[EM] Free riding

[Markus Schulze]

Dear Raph Frank,

you wrote (31 Aug 2008):

 > Btw, is there a simplified explanation of
 > your PR-STV method somewhere?
 >
 > From what I can see, it compares possible
 > outcomes pairwise like CPO-STV (but only
 > compares outcomes that differ by 1 member).
 >
 > Ideally, the simplified version would just
 > need to explain the way to perform that
 > comparison (if each comparison doesn't
 > depend on any of the others)

Suppose that M is the number of seats. Suppose
that _A_ and _B_ are two sets of M candidates
each. Suppose that sets _A_ and set _B_ differ
in exactly one candidate. Suppose that
candidate B is that candidate who is in
set _B_ but not in set _A_.

Suppose that each voter casts a complete
ranking of all candidates. Then the strength
of the win of set _A_ against set _B_ is the
maximum value X such that each candidate in
_A_ has a separate quota of X votes against
candidate B.

In mathematical terms:

X is the maximum value such that the voters
can be partitioned into M+1 disjoint
sets T(1),...,T(M+1) such that:

1. For all i = 1,...,M: |T(i)| >= X.

2. For all i = 1,...,M: Each voter in T(i)
prefers candidate A(i) to candidate B.

*********

There are mainly two reasons why I define
the strength of a win in the manner above.

First: The strength of the win of set _A_
against set _B_ should only depend on which
candidates of the set _A_ the individual voter
prefers to candidate B. But it should not
depend on the order in which the individual
voter prefers these candidates to candidate B.

The reason: Voters, who have understood STV
well, will give insincerely low rankings to
"strong winners" (i.e. candidates who are
elected quite certainly). I call this strategy
"Hylland free riding". However, it is also
clear that, when a strong winner is one of
the favorite candidates of a voter, then this
voter will not rank this strong winner below
candidates he despises. Therefore, to minimize
the vulnerability to Hylland free riding, the
order in which the individual voter ranks
the strong winners should not have any impact
on the result of the elections.

Second: The above definition for the
strength of a win corresponds to the fact
that, in real life, the probability, that
a vote management strategy works, depends
only on the number of votes for the weakest
candidate who participates at this vote
management.

Markus Schulze