[EM] Some Voting Tables

[Forest Simmons]

Correction on my previous posting, followed by (interlinear) comments on
Richard's posting below.

Correction: Borda Strategy would give the win to A. But sophisticated IRV
strategy would likely give the win to C. So Borda's OK on this one.

On Thu, 13 Dec 2001, Richard Moore wrote:

> Forest Simmons wrote:
> 
> > Here's an example that turns out to be more interesting than it first
> > appears to be:
> > 
> > (Sincere intensities or utilities are in parentheses.)
> > 
> > 45  A(100) B(50) C(0)
> > 30  B(100) C(50) A(0)
> > 25  C(100) A(50) B(0)
> 

<snip>

> 
> Assuming you mean "perfect information" about the other voters' utilities,
> rather than "perfect information" about their actual ballots or strategies.

Yes.

> The latter case could exist, e.g., if it's a roll-call vote and you are the
> last name on the roll. But that case wouldn't be very interesting.
> 
> > Who would be the various winners in each of the various methods in the
> > perfect information case (assuming sophisticated voters)?
> > 
> > It seems to me, for example, that under Borda or IRV/STV, the B faction
> > voters would see the handwriting on the wall and be tempted to vote C
> > above B to keep A from winning.

[This wouldn't work in Borda, but would in IRV.]

<snip>

> > Under Approval or Cardinal Ratings candidate B would be the likely winner. 
> > If enough interest is shown, I will give the logic behind this conclusion
> > in a later posting.
> 
> 
> I'd be interested in hearing your reasoning. If I come up with a scenario
> in which B wins, such as
> 
> 19 
> A
> 26 
> AB
> 30 
> BC
> 25 
> C
> (45 A, 56 B, 55 C)
> 
> then one group, the C voters, could get a better result (a win for A) by
> having at least 12 of their voters vote CA:
> 
> 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.


>
> One interesting question is whether there are other stable outcomes
> that favor a candidate other than A (i.e., Is the system bistable, or
> even multistable?).   

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.

Unilateral changes in either of the other two factions will result in no
improvement for either of them. 

By the way, I like B as a winner, since B has the fewest zero utility
ratings.  B is the candidate the greatest number of voters could live with
(assuming anybody could get by on 50% utility, but nobody could get by on
zero utility).

> 
> A full analysis through game theory might be interesting. But what types
> of strategic cooperation need to be considered? I can see that an analysis
> would be feasible if cooperation within factions is allowed (there would
> be 46x31x26 combinations to consider, an easy task for a computer), but
> if cooperation between factions is to be considered, it would be extremely
> complicated.
> 
> The intra-faction cooperation strategy could be applied by individual voters
> who cannot communicate with other members of their faction, by calculating
> the optimum strategy for their faction and using an appropriately weighted
> random process to determine whether to include their second choice.
> 
> Inter-faction strategies are complicated by this: If the B and C groups
> want to cooperate with each other, then they would be working against A.
> But the C group won't cooperate with the B group if the outcome leads to
> B being elected, so both groups have to agree to support C. The B group
> won't have a whole lot of incentive to hold up their end of the bargain,
> however. So I don't think this class of strategies is very feasible. It
> just leads to another form of Prisoners' Dilemma.

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. 

Forest

> 
>   -- Richard
> 
>