# [EM] Sincere Condorcet Cycles

Abd ul-Rahman Lomax abd at lomaxdesign.com
Mon Dec 14 19:31:20 PST 2009

```At 07:53 PM 12/13/2009, fsimmons at pcc.edu wrote:
>  Here's a natural scenario that yields an exact Condorcet Tie:
>
>A together with 39 supporters at the point (0,2)
>B together with 19 supporters at (0,0)
>C together with 19 supporters at (1,0)
>D together with 19 supporters at (4,2)
>
>D is a Condorcet loser.
>A beats B beats C beats A, 60 to 40 in every case.

There are a number of aspects this that are worthy of study. First,
-- and many such election scenarios -- is an assumption of candidates
across a spectrum. However, in some "election games," this might be
reasonable. The Choice of State Capitol game used on some Wikipedia
articles is an example where each proposed candidate does come with a
set of voters who are at the position of the capitol.

To put the capital game into perspective, it's entirely possible that
the optimal solution to the State Capitol problem is "None of the
Above," and it could be optimal to choose some minor town that
actually optimizes the sum of travel distances. Or to set up
subassemblies for regions, with proportional delegation from them to
a smaller state-wide assembly. Or other possibilities.

However, setting this aside, supposed that we accept both the
candidate set and the positions. How can we study this scenario?
Well, we've been given the presumed utilities, and we can assume that
they are commensurable. Optimizing social utility, then, is
accessible, and all they would need to do is use Range or Score
Voting. Let's start by assuming "sincere" votes, which means that
they simply vote the distance between their position and that of the
relevant candidate. (This is inverted Range Voting, the winner is the
candidate with the lowest vote.)

A:      A voters, 39*0 =        0
B voters, 19*2 =        38
C voters, 19*2.236 =    42
D voters, 19*4 =        76
sum A = 156

B:      A voters, 39*2 =        78
B voters, 19*0 =        0
C voters, 19*1 =        19
D voters, 19*4.472 =    85
sum B = 182

C:      A voters, 39*2.236 =    87
B voters, 19*1 =        19
C voters, 19*0 =        0
D voters, 19*3.606 =    69
sum C = 175

D:      A voters, 39*4 =        114
B voters, 19*4.123 =    78
C voters, 19*3.606 =    69
D voters, 19*0 =        0
sum D = 261

I rounded off the total votes to the nearest vote. I'm not bothering
with being more exact; what this analysis shows is that the real
contest, in terms of social utility, is between B and C, and this is
close. The "social preference strength" is low between A and C. C is
the SU winner, though.

Suppose this were an approval election. With bullet voting (which
reduces to plurality), A would win, of course, with 39/96 votes. But
that's not a majority. Suppose D were eliminated. Again, bullet
voting in approval, A:B:C is 39:19:38. A still wins, but very
narrowly, and it's still 39/96 votes, not a majority. If a runoff is
held, it would be between A and C. The B and C voters prefer C, and
the A voters match them and overpower them with one vote. Very close.
How do the D voters vote? The difference is between a distance of 4
and 3.606. They prefer C. If they turn out, C wins, the SU maximizer.

Now, what would be the turnout in the runoff election? It is this
kind of factor that has largely been neglected, even among social
utility proponents. Turnout would depend on the absolute importance
of the decision to the various groups of voters. In the A/C runoff,
the voters with the least incentive to turn out would be the D
voters. If none vote, the result is A, if even one of them votes, the
result is tied, and if two vote, it's C.

Now, consider Deliberative Process as an election method. What would
deliberative process choose? Quite possibly, the negotiators would
toss another variable into the mix. Suppose that there is a Clarke
tax. Suppose that this tax is designed to be the value to each
participant of optimization of choice. If these are travel distances,
it's the cost of that travel, paid to each voter, there is a travel
subsidy given with the proceeds of the tax to those who "lose" the
vote, and there is a net tax paid by those who "win." (I won't give
the details.)

What this would do is to even out the result so that voters don't
care what result takes place, and this would make individual
incentive match social incentive, there would be only the goal of
minimizing overall travel. If that location is chosen, the average
tax would be lower, which would benefit all voters. Literally, they
would all have more in their pocket, each one of them, if the best
overall candidate won.

Deliberative process could go further than this. The capitol might be
sited at a new location designed to minimize travel distance overall.
The state government could also be structured so as to make travel
distance largely irrelevant, and there are plenty of ways that this
might be done.

I'd like it noted that top two runoff in this scenario rather easily
chooses, with high probability, a social utility winner (which would
be either one of A or C). It is about time that the value of iterated
voting is recognized by voting systems experts. Because it hasn't
been valued, top two runoff has been neglected and not supported
against attack by instant runoff voting, which is inflexible and
chaotic, as we well know. Top two runoff, when the voting method is
vote-for-one, is subject to an obvious flaw, Center Squeeze, but that
is easily fix by using a better method, such as Bucklin or Range; if
Range is to be used, the method should include an explicit approval
cutoff (I suggest mid-range).

Note that in some top two runoff implementations, write-in votes
continue to be allowed in the runoff, which under some conditions
fixes, to some extent, Center Squeeze, if there is sufficient
preference strength in the electorate, and with a good method, the
possible spoiler effect in the runoff can be avoided. TTR with good
voting systems is really quite close to ideal. That is how powerful
even a single extra ballot can be; the requirement that voting
systems be deterministic is, essentially, a paralyzing restriction,
and completely at variance with standard democratic process, which
iterates forever until a majority determines a result.

One of the historical events I noticed in researching top two runoff
is that in California, write-ins are allowed in all elections per the
state constitution. This included runoff elections. However, in 2004,
the last top two runoff election in San Francisco, San Francisco had
just passed a measure prohibiting write-in votes in the runoffs, and
this was challenged by a write-in candidate who might actually have
won. (My guess is that the ordinance was passed for that reason! --
but I don't know). The state supreme court ruled that a runoff wasn't
a separate election, and that write-ins being allowed in the primary
was sufficient. That is tantamount to deciding that if write-ins are
allowed in political party primaries, then they need not be allowed
in the final election..... Very bad decision, and where were the
voting systems experts? Did they even notice? The lack of attention
to top two runoff, both in theory and in practice -- the reality is
quite surprising, I've found -- is harming democracy, through the
elimination of write-in votes in runoffs, as well as through
replacement of top two runoff by IRV. From the point of view of
simplistic voting system analysis, the kind that has been too common,
the methods are about the same. It's not true in actual practice.

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