[EM] Is "sincere" voting in Range suboptimal?

[Abd ul-Rahman Lomax]

It is commonly claimed, by some, that strategically optimal voting in 
Range, maximizing the voter's personal expected satisfaction, is 
always Approval style, i.e., only minimum or maximum rating, one 
critic going so far as to claim that Range voters who vote 
intermediate ratings are "suckers."

Is it true? How can we determine optimal voting in Range?

While I've seen the above claim many times, I've never seen a proof 
of it. What I have seen, instead, are examples showing how, in some 
situation, a voter would regret having voted sincerely. However, it 
is possible that a voter could regret having voted insincerely. Which 
is more probable? This is the question that the critics of Range 
don't seem to be asking.

To test this, I settled on considering, first, an election with two 
voters. The second voter has no knowledge regarding the vote of the 
first voter, who has already voted and who had no knowledge regarding 
the planned vote of the second voter, whom I will now call just "the 
voter," unless I specify "first."

For the first pass, I will consider that all votes by the first voter 
are equally probable. There are three candidates, A, B, and C. I will 
later restrict which votes are being considered possible, for reasons 
that will be explained (and which accomplishes a generalization of 
the analysis to many-voter elections.)

The election is Range 2, or what some call CR3, each voter may cast 
up to two votes for each candidate, so there are three possible 
votes: 0, 1, and 2.

We can then express the 27 possible vote patterns as a list of the 
trinary numbers from 000 to 222. (The moot votes of 000 and 222 are 
initially left in; I'll note that these kinds of votes actually occur 
in elections, they are not uncommon.)

Then we look at how our voter votes. The voter has preferences A>B>C, 
with the A>B preference strength being equal to the B>C preference 
strength. We can derive from this a "sincere" Range vote of 210.

We also want to see how the results fall if the voter votes "Approval 
style." There are two possible Approval style votes in this example, 
being 220 and 200. Because of the symmetry of the scenario, however, 
I expect that the expected satisfaction is the same between the votes 
of 220 and 200, and I want to keep things simple, so, if someone 
considers it necessary, I can say that the voter decides that he 
"slightly" prefers B additionally such that he will vote that way, 
but this does not significantly affect the utilities.

One more necessary point: ties are resolved by random choice between 
the tied candidates, all being equally likely. So the utility 
assigned to a tie is the average of the utilities of those tied.

One more point before proceeding to the results: I mention 
"utilities," but this analysis does not depend on any assumption of 
compatible, comparable interpersonal utilities. Rather, we are 
*assuming*, for one voter only, a set of values, which could have any 
meaning whatever, such that the voter's satisfaction increases just 
as much by the selection of A over B as it increases by the selection 
of B over C.

What happens? In another post, when I have time, I will express the 
complete matrix. But this is the summary:

For each vote by the first voter, I add the votes described above, 
220 and 210, considering them base-5 numbers (each digit is now 0-4), 
determine the winner(s), and assign a utility to the result, being 
the utility of the winner, or, if a tie, the average of the winners' utilities.

With the Approval-style vote, the expected utility of the election is 
40/27, or 1.48, compared with an expected utility of 1.0 if the voter 
does not vote at all. With the "sincere" Range vote, the expected 
utility is 46.5/27, or 1.72.

(Note that the maximum utility of the election is 2.00.)

What if the first voter votes "strategically," i.e., approval style, 
as claimed would harm the second voter unless the second voter 
"retaliates" by voting in the same way?

If I eliminate all but the Approval style votes, i.e., no use of 
intermediate ratings, and also the moot votes of 000 and 222, I'm 
left with six initial vote patterns. And the result with the Approval 
style vote of 220 is 8.5/6, or 1.42. The "sincere" vote of 210 yields 
a utility expectation of 9.0/6 or 1.50.

I find it fascinating that approval-style voting by the first voter 
does in fact reduce the expected outcome for the second voter, but it 
does *not* change the optimum strategy. The optimum strategy is still 
the sincere vote.

Okay, but what about an election with many voters?

Well, we can consider that in most elections, the vote of the voter 
is moot, it does not change the outcome. What is of interest to us, 
though, are the circumstances where it *does* change the outcome. I 
will elsewhere argue that this is actually what voters should 
consider: they should vote, *always*, as if their vote could change 
the outcome!

If we subtract the maximum number of moot votes (votes balanced by 
counter-votes, or votes that cannot be part of a winning margin), the 
election reduces to the two-voter election, if we assume that the 
voter's vote can swing the outcome! However, some single-vote 
patterns are very rare compared to others. In particular, with a 
large number of voters, this "affects the outcome" assumption means a 
near-tie, which is, of course, vanishingly rare. So *two* ties would 
be far more rare, so we can eliminate all vote patterns where there 
is a three-way tie, or, in fact, where there are more than two 
candidates with *any* votes.

I also need to "split" some of the vote patterns with two zeros to 
reflect which 0 is involved in the tie and which is not.

Here I have no more time tonight, it's already ridiculously late....