[EM] 02/06/02 - Adam's 02/02 example of three equal candidates:

[Richard Moore]

Richard Moore wrote:

 > In Approval, with all three candidates running equal in the polls (as
 > stated
 > in the premise, they are "neck and neck"), you have as much chance to be
 > instrumental in making your favorite win over your compromise as you
do to
 > be instrumental in making your compromise defeat your least favorite.
With
 > equal opportunities, which strategy do you choose: favorite only, or
 > favorite-plus-compromise? The answer can be found by looking at the
cost of
 > guessing wrong. If you choose the first strategy, and it turns out to be
 > the wrong choice (meaning you end up either pushing your compromise to a
 > loss from a tie with your least favorite, with 50/50 odds the tie will
 > resolve to elect your least favorite, or you end up pushing your
compromise
 > to a tie with your least favorite when otherwise your compromise
would have
 > won), then the cost is the potential regret of causing your
compromise lose
 > to your least favorite. Likewise, if you choose the second strategy, and
 > that strategy turns out to be the wrong choice, the cost is the equally
 > probable regret of causing your favorite lose to your compromise. Since
 > both strategies are equally likely to fail, the correct strategy is the
 > one which you would regret less in the event of failure. If your
preference
 > for your favorite over your compromise is weaker than your
preference for
 > your compromise over your least favorite, then vote for your top two.
 > Otherwise, vote for your favorite only.


Lest Donald counter that he was referring to factional strategies rather
than individual strategies, let me point out that such strategies may look
good on paper but are very risky in practice. Case in point: Forest posted
an example a few weeks back of an Approval election with the following
utilities:

45 A(100) B(50) C(0)
30 B(100) C(50) A(0)
25 C(100) A(50) B(0)

The question was, what group strategies are optimal for this situation?
The most stable strategy turned out to be that the A voters could cast
just enough B votes to make C unwinnable. C can expect at most 55 votes.
If 26 voters from the A faction vote AB, then with the 30 B voters, B
will always beat C. Thus you have

19 A
26 AB
30 B or BC
25 C or CA

Now, since C cannot win, the C voters will have a strong incentive to vote
with the A voters. That gives A a win. The B voters will then vote BC since
both B and C are better than the sure winner. Final (stable) tally:

19 A
26 AB
30 BC
25 CA

A = 70, B = 56, C = 55

That's how it works on paper, anyway.

First, I should point out that there is nothing wrong with this outcome.
A is the sincere CR winner, the IRV winner, and while not a CW, I suspect
would win most Condorcet methods (though I haven't worked them out). A
would also win Approval with individual strategies, if the voters place
optimal ballots based on accurate polling projections (which would name
A as the front runner and B as the runner-up).

Second, this example proves that the optimal group strategy *isn't*
the first-choice-only strategy advocated by Donald.

But will this strategy play out in exactly this way in real life?

Consider this scenario: Due to inaccurate polling information, the A
faction designates an insufficient number of its members to vote AB.
The C voters suspect the A voters aren't perfect strategists and all
vote C only. The final tally could then be

21 A
24 AB
30 BC
25 C

A = 45, B = 54, C = 55

So the A voters would have done better using individual strategies.

In fact, making a group strategy work in a large-scale Approval
election involves a lot of ifs.

IF your polling information is very accurate and very detailed,
and IF you have enough numbers in your faction to dictate a dominant
strategy,
and IF your factions members have similar utilities so they can be
counted on to act in solidarity with the group strategy,
and IF you can accurately predict how the other factions will react,
and IF you can communicate your group strategy to the (millions of)
members of your faction,
and IF you can guarantee a sufficient turnout from your faction,
and IF no more than a tiny percentage of your group members foul up
your numbers by going with their individual strategies instead of
the dictated ballots,
and IF the other faction(s) whose cooperation is necessary does (do)
in fact cooperate,
THEN your group strategy might work.

And if it does work, then all your efforts might just be rewarded
with the result that would have been attained if you had let the
voters exercise individual strategies. If it backfires, you
might end up with a much worse result.

My conclusion is that Approval will discourage factional strategies
for these practical reasons. I don't know which other methods would
also have this tendency. Some might discourage such strategies more,
some might discourage them less.

And with regards to Donald's statement, it is very likely that any
group strategy will involve at least some members of your group
voting for more than one candidate, especially in close races. So
nobody should believe Donald when he says the best strategy in
Approval is to vote for only one candidate. Clearly that isn't true.

     -- Richard