[EM] voter strat & 2-party domination under Condorcet voting

[Rob Lanphier]

Hi Warren,

I'm eagerly awaiting your reply on this message.

Rob

On Sat, 2005-08-13 at 14:12 -0700, Rob Lanphier wrote:
> Hi Warren,
> 
> I'm not following your theorem.  Can you give an example of what you are
> referring to, showing a set of sincere preferences, followed by a set of
> tactical ballots which illustrate your point? 
> 
> E.g.:
> Sincere preferences:
> Group 1 - 18 votes: A>B>C
> Group 2 - 18 votes: A>C>B
> Group 3 - 16 votes: B>C>A
> Group 4 - 16 votes: B>A>C
> Group 5 - 16 votes: C>B>A
> Group 6 - 16 votes: C>A>B
> 
> Tactical ballots:
> Group 1 - 18 votes: A>B>C  #sincere
> Group 2 - 18 votes: A>C>B  #sincere
> Group 3 - 16 votes: C>B>A  #B>C>A voters that bury B to help C win
> Group 4 - 16 votes: B>A>C  #sincere
> Group 5 - 16 votes: C>B>A  #sincere
> Group 6 - 16 votes: C>A>B  #sincere
> 
> This particular example isn't a good one, since Group 3's strategy
> doesn't affect the outcome of the election (A wins no matter what).  I
> hoping to see an example that's more illustrative of the point you are
> making.
> 
> Thanks
> Rob
> 
> On Sat, 2005-08-13 at 09:39 -0400, Warren Smith wrote:
> > On the probability that insincerely ranking the two frontrunners max and min, is
> > optimal voter-strategy in a Condorcet election.
> > ----------Warren D. Smith Aug 2005----------------------------------------------
> > 
> > MATHEMATICAL MODEL: 3-candidate V-voter Condorcet elections
> > with random voters (all 3!=6 permutations=votes equally likely).
> > 
> > QUESTION: Is there a subset of identically-voting voters, who, by changing their vote
> > to rank the two "perceived frontrunners" max and min ("betraying" their 
> > true favorite "third party" candidate) can make their least-worst frontrunner win
> > (whereas, their true favorite cannot be made to win no matter what they do)?
> > 
> > THEOREM: The answer to the above question is "yes" with probability
> > at least  25% = 1/4  in the V=large limit.
> > 
> > PROOF SKETCH:
> > 1. The probability of a condorcet cycle is exactly 25% if we 
> > ignore situations that include exact ties.  
> > (Since: Assume A>B wlog.  Then B>C with prob=50%, then with prob=50% C>A.)
> > 
> > 2. The probability in the V=large limit tends to 1 that all the pairwise
> > victory margins are of order approximately sqrt(V), and that all the 6 kinds of voters 
> > occur with counts approximately V/6 each, i.e. much larger than sqrt(V).
> > 
> > 3. So assume there is a condorcet cycle, wlog it is A>B>C>A, and wlog the smallest
> > margin of victory is C>A so that A is the winner (according to all the usual
> > Condorcet methods, since they all are the same in the 3-candidate case).
> > 
> > 4. Choose a subset, of cardinality of order sqrt(V), of the voters of type "B>C>A".
> > (More precisely, we must choose the cardinality*2 to lie above the
> > previously-mentioned victory margin.)
> > If they betray their favorite B by insincerely switching to "C>B>A", 
> > then C becomes the Condorcet winner,
> > which from their point of view is a better outcome.
> > Q.E.D.
> > 
> > STRENGTHENING.
> > Note our "B>C>A"-type voter subset can argue that obviously, nothing they can do
> > will elect B, since when they rank B top honestly that fails to do it. Therefore,
> > their only chance for an improvement is to go for electing C.  And the only
> > way they can try is to raise C in the rankings.  As we've seen, this reasoning
> > yields success for them, 25% of the time.  However, given their preconception
> > that B has essentially no chance of victory, it actually makes sense
> > for them to rank C top 100%, not 25%, of the time, even though we know
> > this will only be successful for them with probability 25%.  Because given
> > their belief B has no chance, this cannot hurt them, and they know there is a
> > 25% chance it will help them.  So we conclude from this that in fact, the "betray B"
> > strategy is plausibly better than honesty, 100% of the time.
> > 
> > Summary.
> > Adam Tarr in previous posts had questioned my claim that this this plurality-like voter
> > strategy could ever be optimal in Condorcet elections.  He said
> > "I can't easily imagine a scenario where it is useful in Condorcet."
> > He demanded that I "make some simulations that demonstrate
> > this, or at least show some examples."  He apparently had not noticed the
> > fact I had already exhibited an example on the CRV web site, 
> >    http://math.temple.edu/~wds/crv/RangeVoting.html
> > "Why range is better than Condorcet" discussion, perhaps because said example was in the
> > subpage  http://math.temple.edu/~wds/crv/IncentToExagg.html.
> > But now the present discussion shows that Tarr was maximally wrong: this strategy
> > is ALWAYS the right move.
> > 
> > Given that this is the case, we now can take it to be 100% certain that 
> > Condorcet voting methods will lead to 2-party domination, just like the flawed 
> > plurality system those methods were supposed to "fix", and just like experiemntlly
> > is true with IRV.  So anybody who is interested in third parties ever having
> > a chance, would be advised NOT to foolishly advocate either IRV or Condorcet,
> > but insetad would be advised to advocate RANGE VOTING (which experimentally
> > favors third parties far more than either plurality or approval, incidentally,
> > see the CRV web site).
> > -wds
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