Medians (was Re: [EM] Absolute Utilities)

[Forest Simmons]

Thanks for the example Bart. I had found a similar one myself. But I'm not
convinced that the majority candidate is more democratic than the median
candidate, just as I am not convinced that the majority candidate is
better than the Approval candidate.

Presumably in the example below there are other candidates, otherwise
there would be no reason for the voters to vote these two candidates at
positions other than the extremes (same as in Approval) in which case the
median winner would be the same as the majority winner. [As you know
in a two way race approval always picks the majority winner. So does
Median.]

Assuming that these two candidates are part of a larger race, which of
these two candidates is likely to have greater approval in that race? Note
that B is the only candidate with more than 50% of the ratings above 50%.

I will argue in another posting that in general maximizing mean utility is
less democratic than maximizing median utility, which in turn is less
democratic than maximizing (number of voters receiving) acceptable utility
(which corresponds to Approval). 

Think of three ambassadors extolling the economic virtues of their
respective countries. The first brags that of the three his has the
highest average income. The second says, "That's because of a few
billionaires that bring up the average. Most people in your country live
in abject poverty. My country has the greatest median income." The third
says, "That's because 51% of your country holds the 49% ethnic minority in
virtual slavery, not even making a living wage. My country has the
greatest percentage of the population making a living wage or better." 

The calculation of mean income allows the luxuries of the rich to cancel
out some of the misery of the poor, even if the promised trickle down
doesn't pan out.  [Some of the rich are pretty good at caulking those
pesky leaks.]

Similarly, the calculation of mean utility allows high utility for some
voters to cancel out low utility for others, even if there is no system
for spreading the utility around after the winning candidate gets in
office. 

However, it turns out that median winner is usually somewhere in between
the Approval winner and the mean winner, and that under normal conditions
they are all pretty close if not the same. 

For that reason I am currently advocating Grade Ballots or Olympic CR
ballots with Approval cutoff markers (whether in the form of virtual
Minimum Acceptable Candidates or method specified values) as a way of
getting Approval candidates elected while giving the voters the
expressivity they want at the same time. 

If the Approval winner is not the same as the mean rating winner, median
rating winner, Condorcet winner, Borda Count winner, IRV winner, etc. all
of whom can be calculated (or estimated with high accuracy) from the CR
ballots, then no problem; the Approval Winner is more democratic than any
of the others. 

What do you think?

Forest


 On Mon, 30 Apr 2001, Bart Ingles wrote:

> 
> 
> Forest Simmons wrote:
> > 
> > Medians are more democratic measures of general utility than are means.
> 
> 
> A problem regarding medians was pointed out to me a couple of years ago,
> when I had claimed that,
> 
> >> Medians are a natural way of evaluating rated examples,
> >> since a candidate with the highest median rating is by
> >> definition the candidate rated higher than all others 
> >> *by a majority of voters*.  
> 
> 
> The response I received pretty well convinced me otherwise.  Claiming
> that medians are more democratic than averages probably runs into the
> same problem (depending on what you mean by 'democratic':
> 
> 
> > From:  "Steve Eppley" <SEppley at a...>
> > Date:  Tue May 18, 1999  4:49 pm
> > Subject:  [EM] Bart's "Median Rating" method?
> >
> > I think the definition of Bart's Median rating method needs 
> > clarification, since Bart's claim about highest median rating 
> > and majority appears dubious.  Here's an example to illustrate 
> > the problem:
> > 
> > voter 1:  A=95, B=65
> > voter 2:  A=85, B=60
> > voter 3:  A=50, B=20
> > voter 4:  A=40, B= 0
> > voter 5:  A=45, B=55
> > 
> > There is a majority (80%) who rank A ahead of B.
> > 
> > Average rating for A = 63
> > Average rating for B = 40
> > 
> > Median rating for A  = 50?
> > Median rating for B  = 55?
> > 
> > If I've interpreted correctly how Bart intends it to be 
> > tallied, B is the candidate with the highest "median rating."  
> > But I wouldn't agree that B is rated higher than A by a 
> > majority of voters.
> 
>