[EM] Ordering defeats in Minimax

[Juho Laatu]

> On 07 May 2017, at 12:39, Kristofer Munsterhjelm <km_elmet at t-online.de> wrote:
> 
> On 05/07/2017 10:58 AM, Juho Laatu wrote:
>>> On 06 May 2017, at 23:26, Kevin Venzke <stepjak at yahoo.fr
>>> <mailto:stepjak at yahoo.fr>> wrote:
> 
>>> Legislators usually don't use election
>>> methods. I would say essentially they figure out among themselves what
>>> the outcome likely must be and then "elect" it, typically by a majority
>>> vote. If voters could do that directly I think that would be the ideal
>>> situation, as far as minimizing the need for strategy.
>> 
>> That's an interesting viewpoint. I'd like to see Condorcet tested in
>> some parliaments. Although negotiations and Plurality can be used to
>> solve many cases, also Condorcet and the idea of having multiple
>> candidates could have some benefits. The first one in my mind is
>> openness. Even when the "big boys" have agreed something behind the
>> screens, and then bring that decision into a majority vote, some smaller
>> groupings could add some alternatives in the election that they consider
>> better. This would open up the available alternatives to the world
>> (media, public), and all voters would have to take position on the
>> presented alternatives. Small parties or groupings could thus make
>> sensible compromise proposals that might even win, if they are good enough.
> 
> I think some parliaments use the following procedure for multiple proposals or amendments:
> 
> Some group decides the agenda (I'm not sure who; it probably depends on the parliament in question).
> 
> Then the parliament discusses the first two options, and does a majority vote between the two. It then does a majority vote between the winner so far and the next option, and so on until all options have been considered. The winner at the end of the procedure is the option that is accepted.
> 
> This method is Condorcet (because the CW beats everybody pairwise, and so whenever it becomes the winner so far, it'll stay the winner so far; and whenever it's the challenger, it will win against the winner so far and become the next winner so far). Furthermore, it's Smith. How cycles are broken depends on the agenda.

Yes, this is some sort of "serial Condorcet".  The biggest problem of this approach is probably the agenda based cycle breaking that you mentioned.

If there is a Condorcet winner, it will win. But if there is any chance of having a cycle, then the chairman (if he decides the agenda) may arrange the agenda so that his favourite option is the last one in the queue of options. With three candidates this already determines the order of all pairwise votes.

With three candidates the cycle could be either A>B>C>A or A>C>B>A. The chairman wants A to win. The first vote will therefore be between B and C. If we have cycle A>B>C>A, B will win the first round, and A will win the second round (against B).  If we have cycle A>C>B>A, C will win the first round, and A will win the second round (against C). The chairman could thus pick the preferred winner in the case that there is no Condorcet winner.

The voters could also try to use some strategies. This gives us some quite interesting strategy patterns, since after the agenda has been set, the method is no more symmetric with respect to the different options. A and B supporters could also join forces and vote against A at the last round, if they don't like the chairmans idea of constructing a strategic agenda.

Juho