[EM] median rating / lower quartile

[Jobst Heitzig]

Dear Folks!

Let me summarize before giving a new argument in favor of the median: 

After Mike brought up the term "social utility" and seemed to identify 
it with "sum of individual utilities", I wrote:
> The median is a simpler, more accurate, and more robust measure 
> of social utility than the sum!

Let me restate the problem I addressed by this: 
How can we answer the question of how much a candidate is worth for society (="social utility")
by means of the answers the voters gave to the question of how much that candidate is worth for them 
(="individual utility").

I argued that the social utility of a candidate should not be defined by the sum of the individual utilities 
of that candidate but by a more sensible statistics of their distribution. I suggested to use the median,
that is: social utility of X := that value a where half of the voters say that X's utility is at least a and 
half of the voters say that X's utility is at most a.

There are many good arguments to use such a definition of social utility instead of the one which involved the sum:
1. The median is more robust, that is, depends less on extreme values of few voters than the sum. It is not possible for
few voters to increase or decrease the median arbitrarily by increasing or decreasing their individual utility for that 
candidate.
2. When a majority says A's utility is above the value x, then the median is indeed above x. Similarly, when a majority 
says A's utility is below the value x, then the median is indeed below x.
3. When using the median, we need not assume an additive scale, only an ordinal scale. In other words, the median 
is invariant under monotone transformations, not only under affine transformations. For example, it makes not only no
difference when we add 1000 to all individual utilities or multiply everything by 2, but it makes also no difference when we
take the logarithm or square or whatever monotone function of all values. That's again a robustness property.

James commented:
> Scoring candidates by the median rather than the mean might be an
> improvement on standard cardinal ratings. Has this been discussed before?

And indeed, Rob had suggested in April 1998:
> letting people rate candidates
> on a scale of 0 to 100 (or whatever range you like) and merely taking the
> median rating of all of the candidates, and declaring the candidate with
> the highest median rating the winner.

Not surprisingly, he also found an example where only one out of 99 voters
preferred B to A, and 49 gave B no utility at all, but B had a larger median utility:
49: A100 B52
1:  A50 B51
49: A49 B0
medians: A50 B51

Of course, there is also such an example with the sum:
1:  A0 B100
99: A1 B0
sums: A99 B100
Here also only one voter prefers B, and even all other voters gave B no utility at all, 
but still B wins.

My guess is that such a problem will ocurrs with every kind of election by maximizing
"social utility", no matter how social utility is defined, as long as the definition of the 
social utility of A does only look at the individual utilities of A.


Still, in my opinion the median is better that the sum, also for the following reason:
When using the sum, there is often incentive for "exaggerating" and voting only extreme
utility values (0 and 100, say). But with the median, there are by far more strategic equilibria:

For example, when A is the CW, then the following is always a group strategy equilibrium when
we use the median: Rate A at 50, all candidates you prefer to A at at least 50, and the rest below 50.
Then no group can alter the outcome by voting strategically since A's median will always remain 
at least 50 and the group's preferred candidate's median will remain below 50. And: This is no special
property of the number 50 but works for any rating from 1 to 100. 

In other words: When all voters rate the CW at (about) the same value and rate in accordance with their
preferences, then the CW must win. This is not at all the case when we use the sum!


Mike: You wrote
> ...and would it keep CR's social optimization advantage, that if people vote 
> to maximize their utility expectation, then, with a few plausible 
> approximations, CR maximizes the number of voters who will be pleasantly 
> surprised by the outcome--the number of voters for whom the outcome will be 
> better than their pre-election expectation for the election?

Is that so with CR? That would be a big surprise for me. I would rather guess that this is 
something which is *much* better fulfilled by using the median, since the sum does not care
about numbers of voters, but the median does! Still, I don't know what you mean by 
"vote to maximize their utility expectation". And what if people vote more safely and try to 
maximize a different location parameter of the utility distribution?


Yours, Jobst

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