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

[Chris Benham]

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.

>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.

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.

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.

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.


Chris Benham