[EM] Some Approval strategies

[Michael Ossipoff]

WS, in his paper,mentioned a strategy that Rob LeGrand calls Strategy A, and 
which I call BF-Certain.

"BF" stands for "Best Frontrunner".

I'll get to this more, soon, but, for voting in an Approval election, that 
strategy is only any good if it's fairly certain which of the two expected 
frontrunners will outpoll the other. Maybe sometimes you know that the 
biggest votegetter will, more likely than not, outpoll the runner-up, in the 
next election. But you mgiht not know how probable that is. Is it a sure 
thing, or is he just barely more likely to outpoll? That's also the 
situation for the simulated voters who  use previous vote totals to inform 
their strategy in a simulated series of Approval elections.

BF-Uncertain (to be defined below) is optimal for the extreme case when you 
have no idea which frontrunner will outpoll the other. Then, of course, 
BF-Uncertain is a much better choice than BF-Certain.

But, when you don't know how likely it is that the biggest votegetter will 
outpoll the runner-up in the next election, and if you had to choose 
BF-Certain or BF-Uncertain, which would be better?
BF-Uncertain would be much better. That's because BF-Certain is optimal only 
under the conditions under which your vote has the smallest probability of 
influencing the election--when it's nearly certain which candidate will win.

I've been talking about these BF strategies without defining them.

What they all have in common is their assumption that it's nearly certain 
that the winner will be one of the two expected frontrunners (determined 
maybe by the previous election's vote totals, or from polls, if you trust 
the polls--a big "if"). So, in their voting, what they all have in common 
is: Vote for the frontrunner you like better, and don't vote for the other 
frontrunner. Also, vote for everyone you like better than the frontrunner 
you like better. (That last sentence is for the tiny chance that someone 
other else could be in a tie or near-tie with a frontrunner. Of course, 
also, it's desirable to show support for the best candidates).

Where they differ is in which, if any, of the "inbetween candidates" you 
vote for. The inbetween candidates are the candidates whose merit is between 
those of the two frontrunners.

Why vote for inbetweens at all, or for anyone other than your favorite 
frontrunner, if it's certain that one of the frontrunners will win? I said 
only that it's _nearly_ certain that one of the two frontrunners will win. 
The important thing is voting for your preferred frontrunner and not for the 
other. From BF's main assumption, it's nearly irrelevant whether you vote 
for an inbetween. But, as a refinement, one can vote for some inbetweens 
just for the tiny chance that one of them might be in a tie or near-tie with 
one of the frontrunners.

That's why there are several BF versions.

Of course, since the inbetweens matter so little, there's nothing wrong with 
this advice to voters:

Vote for whichever frontrunner you like better than the other, and for 
everyone better than hir. Or, which almost amounts to the same thing, vote 
for the candidate you'd vote for in Plurality, and for everyone better than 
hir. The first of those could be called BF-Plain, and the 2nd could be 
called Plurality-And-Better. They're probably the best BF versions to offer 
the public.

Now to define the other BF versions, in terms of votes for inbetweens:

BF-Certain:

If your favorite frontrunner is the one almost sure to outpoll the other, 
then don't vote for any inbetweens. If your less-favorite frontrunner is the 
one almost sure to outpoll the other, then vote for all the inbetweens.

(This is equivalent to the strategy that WS described, and that Rob LeGrand 
calls Strategy A, except that it's defined when the two frontrunners have 
polled equally, and there's no top votegetter).

BF-Certain is optimal when it's certain which frontrunner will outpoll the 
other.

BF-Uncertain:

Vote for all the inbetweens whose merit is better than the mean of the two 
frontrunners.

That can be shown to be optimal when you have no information about which 
frontrunner will outpoll the other.

BF-Intermediate:

Use an Approval cutoff halfway between those that would be recommended by 
BF-Certain and BF-Uncertain. It could be argued that this minimizes how far 
off your Approval cutoff can be from where it should be. But, because 
BF-Certain is optimal when your vote counts the least, there's a case for 
saying that BF-Uncertain, or something closer to it would be better. Maybe, 
when all you know is that your favorite frontrunner's probability of 
outpolling the other frontrunner is not 0 or 1 or .5 (it isn't .5, because 
he's ahead or behind in the polls), the best place for the Approval cutoff 
is somewhere between BF-Intermediate and BF-Uncertain.

BF-Probability:

Say you have an estimate of the probability that your favorite frontrunner 
will outpoll the other. I'll call that probabililty Px.

It can be shown that, then, it's optimal to vote for all the inbetweens 
whose merit is Px of the way from that of the worse frontrunner to the 
better frontrunner. In other words, say you feel that Px is .67  There's .67 
probability that your favorite frontrunner will outpoll the other. Then vote 
for all the inbetweens whose merit is at least 67% of the way from that of 
the worse frontrunner to that of the better frontrunner.

Px could be estimated, or it could be calculated based on the typical 
dispersion of candidates' vote totals in previous elections, and the 
difference between the frontrunners' vote totals in the previous election.

Or, alternatively, maybe a table of Px estimates could be prepared from a 
simulation. The applicability of the simulation to actual conditions would 
of course be very questionable, but it might be better than nothing. 
Likewise, a simulation might be used to estimate where the Approval cutoff 
should be positioned between that of BF-Uncertain and BF-Intermediate.

The BF strategies are good when you have evidence or a feel for who the 
frontrunners are. When you don't, you could use the Better-Than-Expectation 
strategy:

Vote for each candidate who is so good that you'd rather have hir in office 
than hold the election.

(In other words, vote for everyone who is better than your expectation for 
the election).

That's been shown to be optimal given certain fairly reasonable assumptions. 
At least two different sets of assumptinos lead to that strategy, giving it 
extra confirmation. i posted about that around February of 2005.

But I emphasize that usually you'll know who deserves your approval and who 
doesn't. Rarely would you need these strategies. Who is good enough, 
deserving enough, honest enough? Or who do you just feel you need to 
compromise down to? The strategies described above are only for when you 
don't have such a feel. Maybe unfamiliar candidates or unimportant election. 
Or maybe the strategies could help suggest the right vote configuration for 
the participants in the co-operation-defection dilemma that I'll post about 
soon. The co-operation-defection dilemma is probably the nearest thing to a 
problem for Approval, but I tell why i claim that it isn't a problem.

Mike Ossipoff