z , w ∈ X2 , whenever the terms are defined. Falmagne’s paper is in a random structure (A, X , U ) setting. Here, A and X are non-degenerate real intervals, U = {Uxy (a) | x, y ∈ X , a ∈ A} is a family of random variables with uniquely defined medians. The system is a random structure for additive conjoint measurement if, by definition, for all x, y ∈ X and a ∈ A, the medians M, Mxy (a) = median(Uxy (a)), has the following properties: (i) is continuous in all three variables; (ii) is strictly increasing in a, x and strictly decreasing in y; (iii) maps A into A; (iv) satisfies the cancellation rule Mxz (a) = Mxy (Myz (a)); (v) satisfies the commutativity rule Mz w (Mxy (a)) = Mxy (Mz w (a)). His Figure 2, which is used to illustrate the commutativity rule, resembles the following

where b := Mxy (a), d := Mz w (a), and c := Mz w (b) = Mxy (d). The four lines mark the indifferences (‘isoloudness curves’) (a, x) ∼ (b, y), The additivity of the following four arrays 12

17

18

⎢11 ⎢ ⎢9 C1 := ⎢ ⎢3 ⎢ ⎣2

15

16⎥

⎡

⎡

(d, w ) ∼ (a, z),

12

17

18

15

16⎥

5

⎢11 ⎥ ⎢ ⎢ 14⎥ ⎥ , C2 := ⎢ 9 ⎢4 10⎥ ⎥ ⎢ ⎣2 8⎦

1

4

6

13

17

18

⎢11 ⎢ ⎢9 C3 := ⎢ ⎢4 ⎢ ⎣2 1

13 7

15

⎤

⎤

⎡

⎡

6

⎥ 14⎥ ⎥, 10⎥ ⎥ 8⎦

1

3

5

12

16

18

5

⎢11 ⎢ ⎢9 , C4 := ⎢ ⎢4 10⎥ ⎥ ⎢ ⎣2 8⎦

3

6

12 7

⎤

13 7

⎤

16⎥

15

17⎥

14⎥ ⎥

13

14⎥ ⎥

⎥

1

7 5 3

⎥

10⎥ ⎥ 8⎦ 6

(b, z) ∼ (c , w ), (c , y) ∼ (d, x). The four-term balanced sequence underlying the comparison is ((a, x), (b, y)), ((d, w ), (a, z)), ((b, z), (c , w )), ((c , y), (d, x)). The commutativity rule is the statement that when three of the four indifferences hold, the fourth indifference must also hold. Falmagne’s illustration is matching our Figure 1B of Section 5.1. Of course, the illustration is dependent upon how a, b, c , d and x, y, z , w are ordered in the coordinate spaces. Luce and Steingrimsson call the property conjoint commutativity. What other figures include conjoint commutativity as a special case? To see this we observe that in the balancing of the sequence, the permutation

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

on X1 is the 4-cycle πX1 = (abcd), and the permutation on X2 is the disjoint double 2-cycle πX2 = (xy)(z w ). The structure of the permutations are independent of the ordering on the coordinate spaces. We can judge if a figure includes conjoint commutativity by examining the nature of the permutations –a 4-cycle in the first coordinate space and a disjoint double 2-cycle in the second coordinate space. Figures so identified are 1B,1D, 2B, 6C and 6D. Definition 14 is asymmetric in X1 and X2 . Let us qualify the property as conjoint commutativity in the second component X2 . A ⋄ ⋄ ⋄ ⋄ parallel version is Mcd (Mab (x)) = Mab (Mcd (x)), where M ⋄ is defined ⋄ by Mab (x) = y if (a, x) ∼ (b, y); and we shall call this property conjoint commutativity in the first component X1 . Arrays satisfying conjoint commutativity in the second component are not detected by Figures 1B, 1D, 2B, 6C and 6D. Arrays satisfying conjoint commutativity in the first component are not detected by Figures 1A, 1C, 2A, 6A and 6B. As conjoint commutativity involves the checking of four pairs of indifferences, only non-linear weak orders can violate that. In particular, every (216) critical-to-inspect linear 4 by 4 canonical array satisfies conjoint commutativity in both components. An example of a critical-toinspect array which violates conjoint commutativity in the second component is

⎡

8 ⎢3 ⎣2 1

11 6 5 2

12 8 7 4

9

Appendix A. Cancellation violation sequences for 3 by 3, 3 by 4 and 3 by 5 products

Figure 1 detects all non-additive 3 by 3 arrays and constitutes B(3, 3). Figure 2 and Figure 3 detect all non-additive 4 by 3 arrays and constitute B(3, 4). The three figures also detect all non-additive 5 by 3 arrays because B(3, 5) is empty. Appendix B

⎤

13 10⎥ . 9⎦ 7

It is detected by Figure 1B, recorded in ‘‘D2D7D8criticaltoinspect.txt’’. Appendix C. Computer memory issue handling 9. Axiom testing and future works The triple cancellation (a, x) ≿ (b, w ) and (b, y) ≿ (c , x) and (c , z) ≿ (d, y) H⇒ (a, z) ≿ (d, w ) is an axiom tested in Karabatsos (2005) (table 3, page 58). It is tied to the figure below. This figure does not get entered in Section 5.1 because it is less fundamental than Figure 1. To see that, draw an auxiliary line linking the points (1, 3) and (3, 1), and consider the two auxiliary figures on its right. A violation of the triple cancellation is necessarily a violation of (at least) one of the two double cancellations carried by Figure 1.

Figures 2, 3, and the figures of Section 5.1 offer a rich family of triple cancellation conditions which are testable deterministic axioms worthy of testing. In particular Domingue (2014) mentions the testing of conjoint commutativity as a potential avenue to extend his work. It is also natural to conduct the data testing, as laid by Birnbaum (1968) and Luce and Steingrimsson (2011), to cancellation conditions of higher orders emerging from the figures of Section 6.1, and from those in Li and Ng (2016).

Because of the larger number of 5 by 4 linear canonical arrays, the author cannot complete that in a single run using ‘‘generating5by4.mw’’ on his small desktop computer. The execution is interrupted half way, with the (4164) partially filled output arrays saved to file ‘‘First10list.txt’’. That file is divided into smaller segments, each one small enough for ‘‘generating5by4.mw’’ to finish without a crash (detachment from kernel). Because of that need, the final (1662804) linear canonical arrays are held in batches ranging from ‘‘1to10ofFirst10Linear.txt’’ to ‘‘4101to4164ofFirst10 Linear.txt’’. Particular care should be taken in the generation of non-linear data when there is need to interrupt the execution of a do loop. If we interrupt the do loop ‘for i from 1 to 20’ and change it to ‘for i from 1 to 10’ in round one, we do not always continue with ‘for i from 11 to 20’ in round two. For example, to generate D10 files, the parameter list is [1,2,3,4,5,6,7,8,9,10,10,11,12,13,14,15,16, 17,18,19]. In that case, ‘for i from 1 to 10’ in round one should be continued with ‘for i from 12 to 20’ in round two. This is because the subroutine is coded to execute the parameter list by contiguous segments and so ‘ for i from 1 to 10’ has the same effect as ‘ for i from 1 to 11’. The ‘‘findcriticaltoinpectlist5by4.mw’’ worksheet is then applied to each batch, extracting the relatively few critical-to-inspect arrays, which are saved to files from ‘‘1to10ofFirst10Linearcriticalto inspect.txt’’ to ‘‘4101to4164ofFirst10Linearcriticaltoinspect.txt’’. The union of these are saved to ‘‘Linearcrticaltoinspect.txt’’ which holds the (28) critical-to-inspect arrays.

10

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

Appendix D. List of critical-to-inspect linear 5 by 4 arrays

⎡

10

⎤ ⎡

1

17 12 9 8 4

19 15 14 11 6

20 10 18⎥ ⎢ 5 ⎥ ⎢ 16⎥ , ⎢ 3 13⎦ ⎣ 2 7 1

⎡

10 ⎢6 ⎢ ⎢5 ⎣3 1

11 9 7 4 2

18 15 13 12 8

20 19⎥ ⎥ 17⎥ , 16⎦ 14

⎡

12 8 6 5 2

17 15 14 11 9

20 12 19⎥ ⎢ 9 ⎥ ⎢ 18⎥ , ⎢ 5 16⎦ ⎣ 2 13 1

⎢5 ⎢ ⎢3 ⎣2

10 ⎢7 ⎢ ⎢4 ⎣3 1

⎡

8

⎢6 ⎢ ⎢5 ⎣2 1

11 10 7 4 3

18 16 13 12 9

⎤ ⎡

⎤ ⎡

18 14 11 9 3

10 20 19⎥ ⎢ 6 ⎥ ⎢ 17⎥ , ⎢ 4 16⎦ ⎣ 2 1 12

⎡

17 14 8 5 2

18 15 11 6 4

20 19⎥ ⎥ 16⎥ , 12⎦ 9

⎤ ⎡

18 12 10 7 3

19 16 13 11 6

9 20 17⎥ ⎢5 ⎥ ⎢ 15⎥ , ⎢4 14⎦ ⎣2 1 8

⎡

12 10 7 6 4

19 16 15 13 9

20 18⎥ ⎥ 17⎥ , 14⎦ 11

8

⎢6 ⎢ ⎢3 ⎣2 1

⎡

11 10 7 5 4

18 16 15 12 9

19 16 13 7 4

20 18⎥ ⎥ 14⎥ , 11⎦ 8

18 14 11 7 3

19 16 12 9 5

20 17⎥ ⎥ 15⎥ , 13⎦ 8

⎤

⎤

18 11 10 7 3

19 16 13 12 6

⎤ ⎡

20 14 19⎥ ⎢ 8 ⎥ ⎢ 17⎥ , ⎢ 5 14⎦ ⎣ 3 13 1

19 15 13 9 5

20 18⎥ ⎥ 16⎥ , 12⎦ 7

⎡

13 ⎢11 ⎢ ⎢5 ⎣2 1

16 14 10 4 3

18 17 12 8 6

20 8 19⎥ ⎢7 ⎥ ⎢ 15⎥ , ⎢4 9 ⎦ ⎣2 7 1

⎡

16 12 8 6 2

17 13 11 7 4

20 19⎥ ⎥ 18⎥ , 15⎦ 10

⎤ ⎡

⎤

13 ⎢9 ⎢ ⎢4 ⎣2 1

17 14 11 6 3

18 16 12 8 5

20 12 19⎥ ⎢11 ⎥ ⎢ 15⎥ , ⎢ 6 10⎦ ⎣ 2 7 1

⎡

14 ⎢12 ⎢ ⎢6 ⎣2 1

15 13 9 4 3

18 17 11 7 5

20 19⎥ ⎥ 16⎥ , 10⎦ 8

⎡

13 ⎢11 ⎢ ⎢7 ⎣3 1

16 14 8 4 2

18 17 12 6 5

20 13 19⎥ ⎢ 8 ⎥ ⎢ 15⎥ , ⎢ 6 10⎦ ⎣ 4 9 1

⎡

15 9 7 6 2

17 13 12 10 4

20 19⎥ ⎥ 18⎥ , 16⎦ 11

14 ⎢8 ⎢ ⎢5 ⎣3 1

⎡

7 ⎢5 ⎢ ⎢4 ⎣2 1

13 9 8 6 3

19 17 14 12 10

⎤ ⎡

⎤

16 15 9 4 3

19 17 13 7 5

20 18⎥ ⎥ 14⎥ , 10⎦ 8

16 12 9 5 2

18 14 10 7 3

20 19⎥ ⎥ 17⎥ , 15⎦ 11

⎤

⎤ ⎡

⎤

⎤

⎤ ⎡

7 20 18⎥ ⎢6 ⎥ ⎢ 16⎥ , ⎢4 15⎦ ⎣2 1 11

12 9 8 5 3

18 17 14 11 10

⎤

20 19⎥ ⎥ 16⎥ , 15⎦ 13

⎤

⎡

13 ⎢7 ⎢ ⎢6 ⎣4 1

15 9 8 5 2

18 14 11 10 3

20 19⎥ ⎥ 17⎥ , 16⎦ 12

⎡

16 13 9 5 3

18 15 10 7 4

20 19⎥ ⎥ 17⎥ . 12⎦ 8

14 ⎢11 ⎢ ⎢6 ⎣2 1

⎤

Appendix E. A theorem of Fishburn

15 10 7 6 2

17 13 12 9 4

⎤

20 19⎥ ⎥ 18⎥ , 16⎦ 11

⎤

17 14 10 8 4

⎡

⎤

20 17⎥ ⎥ 15⎥ , 14⎦ 8

⎤

11 ⎢6 ⎢ ⎢3 ⎣2 1

14 ⎢9 ⎢ ⎢5 ⎣3 1

17 15 10 6 3

⎤

9 ⎢5 ⎢ ⎢4 ⎣2 1

⎡

20 18⎥ ⎥ 16⎥ , 13⎦ 7

20 19⎥ ⎥ 17⎥ , 15⎦ 14

15 10 8 5 2

8 ⎢5 ⎢ ⎢3 ⎣2 1

19 15 14 12 6

⎤

13 ⎢7 ⎢ ⎢6 ⎣4 1

⎡

17 11 9 8 4

⎤

⎡

13 ⎢10 ⎢ ⎢7 ⎣3 1

⎤

Definition 15. We call a sequence of distinct pairs (x1 , y1 ), (x2 , y2 ), . . . , (xK , yK ) in X = X1 × X2 a unitary balance sequence if

∑

(xk , yk ) ∈ CK ,

k

and that every xk and yk pair is distinct. A unitary balance sequence (x1 , y1 ), (x2 , y2 ), . . . , (xK , yK ) is irreducible if no member of CJ for 2 ≤ J < K has J ordered pairs in {(x1 , y1 ), (x2 , y2 ), . . . , (xK , yK )} with positive ak for those J pairs (cf. Definition 1). The following is a redacted version of Theorem 3.1 in Fishburn (2001).

12 9 6 5 3

17 16 14 11 10

⎤

20 19⎥ ⎥ 18⎥ , 15⎦ 13

Theorem 16. Let X = X1 × X2 with |X1 | = m and |X2 | = n. Suppose X has an irreducible unitary balance sequence (x1 , y1 ), . . . , (xK , yK ) for which there are real-valued functions u1 on X1 and u2 on X2 such that, for all distinct x = (x1 , x2 ) and y = (y1 , y2 ) in X , u1 (x1 ) + u2 (x2 )

= u1 (y1 ) + u2 (y2 )⟨=⟩{x, y} ∈ {{x1 , y1 }, . . . , {xK , yK }}. Then f (m, n) ≥ K . More specifically, define ≺0 on X by (1), x≺0 y iff

u1 (x1 ) + u2 (x2 ) ≤ u1 (y1 ) + u2 (y2 ).

C.T. Ng / Journal of Mathematical Psychology 82 (2018) 1–11

Form a linear order ≺ on X by taking x ≺ y if x≺0 y whenever {x, y} ̸∈ {{x1 , y1 }, . . . , {xK , yK }} plus xk ≺ yk for k = 1, . . . , K . Then ≺ violates C (K ) and satisfies C (J) for 2 ≤ J < K . In fact, every violation of cancellation for ≺ must involve x1 ≺ y1 , . . . , xK ≺ yK . References Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord, & M. R. Novick (Eds.), Statistical theories of mental test scores. Reading: Addison-Wesley. Domingue, B. (2014). Evaluating the equal-interval hypothesis with test score scales. Psychometrika, 79, 1–19. Falmagne, J. C. (1976). Random conjoint measurement and loudness summation. Psychological Review, 83, 65–79. Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353–365. Fishburn, P. C. (2001). Cancellation conditions for finite two-dimensional additive measurement. Journal of Mathematical Psychology, 45, 2–26. Karabatsos, G. (2005). The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement. Journal of Mathematical Psychology, 49, 51–69.

11

Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of measurement (Vol. 1). New York: Academic. Li, L., & Ng, C. T. (2016). A minimal set of cancellation violating sequences for finite two-dimensional non-additive measurement. Publicationes Mathematicae Debrecen, 89, 389–398. Luce, R. D., & Steingrimsson, R. (2011). Theory and tests of the conjoint commutativity axiom for additive conjoint measurement. Journal of Mathematical Psychology, 55, 379–385. Ng, C. T. (2016a). On fishburn’s questions about finite two-dimensional additive measurement. Journal of Mathematical Psychology, 75, 118–126. Ng, C. T. (2016b). Replication data for: On Fishburn’s questions about finite twodimensional additive measurement, http://hdl.handle.net/10864/11419. Ng, C. T. (2017). Replication data for: On Fishburn’s questions about finite twodimensional additive measurement, II, http://dx.doi.org/10.5683/SP/FBMNWQ. Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1, 233–247. Slinko, A. (2009). Additive representability of finite measurement structures. In S. J. Brams, et al. (Eds.), The mathematics of preference, choice and order: essays in honour of Peter C. Fishburn, Studies in choice and welfare. ISBN: 978-3-540-791270, (pp. 113–133). Springer-Verlag Berlin Heidelberg.