[Election-Methods] Strategy/polling simulation for simple methods

[Kevin Venzke]

Hello,

I've implemented MCA (the version where only one favorite is allowed) and
Condorcet//Approval (with truncation being the only available strategy, and
voters only understanding "better than expectation" strategy). I also
complete rewrote the voter AI for VFA and TTR and came closer to
understanding some odd strategies.

Here again is the scenario I was using:

> 2: D>A>B>C>E 99 80 40 10 1 (utilities)
> 2: A>B>D>C>E 90 60 50 20 1
> 1: B>A>C>D>E 90 60 30 12 1
> 1: B>C>A>E>D 90 60 30 12 1
> 2: C>B>E>A>D 90 60 50 20 1
> 2: E>C>B>A>D 99 80 40 10 1
> (quantities are of factions, not voters)

MCA:

The strategy is not too hard. "Better than expectation" still applies for
the approval votes, but one only regards the polls that resulted in a
winner being chosen according to approval. Then for your "favorite" vote
you consider the odds that you can make that candidate the majority
favorite, and what you would gain from this over accepting the approval
result. You can see that this leaves no chance that a voter would vote as
first preference any candidate who is worse than the approval expectation.

A nice thing about the majority requirement is that if it's assumed that
you are going to vote, there's no way that your vote can move the majority
from one candidate to another. Makes it a little easier to think about the
problem.

So if we run the above situation through this method, the votes start out
as:

2 : vote D>DA
2 : vote A>ABD
1 : vote B>BA
1 : vote B>BC
2 : vote C>CBE
2 : vote E>EC

This initially yields few polls showing an MF. A, B, or C win on approval.
Voters start playing the approval game and don't move their FP. Eventually,
there is some favorite betrayal, in response to one of the viable candidate
obtaining some odds of winning as MF. 

These votes are at equilibrium:
2 : vote D>DA
2 : vote A>A
1 : vote B>B
1 : vote B>BC
2 : vote B>CBE
2 : vote B>ECB

The approval votes are in the same type of configuration they reach under
Approval (one of A and C's supporters vote for B, and the other's don't;
most wins go to B). B wins about 85% of the elections by being MF; A and D
win this way very rarely; A wins 14% or so on approval.

More typically I see this equilibrium (or its opposite):
2 : vote A>DA
2 : vote A>A
2 : vote B>B
2 : vote C>CB
2 : vote C>ECB

A and C win over 14% each by being MF. B wins everything else on approval.
Though B is a useless vote for MF, the B voters don't care because the
other two options are worse than expectation should the win be based on
approval.

Possibly undesirable: If it ever happens that it becomes guaranteed (from
the polls) that the race will be won by a majority favorite, massive vote
changing ensues as everybody "forgets" what they expected from the approval
outcome (since it never occurred). They also cease to find it useful to
strategize with their first preference for the same reason, but also
because if it's guaranteed that there will be a MF, there is no probability
that your first preference vote will help anybody become MF.

Condorcet//Approval:

Pretty simple change to the MCA code. The voters decide where to truncate,
and act as though they believe the method is simply Approval.

Initial votes:
2: vote D>A
2: vote A>B>D
1: vote B>A
1: vote B>C
2: vote C>B>E
2: vote E>C

This results in fairly even win odds for ABC, although B trails a bit.
Voters start picking up and dropping compromises.

Typical equilibrium:
2: vote D>A
2: vote A>B
2: vote B
2: vote C>B
2: vote E>C

B wins about half the time. There's a CW 62% of the time where ABC are more
even. If it goes to approval resolution, though, B wins 28% (of all
elections) while A and C only get 5% each.

It's almost stable (though not as good for B) with the B voters voting for
their second preference also. But it seems eventually the B voters drop the
preference.

My main reservation with this implementation is of course that the voters
are pretending it's Approval and not considering e.g. the fact that if it's
predicted that the election will always have a CW, there would be no reason
to truncate at all.

VFA:

Rewrote the AI here. Still not sure how to get voters to consider the odds
that a candidate will be disqualified (due to having a majority of the
"against" votes) without them voting against totally unviable candidates
simply because there's a chance of disqualification. They still mainly
consider which candidate it would be most advantageous to remove. (And
since VFA always elects either the 1st- or 2nd-place candidate, voters
should naturally approach a majority when everyone thinks simply like
this.)

But the logic for the "for" vote now correctly takes into the consideration
that a vote for A vs. B could be pivotal even when neither comes in first.

The scenario still unfolds the same way as previously described. The field
gradually narrows (though it's chaotic) and looks like it will be a two-way
race. But as soon as a disqualification is guaranteed, people want to vote
for their favorites.

I figured out a cause of voter desire to vote for and against the same
candidate, and I'm not sure it is actually illogical. Consider a situation
where your favorite, B, wins half the time and is disqualified the other
half of the time. When B is disqualified, A (your second favorite) wins
most of the time, but the worst candidate C rarely is able to reach second
place on "for" votes, defeating A.

In this case you might want to vote both "for" A and "against" A. You vote
against A because this is how B wins. (I.e. A is easily seen to be the best
candidate to remove, given a choice.) But if B is disqualified then it's
worthless to vote for him; you may as well vote for A to help him defeat C.

I still need to investigate some odd voting choices, though.

Top-Two Runoff:

Rewrote this AI as well, to take advantage of a lot more information. I
think it may actually be unrealistically good, as I believe I see voters
trying to use pushover strategy. I don't know if the real world has
examples of voters using pushover strategy under TTR.

Currently this simulation is too chaotic to describe well. I'll say a bit
about the strategy implemented, though.

The value of a vote for A is the sum of the following for every other
candidate B: Difference in expectation (I'll define in a second) times the
odds that A comes in second and B comes in lower, times the odds that B
comes in second and A comes in lower.

"Difference in expectation" is the following for A minus the following for
B: The utility expectation (from past polls) of those elections where A/B
placed 2nd in the first round and the other candidate didn't place at all.
If there were not such scenarios (for either candidate) then "difference in
expectation" is defined to be 0.

Theoretically one could see value in pushover strategy if one relatively
liked the observed results of a poor candidate making the second round.

Of course my factions are relatively stupid in that they are so large and
should realize that when they alter their votes, they have a big effect on
the outcome. This can be helped somewhat by running with more factions.

That's all I've got for now.

Kevin Venzke


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