# [EM] Re: group strategy equilibria: no sincere CW

Anthony Duff anthony_duff at yahoo.com.au
Tue Aug 24 20:10:35 PDT 2004

```Warren,

In both scenarios you have assumed a cyclic property of the
electorate in order to demonstrate a cyclic result.  You therefore
are not demonstrating very much.

Anthony

--- Warren Schudy <wschudy at WPI.EDU> wrote:
> On Wed, 25 Aug 2004, [iso-8859-1] Anthony Duff wrote:
> > I am interested in the question of the frequency of non-existance
> of
> > a sincere CW.  I personally do not know that it is probable.
>
> Here are two scenarios where the classic 3-voter 3-candidate
> Condorcet
> cycle arises naturally. They are both a bit contrived, but I think
> they
> show that, at least for small elections, Condorcet cycles aren't
> implausible.
> -------------
> Scenario 1:
> Suppose that voters and candidates have opinions on two issues,
> call them
> x and y (for example social and economic liberalism). Suppose also
> that
> voters rank candidates based on how much they disagree on the
> issues, in
> the sense of ordinary pythagorean distance.
>
> Suppose that the 3 candidates A,B,C are evenly spaced on a unit
> circle, at
> 0, 120 and 240 degrees respectively. Suppose the 3 voters are also
> evenly
> spaced on a unit circle, but at 10, 130 and 250 degrees (please
> draw
> a picture). It is easy to show that the voters preference orders
> are the
> classic:
>
> 1 A>B>C
> 1 B>C>A
> 1 C>A>B
>
> -----------------
> Scenario 2:
> Consider the same two axis. Suppose that this time the
> voters are the same people as the candidates, and their coordinates
> are
> the same as the previous scenario's voters:
>
> A:  0.98,  0.17     (10 degrees)
> B: -0.64,  0.77     (130 degrees)
> C: -0.34, -0.94     (250 degrees)
>
> Consider the plausible case where that the voters do not agree on
> the
> relative importance of the issues. Suppose A and C care a bit more
> than x, but B cares a bit more about x than y. Then:
>
> A: A>B>C
> B: B>C>A
> C: C>A>B
>
> This is again a Condorcet cycle.
>
> ----
>
> These scenarios aren't exactly normal, but they aren't implausible
> either.
>
> -wjs
>
> /-----------------------------------------\
> | Warren Schudy                           |
> | WPI Class of 2005                       |
> | Physics and computer science major      |
> | AIM: WJSchudy  email: wschudy at wpi.edu   |
> | http://users.wpi.edu/~wschudy/          |
> \-----------------------------------------/
>
>
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