[EM] [CES #4445] Re: Looking at Condorcet
Kevin Venzke
stepjak at yahoo.fr
Tue Feb 7 11:07:08 PST 2012
Hi Robert,
I think that the basic claim of "Condorcet doesn't necessarily pick the option whom the elecotorate prefers" (in terms of
total utility) won't be too controversial. Any kind of model usually assumes internal utilities (such as based on distances in
issue space) because we need these to figure out how voters prioritize. One could try to assume that some set of
internal utilities might have some absolute, aggregable value. In that case it is really easy to produce a scenario where the
majority favorite isn't the utility winner. All you need is one case and you get Clay's "not necessarily."
You ask how we can decide, then, not to elect voted majority favorites. Assuming voters are strategic I don't know
of a good answer to this.
You suggest a model where there are only two candidates and the voter-for-candidate utilities are all either 0 or 1. If
that's an accurate model then Clay's claim doesn't work. But with virtually any other model it will be true sometimes
that the voted majority favorite isn't the utility maximizer.
Kevin
De : robert bristow-johnson <rbj at audioimagination.com>
À : election-methods at lists.electorama.com
Envoyé le : Lundi 6 février 2012 21h31
Objet : Re: [EM] [CES #4445] Re: Looking at Condorcet
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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