I told Rob Richie, use "IRV", its catchy (was Re:[EM]Approval Elections & Effective Weights
Martin Harper
mcnh2 at cam.ac.uk
Thu Apr 12 11:09:20 PDT 2001
I was going to write a longer reply, but this seems to be the crux of the matter:
Craig wrote:
> The test [for the power of a vote] is clearly stated but it does not apply to
> methods outside of the topic of preferential voting (i.e. checkbox methods,
> e.g. the ASVB sub-vote Variant Block Vote voting method. I call it Approval).
Now, previously you stated that Approval wasn't a preferential voting method,
right? So the test doesn't apply to Approval, right? Great! So why exactly were we
discussing whether Approval passes or fails the test? :-/
What have I missed? Something important clearly...
Anyway, since you asked me some questions about the email I sent to you - here are
some answers.
--------------------------------
ORIGINAL TEXT
> What ought be done is to have some of these weights be calculated.
Well, first up we need to decide exactly what these weights are, hmm? You don't
say yourself, so I'l trundle merrilly off with a definition of my own.
-- begin definition
Let C1, C2, ... Cn be the candidates.
Let V be a voter, and B be his/her/its ballot paper.
Let U1, U2, ... be V's normalised sincere utilities of the respective candidates.
Let P1, P2, ... be the prior probabilities of the respective candidates winning.
Let P1', P2', ... be the probabilities after the submssion of the ballot paper B.
The Prior utility (U) of the election is the sum of Ux times Px over all x.
The utility after the submission of B (U1) of the election is the sum of Ux times
Px' over all x.
U' - U is the effective weight of the voter V with ballot paper B.
Find the ballot paper B which maximises U' - U. The effective weight of this
ballot paper for V is the effective weight of V. Note that this ballot paper need
not be sincere. The Sincere Effective Weight of V is the maximum effective weight
available to V on any sincere ballot paper.
-- end definition
The justification is that the point of an election for a voter (ignoring 'sending
a message' and such) is to get the best outcome from their perspective, and this
'bestness' is best measured from utilities, cos that's what they are. Every single
election method in the world penalises voters who don't vote or who spoil their
ballot papers or who vote for someone they hate, so we should concentrate on the
effective weight of the *voter* not the *vote*.
In an ideal world the sincere effective weight of any voter would be equal to the
effective weight of that voter, and non-negative. In addition, the effective
weights of each voter should be equal. Dictatorship or Random satisfies this, as
all effective weights are zero. Random Ballot, Approval, Condorcet, IRV, etc, all
fail utterly. In general, A>B=C voters will have no effective weight unless A is a
contender for first place.
--
Anyway, here's an example to see how the concept works in practice, for those who
aren't keen on maths without examples. Skip otherwise.
100 voters have the following utilities, and represent the 'A' faction:
1.0 A
0.1 B
0.0 C
zero or one voters (equiprobably) have the following utilities, and represent the
'B' faction:
1.0 B
0.0 A,C
100 voters have the following utilities, and represent the 'C' faction.
1.0 C
0.1 B
0.0 A
--
Condorcet
If we are a new member of the A faction, we get the values shown for our only
possibly sincere vote:
Initial: P(A) = 0, P(B) = 0.5, P(C) = 0, P(A,C) = 0.5, U = 0.3
After: P(A) = 0.5, P(B) = 0, P(C) = 0, P(A,B) = 0.5, U' = 0.775
So our effective weight is 0.475
A new member of the C faction has the same weight.
If we are a new member of the B faction, we get the values shown for our only
possible sincere vote:
Initial: P(A) = 0, P(B) = 0.5, P(C) = 0, P(A,C) = 0.5, U = 0.5
After: P(A) = 0, P(B) = 1, P(C) = 0, U=1
So our effective weight is 0.5
Hence, the Condorcet effective weights are 0.475 and 0.5.
--
Approval
Presume that at least 60% of the A and C factions bullet vote.
If we are a new member of the A faction, we get the values shown for an AB vote
and an A vote (the two sincere votes available)
Initial: P(A) = 0, P(B) = 0, P(C) = 0, P(A,C) = 1.0, U = 0.5
AB: P(A) = 1, P(B) = 0, P(C) = 0, U' = 1.0
A: P(A) = 1, P(B) = 0, P(C) = 0, U' = 1.0
So our effective weight is 0.5
A new member of the C faction has the same weight.
If we are a new member of the B faction, we get the values shown for our only
possible sincere vote (bullet vote for B):
Initial: P(A) = 0, P(B) = 0, P(C) = 0, P(A,C) = 1.0, U = 0.5
After: P(A) = 0, P(B) = 0, P(C) = 0, P(A,C) = 1.0, U = 0.5
So our effective weight is zero.
Hence the Approval effective weights are 0.5 and 0
--------------------------
QUESTIONS
1) what the "P" function is.
Answer: I was abbreviating to save space. P(A) refers to "The probability that the
candidate named 'A' will win the election". Similarly, P(A,B) refers to "The
probability that the candidates named A and B will draw the election between
them". It's not some kind of function. Sorry for the misunderstanding - I thought
that this was standard notation.
2) derivation of the effective weight of 0.5 for the Approval example.
Answer: The effective weight, as I defined it, is U' - U.
U' = 1.0,
U = 0.5,
1.0 - 0.5 = 0.5
Hence the effective weight is 0.5
3) derivation of U=0.5 for the Approval example.
Answer: There is only one outcome in the initial case: a draw between A and C.
This draw is resolved randomly. Therefore there is a 50% chance of electing A,
with utility (for A voters) of 1.0, and a 50%, with a utility (for A voters) of
0.0
50% of 1.0 + 50% of 0.0 = 0.5
Therefore U=0.5
4) How the 60% figure affects things for the Approval example
Answer: The requirement for greater than 60% bullet-voting is just to ensure that
enough people bullet vote that B doesn't get elected. Given the utilities of A, B,
C for the varying factions, this seemed a reasonable assumption to make. In
practice I'd expect the proportion to be much, much higher than 60%.
5) What does the "if we are a new member of the B faction ... our sincere
vote" sentence mean?
Answer: If we are a member of the B faction, with the utilities given, then the
only way we can sincerely vote in approval in the example given is to bullet vote
for B. Any other vote would be insincere.
6) What's the definition of sincerity?
My definition is:
[subdefine] A greater insincere preference is a case where we prefer A to B, but
express a preference on our ballot paper for B over A. A lesser insincere
preference is a case where either we prefer A to B but do not express a
preference, or we have no preference between them but we express one.
[maindefine] A sincere vote is a vote which does not have any greater insincere
preferences unless voting them is necessary to avoid other greater insincere
paiwise preferences. In addition, a sincere vote does not have any lesser
insincere preferences unless voting them is necessary to avoid other lesser
insincere paiwise preferences
Note that other people have other definitions - but I like mine. My definition of
effective weight works with any definition of sincerity. Note that my defintion
should not be applied to determine the sincerity of more complicated vote types
than those used in Condorcet, Approval, Plurality - such as those used in Dyadic
Approval or Average Ratings.
7) Invitation to prove that the values of P(...) are correct.
Answer: That would be rather tedious, so I shall decline your invitation. However,
for your benefit, I'll rewrite the stuff to not use the P() notation that you
don't like.
Initial Condorcet outcome - If there is one B voter, then B will win. Otherwise A
and C will draw. [equiprobable]
Condorcet outcome after an A vote - If there is one B voter, then A and B will
draw. Otherwise A will win. [equiprobable]
Condorcet outcome after a B vote - B will win.
Condorcet outcome after an C vote - If there is one B voter, then C and B will
draw. Otherwise C will win. [equiprobable]
Initial Approval outcome - A and C draw
Approval outcome after an A vote - A wins
Approval outcome after a B vote - A and C draw
Approval outcome after a C vote - C wins.
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