The reason I am comparing only the diagonal (T/T vs. NT/NT) is that the
A and C sides can't know which they are in advance of the election (in
other words, which is the majority faction). So whatever strategy
applies to one applies to both; in fact there is no way for the two
sides to distinguish themselves on your matrix in advance of the
election.
You're totally missing the point. It doesn't matter whether they are the stronger or weaker faction. If they turn out to be the faction with more votes, then truncation does not help them. If they turn out to be the faction with less votes, truncation ACTIVELY HURTS them by handing the election to the other side.
The only time truncation changes the outcome is when it hurts your faction (and thereby helps the other faction, which is what is confusing you).
Comparing the top left to the bottom right is, once again, totally invalid. In my most recent message I showed two matrices. One applies when CBA has more votes than ABC, and one applies when the opposite is true. If you want to compare how truncation affects your outcome in the face of uncertainty, then you need to combine these two matrices in a probabilistic sense. Here, I'll do it. before the slash is the result if ABC has more votes than CBA, and vice versa after the slash.
xxx| T | NT |
---|-----|-----|
T | A/C | A/B |
---|-----|-----|
NT | B/C | B |
----------------
So for the ABC faction, truncation either takes you from B to C, from A to A, from B to B, or from B to C. None of those are an improvement. For the CBA faction, truncation either takes you from B to A, from C to C, from B to A, or from B to B. Again, none of these are an improvement. There's no need to know the utilities (other than their ordinal rankings) or the probabilities, since not a single outcome supports truncation.
In effect, the two sides combine as a "pool" of votes, and don't know
which side they are on until after the election. In fact by truncating
they are voting for an AC lottery over a probable B win.
This amounts to voter collusion, and is theoretically possible. Given the right candidate utilities, this is indeed the prisoner's dilemma, but as we well know the prisoner's dilemma usually ends with both criminals picking the Nash equilibrium and ratting out their pal. In the same way, both factions would have every reason to betray their collusive rival and vote sincerely once they got into the voting booth. After all, voting their full preferences could only help them, just as the truncation of their opposing faction can only help them.
Really, I don't know how many more ways I can say this. This is a factual, empirically obvious point, and I'm pretty sure you will agree with me if you just take a step back and work through each individual possibility yourself. No faction has any logical reason to truncate. I haven't proven this in a general case, nor have I checked every possibility, so there is a chance that some obscure electoral arrangement could provide incentive to truncate (although I doubt it). But this example is definitely not such a case.
-Adam
