[EM] Some Voting Tables

[Richard Moore]

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)

...

> In this "zero information" case Approval voters have to decide whether to
> use above mean inclusive strategy or above mean exclusive strategy. i.e. 
> should they approve their mean candidates or not?


In true zero-info cases, it's above mean (exclusive). There's no basis for
believing that any particular candidate has a better chance than any other,
given the lack of information, and ignoring your own ballot. So you assume
all candidates have equal chances. But, when you consider your own ballot,
which will certainly include your favorite, the scale is tipped, ever so
slightly, towards your favorite. So there's no incentive to vote a second
choice if that choice is right on your approval threshold.

This strict zero-info strategy leads to A winning.

With large populations, your first choice is already diluted so that the
effect of your second choice on the expected utility of the outcome is
negligible. In such cases you could just as well flip a coin over your
middle choice. But with a small number of voters, the first-choice effect
is significant.


> If approximately half of each faction goes each way, then candidate A will
> also be the Approval winner. This could come about through coin tossing or
> by pro-active collusion within each faction.
> 
> However, that much potential for vote coordination would definitely take
> us out of the zero information case.
> 
> So let us now consider the other (more interesting) extreme, the perfect
> information case.


Assuming you mean "perfect information" about the other voters' utilities,
rather than "perfect information" about their actual ballots or strategies.
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.
> 
> Although I haven't done a precise game theoretic analysis on this yet, it
> seems to me that B has the power and incentive to keep A from winning, and
> that C would know this and not mess things up by supporting A.
> 
> With perfect information it seems likely that under IRV and Borda the
> voters would end up picking the consensus sincere last choice C. 
> 
> In the perfect information case how would the various candidates fair
> under Plurality, Ranked Pairs, Coombs, etc.?
> 
> 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).

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?).

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.

  -- Richard