## Abstract

To analyse paired comparison outcomes, such as with non-standard probabilities, a basis is created for the space of all binary interactions, whether from probabilities, correlations, etc. In this manner, the source of all transitive and non-transitive behaviours (e.g. path dependencies) can be identified.

## 1. Introduction

In an increasing number of disciplines, ranging from the physical (e.g. quantum mechanics) to the social (e.g. psychology, economics) sciences, natural issues arise where standard Kolmogorov probability arguments do not suffice. A unifying structure for these approaches is created here with an easily used basis for paired comparisons (whether for probabilities, correlations or other effects), which separates well-behaved terms satisfying transitivity from those causing path dependencies, cycles and other mysteries. Information about the paired comparisons (e.g. probabilities) may be known, but what causes the behaviour may not. This creates an inverse problem of understanding what kind of modelling might support these outcomes, whether it be the structure of phase space for physical systems, social norms for groups or brain processing for individuals. The basis helps to identify what may be needed.

This basis is illustrated with a surprising ‘order effects’ result [1] that was motived,^{1} in part, by a 1997 Gallup poll about the perceived honesty of President Clinton and Vice President Gore. The order in which the question was posed, about the honesty of first either Clinton or Gore and then about the other, affected the conclusion. Wang *et al.* used a quantum probability model to derive QQ (quantum question) equalities that connect differences. This QQ relationship is empirically supported with three examples.

These QQ equalities (§3e) are valuable contributions for understanding order effects. But verifying the title's claim that their results ‘… reveal the quantum nature of human judgements’ requires showing that these equalities always hold and that they constitute the only explanation when they do. The first, an empirical issue, is beyond the scope of this paper. As for the second, it follows from the basis that:

(i) A measure is needed to determine when the QQ relationships hold. But rather than a ‘quantum nature’ explanation, it follows from the basis that the QQ relationships are equivalent to a simple transitivity condition that just determines whether the relevant variables admit cyclic effects.

(ii) This basis identifies properties and unspecified variables involved in the QQ expression.

Notions needed to describe the space of binary interactions are introduced (§2) by describing pairwise voting difficulties. Resolving these problems requires identifying what causes them, which is the theme of §2b. This material is then generalized (§3) to handle widespread paired comparison difficulties.

## 2. Pairwise voting

The voting concerns are motivated by my fictional boast [2]:
Before your next election, tell me who you want to win. For a price, I will visit your group, talk with your colleagues, and design a fair (e.g. all candidates are considered) election method. Your candidate will win.

To illustrate, suppose a 15 member department is to select one out of five candidates for a tenure track position. Their preferences (where ‘≻’ means ‘strictly preferred to’) are
2.1
These voters *unanimously* prefer *C*≻*D*≻*E*, so *E* *cannot* be their top choice. Yet, *E* can be ‘convincingly’ elected with a method endorsed by *Robert's Rules of Order*—an agenda. This approach shares features of a tournament where, with a specified ordering of the alternatives, the winner of the first two is compared with the next listed candidate, that winner with the next one, etc. The method's outcome is the winner of the last pair. Thus, the 〈*D*,*C*,*B*,*A*,*E*〉 agenda advances the {*C*,*D*} majority vote winner to a vote with *B*, that winner with *A*, and that winner with *E* to obtain the final outcome.

With equation (2.1) and this agenda, *C* unanimously beats *D* to be compared with *B*, *B* beats *C* with a two-thirds vote (the 10 voters on the top line) to be compared with *A*, *A* beats *B* with a two-thirds vote (the 10 voters in the first column) to be compared with *E*. *E* is the overall winner by convincingly beating *A* with a two-thirds vote (the 10 voters in the second row and column).^{2}

All votes are unanimous or of landslide (two-thirds) proportions, which (incorrectly) identifies *E* as the voters’ overwhelming choice. Translating this example into probabilities highlights a realistic concern (which is extended to general settings with the §3 constructs) that *a collection of pairwise probabilities can seriously misrepresent the actual structure of an underlying source space.* An accompanying issue is to explain why this is so; a second is to discover how to combine paired outcomes to combat such difficulties (§2b).

Natural objectives suggested by this example include finding all possible paired comparison outcomes, and identifying all structures in the source space—the space of voter preferences—that cause such conclusions; this includes finding what causes cycles and all path dependency concerns. These goals have been answered for any number of alternatives, and even for triplets, etc. [3–4]. While any collection of paired rankings can emerge from voting [5], it is of more interest to determine all possible tallies. Constraints on all possible three-candidate, pairwise tallies (from complete transitive preferences) are identified by Saari [2,6]. As indicated next (e.g. theorem 2.1; generalized in theorem 3.5), answers for *n*≥3 alternatives involve identifying the profile structures with extreme outcomes.

### (a) Structure of the source space

The source space uses standard social choice assumptions: each voter has a complete (each pair can be compared) transitive ranking of all *n*≥3 alternatives. Assign each of the *n*! strict (i.e. no ties) rankings to a particular axis. A profile (a list of how many voters prefer each ranking) becomes a point in .

To emphasize differences in tallies, let *P*(*X*,*Y*)=*n*(*X*)−*n*(*Y*), where *n*(*X*),*n*(*Y*) are, respectively, *X*'s and *Y* 's vote in an {*X*,*Y* } paired comparison. So if *X* receives 60 votes and *Y* has 40, then *P*(*X*,*Y*)=20=−*P*(*Y*,*X*). The way in which new results are obtained is by orthogonally dividing into three subspaces. The first is the *kernel*, consisting of all profiles where, for each *X* and *Y* , *P*(*X*,*Y*)=0. The second is the *strongly transitive* space, , where, far beyond transitivity, any set of *k*≥2 alternatives {*X*_{1},*X*_{2},…,*X*_{k}} satisfies the demanding
2.2
A basis for the (*n*−1)-dimensional is given in [3].

The final subspace is the orthogonal complement of the kernel and strongly transitive spaces. The importance of this *Cyclic space*, , is that it consists of all possible profiles and profile components causing cycles, path dependencies and all other complexities that can occur with majority votes.

The central construct for a basis is what I call a *ranking wheel*. As indicated in figure 1, this rotating wheel lists ranking numbers, 1 to *n*, uniformly along the wheel's edge. Place the names of the *n* alternatives on the wall. The ranking wheel number adjacent to a name identifies its ranking position. To obtain the next ranking, rotate the wheel to place ‘1’ by the next alternative. Continue until ‘1’ has been by each name; this defines a set of *n* rankings. Illustrating with figure 1, the set is
2.3
That these terms must play a central role in developing a comprehensive theory for voting is suggested by the fact that the top row of equation (2.3) was used to create the equation (2.1) example with its cyclic behaviour.

A ranking wheel array is a *Z*_{n} orbit of a ranking. By construction, each candidate is in first, second,…,last place precisely once, so no candidate is favoured: The outcome should be a tie. But by ignoring the full symmetry structure, pairwise outcomes define a cycle. The *D*≻*E* ranking, for instance, appears in the first four equation (2.3) rankings; only in the last is *E*≻*D*, which leads to an overwhelming *D*≻*E* victory of 4:1. (So *P*(*D*,*E*)=3.) Reflecting the cyclic action of the ranking wheel, the cyclic majority vote outcome is
2.4
where each has the 4:1 landslide victory. With *n*≥3 alternatives, the tallies in this cycle are (*n*−1):1 where each tally is just a single vote away from unanimity.

Restating equation (2.4) in terms of (signed) probabilities, a representative term becomes . Instead of the equation (2.2) equality, the example has
In general, the probabilities associated with a ranking wheel defined by *X*_{1}≻⋯≻*X*_{n} significantly violate the strongly transitive constraint equation (2.2) by satisfying *p*(*X*_{i},*X*_{i+1})=((*n*−1)−1)/*n*=(*n*−2)/*n* and the equality
2.5

According to the decomposition, paired comparison outcomes fail the strongly transitive condition (equation (2.2)) if and only if the supporting profile includes ranking wheel components. Extremes follow:

### Theorem 2.1

*If k voters have complete transitive preferences over n*≥3 *alternatives, then for p*(*X*_{i}*,X*_{j})=*P(X*_{i}*,X*_{j})/*k*
2.6
*Equality holds iff all of a profile's (at least two) rankings come only from the ranking wheel defined by X*_{1}*≻⋯≻X*_{n}.

The next step is to discover what else affects pairwise rankings. According to theorem 2.2, this is it: All pairwise behaviour is captured just by the cyclic and strictly transitive terms. (Theorem 2.1 is a consequence of theorem 2.2.)

### Theorem 2.2 (Saari [2,3])

*With pairwise voting, the space of profiles* *can be orthogonally decomposed into a kernel, where P*(*X*_{i}*,X*_{j})=0 *for all i,j and all profiles, the* (*n*−1)-*dimensional* *, and the* (*n*−1)!/2-*dimensional* *space spanned by ranking wheel configurations.*

According to theorem 2.2, the last theorem 2.1 statement requires the profile to be the sum of the ranking wheel configuration (which gives the equation (2.6) equality) and strongly transitive terms. Transitive outcomes can satisfy equation (2.6) iff *all preferences* come from the specified ranking wheel. To illustrate with equation (2.3), if *α* voters (rather than just one) have the *A*≻*B*≻*C*≻*D*≻*E* preferences, then *p*(*A*,*B*)=*p*(*B*,*C*)=*p*(*C*,*D*)=*p*(*D*,*E*)=(*α*+3−1)/(*α*+4) and *p*(*A*,*E*)=(*α*−4)/(*α*+4), so the left-hand side of equation (2.6) equals (4(*α*+2)+(4−*α*))/(*α*+4)=5−2 with the transitive *A*≻*B*≻*C*≻*D*≻*E* outcome for *α*>4. As illustrated by equation (2.1) and its equation (2.6) equality, not all *Z*_{n} rankings are needed. But adding a ranking to equation (2.3) not from the equation (2.3) list forces an equation (2.6) inequality; e.g. adding a voter with *E*≻*D*≻*C*≻*B*≻*A* preferences changes the above to *p*(*A*,*B*)=*p*(*B*,*C*)=*p*(*C*,*D*)=*p*(*D*,*E*)=(*α*+3−2)/(*α*+5) and *p*(*A*,*E*)=(*α*−5)/(*α*+5), so the left side of equation (2.6) equals (4(*α*+1)+(5−*α*))/(*α*+5)<3. Differences from the equation (2.6) equality, then, are due to rankings that differ from those in the specified ranking wheel configuration (e.g. [7]).

### (b) Unavoidable problems

Surprisingly, the *sole source* of all majority and supermajority pairwise voting problems are the ranking wheel configurations (theorem 2.2). Thus, this structure can be used to simplify, subsume and extend the large social choice/voting theory literature that examines these complexities [7]. To illustrate with the seminal Arrow's Theorem [8], consider the challenge of designing a voting rule for voters with complete transitive preferences over *n*≥3 alternatives; the rule must produce a complete, transitive societal (i.e. group) ranking. A natural approach is to use the reductionist method where, to reduce the complexity, paired comparisons are emphasized; i.e. for each pair of alternatives, design a decision rule. Manifesting the gained simplicity is that the rule uses only information about how each voter ranks this particular pair. (This reductionist step corresponds to Arrow's *Independence of Irrelevant Alternatives*, or IIA.)

To eliminate useless choices, the rules cannot have a constant outcome. (Arrow uses a Pareto condition where, if everyone has the same ranking of a pair, that is the pair's societal outcome. To see how his condition satisfies mine, use two unanimity profiles with different rankings of the pair.) Finally, to capture the intent that the outcome reflects information from more than a single source, not all rules can depend just on the preferences of a single voter; there are settings where a different voter changing preferences can alter at least one rule's outcome. (This includes Arrow's ‘no dictator’ condition.)

The approach seems simple, but the objective is impossible to attain: no such method exists. This is the conclusion of Arrow's result (which played a role in his 1972 Nobel Prize) and of my above generalized formulation [6, pp. 83–100, 9]. A common branding of Arrow's Theorem is that ‘with three or more alternatives, no voting method is fair’. But, as the above description demonstrates, this interpretation is incorrect. Instead, Arrow's Theorem is a negative commentary about a *methodology*: the reductionist approach used with paired comparisons guarantees that situations exist where the outcome violates transitivity.

This difficulty is strictly caused by the ranking wheel structure. By forming the basis for the Cyclic subspace, these configurations identify natural connecting links for pairs. But paired comparisons emphasize ‘parts’, so rather than involving these critical linking structures, paired comparisons *sever* them. By doing so, they generate cycles, which now must be expected because they reflect the ranking wheel's circular construction. Indeed, a profile free from Cyclic terms (it is strongly transitive) has no links to be cut, which then allows Arrow's conditions to be satisfied by pairwise voting. Thus Arrow's Theorem and all of the problems described in [7] share the same explanation: by severing these crucial links, paired comparisons distort critical information about the true underlying structure of the source space (ranking wheel configurations); this distortion is what generates all possible pairwise difficulties. Paired comparison outcomes, then, can provide a distorted image of the source space by ignoring—actually, cutting into pieces—the actual underlying structure: as developed in §3, this can happen in general.

If ranking wheel components, which should create complete ties, cause all of the difficulties, a natural resolution is to strip a profile of these terms and use what remains: this is equivalent to projecting the profile to the strongly transitive subspace. An easier way to accomplish this objective is with the following:

### Theorem 2.3

*With alternatives* {*X*_{1}*,X*_{2},…,*X*_{n}}, *let*
2.7
*If profile* , *then B*(*X*_{j})=0 *for all j. Thus the B*(*X*_{i}) *rankings depend only on the strongly transitive portion of a profile. If* *, the majority vote rankings are complete, transitive and agree with the ranking of B*(*X*_{j}) *values.*

This *B*(*X*_{i}) approach is equivalent to the Borda Count [2], which tallies ballots by assigning *n*−*j* points to the *j*th positioned candidate. To indicate why equation (2.7) serves as a projection, notice from equation (2.3) that
2.8
which sum to zero. As this relationship holds for all candidates and any *n*, the *B*(*X*_{i}) tally drops a profile's Cyclic components and retains information only about the strongly transitive terms.

## 3. Space of paired outcomes

To address most ways to make paired comparisons [10], use to compare the {*X*_{i},*X*_{j}} pair. Similar to the *P*(*X*_{i},*X*_{j}) function, let
3.1
All of the *n*(*n*−1) *d*_{i,j} values are determined by the {*d*_{i,j}}_{i<j} terms (equation (3.1)), so vector
3.2
catalogues all needed information. Changes in the first index are indicated by semicolons.

No interpretation is assigned to *d*_{i,j}; it can represent marginal probabilities, *P*(*X*_{i},*X*_{j}) values, correlation indices, differences from an average, or whatever paired measure is desired. With projections, normalizations, linear transformations, the *d*_{i,j} could be restricted to the {−1,0,1} values, the interval [−1,1] (e.g. marginal probabilities) or (e.g. differences in vote tallies); the *d*_{i,j} values are observables. A goal is to develop appropriate structures that uncover properties of **d**^{n} vectors and assist in determining features of an associated source. Most of what is described in §2 extends.

### (a) A basis for

The basis identifies which **d**^{n} components lie in well-behaved (transitive) and ill-behaved subspaces.

#### (i) The strongly transitive space

What assumes the role of equation (2.2) is the *strongly transitive* requirement
3.3
As with equation (2.2), this expression resembles sums of signed distances where the distance from *i* to *j* plus that from *j* to *k* equals the signed distance from *i* to *k*. In this way, it represents a strong version of transitivity.

Equation (3.3) restricts **d**^{n} to an (*n*−1)-dimensional^{3} *strongly transitive plane:*
3.4

An basis follows:

### Definition 3.1

For each *i*=1,…,*n*, let be where *d*_{i,j}=1 for *j*≠*i*, *j*=1,…,*n*, and *d*_{k,j}=0 if *k*,*j*≠*i*. is called the ‘*X*_{i} basic vector’.

To illustrate with *n*=4 where **d**^{4}=(*d*_{1,2},*d*_{1,3},*d*_{1,4};*d*_{2,3},*d*_{2,4};*d*_{3,4}), the *n*−1=3 basic vectors are
3.5
To explain the negative components, definition 3.1 mandates *d*_{3,1}=*d*_{3,2}=*d*_{3,4}=1. But the **d**^{4} representation (equation (3.1)) requires using *d*_{1,3}=−*d*_{3,1}=−1 and *d*_{2,3}=−*d*_{3,2}=−1. Because , it follows that , which means that the basic vectors define a three-dimensional space.

A direct computation proves that vectors in this space satisfy equation (3.1). Illustrating with
it must be shown that *d*_{i,j}+*d*_{j,k}=(*a*_{i}−*a*_{j})+(*a*_{j}−*a*_{k}) equals *d*_{i,k}=(*a*_{i}−*a*_{k}), which is immediate.

### Theorem 3.2 [10]

*A basis for* *is given by* .

While is a basis for , applications emphasize appropriate subsets.

### (b) ; the normal space

The plane of strongly transitive entries and its normal space (denoted by for ‘cyclic’) define an coordinate system. The importance of is that (similar to in voting) its terms cause *all* paired comparison peculiarities. A basis for this *cyclic space* resembles the ranking wheel construction of figure 1.

As indicated in figure 2, list, in any order, the *n* indices along the edge of a circle. Moving clockwise about the circle, each integer is preceded by an integer and followed by a different one. In figure 1*a*, for instance, 1 precedes 6 and follows 3.

### Definition 3.3

Let *π* be a specified permutation, or listing, of the indices 1,2,…,*n* around a circle. Define as follows: If *j* immediately follows *i* in a clockwise direction, then *d*_{i,j}=1. If *j* immediately precedes *i*, then *d*_{i,j}=−1. Otherwise *d*_{i,j}=0. Vector is the ‘cyclic direction defined by *π*’.

With figure 2*a*, *d*_{1,6}=1 as 6 follows 1, while *d*_{1,3}=−1 because *d*_{3,1}=1. Figure 2*b* defines and figure 2*c* defines The vectors have obvious properties: rotating a listing defines the same vector, so there are *n*!/*n*=(*n*−1)! different 's. Reversing a listing defines . (For instance, is the negative of )

To prove that each is orthogonal to each , notice that, in , all *d*_{j,k}=1 and all other entries are zero. But has only two non-zero *d*_{j,k} values, where one equals −1 and the other equals 1, so and are orthogonal. A simple proof that the vectors span the normal space for is in [10]. Applications may emphasize specific Cyclic subsets.

### Theorem 3.4

*[*10*] The* *normal bundle, denoted by* *, is spanned by the set of all* *vectors. Thus* *is orthogonally decomposed into the* (*n*−1)-*dimensional subspace* *and the* *-dimensional* .

The importance of theorem 3.4 is that it generalizes theorem 2.2 to all settings. It asserts that if **d**^{n} is not strongly transitive, then the *only* other **d**^{n} components are cyclic terms—nothing else need be considered. For instance, as all **d**^{4}=(4,6,8;4,6;4) entries are positive, the pairs define the transitive ranking *A*≻*B*≻*C*≻*D*. But *d*_{1,2}+*d*_{2,4}=4+6≠8=*d*_{1,4}, so **d**^{4} is *not* strongly transitive. According to theorem 3.4, **d**^{4} must be the sum of strongly transitive and cyclic terms; indeed,

So, nothing other than strongly transitive and cyclic components (which have very different properties) need be considered. An illustration of the value theorem 3.4 adds is that if a given **d**^{n} exhibits unexpected features, then **d**^{n}'s cyclic component must be the cause! Thus, for instance, developing conclusions similar to Arrow's result (§2b) now are easy and immediate [9]. Also, theorem 3.4 identifies source space properties needed to generate certain behaviours; i.e. the properties must be able to create appropriately required terms.

### (c) Properties

As with *P*(*X*_{i},*X*_{j}), non-transitive outcomes require terms. Extreme settings are defined by these components. For instance, equation (2.5) defined the maximum sum of marginal differences, which come from a ranking wheel configuration. With defined by the listing *π*=(1,2,…,*n*), we have the following:

### Theorem 3.5

*Let the space of* *vectors be normalized so that d*_{i,j}∈[−1,1] *and d*_{i,j}=1 *represents the maximum dominance of X*_{i} *over X*_{j}*. The maximum deviation from satisfying the strongly transitive condition is*
3.6
*which is defined by* *. If these* *components do not come from a multiple of* *, then equation (3.6) is a strict inequality.*

With the basis and theorem 3.4, the proof is immediate: with only *n* components, each *d*_{i,j} must equal unity. These *d*_{1,2}=⋯=*d*_{n−1,n}=*d*_{n,1}=1 values uniquely define , which (theorem 3.4) is the only source of non-transitivity. Equation (3.6) holds independent of what the *d*_{i,j} terms model, whether marginal probabilities, correlations, etc. The particular choice, however, may impose constraints whereby *βn*, *β*<1, is the actual equation (3.6) upper bound. (With standard neutrality conditions, where each *d*_{i,j} can have the same value, generates this maximum value.)

Because *β* depends on *d*_{i,j} constraints, it offers useful information about the source space structure. To explain with the *k*-voter material of §2, as *d*_{i,j}=*P*(*X*_{i},*X*_{j})/*k*, *d*_{i,j}=1 means that *X*_{i} unanimously beats *X*_{j}. Thus, the equation (3.6) equality with *n*=4 occurs iff *d*_{1,2}=*d*_{2,3}=*d*_{3,4}=*d*_{4,1}=1, which requires *everyone* to have the *A*≻*B*,*B*≻*C*,*C*≻*D*,*D*≻*A* cyclic preferences. So, if cyclic preferences are not permitted, then equation (3.6) cannot hold. In turn, *β*<1 means that not all voters can have cyclic preferences. The equation (2.5) upper bound of *β*=(*n*−2)/*n*, of one less than unanimity, leads to the ranking wheel construction establishing that the source space structure allows transitive inputs (preferences) to cause cyclic outcomes. In this manner, theorem 3.5 extends theorem 2.1 to all settings, and offers insights into the structure of source space.

Assuming the *B*(*X*_{j}) role (equation (2.7)) is the *Borda assignment rule* (BAR) defined as
3.7

Similar to *B*(*X*_{j}), BAR (equation (3.7)) eliminates cyclic components. Indeed, the value of any is zero because has only two non-zero *d*_{j,k} terms where one is positive and the other negative, so they cancel in the summation. Thus, serves as a projection of **d** from to its strongly transitive space. The way in which the functions handle strongly transitive terms is specified in the following.

### Theorem 3.6 [10]

*For any n*≥3 (*where a*_{n}=0), *let* *Then* *, and*
3.8

Thus, the BAR ranking for **d**^{n} reflects the ordering of the coefficients in its strongly transitive component; cyclic terms are ignored. If *a*_{i}>*a*_{j}, then, independent of any cyclic terms, it must be that

### (d) An interesting phenomenon

It is reasonable to expect that dropping an alternative has a minimal effect on a ranking. For instance, with a *X*_{1}≻*X*_{2}≻*X*_{3}≻*X*_{4} ranking, if alternative *X*_{4} is dropped, the *X*_{1}≻*X*_{2}≻*X*_{3} outcome would be anticipated. But this need not be the case! The sole source of this effect is the cyclic components.

### Theorem 3.7 [10]

*Let* *be the projection mapping defined by dropping the kth alternative. Then*
3.9
*But* *is the sum of a multiple of* *and non-zero strongly transitive terms, where π** *is the listing obtained from π by removing index k. Thus,* *iff*

Proving equation (3.9) is a simple computation. To illustrate the cyclic term assertion, notice that Dropping *X*_{4} removes all *d*_{j,4} terms from to create **d**^{3}=(1,0;1), where
3.10

As equation (3.10) illustrates, dropping alternatives from cyclic terms creates cyclic terms *plus* strongly transitive terms. (The term in equation (3.10) defines the *X*_{1}≻*X*_{2}≻*X*_{3} ranking.) These unexpected components can cause a rule's ranking (definitely its weights) to differ. The BAR ranking for is *X*_{1}≻*X*_{2}≻*X*_{3}≻*X*_{4}, for instance, but by dropping *X*_{4} (see equation (3.10)) it becomes the reverse, *X*_{3}≻*X*_{2}≻*X*_{1}. What adds concern to this feature is that *all major decision rules involve data components*, so all of these rules are affected by this theorem 3.7 feature whereby dropping alternatives can scramble the resulting ranking.

### Theorem 3.8 [10]

*If* *, then the rankings defined by d*_{i,j} *terms and the BAR outcome for* *agree with the outcome obtained after projecting* *to any subset of n alternatives*. (*The projection is in* .) *Conversely, if* *is the projection of* *, then* .

*If a BAR ranking changes after dropping an alternative, then* *has* *components. With a sufficiently strong* *component, the rankings for n and n*+1 *alternatives can differ in any desired manner. However, the* *value for* (*n*+1) *alternatives is* 1/(*n*−1) *times the sum of the* *values over all subsets of n alternatives.*

This difficulty, strictly caused by the terms, can be expected to affect *all* major paired comparison rules. But BAR retains some level of regularity because (theorem 3.8) the averaged outcome over all ways to drop an alternative remains consistent. The dimension of exceeds that of for *n*≥5, which means that these difficulties can become commonplace with larger *n* values.

### (e) A quantum question example

To conclude, the basis is illustrated with the 1997 Gallup poll information (as reported in [1]) concerning the perceived honesty of President Clinton and Vice President Gore. The four alternatives are

—

*X*_{1}is a positive opinion of Clinton,*X*_{2}is a positive opinion of Gore,—

*X*_{3}is a negative opinion of Clinton,*X*_{4}is a negative opinion of Gore.

Let *s*_{i,j} be the fraction of people with *X*_{i} as the first outcome and *X*_{j} as the second. For instance, *s*_{1,4}=0.0447 is the reported fraction of people who view Clinton as honest (*X*_{1}) and Gore as dishonest (*X*_{4}) when asked in that order. By contrast, *s*_{4,1}=0.0255 is the fraction with these opinions when asked about Gore first and Clinton second. To capture order effects, let *d*_{i,j}=*s*_{i,j}−*s*_{j,i}. Thus *d*_{1,4}=0.0447−0.0255=0.0192.

There are no order effects iff the defined **d**^{4}=**0**, so how **d**^{4} deviates from zero captures ways in which the order matters. The Gallup information [1] defines the context vector
3.11
The zeros represent unavailable, but presumably zero, values for changed opinions about the same person; e.g. *d*_{2,4} is where Gore is viewed as honest and then dishonest. The BAR values are , defining the *X*_{2}≻*X*_{3}≻*X*_{1}≻*X*_{4} ranking. The strongly positive opinion of Gore is reflected by how the *X*_{2}≻*X*_{4} inequality is separated by the *X*_{3}≻*X*_{1} negative assessment of Clinton.

Beyond *d*_{1,3}=*d*_{2,4}=0, there is another **d**^{4} constraint. Namely, the sum of the probabilities listing Clinton first must equal unity, [*s*_{1,2}+*s*_{1,4}]+[*s*_{3,2}+*s*_{3,4}]=1; similarly for Gore, [*s*_{2,1}+*s*_{2,3}]+[*s*_{4,1}+*s*_{4,3}]=1; so
3.12

Any other independent equality with these variables could split equation (3.12) into two equalities. A natural choice, which can be expected to hold by merely being a standard transitivity condition asserting that the relevant variables have no cyclic effects, is the orthogonality condition
3.13
(For , the right-hand side of equation (3.13) equals 0.0092.) Solving equations (3.12) and (3.13) leads to *d*_{1,2}+*d*_{3,4}=0, which means that YY path effects are countered by NN effects, and *d*_{1,4}=*d*_{2,3}, which means that NY and YN values agree. *These are the QQ equations!* As proved next, this transitivity equation (3.13), which just mandates an absence of cyclic effects in the relevant data, is equivalent to the QQ expressions.

### Theorem 3.9

*If* *is orthogonal to* (*equation 3.13*), *then* *has the idealized QQ vector form*
3.14
*where the QQ equalities hold. Conversely, if QQ holds, then so does equation* (*3.14*); *its decomposition is*
3.15

*so* *is orthogonal to* . *Equation* (*3.14*) *BAR values are*

As it is easy to compute, the essence of theorem 3.9 holds even without the *d*_{1,3}=*d*_{2,4}=0 equations. But to show their influence, use the decomposition
with these two equations to obtain . Substituting into equation (3.12) leads to the *a*_{1}−*a*_{2}+*a*_{3}=0 equality, which defines the form
3.16
As the *c*_{2} and *c*_{3} values reflect a cyclic effect induced by *d*_{1,3}=*d*_{2,4}=0, the only cyclic term of interest for the analysis is . According to equation (3.16), the QQ equalities hold iff *c*_{1}=0 and iff **d**^{4} and **C**_{1,2,3,4} are orthogonal. Thus, a measure of the QQ relationship is how closely equation (3.13) is satisfied. (So an answer to (i) in §1 is that a QQ setting occurs iff there is a (*Clinton*_{yes},Gore_{yes})+(Gore_{yes},Clinton_{no})+(Clinton_{no},*Gore*_{no})−(Clinton_{yes},Gore_{no}) cyclic cancelling relationship.) The decomposition is
3.17
with a small component. The cyclic term, then, measures whether QQ holds. When it does, this term (which *never appears* with an idealized QQ vector (theorem 3.9)) represents noise.

## 4. Summary

Guided by structures developed to analyse pairwise voting, a simple basis can be created to examine paired comparisons. An advantage of this basis is that, by involving familiar linear algebra and vector analysis concepts, only a minimal learning curve is needed to use it. As the basis can handle all paired relationships, these simple tools recapture conclusions based on more complicated assumptions and structures, while offering simpler, alternative interpretations. For instance, the QQ equalities are based on the ‘law of reciprocity’ from quantum theory. According to theorem 3.9, the effect of this reciprocity law can be replaced by the simpler transitivity condition that the relevant variables do not experience a cyclic effect (equation (3.13)).

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

My thanks to two referees for their useful suggestions!

## Footnotes

One contribution of 14 to a theme issue ‘Quantum probability and the mathematical modelling of decision making’.

↵1 From [1] and Joyce Wang's informative presentation on 11 March 2015 at a Field's Institute conference on ‘Quantum probability and the mathematical modeling of decision making’.

↵2 Equation (2.1) generates a cycle, which creates the path dependency phenomenon where

*any candidate*can be elected with an appropriate ordering! To elect*D*, for instance, use 〈*C*,*B*,*A*,*E*,*D*〉. For*C*, use 〈*B*,*A*,*E*,*D*,*C*〉.↵3 The

*d*_{i,k}values satisfying equation (3.3) can be determined from

- Accepted July 2, 2015.

- © 2015 The Author(s)