# [EM] Basic Arrow's Theorem Question

Alex Small asmall at physics.ucsb.edu
Tue Apr 22 17:22:11 PDT 2003

```I'm not sure what your question is, but your example focuses on
competition between candidates and the dynamics of a debate.  Arrow's
Theorem doesn't address the fact that adding more candidates to a debate
will change the types of questions discussed and perhaps change people's
preferences.

Arrow's Theorem addresses this situation:  "OK, assume that every voter
has made up his or her mind about the relative merits of WN, PS, and BBQM.
If you only give them ballots with PS and BBQM listed, say that PS wins.
Then say we add WN and they all vote again.  This time, BBQM (the old
loser) wins, rather than PS (the previous winner) or WN (the new
challenger)."

Arrow proved that you can't design a system that avoids that problem if
you use ranked ballots (assume no equal ranking or truncation for now),
you don't have a dictator (sorry, Ashcroft...), and your method always
picks the unanimous first choice of the voters if such a candidate exists.

Arrow says nothing about that third candidate perhaps changing the terms
of the debate, because that would change voter preferences.  The
assumption is that voter preferences stay fixed.

>
> OK, If everyone thinks A is better than B, A beats B. Fine.
>
> If C enters the race, no ones opinion of the A-B pairing changes.
>
> Hmmm.
>
> Sample Race, with imaginary figures.
>
>
> 1.
> PloddingSafe and BrashBrightQuestionMark enter a race.
>
> 2.
> PS and BBQM debate.
>
> 3.
> People decide who they like in the PS-BBQM pairing.
>
> 4.
> WingNut enters race.
>
> 5.
> PS, BBQM, and WN debate.
>
> 6.
> WingNut makes PloddingSafe lose cool! red in the face! loss of words!
> more?
>
> 7.
> Some PS supporters think...
>     hey, what gives!
>     Maybe BBQM is really the PS candidate!
>     After all, BBQM doesn't blow up around WN
>
> The entry of C(WingNut) into the race has changed the A-B pairing? Or is
> that something else entirely.
>
> Thanx.
>
> Thing is, remember, third party candidates tend to be extreme and always
> hurt the candidate from their side.
>
>
>
>
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```