Arrow and Gibbard-Satterthwaite

David Marsay djmarsay at
Fri Sep 19 02:18:55 PDT 1997

>From a newbie. Markus Schulze provided some interesting examples of 
methods that are not Pareto (do not respect a consensus).

> Smith//Random fails to meet Pareto.
> Example:
>    40 voters vote ABCD.
>    35 voters vote CDAB.
>    25 voters vote DABC.
> <SNIP>
>    Thus, B is elected with a probability of 25%, although
>    every voter prefers A to B.

It would be better to break the tie by choosing a voter at random, 
then taking their best option. This is Pareto for your example. Any 
method that never leaves an option A while removing an option B that 
the majority prefer, followed by Gibbard Random, is Pareto.
 SEE:  A. 
Gibbard Manipulation of Voting Schemes: A general result, 
Econometrica 47, 1973  

> Smith//Condorcet[EM] with the subcycle rule fails
> to meet Pareto.
> I don't want to criticize Smith//Condorcet[EM] with the
> subcycle rule. In the election methods list, it has never
> been said, that Pareto is important.
> Even Smith//Random, which fails to meet Pareto very obviously,
> has never been criticized for that.

I just have! If you are saying that Pareto is unachievable, then I 
consider this to be an important insight! I'd certainly like to know 
how other methods fare with your examples.

> Thus: Arrow's theorem in its original version, which says,
> that every method, that meets Pareto and some other suppositions,
> fails to meet IIAC, has a dubious relevance.

Historic importance, surely?
> Thus: To have a constructive discussion, we have to investigate,
> whether there are other criteria such that, if a method
> meets these criteria, it fails to meet IIAC.
> In my e-mail "Arrow and Gibbard-Satterthwaite", I demonstrated,
> that every method, that meets PMC, fails to meet IIAC.
> A method meets PMC if & only if:
>   If there are only two candidates, then that candidate is
>   elected, who is preferred by more voters.

If I've understood, then where  IIAC + PMC -> Pareto -> PMC, so your 
result is equivalent to Pareto -> not IIAC, a considerable 
simplification of Arrow's result. Has anyone checked it out? 
Sorry folks, but apparently I have to do this. :-(
The views expressed above are entirely those of the writer
and do not represent the views, policy or understanding of
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