[EM] Condorcet-Winner/Borda-Winner

[Alex Small]

Steve Barney said:
> _Basic Geometry of Voting_. Does anyone here know the proportion of all
> possible profiles (collection of fully ranked ballots) which
> yield a cyclic or intransitive Majority Vote outcome?


For 3 candidates there are 3 pairwise contests.  Each pairwise contest has
2 possible outcomes, so there are 8 possible combinations of pairwise
results, each occuring with a frequency of 12.5%.  (I'm not counting
pairwise ties;  I believe that mathematicians would say that the set of
all profiles which give at least one pairwise tie is a "set of measure
zero.")

There are 2 combinations of pairwise results that give a cyclic amibuity,
so 25% of the time we'll see a cyclic ambiguity.  Among the combinations
of pairwise results that give a Condorcet winner, there are 3 possibile
CW's and for each CW there are 2 possible relative rankings of the
candidates whom he defeated (i.e. A>B>C or A>C>B).  So each possible
result (A is CW, B is CW, C is CW, or no CW) covers 25% of the collection
of profiles.

Although we can't equate proportions of profile space with probabilities
of outcomes in public elections, it is clear that there is a wide range of
circumstances yielding cyclic ambiguities.  Adam has given a nice argument
about this in terms of the geometry of issue space.



Alex


----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em