[EM] Re: Approval cutoff AKA "None of the Below"

[Ted Stern]

On 12 Mar 2005 at 05:04 PST, Jobst Heitzig wrote:
>Ted wrote earlier:
>> -- the Nash equilibrium will occur with
>> 
>>       40: A>>B>C
>>       35: B>>C>A
>>       25: C>>A>B
>> 
>> which again has B winning.
>
> Why is that? Can you argue for this equilibrium in more detail? It was
> always my impression that in most situations there is either no
> equilibrium of an interesting kind, or many of them, so why is there
> exactly one in this situation? And what definition of "Nash equilibrium"
> do you use, are the players (i) the individual voters, (ii) groups of
> voters with identical sincere preferences [and approvals], or (iii)
> arbitrary groups of voters?

I'm assuming (ii), though of course this isn't entirely realistic.

Starting from the example with more generous approval cutoff, and examining
what happens if one block (say the A's or C's) attempt to gain an advantage
by not approving their 2nd choice.

If the election were repeated, the two other blocks would have no reason to
trust either of the other groups.  Like 3-way rock/paper/scissors.  Each of
them would withdraw to the more mistrustful position and the result would be
no better than before.  But at least no worse.

I call it a Nash equilbrium because it's the strategy that eventually will
stabilize due to blocks attempting to gain best advantage for themselves.
Kind of like the way bidders on eBay have an incentive to bid at the very last
second ("sniping") to avoid betraying their interest.

Any time you want to augment Condorcet with Approval, you need to consider how
to prevent truncated approval, since that is its weakest point.

In my just-proposed Total Approval Beatpath (which may have been proposed long
ago), there is no benefit to "bullet" approval cutoff.  Individuals or Blocks
actually have a stronger incentive to put the cutoff below the 2nd or 3rd
choice -- you want candidates you rank slightly lower to be about as high in
the ratings as your favorite, in order to get your softest opponents into
the approval ranking nearby.

Ted
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	ted stern at u dot washington dot edu

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