# [EM] Round Robin Tournament Showings (correction)

Susan Simmons suzerainsimmons at outlook.com
Wed Aug 4 17:57:28 PDT 2021

```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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