[EM] The Green scenario, and IRV in the Green scenario, is a new topic here. Hence these additional comments. Clarification of position and why.

[Kristofer Munsterhjelm]

On 02/05/2013 12:52 AM, Peter Zbornik wrote:
> Kristoffer,
>
> no the example below applies for my two-round proposal as well, thus
> rapidly sinking what I previously proposed :o)
> Nice to having had done away with the two-round variant of IRV.
> Now I don't have to bother about it any more.
>
> For Condorcet I am not sure.
> I guess, there might even be a new criterion invented: multiple-round
> strategy-proof , but I don't know of any method satisfying this
> criterion.

I don't think any ranked nonprobabilistic method can pass that. To 
repurpose the proof of the previous posting.

Say you have a multiple-round strategy-proof method (by which I imagine 
you mean that taken as a whole, the method is strategy-proof, even if 
some of the rounds by themselves aren't; I'll get to the other option 
later).

Then this multiple-round method works by that the voters do something in 
the first round, then the method proceeds, then the voters do something 
in the next, and so on.

So invent an algorithm so that you can replace the voters with this 
algorithm for everything but the first round. This algorithm then 
emulates how the voters act if their internal preferences don't change 
between rounds and their internal preferences are ranked.

By replacing the voters with an algorithm, you make a DSV ranked method. 
This ranked method must by necessity be subject to Gibbard-Satterthwaite 
and to Arrow's impossibility theorem. In particular, there are times 
when dishonesty pays for the voters using this method.

Consequently, by doing the transformation in reverse, if there's an 
election where all the voters act according to the algorithm, it must 
sometimes pay for a voter to act as if his internal preferences were 
different. Thus, the multi-round method also is subject to G-S and Arrow 
in the worst case. It might not be subject to Arrow if the internal 
preferences are never only ranked, but it would still be subject to G-S.

Thus, the only way in which a method would be strategy-proof would be if 
the voters never acted like any of the (numerous) sets of algorithms 
that would make the transformation above work.

(A similar DSV construction can be used to show that Approval, Range, 
and MJ are subject to Balinski & Laraki's Arrowian objection when voters 
act in a comparative manner -- at least if we define "comparatively" 
properly.)

On the other hand, some runoff systems have equilibria that elect the 
honest Condorcet winner whenever there is one. That is, there's no 
incentive for any of the voters to strategize because it can only lead 
to counterstrategy that makes things worse for them. However, these 
equilibria usually require communication and so may not be very 
relevant. See for instance Messner et al. (2002-11-01), "Robust 
Political Equilibria under Plurality and Runoff Rule", 
http://politics.as.nyu.edu/docs/IO/4753/polborn.pdf

> The two-round method would however be suitable when trying out which
> of two methods is the best by letting the winners meet in the second
> round (like plurality vs. IRV winner), in order to gather political
> support, but that's an other topic.

I think that some kind of demonstration of or experiment to determine 
the voting method's accuracy could also be useful. For instance, one may 
have a game where a large group of voters decide what to do next - 
either from suggestions given by the players, or directly - and then the 
game proceeds. The better play ensues, the better the method. Such 
demonstrations could also be used to determine if one can make direct 
democracy that outperforms representative democracy, or if asset voting 
or liquid democracy can work better than both. It might not capture the 
principal-agent problem of real politics, though, unless there's some 
kind of "side benefit" (e.g. the player and/or voters whose suggestions 
was picked the most get a bonus).