[EM] minmax

[Kevin Venzke]

Mike,

--- MIKE OSSIPOFF <nkklrp at hotmail.com> wrote:
>> When this rule is used even with partial rankings, this is what I call
>> "MinMax (pairwise opposition)" or "MMPO."
>
> I reply:
> 
> Well, the people who have already defined minmax (including Nalebuff, Levin, 
> and Fishburn) have given us only definitions that don't apply when there are 
> truncated ballots. So can you really call the above-quoted method "minmax"?
> Even if you say that minmax means something when there are truncated 
> ballots, that doesn't mean that we can say _what_ it means with truncated 
> ballots, because those who give us our definition of minmax won't say.

Well, the definitions Markus quoted don't agree with each other. If you look
at Levin and Nalebuff's definition:

>> A candidate's "max" score is the
>> largest number of votes against that candidate across all
>> head-to-head matchups. The rule selects the candidate with
>> the minimum max score.

This is obviously MMPO, if we use this definition as written even with partial
rankings. (I don't really accept that this is meaningless in the context of
partial rankings, just because the authors chose to use full rankings.)

But Fishburn's definition:

Page 471:
>>> For any situation (A, p) and alternatives x, y e A, we shall let
>>> p(x,y) be the number of terms in p that have x preferred to y.
>>> Hence if p = (p1,p2,...,pn) then p(x,y)+p(y,x) = n when x <> y.
>>
>>Page 471:
>>> f(x,A,p) = min { p(x,y) | y e A\{x} }.
>>
>>Page 472:
>>> C6: Condorcet's function. (Also minimax function.)
>>> C6(A,p) = { x e A: f(x,A,p) >= f(y,A,p) for all y e A }.

This isn't MMPO, since it elects the candidate for whom the fewest votes for
him in a pairwise contest is the greatest.

> But, aside from that, if we accept that the above is minmax with truncation, 
> then, Kevin must agree that minmax isn't PC, since PC meets Condorcet's 
> Criterion.

I consider "Minmax" to refer to several methods, but not to actually be a
method itself.

Kevin Venzke



	

	
		
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