[EM] strategy-free Condorcet method after all!

[Juho]

Very nice construction.

The first strategic thought in my mind is to give false poll  
information since the method relies on that information to be  
available. Let's see what happens with a simple loop of three.

1: A>B>C
1: B>C>A
1: C>A>B

The A supporter is strategic, so the poll results could be as follows.
1: B>A>C or B>C>A
1: B>C>A
1: C>A>B
=> the falsified strategic preferences indicate B>C>A

Depending on the ordering any one of the candidates could be the one  
that will be checked first. The A supporter will not know which one  
when giving the poll answer IF the ordering will be decided only just  
before (the first round of) the election.

- A will be checked first => A will be elected (since the C supporter  
is afraid that B would win A)
- B will be checked first => B will be elected (the A supporter votes  
for B since A would lose to C if B would not be elected)
- C will be checked first => C will not be elected (since the B  
supporter thinks that B will win A) => A will win since A is preferred  
over B

In this example the A supporter was able to improve the results. There  
could thus be some false information in the polls (or in the  
discussions between these three voters).

Juho


P.S. The A supporter could also try C>B>A in the poll.



On Nov 14, 2009, at 2:32 PM, Jobst Heitzig wrote:

> Dear folks,
>
> it seems there is a stragegy-free Condorcet method after all -- say
> good-bye to burying, strategic truncation and their relatives!
>
> More precisely, I believe that at least in case of complete  
> information
> (all voters knowing some details about the true preferences of all  
> other
> voters) and when all voters will follow dominating strategies, then  
> the
> following astonishingly simple method will always make unanimous  
> sincere
> voting the unique dominating strategy, and it will always elect a true
> beats-all winner (=Condorcet winner):
>
>
> Method: Reverse Llull
> =====================
>
> 1. Sort the options into some arbitrary ordering X1,...,Xn (e.g.
> alphabetically or randomly), publish this ordering, and put i=n.
>
> 2. If already i=1, then X1 is the winner. Otherwise, ask all voters
> whether they prefer Xi or the option they expect to be the winner of
> applying this method to the remaining options X1,...,X(i-1).
>
> 3. If more voters prefer Xi, Xi is the winner. Otherwise, decrease i  
> by
> 1 and repeat steps 2 and 3.
>
>
> Why should this be strategy-free?
>
> If n=2, the question in step 2 is whether X1 or X2 is preferred and  
> the
> method is traditional majority choice in which sincere voting is known
> to be the dominant strategy in case of 2 options.
>
> For n>2, we prove strategy-freeness inductively, assuming it has been
> proved for n-1 options already: Since we assume that each voter  
> follows
> dominant strategies and knows enough about the other voter's
> preferences, and since each voters knows that sincere voting is the
> unique dominant strategy for all cases of at most n-1 options, she  
> will
> know in step 2 which option Xj would win if the method was applied to
> X1,...,X(i-1), and she will also know that her vote at this step does
> not influence which option Xj is but only whether Xi or Xj will win.
> That is, in step 2 all voters face a simple majority choice between  
> two
> known options Xi and Xj, so again voting sincerely in this step is the
> unique dominant strategy. By induction, the whole method is strategy- 
> free.
>
>
> The method is in some sense the reverse of Llull's famous earliest  
> known
> "Condorcet' method from the 13th century (cited recently on this  
> list):
> In the classical Llull method, voters would first make a majority
> decision between X1 and X2, then a majority choice between the  
> winner of
> the first choice and X3, and so on working thru the whole list of
> options, always keeping the last winner and comparing it with the next
> option in the list. The overall winner is the winner of the last  
> comparison.
>
> So, the only difference between classical Llull and Reverse Llull is  
> the
> order in which these pairwise comparisons are done. If we assume all
> voters vote sincerely in classical Llull, both method would be
> equivalent. But with strategic voters, the difference is important: In
> classical Llull, a voter's voting behaviour in one step can influence
> the results of the later steps (because it can influence which  
> candidate
> "stays in the ring"), whereas in Reverse Llull it cannot.
>
>
> In practice, the method can be sped-up by using approval-style ballots
> on which each voter marks after step 1 every option Xi which she  
> prefers
> to the expected winner of the subset X1,...,X(i-1).
>
> As for additional properties, Reverse Llull is Pareto-efficient,
> Smith-efficient (i.e. elects a member of the Smith set), and  
> monotonic,
> but not clone-proof.
>
> I wonder if we can also find a clone-proof version of this... Any  
> ideas?
>
>
> Yours, Jobst
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