[EM] The chicken dilemma, median systems, and the pivotal voter

[Jameson Quinn]

I have argued before that median systems like
MJ<http://wiki.electorama.com/wiki/MJ>and
GMJ <http://wiki.electorama.com/wiki/GMJ> are more resistant to chicken
dilemma pathologies than most other systems*. The various arguments I've
made all come from the same underlying dynamics, but they express the
matter in different terms. Here's yet another argument along those lines:

Imagine a median system with an infinite (continuous) spectrum of possible
ratings, say, rational numbers from 0 to 100. To make sure there's no silly
"who can name the smallest number" dynamics, disallow votes between 0 and
1, so that 1 is the unique second-lowest rating (in other words, like any
seriously-proposed median system, the set of ratings minus the lowest
ratings is a closed set.) Now take a simple chicken dilemma scenario:

40: X>Y=Z
25: Y>Z>X
35: Z>Y>X

Let's define the "honest" votes of the latter two groups as rating the
favorite candidate at top, X at bottom, and the other candidate somewhere
in the middle. But because there are an infinite number of possible votes
in the middle, we can assume that no two voters will put the middle
candidate at exactly the same place.

If there's an odd number of voters, then, the precise value of Y's and Z's
medians will be chosen by exactly one pivotal voter each, while X will have
a median of 0.

Also, note that if the distributions of preference strength are similar,
then the pivotal voter for Z will have a higher rating than the one for Y.
For instance, if it were an even distribution 50-100, Z would get 70, while
Y would get 64.29 (9/14)

What does that mean for the strategic dynamics of the chicken dilemma? It
means that, in a very real sense, those two pivotal voters are the only
ones under "strategic pressure". Anybody who cares about the Y/Z
distinction more than the pivotal voter in their faction does not matter,
because those people can strategize all they want without changing the
medians. And anyone who cares about the Y/Z distinction less than the
pivotal voter in their faction, has less of a motivation to strategize than
the pivotal voter in their faction, so they would look to that pivotal
voter to start to strategize before them.

So what is the strategic situation for those two pivotal voters? Well, as
long as their honest vote is above the second-lowest rating "1", they can
safely strategize, lowering their middle rating in order to improve their
favorite's chances. But once their honest vote falls to 1, they are in a
game of chicken: they want to defect if they expect the other one to
cooperate, but cooperate if they expect the other one to defect.

So, that's just the standard chicken dilemma then, right? No; not at all.
By reducing the situation from a collective game of chicken into a game of
chicken between two, individual but anonymous, pivotal voters, the
strategic characteristics are vastly improved. In order to get a "healthy"
resolution to the chicken dilemma with Score or Approval with rational,
strategically-inclined voters, you must assume strategic fractional rating
(SFR), which requires voters to have both very precise measures of the
percentages involved, and a uniformly high level of mathematical
sophistication in order to calculate and carry out the SFR strategy. I
mean, I bet if you asked any three people on this list to do the
calculations, at least one of them would make slightly different
assumptions or some mistake in the calculations; and honestly, the three
most-mathematically-sophisticated members of this list are easily in the
top 1 or 2 percent of math for the population. Any such difference could
easily lead to a strategic breakdown. Meanwhile, if it's just a two-person
game, the simple zero-knowledge strategy is to cooperate if your honest
rating is over 50.

So with this continuous-spectrum median system, the strategic result would
be a tie for Y and Z with a median of 1. What does that translate to in
real proposed median systems such as (G)MJ? Well, first, the idea of
"pivotal voter" still has meaning. Of course, there will be a group of
voters who give the same median grade to Y, and another group for Z. But
among that group, there will be some who are "more motivated" to strategize
because their honest rating is on the low side of that grade, and others
whe are "less motivated". So if they committed to strategize in order of
motivation, there would still be one voter whose strategy could drop the
candidate's median to 0 and thus risk a win by X.

Second, in (G)MJ, the tiebreaker would function to make Z win if all voters
used subcritical (second-to-bottom) strategy. And so voters might
reasonably ask themselves: what's the point of all that strategy? If Z is
going to win with or without strategy, or indeed with any symmetric level
of strategy, why not just save myself the trouble of strategic thinking and
vote all candidates honestly? This is what I believe, and suspect my
research will show, will happen.

I'd be happy to further explain this argument if anyone has questions. And
I'm wondering if people think that this could be simplified enough to make
an argument like this with non-voting-system-enthusiasts. I'd love to hear
any comments.

Jameson

*By "most other systems", I mean of course Approval and Score most
obviously. But I'd actually include a number of other systems there. The
key idea in that respect is that many voting systems have strategic
dynamics which lead significantly different situations to give
indistinguishable ballots. I'm talking about the "honest chicken dilemma",
with utilities something like:
40: X100 Y0 Z0
25: X0 Y100 Z80
35: X0 Y80 Z100

versus the "one-sided pseudo-chicken", which is the same except that Y
voters like Z significantly less than vice versa, and so Y instead of Z is
the utility maximizer:
40: X100 Y0 Z0
25: X0 Y100 Z10
35: X0 Y80 Z100

Even systems that do not exhibit any strategic Chicken pathologies do not
handle both of these situations correctly. For instance, in IRV or
Condorcet-margins, there is no way for Y to win, even in the second
scenario where they "should". Meanwhile, (G)MJ can get the right answer
with "honest" votes, and is unlikely to show a vicious cycle of strategic
escalation for the reasons above. So although bad behavior under
pseudo-chicken is not the same as the chicken dilemma per se, to me it's
related, and so I think that MJ has a valid claim to beat even IRV in this
sense.
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