# [EM] Borda Sorted Margins

Susan Simmons suzerainsimmons at outlook.com
Thu Aug 5 18:06:32 PDT 2021

```Borda is no good as an election method because (like Kemeny-Young) it is vulnerable to clone distortions... it is not "clone free." But what about the sports tournament context? How could clone dependence rear its ugly head in that context?

Off the top of my head the only thing that comes to mind is that clone dependence can easily cause a CW to lose ... the team that beats all of the others pairwise can lose a tournament tallied by the Borda Count. Of course, there are many makeshift tweaks to fix this problem, but how about a
way of getting a tournament "team finish order"  that takes care of this problem seamlessly?

Enter Borda Sorted Margins: list the teams in the order of their Borda scores ... then while any adjacent pair is out of order pairwise, transpose the offending pair with the closest Borda scores.

So this gives us a suitable tournament method Borda Sorted Margins that respects pairwise wins, and (it turns out) has the same reverse symmetry property enjoyed by RSM (ratio sorted msrgins) and ASM (approval sorted margins).

You may wonder what the Borda score even means in this ballot-less context. Well, it turns out that even in the ballot context the pairwise matrix is all we need to calculate  Borda scores; up to a monotone affine transformation the Borda score for team X is the sum of its row entries minus the sum of its column entries.

Remember the entries of X's row are X's scores against the other teams ... in aggregate a measure of X's offensive prowess.

Similarly, the entries in X's column represent the points scored against X by the other teams .. in aggregate a measure of X's lack of defensive prowess.

If the column sum ColSum represents lack of defensive prowess, then the opposite of that sum, -ColSum, represents defensive prowess. Therefore, the sum of defensive prowess and offensive prowess is given by

-ColSum + RowSum, which is the same arithmetically as RowSum - ColSum,

which we recognize as the Borda score for the candidate that goes with the row and column sums we have been talking about.

In summary, the Borda score for a candidate is a combination of its offensive and defensive prowess shown by its performance in the tournament.

How is this Borda score RowSum - ColSum related to the R ratio used in Ratio Sorted Margins?

Compare ...

Borda = RowSum - ColSum

with

R = RowMin ÷ ColMax

The subtraction is replaced by division, and only the smallest term is kept in the row sum, and only the largest term from the column sum.

And since logarithms transform division to subtraction, we have ...

Log R = log(RowMin) - log(ColMax)

What is the purpose of this transformation?

Answer: to change the tournament method Borda Sorted Margins into a Clone Free method ... namely Ratio Sorted Margins ... that is suitable for elections, unlike BSM, which is clone defective like Kemeny-Young. [However also note that unlike K-Y, Borda Sorted Margins is easily computed from the pairwise matrix.]

I hope that this monologue elucidates some relationships among concepts that some EM lurkers were too shy to ask about:-)

Sent from my MetroPCS 4G LTE Android Device
-------- Mensaje original --------
De: Susan Simmons <suzerainsimmons at outlook.com>
Fecha: 4/8/21 5:57 p. m. (GMT-08:00)
A: election-methods at lists.electorama.com
Asunto: Re: Round Robin Tournament Showings (correction)

Reverse symmetry of RSM ...

ASM has the following reverse symmetry property:

If you reverse the ballot rankings of the candidates including the (virtual) approval cutoff candidate (so that the approval order is reversed), then the ASM output will be the same list of candidates, but in reverse order.

Since there are no ballots in the tournament context, how do we formulate this property in that context?

The key to this mystery is that (in the ASM context) reversing the ballot rankings corresponds to both (1) reversing the tentative pre-sorted order and (2) transposing the pairwise matrix.

In the RSM context this means (1) listing pre-sort in order of 1/R(X) instead of R(X), and (2) transposing the pairwise matrix, both of which are equivalent to switching team names on the score board at each game ... giving team i team j's score and vice versa.

So if, in every pairwise contest, the scores are switched, then the final tournament placings will be reversed.

Put this reverse symmetry property together with the property that the order of adjacent teams in the RSM output is consistent with the pairwise win order, and you have a pretty decent result!

Anybody know of a better Round Robin Tournament results method?

Sent from my MetroPCS 4G LTE Android Device

-------- Mensaje original --------
De: Susan Simmons <suzerainsimmons at outlook.com>
Fecha: 4/8/21 3:09 p. m. (GMT-08:00)
A: election-methods at lists.electorama.com
Asunto: Re: Round Robin Tournament Showings (correction)

The ratio R(X) should be ...

... the smallest pairwise support for X divided by the greatest pairwise opposition to X.

In the sports context that would be the ratio of X's smallest score to the greatest score against it.

In terms of the pairwise matrix ... it is the smallest entry in X's row divided by the largest entry in X's column.

The entry in row i of column j is simply the number of points team i scored in the game against team j.

In the election context it is the number of ballots on which candidate i was strictly preferred over candidate j ... plus the number on which they were both voted equal Top and half the number on which they were voted equal but not Top or Bottom.

I hope everybody gets this corrected definition of the ratio R.

Sent from my MetroPCS 4G LTE Android Device

-------- Mensaje original --------
De: Susan Simmons <suzerainsimmons at outlook.com>
Fecha: 4/8/21 11:30 a. m. (GMT-08:00)
A: election-methods at lists.electorama.com
Asunto: Re: Round Robin Tournament Showings

Note if we replace A(X) approval of X with
Log R(X), where R(X) is the ratio of X's best pairwise score to X's worst pairwise score, then ASM and RSM are the same

Sent from my MetroPCS 4G LTE Android Device

-------- Mensaje original --------
De: Susan Simmons <suzerainsimmons at outlook.com>
Fecha: 4/8/21 10:31 a. m. (GMT-08:00)
A: election-methods at lists.electorama.com
Asunto: Round Robin Tournament Showings

After a Round RobinTournament concludes its pairwise contests how should we decide the finishing order (1st place, 2nd place, 3rd place, etc.) of the participating teams?

Here's a solution that's reminiscent of Approval Sorted Margins:

Since there is no precise analogue for a team's approval in this context we use the ratio R of its best score to its worst score to determine a tentative list order.

Then as long as some adjacent pair of teams is out of order pairwise, among such pairs transpose the one whose members' R ratios are closest,  i.e. with the smallest absolute value of log(R1/R2).

The CW and CL (when they exist) will appear at opposite ends of the sorted list.

And there is the same kind of reverse symmetry that ASM provides in the context of elections.

In fact, we could call this method RSM or Ratio Sorted Margins, where the margins are the absolute differences of form
|log R1 - log R2|
in analogy to the approval margins of form |A1 - A2|.

People that are uncomfortable with approval cutoffs can use RSM instead of ASM ... no approval necessary ... ranked preference style ballots are perfectly adequate ...in fact, since it is a tournament method, the pairwise vote matrix is adequate by itself.

No more excuses for clone dependent, intractable Kemeny-Young: just use RSM!

Sent from my MetroPCS 4G LTE Android Device
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