[EM] Sincere Condorcet Cycles

[Abd ul-Rahman Lomax]

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, 
is this a "natural scenario"? What is *highly* unnatural about this 
-- and many such election scenarios -- is an assumption of candidates 
and supporters at the same exact positions, instead of being spread 
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.