[EM] Re: simulating an Approval campaign/election

[Rob LeGrand]

Russ Paielli wrote:
> Here's what I modeled. I have three candidates only. I randomly
> generate votes, with equal probabilities for all six possible
> preference orders. The only control variable for each vote is
> where the voter "draws the line." In this case, that amounts to
> whether or not the voter approves the middle candidate of his
> preference list. I initialized the middle-candidate state of each
> vote randomly, with an expected mean of half approved and half
> not.
>
> Then I started an iterative simulation of polling cycles and
> voter re-evaluation of his vote. I simply assumed that complete
> and perfect polling data is available to every voter. Then I have
> each voter re-evaluate his approval/disapproval of his middle
> candidate based on Forest Simmons elegant strategy rule (special
> case for three candidates only): if the voters first choice has
> more votes than his third (last) choice, the middle candidate
> does not get approved, but if the third choice has more votes
> than the first choice, the middle candidate gets approved (if
> they are equal I leave it unchanged).

You're simulating a DSV (Declared-Strategy Voting) election with
Approval.  My current research is on just that topic, though I'm
also interested in using DSV with other point-count systems such as
plurality, Borda and several others.  That Approval strategy is
identical to strategy A in the 3-candidate case.

> I then let the system iterate until the vote counts stabilize
> (i.e., repeat without changing).

It's not clear when you update the poll that the voters see.  If
all voters vote at once, then the poll is updated, then all voters
vote again, etc., that's equivalent to Approval DSV in batch mode.
If the voters are given chances to change their votes in some order
(random or not) and the poll is updated after each vote change,
that's like Approval DSV in ballot-by-ballot mode.

> I varied the number of votes from 100 to 10,000. The program runs
> very fast, and I could increase the number of votes much higher
> eventually. If the counts do not converge within 100 iterations,
> I arbitrarily declare non-convergence (for now).

With 3 candidates, that's a safe declaration.  DSV in cumulative
batch mode can sometimes take a lot longer to converge but it gets
trapped in a loop less often (if at all).

> The first few runs I tried showed rapid convergence within a
> cycle or two. Then I wrapped the whole thing in another loop to
> simulate many separate elections. I found that most of them
> converged within 2 or 3 iterations. However, roughly 1 in 10
> fails to converge either to a stable vote count or a stable
> winner.

1 in 10 agrees with Merrill's figure: 91.6% of random elections
with 3 candidates and 25 voters have a Condorcet winner.  You used
more voters, but that would decrease the percentage only very
slightly.  Actually, Approval DSV in batch mode using strategy A
doesn't always converge even when there's a Condorcet winner, but
the examples are quite contrived and require more than 3
candidates.  Ballot-by-ballot mode, when the voter order is weakly
fair (no voter is shut out for more than 2n steps, say), always
finds an equilibrium eventually in my simulations.  There's always
a path of changes that leads to an equilibrium, anyway.  When no
Condorcet winner exists, strategy A can't lead to an equilibrium
because any poll leader can and will be toppled.

> So the bottom line is that, even in the simplest, most idealized
> case, Approval Voting can be unstable. In such cases, the
> ultimate winner would essentially be a random function of when
> the election happened to be held. A sort of random lottery. And
> many voters would regret their decision.

Any voting system for which you can't say the same (like plurality)
is easily manipulated and leads to multiple equilibria, some of
which may not elect an existing Condorcet winner.  If you find
convergence more important than competitive elections and sincere
voting, you may prefer plurality to Approval.  But I see
plurality's many equilibria as false ones that hide much about the
electorate's wishes.  Approval only fails to converge when the
electorate's wishes are collectively irrational, in a sense, and in
that case Approval will eventually cycle only among the sincere
Schwartz set.

Note that all Condorcet-compliant ranked-ballot voting systems are
sometimes manipulable and nonconvergent when there's no Condorcet
winner.  Some prefer Condorcet methods to Approval because they see
them as harder to manipulate and thus more stable, but I'd rather
voters know the rules of the game they're playing.  Alex Small
wrote on the ApprovalVoting list:

> Legitimacy should come from a transparent connection between the
> decisions people make in the voting booth and the final outcome.
> If it takes a game theorist to sketch out a flow chart and
> explain why voting for A allowed B to win, how much respect will
> the system command?
>
> That's actually one reason why I like Approval Voting:  Although
> there are sometimes risky decisions to be made (do I approve my
> second choice or only my first?  Do I risk my least favorite
> winning or risk hurting my favorite?), at least the cause and
> effect is clear.  We won't need a game theorist with a flow chart
> to explain things to us the next morning.

I second that.  Besides, Approval can make a sincerity guarantee
that no ranked-ballot system can:  You should always vote the
maximum for your favorite candidate and the minimum for your least
favorite.  If all you're given is poll information, you should
never vote for B and not for A when you prefer A to B; it never
pays to express a false pairwise preference.  I still haven't found
another system that has that property of weak sincerity.

Anyway, the point is that I think Approval has the best combination
of manipulation-resistance, convergence and quality of winners, not
to mention simplicity.  A little divergence is worth the better
equilibria.

=====
Rob LeGrand, psephologist
rob at approvalvoting.org
Citizens for Approval Voting
http://www.approvalvoting.org/

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