[EM] [CES #4445] Re: Looking at Condorcet
robert bristow-johnson
rbj at audioimagination.com
Mon Feb 6 19:31:10 PST 2012
one thing i forgot to mention...
On 2/5/12 5:07 PM, Kristofer Munsterhjelm wrote:
> On 02/04/2012 06:14 PM, robert bristow-johnson wrote:
...
> that is not well defined. given Abd's example:
>>
>>> 2: Pepperoni (0.61), Cheese (0.5), Mushroom (0.4)
>>> 1: Cheese (0.8), Mushroom (0.7), Pepperoni (0)
>>
>> who says that for that 1 voter that the utility of Cheese is 0.8?
>
> The voter does. In this thought experiment, one simply assumes the
> 1-voter's utility of Cheese is 0.8 so as to show the point. The point
> is that there may be situations where utilitarian optimization and
> majority rule differs.
>
so my question, when running simulations or trying to construct a
quantitative case of maximizing utility, it depends of course on how
utility is quantitatively defined. and we understand that the aggregate
utility is some combination of every voter's individual utility, and,
for the sake of argument (and because it sounds reasonable), we'll say
that the metric of aggregate utility is equal to the sum of the
individual metrics of utility. so maximizing the sum is the same as
maximizing the mean.
but there is still no model of individual utility other than "one simply
assumes". 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. when you guess at a model that is part
of your "proof", it doesn't make for a very rigorous proof. a *real*
proof is that the Devil hands you the model (that's within the domain of
possible models) and you make your proof work anyway. *you* don't get
to cook up heuristics like "the utility to voter X that Candidate A is
elected is equal to 0.8".
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.
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?
sorry that i am belaboring what i would have thought were simple axioms,
but i can't tell that they are widely accepted and i want to probe how
they are not widely accepted. how can it be that when Candidate A gets
more votes than Candidate B (and they are the only choices) that anyone
would advocate awarding office to Candidate B? something has to be
anomalous to come to such a conclusion. perhaps the votes for Candidate
B count more than the votes for Candidate A (violating one person, one
vote). perhaps we introduce a goofy rule such as tossing in a random
variable (like draw two non-negative random integers within some given
range and add one to Candidate A's total votes and the other to
Candidate B's total votes) and Candidate B got a higher number out of
the lotto. that would make the decision threshold fuzzier, but i don't
think that supporters of Candidate A would consider it fair.
--
r b-j rbj at audioimagination.com
"Imagination is more important than knowledge."
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