[EM] Strategy and Bayesian Regret

[Jameson Quinn]

What makes a single-winner election method good? The primary consideration
is that it gives good results. The clearest way to measure the quality of
results is simulated voter utility, otherwise known as Bayesian Regret (BR).

This is not the only consideration. But for this message, we'll discount the
others, including:

   - Simplicity/voter comprehension of the system itself
   - Ballot simplicity
   - Strategic simplicity
   - Perceived fairness
   - Candidate/campaign strategy incentives


Calculating BR for honest voters is relatively simple, and it's clear that
Range voting is best. But how do you deal with strategy? Figuring out what
strategies are sensible is the relatively easy part; whether it's
first-order rational strategies (as James Green-Armytage has worked
out<http://www.econ.ucsb.edu/~armytage/svn2010.pdf>)
or n-order strategies under uncertainty (as Kevin Venzke does) or even just
simple rules of thumb justified by some handwaving (as in Warren Smith's
original BR work over 10 years ago), we know how to get this far. But once
you've done that, you still have to make some assumptions about how many
voters will use strategy. There are several ways to go about this. In order
of increasing realism, these are:

   1. Assume that voters are inherently strategic or honest and do not
   respond to strategic incentives. Thus, the number of voters using strategy
   will be the same across different systems. Warren Smith's original BR work
   with IEVS seems to have shown that Range is still robustly best under these
   conditions. Although I am not 100% convinced that his definition of strategy
   was good enough, the results are probably robust enough that they'd hold up
   under different definitions.
   2. Avoid the question, and just look at strategic worst cases. I count
   this as more realistic than the above, even though it's just a special case
   of 100% strategy, because it doesn't give unrealistically-precise numbers.
   But of course, if I say that method X has a BR score somewhere between Y and
   Z, and method A has a BR between B and C, if Y<B<X<C I cannot conclude that
   X is better than A. So you lose the ability to answer the important
   question, "which method is better?"
   3. Try to use some rational or cognitive model of voters to figure out
   how much strategy real people will use under each method. This is hard work
   and involves a lot of assumptions, but it's probably the best we can do
   today.
   4. Try to get real data about how people would behave in high-stakes
   elections. This is extremely hard, especially because low-stakes polls may
   not be a valid proxy for high-stakes elections.

As you might have guessed, I'm arguing here for method 3. Kevin Venzke has
done work in this direction, but his assumptions --- that voters will look
for first-order strategies in an environment of highly volatile polling data
--- while very useful for making a computable model, are still obviously
unrealistic.

What kind of voter strategy model would be better? That is, what factors
probably affect a voters' decision about whether to be strategic? I can
think of several. I'll give them in order from easiest explanation to
hardest; the order below has nothing to do with the relative importance.

First, there's the cognitive difficulty of strategizing versus voting
honestly. In a system like SODA, an honest bullet vote is much simpler than
a strategic explicit truncation, so we can expect that this factor would
lead to less strategy. In a ranked system, it is arguably easier to
strategically exaggerate the perceived frontrunners (Warren's "naive
exaggeration strategy" or NES) than to honestly rank all the candidates, so
we might expect this factor to increase strategizing. Note that the
cognitive burden for strategy is reduced if defensive and offensive
strategies are the same. For instance, under Range, exaggeration is always a
good idea, whether it's offensively or defensively.

(Note: This overall cognitive factor is probably most important for "lazy
voters", and such "lazy voters" are also probably open to strategic and/or
honest advice from peers, so the cognitive factor is perhaps not too
important overall.)

Second, there's offensive strategy. The more likely it is that strategy will
be advantageous against honest opponents, and the more advantageous it is
likely to be, the more strategy people will use. The first question has been
addressed by the Green-Armytage
paper<http://www.econ.ucsb.edu/~armytage/svn2010.pdf>;
it appears that IRV is relatively strategy-resistant, Condorcet is middling,
and Range and Approval are likely to be subject to strategy. But remember,
the whole point of this discussion is that strategy is not so much a problem
in itself, as an input to the model for determining BR. If Approval gives
better results under 100% strategy than IRV does with 0%, then Approval is
still a better system.

Third, there's defensive strategy. Basically, this means looking at the
probability that the result will be subject to strategy from some other
group, and seeing if you can defend against that.

Fourth, there's peer pressure. If you feel that everyone else is
strategizing, you are more likely to do so yourself. This raises the
possibility of positive feedback and multiple equilibria.

It is crucially important to understand that defensive strategy is not like
offensive strategy in terms of peer pressure. If you think that your allies
are unlikely to back you up on your offensive strategy, you may decide it's
pointless to attempt it. But some people will use defensive strategy merely
as insurance. Thus, there is more likely to be a "floor" for defensive
strategy, a certain number of people who use it even if nobody else is. But
it is also true that the more people use strategy, the more people will
worry about defensive strategy. Thus, a method where defensive strategies
are likely to be possible is more likely to be driven to a high-strategy
equilibrium, than one where only offensive strategies are an issue.

So, what does all this mean for BR calculations? Well, first, we should try
to characterize the different systems in terms of the first three factors
above. For the cognitive factor, can we develop some objective measure of
how cognitively difficult it is to work out a good strategy under different
systems? For the offensive strategy factor, we can thank Green-Armytage for
making a good first step in giving the *probability* of strategic
vulnerability, but we should follow up by working out the *amount* of
strategic advantage a voter could expect. For the defensive strategy factor,
Kevin Venzke's work gives some interesting clues, but more work is needed to
isolate defensive factors.

But even once we have all that nailed down, we need a voter model to turn it
into a BR measure for each system. Of course, any such model will be open to
accusations of bias, as it will include varying amounts of strategy under
different voting methods. Range voting advocates in particular might be
motivated to assume that strategy percentage will be the same under
different systems. But it's important to undersand that no assumption here
is unbiased; without real-world data, assuming equal strategy is at least as
biased as a model which accounts for the factors above.

So in the end, I'm inclined to bite the bullet, and make arbitrary
assumptions for now. I'd guess that a method where the factors above favor
strategy --- for instance, Range voting, where all three of the
a-priori-quantifiable factors favor strategy --- would lead to a high degree
of strategy, something around 75%. Meanwhile, a method where the factors
favor honesty --- such as, I'd argue, SODA --- might have a low amount of
strategy, something under 25%. Something like Condorcet or Majority Judgment
has the factors somewhere in the middle, but it would be harder to guess
what that would mean in practice; I'd guess that peer pressure feedback
would mean that either <25% or ~75% would be more likely than an unstable
middle value like 50%.

Again, the end "quality" number is not strategy percentage, but the
resulting BR. So even if range does lead to more strategy than any other
system, it could still end up being the best system. I'd like to see real
numbers on this. Any assumptions will be biased, but that doesn't make the
numbers useless.

Jameson
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