[Election-Methods] [EM] Is "sincere" voting in Range suboptimal?

[Abd ul-Rahman Lomax]

At 12:40 PM 7/23/2007, Chris Benham wrote:

>Abd ul-Rahman Lomax wrote:
>
>>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.
>>
>I see you are measuring "satisfaction increase" by absolute units 
>rather than by relative increase.

That's right. I would agree that percentage increase would be a more 
general statement; but for convenience I simply used the Range units. 
To compare these results with others for higher resolution Range, I'd 
need to convert them to percentages of full scale.


>>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.
>"Keep things simple"? The scenario is so simple that it is easy to 
>work out everything exactly.

Yes. Which is what I did. Because I kept it simple!


>According to my calculations the voter whose sincere preferences are
>A2>B1>C0 in your
>0-info 2-voter election gains the greatest "average satisfaction" 
>with the result by voting A2 B0 C0.
>Next best is A2 B1 C0.

I get different results. First of all, A2, B0, C0 should be the same 
as A2, B2, C0, because of the symmetry in preference strength. Now, 
if that isn't true, I'd wonder why, but I'd certainly be open to 
having made some mistake here. However, given that B is exact 
midrange in utility, there is no reason for preferring the vote of B 
0 to B 2, they both deviate the exact same amount from true utility, 
but in opposite directions, so the loss of utility, if any, or the 
gain, if any, would be the same in either case, matching what was 
said by others, including Benham, that it was equally advantageous to 
0 rate B as to max rate B.


>This surprised me. I only considered votes which both max-rate at 
>least one and also min-rate at
>least one (200, 210, 220).  I first considered the voter having 
>sincere ratings (utility) of  A6> B3> C0
>and then got the same result with  A6>B4>C2.

To compare Benham's results with mine, I think he would have to more 
exactly specify what he did. As to what he writes above, I considered 
only two of the three possibilities he lists because two of them, 
I've been led to think, are equivalent by the symmetry of the 
problem. However, it would be simple to add another vote column to my 
spreadsheet.... I just haven't bothered yet.

I also assumed, of course, that the voter will max rate A and min 
rate C, thus utilizing full vote strength.
The utilities of 6, 4, 2 and 6, 3, 0 would be expected to yield the 
same relative results because they all normalize to the same votes 
(and are the same relative utilities). The problem was stated through 
preference strength, not utilities, and the Range votes inferred from 
the equality of preference strength. The Range votes are then treated 
as equivalent to utilities. Since we are not comparing utilities with 
those of other voters, all the questions about the significance of 
that are moot. We are only concerned here with the maximization of 
the individual voter's utilities, which we *define* as involving 
equal preference strength.

>Interestingly there was no possible opposing ballot  where an 
>overall lower-ranked strategy did better
>than a higher ranked one. For example if the opposing ballot is  C2 
>B0 C0, then  A2 B0 C0 creates
>an AC tie  (average "voter satisfaction" 1); while the (seemingly 
>'safer') A2 B2 C0 just creates an ABC
>tie (again giving average "voter satisfaction" of 1).  A2 B1 C0  has 
>the same effect as A2 B0 C0.

I'm glad Chris found it interesting. I certainly did. If I'm correct, 
I may have stumbled across a method of simplifying the study of 
election strategy....

My results don't agree with what Chris reported, but he did not give 
the utilities he found.

Note that the next part of the study generalizes it to large 
elections. It does show that the advantage of sincere voting lessens 
with large elections, and we already know that with other than zero 
knowledge, the optimal vote moves toward voting Approval style -- not 
necessarily all the way, as long as the risk of error in estimating 
the frontrunners exists. At a certain point, though, voting Approval 
style would be simpler and would yield *almost* the same return. It's 
already pretty close zero-knowledge.

Using this method for higher resolution range adds a huge number of 
vote possibilities.... but we *do* have computers.... It shouldn't be 
difficult to go to Range 9. (0-9). The votes are then simply 
three-digit decimal numbers, keeping it simple to manipulate them.