# [EM] step by step explanation of CPO-STV

James Green-Armytage jarmyta at antioch-college.edu
Sat Jul 26 14:37:36 PDT 2003

```David Gamble wrote:
>Could you provide a (relatively simple?!?) worked example of exactly
>how CPO-STV is done. I'm having difficulty working out exactly what
>to do from Tideman's Meek, Warren, Newland-Britton compared paper.

I would be happy to provide a simple example and show step by step how
CPO-STV (comparison of pairs of outcomes by single transferable vote)
decides on a result.

In my example, 5 candidates are competing for 3 seats.
The five candidates are Andre the Giant, M.C. Escher, George Bush, Al
There are 300 voters.
To keep things simple, I will use the Hare quota, which is votes / seats,
or 300 / 3, or 100.
Also to keep things simple, this is not an example where Meek or Warren
would make any difference.
(With CPO-STV you can use Meek, Warren, or Newland-Britain, as you like.
Also, you can use whichever quota you prefer and whichever Condorcet
completion method you prefer. In this example, I have chosen a Hare quota,
but the other two decisions happen to be irrelevant to the result.)

Here are the preference rankings of the 300 voters:

100: Escher
6: Gore, Bush
45: Bush, Gore

(To keep things simple, these are just the preference rankings that
matter to the election. It would be possible to add further preferences to
each of them without it making any difference at all.)

The reason that I have chosen these numbers is that it creates a similar
situation to the classic Condorcet-IRV divergence example. Since the quota
is 100 votes, there is no doubt that Escher and Andre deserve two of the
three seats. It is the third seat that is more up for grabs, in the
competition between Bush, Gore, and Nader.
I included a 10 vote surplus transfer from Andre to Nader and Gore, just
so that I could illustrate how CPO-STV deals with surplus transfers.

So, using plain STV, the winning outcome is Escher + Andre + Bush.
Correct?
That is, first of all, Escher and Andre would win seats, and Andre's
surplus would be transferred, resulting in this situation:

100: Escher [Elected]
100: Andre [Elected]
6: Gore, Bush
(27 total top choice for Gore)
45: Bush, Gore

(Notice that the situation between Bush, Gore, and Nader is now the
classic example of divergence between Condorcet and IRV)
Then, Gore would be eliminated (having the fewest top choice votes),
resulting in this situation:

100: Escher [Elected]
100: Andre [Elected]
51: Bush

Nader would be eliminated, and the tally would be over, resulting in the
outcome Escher + Andre + Bush.
Now, I think that this outcome is unfair for all the same reasons that it
is unfair in the corresponding single-winner case, and I assume that most
Condorcet fans will agree with me.
The appropriate outcome, I believe, is Escher + Andre + Gore.
This is, of course, the outcome that CPO-STV produces, and I will now
show how it reaches that outcome.

To begin with, since there are 3 seats and 5 candidates, there are 10
possible outcomes for the election. These are as follows:

Escher + Andre + Gore
Escher + Andre + Bush
Escher + Gore + Bush
Andre + Gore + Bush

In CPO-STV, each possible outcome is compared with every other possible
outcome in a pairwise competition.
(For the purpose of this post, I am leaving out the possibility of
shortcuts, through which some comparisons can be omitted without at all
compromising the integrity of the result. I have covered some of those in
a previous post.)
These comparisons basically work according to the same logic as any other
Condorcet comparisons, that is they result in win loss or tie, and there
are corresponding numbers to indicate magnitude.
For example, the competition between outcome Escher + Andre + Gore (EAG)
and outcome Escher + Andre + Bush (EAB) can be expressed like this:
EAG:EAB=255:245
Therefore, you can use whichever Condorcet completion method you prefer
to complete CPO-STV as well.

So far so good. But how are these comparisons carried out?
I will try to go through the process step by step.
Let's say that we want to compare the outcome Escher +Andre + Gore to the
outcome Escher + Andre + Bush.
The first step is to eliminate all of the candidates who are not in
either outcome, and transfer their votes. In this case, the only candidate
who is in neither outcome is Nader. If his votes are transferred, then the
situation looks like this:

100: Escher
110: Andre, Gore
39: Gore
6: Gore, Bush
45: Bush, Gore

The next step is to transfer surpluses. In CPO-STV, you only transfer
surpluses of candidates who are in both outcomes. In this case, the only
surplus belongs to Andre. Since he is in both outcomes, you can transfer
the surplus, which results in this situation:

100: Escher
100: Andre, Gore
49: Gore
6: Gore, Bush
45: Bush, Gore

Now it is possible to compare the two outcomes. This is done simply by
summing the (top choice) vote totals held by candidates in each outcome.
In this case, it would end up like so:

Escher + Andre + Gore = 100 + 100 + 55 = 255
Escher + Andre + Bush = 100 + 100 + 45 = 245

So, again the pairwise comparison between Escher + Andre + Gore, and
Escher + Andre + Bush is 255 over 245, that is a victory for EAG over EAB
with a magnitude of 255.

Now I will repeat the process in comparing Escher + Andre + Gore to
First, take Bush out of the mix.

100: Escher
51: Gore

Next transfer the surplus.

100: Escher
51: Gore

Now, sum and compare.

Escher + Andre + Gore = 100 + 100 + 72 = 272
Escher + Andre + Nader = 100 + 100 + 28 = 228

Therefore,

EAG:EAN=272:228

At this point, it should be fairly obvious that EAG will win all of its
pairwise comparisons, since any outcome without both Escher and Andre in
it would be rather absurd, and it has been shown that of the three
outcomes that do contain both of them, EAG is a clear Condorcet winner.
However, I will do one more comparison, in order to demonstrate the
surplus rule, and to help illustrate that no outcome that doesn't include
both Escher and Andre doesn't have a chance.

Take for example Escher + Andre + Gore versus Escher + Nader + Gore.

100: Escher
51: Gore

Now, this is a case where the surplus is not transferred. That is, it is
not transferred because Andre is not in both outcomes. If it was to be
transferred, then it would be counting against Andre in favor of Nader.
Obviously any voter who ranked Andre above Nader would not want this to
happen. (Not that it would make a difference in this case, but the rule is
followed nevertheless.)
So, you skip straight to the summary and comparison.

Escher + Andre + Gore = 100 + 110 + 72 = 282
Escher + Nader + Gore = 100 + 18 + 72 = 190

EAG:ENG=282:190

It should be clear that this isn't going anywhere. Escher + Andre + Gore
is a clear Condorcet winner, and therefore is the final result of the
election.

I hope that this has been a helpful explanation of the CPO-STV method. I
think that it has a lot to offer. My hope is that anyone who seriously
values both Condorcet's method and STV proportional representation will
take CPO-STV seriously.
If anyone can think of a drawback to CPO-STV aside from its computational
cost, that is an example where it produces an unfair result, then I would
like to see it. Also, if anyone can produce Condorcet - STV hybrid that is
as effective as CPO-STV, I would like to see that too. Until that point, I
will continue to assume that CPO-STV is the best proportional method,
computational resources and political attitudes permitting.

-- James Green-Armytage

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