From: kym@ukato.freeshell.org   
      
   Ted Dunning  wrote:   
   > russell kym horsell wrote:   
   > > Dephased  wrote:   
   > > > Hello everyone,   
   > >   
   > > > I have a dataset of observations (about 100 000 observations). Each   
   > > > observation gives me the state of  30 discrete variables at a given   
   > > > time.   
   > >   
   > > ...   
   > >   
   > > Sounds like another case of hittin yor hed up aginst Arrow's Theorem.   
   > > There are some std fixes in "multivariate decision theory", but nothing   
   > > is good. But if it were easy, we'd all be kicking back with a Bud.   
   > While it is true that there isn't any globally best way to compare   
   > instances like this, I don't think that Arrow's theorem applies (which   
   > states that no voting system can meet all of a set of plausible   
   > desiderata).   
   [...]   
      
   I'm not sure whethr you're citing something deep or just google. :)   
   I've been elsewhere told that Arrow only applies to sociology because   
   some mention of "utility function" was made somewhere (!), or that it   
   deals with "preferences but not numbers" (!!).   
      
   The thm applies when one is trying to determine a function that will indicate   
   which point is "best" given more than 2 variates/variables/dimensions.   
   That it's not possible to find which is the "best car" given a set of   
   colour, fuel economy, performanc, availability and option-set data should   
   be pretty obvious.   
   That you also can not determine which car is "closest" to (but not   
   identical with) a given selection maybe not so obvious.   
      
   One proof relies on a simple counting argument -- if the utility function is   
   meant to order N data points there are obviously N! ways to do that.   
   If there are K variables then there are (N!)^(K-1) of arranging them.   
   Obviously the later is much bigger than the former -- hence there can be   
   no guarantee that any utility function will appear to agree to any   
   appreciable degree with the individual variables.   
   That being the case, there is no overall "best" (or even "good") utility   
   function.   
      
   Various generations go on to show there can be no guaranteed lower bound on   
   performance of utility functions, etc.   
      
   As you say, the thm can be applied to (but obviously are not restricted   
   to) voting systems and is then related to   
   the various "voting paradoxes" -- e.g. that the outcome of an election no   
   matter what voting system is used need not actually agree with the actual   
   preference of the majority of the voters. And I'm not talking about Florida.   
      
   It souldn't be surprising that the data from K variables for N data points   
   can't be projected down into a consistent 1-dimensional whole in non-pareto   
   situations. But like other   
   theorems dealing with the limits of rationality, it is often discounted   
   or said to "not apply".   
      
   Various methods can "get around" the axioms of Arrow. One standard way   
   is to drop the "2 variables" limit. If a principal component analysis (e.g.)   
   *can* reduce the dimensionality to 1 or 2 then Arrow definitely doesn't apply.   
      
   Sen's method is another way of dropping the dims to 1 -- but using   
   the "most important" variable and just ignoring the others. (I.e.   
   ignoring the "no dictators" axiom).   
      
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