On 2/7/12 6:30 PM, Juho Laatu wrote:
On 7.2.2012, at 5.31, robert bristow-johnson wrote:
how can Clay build a proof where he claims that "it's a proven mathematical fact
that the Condorcet winner is not necessarily the option whom the electorate
prefers"? if he is making a utilitarian argument, he needs to define how the
individual metrics of utility are define and that's just guessing.
Yes, I think Clay assumes that we know how the "aggregate utility" of a society is to be counted.
There could be many opinions on how to define "aggregate utility" or "electorate
preference", and also opinions that it can not be defined.
It is actually not necessary to talk about those general concepts. It is enough
to agree what the targets of the election are. Maybe Clay should tell
explicitly that in this particular election that he considers the maximal sum
of individual (sincere or given strategic) utilities to be the target.
so he's seeking to maximize a measure of utility that is the sum of
individual utilities and, again, i see no mathematical expression of the
individual utility to sum up. how does Clay maximize this sum of
undefined quantities?
as best as i can tell, we only know what this quantity of individual
utility is for a simple two-choice election. assuming all of the voters
are of equal weight, if the candidate some voter has voted for is
subsequently elected, the utility to that voter is 1. if the other
candidate is elected, the utility to that same voter is 0.
but when there is a multi-candidate race, this is much more poorly
defined. say there are 3 candidates, if the candidate that some voter
votes for is elected, the measure of utility (to that voter) is 1. if
the candidate that this voter ranked last is elected, the utility to
that voter is 0. but what about that voter's 2nd choice? it depends
who it is and who it is to the voter. if we were to always assume that
the utility is 1/2, then it seems like the kind of assumption Borda
makes. but the voter's 1st and 2nd choice could be very close to each
other, or the 2nd choice could be a piece of crap just a little better
than the last choice. we don't know. so how do you put together an
argument that "it's a proven mathematical fact that the Condorcet winner
is not necessarily the option whom the electorate prefers" when you just
don't know whom the electorate prefers because you don't know the
utility metrics for each voter?
if you answer, "we ask the voters what the utility measure is with a
Score ballot", then my response is: "how do you know that this is
accurate? that the voter even knows or that the voter isn't lying on
his ballot to try to bury his 2nd choice or to compromise and forsake
his favorite candidate?" there are so many assumptions made here, it's
like we're pulling numbers out of our butts.
hardly constitutes anything approximating "a proven mathematical fact".
And then he could continue to say that Condorcet is not designed to meet this
target.
how does he know when this target is not even operationally defined.
Condorcet may however perform quite well as a method that approxmates that
target in a highly competitive environment.
For some other election the target could be to let the majority decide, or to
maximize the worst outcome to any individual voter. Clay's target (corrected to
refer to the sum of preferences based target of the election, not to the
ambiguous electorate preference) may thus be valid for some elections but not
all. (Also Range could be used to approximate majority decisions or Condorcet
criterion, but only approximate.)
now, with the simple two-candidate or two-choice election that is (remember all
those conditions i attached?) Governmental with reasonably high stakes,
Competitive, and Equality of franchise, you *do* have a reasonable assumption
of what the individual metric of utility is for a voter. if the candidate that
some voter supports is elected, the utility to that voter is 1. if the other
candidate is elected, the utility to that voter is 0. (it could be any two
numbers as long as the utility of electing my candidate exceeds the utility of
not electing who i voted for. it's a linear and monotonic mapping that changes
nothing.) all voters have equal franchise, which means that the utility of
each voter has equal weight in combining into an overall utility for the
electorate. that simply means that the maximum utility is obtained by electing
the candidate who had the most votes which, because there are only two
candidates, is also the majority candidate.
I wouldn't say that "the maximum utility is obtained" because that is a too much general utility
oriented term. I'd say that "the maximum utility to the society, as agreed, is obtained". Or maybe
"the most reasonable practical result is obtained" (based on the conditions that you gave). I thus
want to see also your conditions as one possible agreed way to define the (in this case maybe only sensible)
targets for the election.
what other conditions could be agreed on? Two-candidate is a given,
High stakes and Competitive are pretty hard to agree to change, they are
just there. if you want to consider a variance to Equal franchise, then
whose ballots are going to be attenuated? will you be able to get those
voters to agree to have their ballots each count less than your ballot?
this is soooo fundamental. all i want to do is get people to agree that
when there are only two choices, that the candidate with the most votes
wins, which is simple enough. if you *don't* agree with this, what are
the conditions you are envisioning for when election to office is
awarded to the candidate with the fewest votes? sometimes when
considering a simplified case like this, you have to ask yourself about
the contra-indication. either you award the election to the candidate
with the greater number of voters or you award the election to the
candidate with the fewer number of votes. i am astonished that anyone
can see this in any more nuanced manner. how would you *ever* award
election to the less-supported candidate?
if Clay or any others are disputing that electing the majority candidate (as
opposed to electing the minority candidate) does not maximize the utility, can
you please spell out the model and the assumptions you are making to get to
your conclusion?
I think he made his assumptions / definition of the general utility of the
society
and what are they? general utility of the society is equal to the sum
of the individual utilities, so how are the individual utilities
defined? i don't see an answer there and i don't see how there *can* be
an answer without making a lot of assumptions. and then if you do that,
i don't see much confidence in the answer arrived at.
and then assumed that this can be set as an universal target also for all
single-winner elections. I wouldn't generalize that approach that much. For
example majority oriented elections are a common practice in most societies. So
we have at least two fundamentally different approaches to defining the targets
of an election. For competitive environments I find your approach to be a very
sensible approach. You can either assume that majority rule is what you want,
or that majority rule is what you must satisfy with in a competitive
environment.
if it's not the majority that rule, what's the alternative?
once we can settle this simple issue, i'll move on to "why Condorcet".
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
----
Election-Methods mailing list - see http://electorama.com/em for list info
