I wanted to consider this afresh.
At 01:58 PM 4/28/2008, Jobst Heitzig wrote:
Hello folks,
over the last months I have again and again tried to find a solution to
a seemingly simple problem:
The Goal
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Find a group decision method which will elect C with near certainty in
the following situation:
- There are three options A,B,C
- There are 51 voters who prefer A to B, and 49 who prefer B to A.
- All voters prefer C to a lottery in which their favourite has 51%
probability and the other faction's favourite has 49% probability.
- Both factions are strategic and may coordinate their voting behaviour.
First of all, a method which elects C unambiguously from these
conditions is problematic. This is a very close election, but the
majority prefer A. Voters, however, may not like taking risks; for
the A voters to purely shoot for an A victory is very dangerous, slip
up and they get B. As stated, there is no way to know who is the best
winner from the SU measure, which, however, Jobst fixes:
Those of you who like cardinal utilities may assume the following:
51: A 100 > C 52 > B 0
49: B 100 > C 52 > A 0
That's down in the noise, however.
Note that Range Voting would meet the goal if the voters would be
assumed to vote honestly instead of strategically. With strategic
voters, however, Range Voting will elect A.
That depends on their knowledge. If this is a zero-knowledge
situation, and they vote according to personal expected outcome
maximization, and it is Range 100, all voters will vote C 52. Which
shows up to problems: human beings don't have preferences that fine,
unless special abstractions allow them to assign precise utilities,
such as known monetary return from outcomes.
However, all voters would vote, in Range 2 and Range 10, C as
midrange. In Range 2, this would give us A 5100, B: 5000, C:4900. A
wins. But with Range 100, and assuming, instead, that 52% is the
*average* sincere rating, which makes more sense than assuming they
are uniform, they would, if it's zero-knowledge and they vote
intelligently to maximize personal utility, elect B. The certainty of
this increases with the number of voters.
Contrary to Jobst's first intuition, Range *does* satisfy this, if
the resolution is adequate.
Imagine a continuum of voters who are described, in sum, as above.
The numbers Jobst gives for the C utility are averages. Given that
all voters vote "strategically," all voters vote the extremes. So the
only question is how they will rate C.
The claim is often made that "strategic" voters in Range will bullet
vote. However, a bullet vote risks the election of the least-desired
candidate. The issue is often complicated by Range critics by
aasuming some weak utility, then assuming that the voter will,
knee-jerk, bullet vote for the favorite, as if the voter is strongly
motivated. What has been done is to assume two contradictory voter
utilities: a weak one and a strong one. A voter who votes, say, a
straight party ticket, without regard for the individual candidates,
has a strong utility for the party's candidate, not a weak one. Very
strong. Only if you grab this voter by the scruff and shake him and
say, "No, you are not going to elect Abe Lincoln. Now that we have
settled that, and stricken Lincoln's name from the ballot, whom do
you prefer, Adolf Hitler or Josef Stalin?" Never mind that the
voter's response may well be a serious abstention, caused by rapidly
leaving the jurisdiction in question. If the voter has any sense and
is able to do so.
If we assume that the voters exist in a curve such that 52 is the
median vote (as well as the average one), and they vote
strategically, using the best game theory to do so, Approval probably
satisfies the problem. At 50% utility, bullet voting and voting for
two are exactly matched, take your pick, if there are enough voters.
(When I studied this in detail, I found that bullet voting was
slightly better in expected outcome, but it was only with very small
numbers of voters. With many voters, they are equal. But if the
expected utility is higher than 50%, in Approval, the zero-knowledge
strategic vote is to vote for both. And a lot of words have been
written to the contrary without actually studying the game matrix and
election probabilities.)
If the median vote is 52, what is the average vote, if voters can
only vote 0 or 100? (Approval). Under the strategic voting
assumption, it would surely be above 50. And probably more than 1%
above, with high certainty, but I haven't done the exact math, and
the shape of the curve is not precisely stated.
If they can vote Range 100, and if they know their utilities and the
math and vote strategically, they vote their average vote, which, if
the bell curve is symmetrical, would probably be 52. C wins.
Why is this contrary to what so many have written? Well, zero knowledge.
So let's look at the opposite extreme? Perfect knowledge. Range 100.
But there is a problem. If every voter has perfect knowledge of all
other voters, we get a loop, the votes are indeterminate. But we can
do it recursively. Suppose 99 voters have voted, and one last voter
votes. The 99 voters voted strategically, zero-knowledge, as described.
If the last voter supports A:
The presenting situation is
A: 5000
B: 4900
C: 5200 - 52 = 5148
The last voter can't swing the election. You have to have two A
voters as the last voters, both of them know the results, and they
both know that they are A voters.
If the last voter supports B:
A: 5100
B: 4800
C: 5148. Again, the last voter can't swing it, period.
Jobst, Range is your method.
Now, what is wrong with this picture? Well, voters don't vote
strategically in their own interest. Or do they? Maybe we just don't
understand their true expectations and utilities.
I think this example shows the complexity of the real issues. We like
to analyze elections based on highly simplified scenarios, which then
allow us to make sweeping generalizations, such as "they all vote
strategically," as if that means that they all bullet vote if they can.
Antiplurality, though, is a fairly simple solution. Vote for one only
to lose. Rationally, nobody would vote for B to lose, everyone
prefers A or B to lose. Sure, they could vote "strategically," but
they'd better be very, very sure, they could easily cause the
election of their least favorite. It makes no sense from game theory.
Of course, a lot of human behavior doesn't make sense from the
simplest versions of game theory.
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