[EM] Nash Equilibria, CR, and Approval

[Alex Small]

Since the issue of CR and Approval being strategically equivalent has come
up here a bit in the last week or so, here's another attempt to resolve
the issue:

If all Nash equilibria for CR involve giving every candidate either zero
or full points then Approval and CR are strategically equivalent.  The
methods are also equivalent if a faction of votes with identical
preferences (treated as a single player for these purposes) is indifferent
between the outcome obtained by giving a candidate an intermediate rating
and the outcome obtained by giving that candidate the top or bottom
rating.

Here's a sketch of the argument, but maybe not a rigorous proof:

Say that we have a Nash equilibrium where candidate B wins, and a faction
which prefers A to B has given A an intermediate rating.  If that faction
alone could cause candidate A to win by giving A the top rating then we
wouldn't have a Nash equilbrium.  Conversely, if that faction cannot cause
A to win by giving him the top rating, then the faction is indifferent
between giving him an intermediate rating versus giving him either the top
or bottom rating.

Also, if candidate B wins, but A would win if a faction preferring A to B
gave B a zero rating instead of an intermediate rating, then we wouldn't
be at a Nash equilibrium.  Moreover, if that faction cannot cause A to
lose by giving B a zero rating then the faction is indifferent between
giving him an intermediate rating versus giving him either the top or
bottom rating.

So, I now concede the strategic equivalence of Approval and CR when
payoffs are considered for a single election only and voters have perfect
information.  However, I still maintain that payoffs calculated over
multiple elections may give incentives to give intermediate ratings, and
that intermediate ratings may be optimal in some rare cases with imperfect
information.

The long-term case:  I used the Greens as an example.  I argued that if
the Greens give the Democrats an intermediate rating, they are a valued
constituency that cannot be written off easily.  Moreover, they have both
a stick to punish transgressions and a carrot to entice further
concessions from the Dems.

Mike argued that if the Greens want to influence the Dems they should use
the biggest stick possible, and what stick is bigger than a zero rating?

My reply:  First, consider a person from a GOP core constituency who
regularly gives money to the GOP.  This person would presumably give the
Dems a zero rating in CR elections.  He is using the biggest stick
possible.  Apart from a few old-school conservative Southern Democrats,
most Dems will write this person off as unreceptive and seek support from
more moderate or liberal voters.

Now consider Greens who consistently refuse to support the Dems.  After a
few elections the Dems may write them off as unreachable, just as they
disregard conservative constituencies.  Greens then have no leverage over
the Dems, because they can't make their stick any bigger and the Dems
think that pursuing the Green carrot is a waste of time.  However, if the
Green give partial support they have a stick that can always be made
larger and a carrot that interests the Dems.

Note:  I use the Greens and Dems only to illustrate a strategic question,
not to take sides in the debate between Greens and Dems.  If a Green
believes that the Dems are unreformable then withholding the carrot and
using the biggest stick is of course the rational and optimal strategy. 
The above argument is predicated on the assumption that both parties are
capable of finding common ground, an assumption that fails if one of the
parties is unreformable.




Alex


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