# [EM] Concerning Chicken Proof Smith compliant methods

Forest Simmons fsimmons at pcc.edu
Fri May 9 17:48:53 PDT 2014

```Suppose that max(y, z) < x < y+z,  and that a sincere summary of the voter
preferences is

x: A>C
y: B>C
z: C>A

These sincere preferences could not constitute an informed ballot profile.
Why not?  Because it would not constitute a strategic equilibrium:  The A
faction could unilaterally truncate C, and thereby win the election.

How do we know this without knowing what election method i being used?
Well, we are assuming that the metho is chicken proof, an if so, candidate
A would be elected wih the following ballot set:

x: A
y: B>C
z: C

And untruncating A in the C faction could not make A lose in any of the
methods we have been considering, even the non-mono-raise ones like Benham
and Woodall.

x: A
y: B>C
z: C>A

But this position is not a strategic equilibrium either, since th B action
could benefit y unilaterally raising C to equal top:

x: A
y: B=C
z: C>A

in which case C would be the winner.

What's more, this position is a strategic equilibrium, as is the posiiion

x:A>C
y:B=C
z:C>A

which is just one move from the sincere preferences, and hence the most
likely equilibrium position.  Under pefect information it is the strongest
game theoretic solution.

In summary, if sincere preferences are

x: A>C
y: B>C
z: C>A,

then rational ballots will be

x: A>C
y: B=C
z: C>B

So the sincere Condorcet preference is also the strategic ballot CW.

In general (at least in the case of three candidates) if candidate X is the
sincere Condorcet preference, candidate X will also be the ballot CW for
ballot voted by rational voters under complete infomation.

In particular, the ballot set

x: A>B
y: B>C
z: C>A

will never be voted by rational voters when there is a sincere Condorcet
preference.  Nor will

x: A
y: B>C
z: C,

Why not?  Because they are not strategic equilibria, except possibly in the
absence of any true Condorcet preference.

So why do we pay so much attention to these non-equilibrium ballot sets?
Precisely because we want to make sure that they are not equilibrium
positions potentially rewarding arm twisting strategy, like the chicken
strategy.

Forest
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