Paired Comparison Links

The paired comparison linksFootnote ⁹ are illustrated with five alternatives and the Figure 1 ‘ranking wheel’. Uniformly mark the N ranking numbers, here from 1 to 5, on a rotating wheel. Place in any order the alternatives’ names on the wall. The number next to an alternative determines its ranking; for example, the Figure 1 ranking is A>B>C>D>E where ‘>’ means ‘preferred to.’ Next, rotate the wheel to place ‘1’ by the next alternative (here, B), and read off the new ranking. Continue until ‘1’ has been next to each alternative. The five Figure 1 rankings are:

(1) $$\eqalignno{ & A\gt B\gt C\gt D\gt E,{\rm }B\gt C\gt D\gt E\gt A,{\rm }C\gt D\gt E \gt A\gt B,{\rm }D\gt E\gt A\gt B\gt C,{\rm }E\gt A\gt B\gt C\gt D $$

The theoremFootnote ¹⁰ crucial for analyzing paired comparisons is that ranking wheel configurations, and only these configurations (and portions), cause all possible paired comparison difficulties. (One version handles non-voting concerns from, say, engineering or statistics.)Footnote ¹¹ Namely, ranking wheel configurations define the links connecting paired outcomes. Similar to how relationships among Rubik cube faces must be incorporated to solve Rubik puzzles, ways to handle these connecting links must be discovered to resolve decision problems. What simplifies the challenge is that only ranking wheel configurations need be considered.

Fig. 1

Ranking wheel for N=5

As each Equation 1 alternative is in each position once, no alternative is favored, which suggests a completely tied outcome. But similar to the Rubik cube fiasco of separately emphasizing each face, pairwise methods sever Equation 1 links and lose critical information. What demonstrates the magnitude of the loss is that, rather than an A=B tie, the majority vote outcome is A>B with a 4:1 vote! Reflecting the wheel’s rotating construction, Equation 1 generates the cycle:

(2) $$A\gt B,{\rm }B\gt C,{\rm }C\gt D,{\rm }D\gt E,{\rm }E\gt A,$$

where each tally enjoys a landslide 4:1 victory. As this cyclic outcome shows, severing connections loses information and destroys an intended order.

Namely, a system’s global properties (for example: data, preferences, etc.) may identify one answer (such as a complete tie), but by ignoring connecting links, the reductionist approach degenerates into those conflicting ‘blind men describing an elephant’ conclusions. As indicated in the supplementary notes, more general assertions follow.

Remedies

Knowing what causes difficulties helps to circumvent them. An example is Theorem 1, which asserts that preferences free of linking terms release Arrow’s Theorem from its described troubles.Footnote ¹² (In tallying an n-candidate ballot, the Borda Count assigns the jth positioned candidate n−j points; so for three-candidates, assign 2, 1, 0 points, respectively, to a ballot’s first, second, and third ranked choices.)

Thus, Arrow’s ‘reductionist’ concerns, and all differences between Condorcet and Borda outcomes, reflect information (Equation 1) ignored by reductionist techniques. All paired comparison concerns, including agenda manipulation (ensuring a personally desired outcome by designing an appropriate agenda), tournaments, etc., are ranking wheel consequences.

Arrow’s theorem

Any substantive way to circumvent Arrovian problems must reject the reductionist’s emphasis on pairs to address ranking wheel interactions. Rather than using pairwise outcomes as societal rankings, one approach combines IIA information to neutralize ranking wheel complications.

To do so, notice that A’s tallies over all Equation 1 pairwise elections are:

(3) $$A\gt B\,{\rm by}\,{\rm 4}\!:\!{\rm 1},A\gt C\,{\rm by}\,{\rm 3}:{\rm 2},D\gt A\,{\rm by}\,{\rm 3}:{\rm 2},E\gt A\,{\rm by}\,{\rm 4}:{\rm 1}.$$

A’s, and each candidate’s total is 10, which defines the desired complete tie and addresses the linkage problem.

As it turns out, summing each candidate’s points over all paired comparisons neutralizes all ranking wheel complications. Notice that this ‘paired summation approach’ retains the spirit of Arrow’s condition; it uses IIA information to counteract troublesome ranking wheel affects. For instance, with the three-candidate majority vote tallies:

(4) $$A\gt B\,{\rm by}\,{\rm 51}\!:\!{\rm 49},A\gt C\,{\rm by}\,{\rm 52}\!:\!\,{\rm 48},C\gt B\,{\rm by}\,{\rm 6}0\!:\!\,{\rm 4}0,$$

the Condorcet winner A receives 51+52=103 points, Condorcet loser B receives 89 points, and C receives 112 points to define the C>A>B ranking.

THEOREM 2 A candidate’s paired summation ranking method’s tally is the sum of points received over all pairwise votes. These rankings are complete and transitive. If voter preferences are free of ranking wheel components, this ranking agrees with Condorcet rankings.

The Equation 4 rankings disagree with the paired summation outcome because the voter preferences defining these rankings must include ranking wheel effects.

Condorcet vs. Borda

Theorem 2 differs from Arrow’s Theorem in that it has a positive conclusion, and, rather than treating IIA outcomes as societal conclusions (Arrow’s condition 2), it just further massages them to restore severed links. Namely, Theorem 2 retains aspects of IIA yet provides a natural, positive version of Arrow’s negative theorem.

But ten candidates (not unusual in primaries) require forty-five pairwise elections! To motivate a simpler equivalent method, the pairwise votes for A>B>C preferences are in the middle three rows of:

(5) $$\matrix{ {{\rm Pairs}} & A & B & C \cr {\{ A,\,B\} } & 1 & 0 & {\minus} \cr {\{ A,\,C\} } & 1 & {\minus} & 0 \cr {\{ B,\,C\} } & {\minus} & 1 & 0 \cr {{\rm Total}} & 2 & 1 & 0 \cr } $$

The sums (bottom row) are the voter’s paired summation contributions and they are the points assigned by the Borda Count. As this holds for any number of candidates,Footnote ¹⁴ the Borda Count is equivalent to the paired summation method, and, according to Equation 5, it is the natural extension of the pairwise vote. Thus (Theorem 2), the Borda Count provides a positive way to salvage Arrow’s basic conditions.

With respect to the Condorcet vs. Borda winners:

• – Arrow’s Theorem is an indictment of Condorcet’s approach. Problems suffered by Condorcet’s method reflect how reductionist methods ignore crucial connecting information.

• – The Borda Count is a positive, usable voting rule that retains the spirit of Arrow’s conditions, but avoids Arrow’s pessimistic conclusion by neutralizing the negativity created by ranking wheel configurations.

Other Settings

Reductionist concerns arise in unexpected places. Laws, for instance, can determine priorities.Footnote ¹⁵ In a loan default, assets are allocated to lenders based on the order in which loans were made. To ensure public knowledge and reduce litigation, loans should be registered: an incentive to do so is to give registered loans priority over unregistered ones. But this condition introduces an opportunity for the knowledgeable to exploit the naïve, which motivates a ‘fairness’ constraint: a registered loan gains priority only without prior knowledge of an unregistered loan.

What unexpectedly subjects these laws to the reductionist argument is that:

1. (1) There is an accepted consistency condition – a transitive ranking of loans.

2. (2) Components are defined by priority conditions for different pairs of lenders.

Linking conditions (which, for pairs, usually resemble ranking wheel constructions) are arrangements among components that violate the intended order. Suppose, for instance, Dave borrows money from Ann to buy a computer to write a novel. Times are bad, so Dave borrows money from Barb; again, the computer serves as collateral. Barb knows that Dave borrowed money from Ann, but, unlike Ann, she registers the loan. Her knowledge, however, ensures Ann’s priority over Barb. Writing is going slowly, so money is borrowed from Carole, who, when registering, knows nothing about Ann.

Dave defaults, so who gets his computer? According this suite of laws, Ann has priority over Barb. By registering first, Barb has priority over Carole. By not knowing about Ann, Carole has priority over Ann, which creates a cycle that destroys the intended order.

This default story is but a tip of a legal iceberg of unintended consequences caused by laws severing unexpected links. Katz and Stearns are among the leaders in exploring legal complications:Footnote ¹⁶ a topic that deserves more attention.

For other examples, search wherever consistency conditions accompany decisions about ‘parts.’ In a particular country, a church’s leader was required to be registered before his group could be treated as a legitimate religion.Footnote ¹⁷ The second component described one way in which the leader could bend legal barriers to complete the registration process: marry a local girl. The consistency condition came from the religion: imagine the difficulties the Catholic Bishop would have encountered by getting married.

A rich source of examples – an area encumbered with ‘silo effects’ (components) – is the design and practice of foreign affairs. Complications reflect violations of consistency conditions, which are desired outcomes. Although not addressed from this perspective, illustrating examples are to be found in Jervis’s book.Footnote ¹⁸

Is the Prediction Method Reliable?

Dark matter is the difference between mathematical predictions and observations, so the first step is to check whether the method can exaggerate mass predictions. If so, this would cast doubt on the reliability of predictions and indicate that reductionist concerns exist.

To answer this question, I created analytic solutions of billion-body problems. The precise mass is known, but a standard prediction method exaggerated exact values by multiples in the millions.Footnote ¹⁹ With certain solutions, it predicted that 70 per cent of the mass lies in an outer fringe region, which has, in reality, less than 5 per cent. These consistent errors indicate that standard predictions about the existence of huge amounts of dark matter are questionable and caused by unanticipated reductionist effects.

Connecting Links

Finding connecting links was surprisingly simple. Rather than pulling a star toward the galaxy’s center, if a faster moving body comes nearby, it pulls the star toward and along with it. This behavior, mandated by Newtonian interactions of discrete particles, defines a rotating connection (with a ranking wheel flavor) where a fast moving particle pulls along a slower one, this body drags an even slower one, and so forth. Trailing streams of stars are supported by Newton’s equations and observed in pictures of galaxies. But dragged stars travel faster than would have been possible had the only force been the interior star-soup. Substituting faster velocities into prediction equations exaggerates mass predictions, which explains the radically distorted mass distributions for analytic solutions.

Another connection with Arrow’s result is found in Theorem 1, which asserts that data with no (or minimal) linkage terms frees Arrow’s and Sen’s conditions from negative consequences. Similarly, in a gravitational setting with minimal dragging links, such as in our Solar System (with the small sizes of planets), the prediction method works reasonably well; it is used to compute the Sun’s mass. As with the above Arrow analysis, dark matter problems reflect how prediction methods sever links.

Resolutions

Ways to resolve difficulties that incorporate linkage properties are technical,Footnote ²⁰ so only suggestions are offered. But it is clear that a holistic, systems approach is needed. Guided by the Borda resolution of Arrow’s problems, what needs to be done is immediately obvious: emphasize dynamic notions that involve rotating streams of bodies – the system’s angular momentum.

By utilizing recently derived properties of the angular momentumFootnote ²¹ along with basic assumptions astronomers have made about the stability of galaxies, new approaches emerge relating mass values with rotational velocities of stars. With this approach, the amount of mass needed for galactic stability is more consistent with observed values. Stated in a different manner, while many believe there exist huge amounts of dark matter, this opinion is not supported by Newton’s actual equations.