[EM] Some Voting Tables

[Richard Moore]

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.

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? Maybe you meant

45 AB
30 BC
16 C
  9 CA

So B wins, and the A voters can drop (at most) 19 B votes without risking
a C win. But this is still not stable, since the C faction could always
increase their A votes.

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