[EM] Approval and cycles

[rob brown]

At first, one might argue that Approval is not subject to cycles as other
methods, such as Condorcet methods are.  From a "vote counting" perspective,
it is not.  You add up all the votes, and unless there is a true tie, it is
completely unambiguous who wins.  Very nice and clean.

After my little "thought experiment" that turned out to be a rediscovery of
a form of DSV, I have tentatively* concluded otherwise.  I think Approval is
still subject to cycles, it has just moved those cycles out of the vote
tabulation system, and put them into each individual voter's head.

Let's start with a few (obviously unrealistic) assumptions.  First, let's
assume that each voter is rational, and, disregarding strategy, is able to
assign each candidate a rating from 0 to 100.   Let's also assume that each
voter has perfect knowledge of what every other voter's preferences are, and
that each voter has perfect intelligence, therefore always picking the
optimum strategy given his knowledge.  Finally, we'll assume that each voter
knows that all the other voters share these characteristics.

While these assumptions are unrealistic in the real world, that's not the
point.  I am trying to model this as a Game Theorist would.  Using the DSV
model, we can force the situation to be -- for all intents and purposes --
true, since it is not the voter's themselves choosing the strategy, it is a
software agent working on the voter's behalf.

So, in this "perfect" model of Approval voting, the system fails to reliably
pick a winner, but can end up in a cycle just like a Condorcet cycle.  In
the real world, Approval voting solves this problem by introducing
semi-random slop into the system....allowing each voter to convert their
preference into a strategic Approval ballot, using their imperfect
information and their imperfect intelligence.

I think the DSV model should always be "preferred" to Approval, since it can
eliminate the unpredictable strategizing that voters' must do to vote
effectively under Approval.  But if it is not able to find solutions to
cycles, then it is flawed.  By extension, Approval has the exact same flaw,
but covers it up with semi-random slop.

*note:  I write this with the assumption that the DSV/Approval method is not
able to eliminate cycles in any straightforward way.  On my previous posts,
I was obviously not convinced this was true. I'm still not sure.  But it
certainly seemed to be where that discussion was going.

-rob
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