# [EM] Spoiler Effect on Wikipedia

Markus Schulze markus.schulze at alumni.tu-berlin.de
Thu Nov 11 13:52:44 PST 2004

```Hallo,

> I'm responsible for the edits to that page that make that claim, but
> if it's wrong please do fix it.  Markus S - I'm very surprised that
> IIA does not imply ICC, could you give an example?  I mean the strong
> version of IIA.

I prefer the following definition for IIA: Suppose, P[X] is the
probability that candidate X is elected. Then adding a candidate B
with B<>A must not increase P[A].

I prefer the following definition for ICC:

Suppose candidate D is replaced by a set of candidates D(1),...,D(m)
in such a manner that for every candidate D(i) in this set, for every
candidate F outside this set, and for every voter v the following two
statements are valid:

(a) v strictly preferred D to F if and only if
v strictly prefers D(i) to F.
(b) v strictly preferred F to D if and only if
v strictly prefers F to D(i).

If candidate D and candidate E were two different candidates,
then replacing candidate D by a set of candidates D(1),...,D(m) in
the manner described above must not increase P[E]. If candidate D
and candidate E were two different candidates and (1) there was at
least one voter who either strictly preferred candidate D to
candidate E or strictly preferred candidate E to candidate D or
(2) P[D] = 0, then replacing candidate D by a set of candidates
D(1),...,D(m) in the manner described above must not change P[E].

This definition of ICC is very long because it is necessary to exclude
those situations where the used election method chooses candidate D
with a positive probability and where each voter is indifferent between
candidate D and candidate E. In this case, candidate D and candidate E
are elected with the same probability [at least when the used election
method satisfies neutrality]. And when candidate D is replaced by a set
of candidates D(1),...,D(m) and each voter is indifferent between the
candidates D(1),...,D(m),E then again each candidate D(1),...,D(m),E is
elected with the same probability so that it is advantageous to run a
large number of candidates.

******

Suppose, N is the number of candidates.

"Random Candidate" says that each candidate is elected with the same
probability of 1/N. Random Candidate satisfies IIA but not ICC.

"David Catchpole's Random Candidate" says that when a Condorcet
candidate exists then this Condorcet candidate is elected with a
probability of 2/(N+1) and every other candidate with a probability
of 1/(N+1) and when no Condorcet candidate exists then each candidate
is elected with a probability of 1/N. David Catchpole's Random
Candidate satisfies IIA but not ICC.

"Random Pairs" says that each pair of candidates A and B is chosen with
the same probability of 2/(N*(N-1)) and the winner is that of these
two candidates who wins the pairwise comparison between them. Random
Pairs satisfies IIA but not ICC.

Markus Schulze

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