[EM] Some Voting Tables

[Forest Simmons]

On Fri, 14 Dec 2001, Richard Moore wrote:

> Forest Simmons wrote:
> 
> >>19 A
> >>26 AB
> >>30 BC
> >>13 C
> >>12 CA
> >>(57 A, 56 B, 55 C)
> >>
> >>As it happens, this is a stable solution, in the sense that the A voters
> >>can't do anything to improve their outcome (their favorite is already
> >>a winner), the B voters can't do anything to improve their outcome
> >>(they've already done everything they can to support their compromise),
> >>and the C voters can't do anything to improve their outcome (they
> >>already did improve their outcome).
> >>
> > 
> > It seems to me that the C voters could improve their outcome by voting
> > straight C, reducing A's approval by 12, and giving the win to C.
> 
> 
> That just takes us back to the first scenario, where B won.

Oops, you're right!

> 
> One reason the A voters might have chosen their 19/26 split, by the way, is
> their intention of keeping C from winning. 26 votes for B from the A faction
> is just enough to lock C out, assuming the B voters and the C voters all
> vote for C (and all the B voters vote for B, of course).
> 
>  
> > Here's the (marginally) stable configuration that I had in mind for a B
> > win:
> > 
> > 45 AB
> > 30 BC
> > 25 CA
> > 
> > If the first faction drops B, then C wins, which is worse for that the
> > first faction.
> 
> 
> Wait, if the first faction drops B, then
> 
> 45 A
> 30 BC
> 25 CA
> 
> 
> A wins, right? 

You're right again! 

> Maybe you meant
> 
> 45 AB
> 30 BC
> 16 C
>   9 CA
> 

Actually, what I had in mind was

45 AB
30 BC
25 C

Any unilateral change made by one voter or one faction will not improve
the outcome for that voter or faction.

In particular, increasing the C faction's support for A will not help A
win without cooperation from the A faction.

> 
> I also don't see a stable solution in which C wins. The A voters can always
> prevent C from winning by voting a sufficient number of B votes. In fact,
> this power gives them a strong influence over the C faction, so they can
> probably persuade the C voters to vote CA.
> 
> 
> > In this example any two groups can work together to defeat the other
> > group.  This requires one of the two cooperating groups to support their
> > compromise candidate. In actual politics this effect is probably strong
> > enough to overcome the unilateral marginal stability of the solution I
> > gave above. The winner will be determined by the two groups that can make
> > the best deal with each other.
> 
> 
> Based on my last observation, I would say the two groups most likely to 
> cooperate
> are the A and C factions. If the A and B voters agree to cooperate to elect
> candidate B:
> 
> 45 AB
> 30 B
> 25 CA
> 
> then it would certainly be tempting for the A voters to back out of the 
> deal at
> the ballot box. If the B and C voters agree to cooperate to elect 
> candidate C:
> 
> 45 AB
> 30 BC
> 25 C
> 
> then the B voters have a lot to gain by backing out of the deal.
> 
> Of course there are other considerations (outside of electoral mathematics).
> First, the deal-breakers may hurt their ability to negotiate compromises in
> future elections, since they won't be trusted. Another possibility is that
> one party could use something besides votes as a bargaining chip. For 
> example,
> if B's party has control of the legislative branch, B's party could 
> negotiate
> a deal with A's party wherein B's party promises passage of a legislative
> package favored by A's party only if A's party helps B win.
> 
>   -- Richard
> 

I think you've described the two way deal dynamics quite well.

Forest