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

[Steve Eppley]

What poll question is Kevin asking his simulated voters?  Who is your 
favorite?  How would you vote if the election were held today?  What is 
your order of preference?  How much would you pay to elect X if it were 
a contest between only X and Y? 

Wouldn't the wording of the poll question, and the allowed answers, make 
a difference?

Regards,
Steve
----------------------------
Kevin Venzke wrote:
> Hello,
>
> I've been writing a group of simulations to see what happens when
> intelligent factions gradually adjust their planned voting strategy in
> response to "polls." Currently I've implemented FPP, antiplurality, any
> combination methods (i.e. grant X points to one candidate and deduct Y
> points from another), Approval, top-two runoff, and VFA. (Though the voter
> AI for TTR and VFA are works in progress...)
>
> The setup is that each faction has the same maximum possible size. What
> introduces randomness is that only a random proportion of each faction will
> show up to vote in each poll. I've been doing 500 polls between each
> opportunity for a faction to reconsider their vote, and sometimes factions
> are forced to wait awhile (so that poll results don't drastically change
> overnight).
>
> For each round of polls I can also note the rate of agreement between the
> poll winner and the sincere CW (if there is one), sincere MF (if there is
> one), and social utility maximizer given the voters who show up.
>
> The strategy used varies by the method. Approval is the easiest:
> Expectation can be perfectly calculated from the previous round of polls.
> Approve above expectation.
>
> FPP, antiplurality, and combinations involve estimating the value and
> likelihood of all ties that could be broken from a given vote.
>
> TTR is similar but harder (a vote is effective if it moves a candidate into
> second place in the first round; tricky to guess the value of the vote and
> also which probabilities to compare; for example if a candidate is 100%
> guaranteed to come in first or second then there is no value in voting for
> him at all).
>
> VFA is quite tricky, even though the "for" and "against" votes can be
> determined independently. The value of a "for" vote for A over B needs to
> consider the ability of each to place first, the likelihood that either
> will be disqualified, and ideally also the likelihood that a different
> candidate is disqualified and A could beat B only placing second (but here
> I face a apples-oranges problem when I want to add this in). The "against"
> vote currently considers the expected benefit from disqualifying the
> candidate, and not even the observed odds of disqualification (though it
> should be related when everyone thinks along the same lines). I used to
> consider the observed odds, but this created a problem where voters felt
> they should only vote against a candidate who might be disqualified, no
> matter how good that candidate was.
>
> For this post, I decided to plug this scenario into each method:
>
> 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)
>
> FPP: This immediately almost seems to be at equilibrium with everyone
> voting for their favorite candidate. With everybody having equal odds,
> nobody feels compelled to move their vote.
>
> Eventually, though, some candidate has a bad run and one of their factions
> goes somewhere else. And from there other candidates drop like flies.
> Ultimately the equilibrium usually ends up being A vs. C, split down the
> middle. But if A or C is the first candidate to fall, B will take their
> place, and have very high win odds.
>
> In one case I saw B take the votes from all the factions except for D and
> E, who continued to vote for their candidates. Eventually D had a bad run
> and those voters moved to B. At this point E's win odds dropped to nothing,
> and factions started to go back to voting for their sincere favorites again
> since there was no reason to vote for a guaranteed winner. In this
> particular case the reversion to the original scenario was total.
>
> Antiplurality: This method is a mess. At first everyone votes against D or
> E, the two most extreme candidates. This results in a large tie in odds
> among ABC. So voters move their votes to the most extreme of these: A and
> C. Then B wins with 100% odds (not everybody changes strategy at once).
> Then B will start to get votes (due to it being one of the only meaningful
> options), or perhaps A and C will start to lose votes (since voting against
> them is pointless if they never win), putting them back in the running.
> Over many runs of polls, B's win odds bounce around from nothing to 100%.
> No equilibrium can be reached.
>
> Approval: Starts out with the factions voting DA, ABD, BA, BC, CBE, and EC.
> B wins 60% of the resulting polls with the rest to A and C. ABC voters
> start dropping a compromise or two in response. The main effect is that A
> and C lose votes from the B voters, causing B's win odds to get even
> higher. A and C voters may also try dropping support for B, but eventually
> one of them won't do very well and they will put the B vote back. This
> ultimately kills their candidate because there's no way to win when all
> their voters back B, and B voters give nothing back.
>
> The equilibrium ends up being DAB, AB, B, B, C, EC, or else the reverse. B
> wins maybe 85% of the polls with the rest going to C in this case.
>
> I should note that with random scenarios, I was a bit alarmed at how
> frequently Approval never reached equilbrium at all, as factions ran around
> in circles offering and retracting compromises however they felt they were
> able.
>
> Top-two Runoff: Not much fun. It starts out at equilibrium, with every
> faction content to just vote for their favorite in the first round. B wins
> 35-40% of the polls, and D and E win well under 10% each.
>
> However, if I drop the number of polls from 500 to 200, we can make it more
> likely that a faction gets scared and moves. This also makes it unlikely
> that we can get a different equilibrium, though. I'm currently looking at a
> fairly stable situation where the D factions vote sincerely, the A and B
> factions vote for A, and the C and E factions vote for E. A wins the polls
> over 80% of the time, D only a few percent; A and E each make the second
> round over 90% of the time.
>
> The tranquility is disturbed when the A>B>D faction moves their vote from A
> to D. It's a combination of confidence that A will place, and desire to
> have D rather than E in the finals if possible. But the result of the
> strategy change is a big hit to A, and the faction changes their vote back
> to A at the next opportunity.
>
> Running through many cases this is the only hiccup I see. In one case, in
> response to this strategy change, one of the C>B>E>A>D factions moves their
> vote from E to A. But it doesn't last long.
>
> Running quickly through over and over from scratch, these situations also
> show up as being at equilibrium for long stretches:
> DDAAACCCEE, DDDDCCCCEE (reverse of above), AAAABBCCCC.
>
> VFA: We start with everyone voting for their sincere favorite, and against
> either D or E. Always either D or E is then projected to be disqualified,
> hurting their win odds. Immediately D and E voters start moving in,
> switching their "for" votes to A and C. This then causes the "against"
> votes to follow to A and C, since it's pointless to vote against candidates
> that no one is voting for. B does not fare well, since the D and E factions
> don't favor B. Thus the B voters are likely to move to A or C as well.
>
> It tends to get close to an equilbrium where only two candidates receive
> "for" or "against" votes. But as soon as this is guaranteed, factions
> (usually multiple) try to move their "for" vote to their sincere favorite,
> since there's no reason not to. So far it seems to me that rather than
> introduce a third contender, this mainly introduces a lot of chaos because
> everybody's trying to do it.
>
> This set of votes seems to be an equilibrium though: "For" votes for
> AAAABBCCCC, "against" votes for CCCCCAAAAA. B hardly ever wins this, but
> his presence is enough to keep any fourth candidate from finding support.
>
> I have noticed just now that with this situation, if A does particularly
> badly in a given run, some of their voters may move their "for" vote to B.
> This makes B dominate, can obliterate A, and makes C voters waffle as to
> which candidate to vote against. But if eventually A is gone and C voters
> decide on voting against B, we again reach the chaotic situation where
> factions try to vote for their sincere favorites again.
>
> I should reiterate that my voter AI code for TTR and VFA may be poor. But I
> see behavior that mostly makes sense. (It does sometimes happen that my VFA
> voters decide to vote for and against the same candidate. The logic that
> goes into a "for" vote vs. an "against" vote is independent and not very
> similar, so I'm afraid this bug may always exist.)
>
> I plan to play around with different scenarios in the future. I'm not sure
> I will implement any new methods. (Rank methods are out unless the range of
> available strategies is extremely limited, and not difficult to evaluate.)
>
> Hopefully this is a somewhat interesting read. Any thoughts?
>
> Kevin Venzke
>