[EM] Two Theorems: Please criticize or point out errors

Craig Carey research at ijs.co.nz
Fri Jul 26 21:42:08 PDT 2002

Help was asked for. But there will quote a lot of difficulty getting
the complex English here, converted into simple quantifier logic
equations. Possibly no one would help.

At 02\07\26 20:06 -0700 Friday, Alex Small wrote:
 >I was thinking about a statement that Mike has made:  That with Approval
 >Voting there is always a Nash equilibrium where the Condorcet Winner wins
 >the election and every voter votes sincerely.  Here's what I came up with:
 >First, definitions:
 >Definition:  A Nash equilibrium for an election is a situation in which
 >each group of voters with identical preference orders follows the same
 >(possibly mixed) strategy and no faction has any incentive to pursue a
 >different strategy when all other factions keep to the same strategies.
 >(This definition has already been defended by me and others in many
 >previous posts.)
 >Definition:  A Sincere Approval Ballot is one on which if a voter approves
 >candidate j he also approve all candidates whom he preferes to j.

A voter is not the same as a ballot paper: the latter have a real number
associated with them. A voter can cast more than one vote in a general
preferential voting election, e.g. a vote held in a living room.

Past that, there is an attempt to create some information on preferences.
The preference information would not be unique. There is no maths here
if the proof vanishes when voters are replaced with machines.

 >Theorem:  If a Condorcet Candidate exists and the electorate uses Approval
 >Voting then there is always at least one Nash Equilibrium in which all
 >voters cast sincere ballots and the Condorcet Winner is elected.

The Nash Equilibrium is a function of preferential voting method and that
"Nash equilibrium function when the method is 'Approval'" returns a
Boolean value.

The theorem starts with "If .. exists" and identifies if a Condorcet winner
exists. Therefore all the weighted preference list information is available
since a unique Condorcet winner is completely independent of voters.

Undoing the correction I applied to the definition of "Sincere Approval
Ballot", now there exist important voters (e.g. 3 digits on a child's
book that has 2 pages listing random numbers). No longer is there a
multivalued poorly described "approval preferences device".

There are serious problems with the Nash definition, especially with the
words "incentive" and "strategy". Those are completely undefined. The
theorem will ignored if Alex Small says that they are properties of
voters after having started the "theorem" with "If a Condorcet candidate".
How can the voters be altered once the votes are already received.

It seems very very uninteresting since right at the outset all the
ballot paper counts are created by a "(There Exists)" quantifier
operator, and then after that, it attempts to construct ambiguously
"human's preferences".

I find the wording incomprehensible, and the mistakes of leaving voters
strewn around the definitions raises doubts

If someone got it defined then it still may fail to correctly handle
internal ties. Sometimes sincerity could not be defined (e.g. if there
is symmetry). But the theorem is not just saying something about the
Nash equilibrium but it is also saying that sometimes sincerity must be
defined if the Condorcet winner is defined. To guess, I assume that
in a fixed definition, the 'flat' boundary internal ties of the
Condorcet winner would not contain all the internal-tie flats of the
sincerity thing.

It is not well motivated:

  why tweak up voters (that don't exist in real mathematical theorems),
   after the weights of the preference lists are known.

It seems like a mistake that is virtually impossible to make, yet one
that is often seen here.

 >(Note that there may also be sincere Nash equilibria in which the CW
 >loses.  For an example, seehttp://groups.yahoo.com/group/election-methods-list/message/9351)
 >Proof:  I will construct a particular set of sincere ballots that elect

 >identical ballots with MCA, except that some voters indicate distinctions
 >among those whom they approved, AND ALL VOTERS WHO APPROVED BOTH THE CW

The Condorcet winner is the wrong candidate to make the winner, in 1 winner
elections, in general. Acceptance could be slow if that purpose was to
prove an untruth. But, alas, this does not handle principles well too.

(I just rejoined)

Craig Carey

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