What are we all about?, etc.

[Richard Moore]

Back from a week's absence I find a lot of EM posts in my inbox. I'll
combine all my comments into a single response.

1. Thanks, Forest, for founding EMAC (July 23). I accept the 
invitation to be a charter member, though I think time constraints 
will keep me from playing a significant role. I hope someone steps up 
to become an organizing force.

2. On the "one person, one vote" argument against approval voting by 
James Gilmour (July 23), I don't know if the person who coined the 
phrase ever gave it an exact meaning. I suspect he (she?) meant "one 
person, one ballot", with the understanding that all ballots should 
have equal weight in the counting of the votes.

I suspect the equal weights of ballots are what Joe Weinstein was 
referring to when he wrote "of equal inherent power". It seems to me 
that "voting power" can be defined in different ways, too, so I would 
like to avoid that term if possible. But to be fair to Joe, his use of 
the word "inherent" points to something that is a property of the 
ballot prior to marking, not something that is dependent on things 
extrinsic to the ballot -- such as the political climate or the 
voter's preferences.

But as long as the "one person, one vote" phrase is being used as an 
argument against approval voting, then we should try to address the 
likely meaning as understood by those who use it in that way. I 
suspect those using this phrase in such a context intend it to mean 
"one person, one ballot mark per race". So first, I would like to ask 
those who believe this is an argument against approval voting to 
explain what is so objectionable about multiple marks by one voter in 
any given race. I would then like them to explain how this argument 
allows any form of voting other than lone-mark plurality.

I'm also familiar with the reasoning that the person who elects to
mark only one candidate (or all but one candidate) has expressed fewer
preferences than one who has marked, say, half of the candidates, and
therefore has exercised less voting power. E.g., with 4 candidates, 
those who mark 1 or 3 candidates express 3 preferences, while those 
who mark 2 candidates express 4 preferences. As I've noted before,
a decision to exercise fewer preferences, made by a knowledgeable 
voter, is made in order to maximize the strategic value of that 
voter's ballot. Surely a voluntary choice made to maximize one's own 
expectation of the outcome, within the constraints set by the voting 
method, can't be held as a reduction of the voter's power. This is 
merely an instrumentality/expressivity tradeoff.

No matter what the election method, the political situation may
empower certain voters over other voters; lucky is the voter who
already has such a political tide in favor of his favorite candidate
that he doesn't feel the need to add more choices. The same inequality
exists in lone-mark plurality elections, but it is exacerbated by the
method's asymmetry: See my previous post at
http://groups.yahoo.com/group/election-methods-list/message/9795.

Besides, how can we avoid applying this argument to ranked methods
where reversal or collapsing of preferences can be used to strategic
advantage?

3. I'd like to suggest that we give the name "Small-Nash Equilibrium" 
  to the Nash Equilibrium defined in Alex's July 26 post (that is, if 
Alex was the first to propose it; I don't recall). This will 
distinguish this special case from other sets of Nash Equilibria that 
result when the electorate is divided into teams ("players") in 
different ways than Alex's division.

In Craig Carey's response to Alex he states "The theorem starts with
"If.. exists" and identifies if a Condorcet winner exists. Therefore
all the weighted preference list information is available since a
unique Condorcet winner is completely independent of voters."

That conclusion depends on how Condorcet winner is defined. Google 
turns up two definitions, one on condorcet.org and the other in Lorrie
Cranor's dissertation. Both refer to an alternative that wins pairwise
against all other alternatives. But they are subtly different.

If we take the condorcet.org definition "An alternative that pairwise
beats every other alternative", nothing is hinted about how the
balloting is done. If a single set of approval ballots is counted, and
the pairwise information is extracted from this, then a Condorcet
winner by this definition will be found, except in the case of exact
ties. If we take Cranor's definition "An alternative that beats or
ties all others in a series of pairwise contests", then the
requirement for a series of contests (which could be met with 
simultaneous contests through ranked balloting) implies that the 
Condorcet winner is something that cannot be determined from a single 
set of approval ballots. A might beat B in the approval contest, while 
B might beat A if a pairwise contest between the two is held. So
Craig Carey's statement is consistent with Cranor's definition -- 
provided that no voter votes inconsistently in any set of three 
pairwise contests, which is a given in ranked balloting -- and not 
with condorcet.org's. The CW Alex refers to in his theorem is probably 
the Cranor CW; if so, Craig's point is correct.

However, Alex was merely developing an idea (in the collaborative 
environment of the list), not submitting it for publication in a 
mathematical journal, so most of Craig's pedantic comments in that 
post are uncalled for at this stage.

4. Craig's anti-approval diatribe (July 27) fails to point out any 
real problems. Examples presented on the EM list and elsewhere lack a 
sufficient number of candidates for his taste. Yet the existence of 
small examples does not imply failure with large numbers of 
candidates. Will Craig point out how increasing the number of 
candidates would make approval deteriorate?

It's simple to do this for IRV: If there are N candidates, and M
candidates are eliminated before there is a pairwise winner against
the remaining candidates, then it is possible that that winner
is a pairwise loser in M contests. This looks very bad for IRV if M =
N - 2 and N is large. IRV needs to test the winner against M 
additional candidates.

I would like Craig to present an argument, at least as coherent as the 
one above, for why Approval fails if the number of candidates is 
large,  and to state what test it is failing.

Also, why is IFPP only formulated for small numbers of candidates, if 
large fields of candidates are such an important consideration?

  -- Richard

----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em