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

[Warren Schudy]

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 about y
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   |
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