[EM] A fix for overlenient equilibrium definition

[MIKE OSSIPOFF]

Richard objected that my definition of voting Nash equilibrium isn't
useful for public elections because members of a faction don't all
have exactly the same sincere rankings and don't all vote the same
way. But I didn't use the word "faction" in my definition. It's common
for there to be some voters whose sincere rankings are the same and
who vote the same.

But the reason why someone could say that my voting Nash equilibrium
definition is overlenient is that maybe, if there are 20 candidates,
and everyone has preferences among all of them and ranks completely,
there are less likely to be groups of same-sincere-ranking and
same-voting voters who are big enough to change the outcome at all.
My definition would call that an equilibrium, and that's too lenient.

So how about this?:

First, for brevity, let me say that a "similar set" is a set of voters
who share the same sincere rankings and who vote the same way.

Since these comparisons are used like a criterion,
and since criteria are usually written so that there must not be
even one failure, and so they're written to emphasize failure situations
instead of successes, why not say that a voting Nash non-equilibrium
is a situation (configuration of candidates and voters' sincere rankngs)
in which that there's at least one sincere set that could change the
outcome if it changed its vote, and in which some similar set could
improve its outcome if it changed its votes.

Now, what I was saying about the margins methods, Plurality, and IRV
could be said:

With the margins methods, IRV, and Plurality, there are situations
in which any votes-configurations that don't include defensive
order-reversal are voting Nash non-equilibria.

That isn't true of Approval or Condorcet(wv).

"Voting-Nash non-equilibria" may be too wordy. Why not just say
"voting-Nash-unstable".

Maybe an outcome where _some_ set of voters could gain by changing
their votes could be called an outcome that is "broadly unstable",
because it's a broader instability definition that calls more outcomes
unstable.

Margins, IRV, & Plurality have situations where all the nonreversed
outcomes are broadly unstable and voting-Nash-unstable.

Mike Ossipoff






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