[Election-Methods] Representative Range Voting with Compensation - a new attempt
Dear folks,
I must admit the last versions of RRVC (Representative Range Voting with
Compensation) all had a flaw which I saw only yesterday night. Although
they did achieve efficiency and strategy-freeness, they did not achieve
my other goal: that voters who like the winner more than the random
ballot lottery compensate voters who liked the random ballot lottery
more than the winner. In short, the flaw was to use the three randomly
drawn voter groups for only one task each, either for the benchmark, or
the compensation, or the decision.
I spare you the details and just give a new version which I think may
finally achieve all three goals: efficiency, strategy-freeness, and
voter compensation.
The basic idea is still the same: Partition the voters randomly into
three groups, let one group decide via Range Voting, and use each group
to benchmark another group and to compensate still another group.
To make an analysis more easy, I write it down more formally this time
and assume the number of voters is a multiple of 3.
DEFINITION OF METHOD RRVC (Version 3)
=
Notation:
-
X,Y,Z are variables for options
i,j,k are variables for voters
f,g,h are variables for groups of voters
Input:
--
All voters give ratings and mark a favourit. Put...
R(X,i) := the rating voter i gave option X
F(i) := the option marked favourite on ballot of voter i
A(i) := balance on voter i's voting account before the decision
Tally:
--
Randomly partition the N voters into three groups of equal size.
The winner is the range voting winner of group 1.
The voting accounts are adjusted as follows. Put...
S := N/3
Q := (S-1)/S
G(i) := group in which voter i landed
T(X,f) := total rating group f gave option X
= sum { R(X,i) : i in group f }
W(g) := range voting winner of group g
= that W with T(W,g)T(X,g) for all X other than W
P(X,h) := proportion of group h favouring X
= probability of X in group h's random ballot lottery
= # { i in group h : F(i)=X } / S
D(f,g,i) := rating difference on voter i's ballot
between the range voting winner of group f
and the random ballot lottery of group g
= R(W(f),i) - sum { P(X,g)*R(X,i) : X }
E(f,g,h) := total rating difference in group h
between the range voting winner of group f
and the random ballot lottery of group g
= sum { D(f,g,i) : i in group g }
For each voter i, add the following amount to her voting account C(i):
If i is in group 1:
deltaC(i) := E(1,2,1)-D(1,2,i) - E(2,2,2) - Q*E(3,3,2) + E(3,3,3)
If i is in group 2:
deltaC(i) := E(3,3,2)-D(3,3,i) - E(3,3,3) - Q*E(1,1,3) + E(1,1,1)
If i is in group 3:
deltaC(i) := E(1,1,3)-D(1,1,i) - E(1,1,1) - Q*E(1,2,1) + E(2,2,2)
(Remark: E(1,2,1) and D(1,2,1) are not typos!)
(END OF METHOD RRVC)
Analysis:
-
1.
The sum of all C(i) remains constant, so voting money retains its
value. To see this, note that
sum { E(1,2,1)-D(1,2,i) - E(2,2,2) : i in group 1 }
= S*E(1,2,1) - E(1,2,1) - S*E(2,2,2) )
= S*( Q*E(1,2,1) - E(2,2,2) )
= sum { Q*E(1,2,1) - E(2,2,2) : i in group 3 }
and analogous for the other terms in the above sums.
2.
Note that the terms E(1,2,1)-D(1,2,i), E(3,3,2)-D(3,3,i), and
E(1,1,3)-D(1,1,i) in the above sums do not depend on voter i's ratings!
Hence the only way in which the ballot of voter i can affect her own
voting account is trough the dependency of W(1) on her ratings, and
this is only the case for voters in group 1, the deciding group.
So, as only voters in group 1 can influence their outcome, an analysis
of individual voting strategy is only required these voters. For such a
voter i the net outcome, up to some constant which is independent of
i's behaviour, is this:
O(i) := sum { R(W(1),j) : j other than i } + U(W(1),i)
where
U(X,i) := true value of X for i.
If voter i is honest and puts R(X,i)=U(X,i), this simply adds up to
O(i) = T(W(1),1) (if i is honest).
Now assume this honest voter i thinks about changing the winner from
W(1) to some other option Y by voting dishonestly. The net outcome for
i after this manipulation would be
O'(i) = sum { R(Y,j) : j other than i } + U(Y,i)
= T(Y,1)-R(Y,i) + U(Y,i)
= T(Y,1)
T(W(1),1) = O(i).
So after all, i has no incentive to manipulate the outcome because she
would have to pay more than she gains from this.
3.
Now consider a large electorate of honest voters, and think about what a
voter can expect, before the random process of drawing the three groups
is applied, of how much her voting account will be adjusted. If I got
it right this time, this expected value of deltaC(i) should be, up to
some constant term which is equal for all voters, just
the rating difference on voter i's ballot
between the random ballot lottery
and the winner of the decision,
Re: [Election-Methods] Representative Range Voting with Compensation -a new attempt
A first typo: It must read C(i) instead of A(i) under Input...
--
Dear folks,
I must admit the last versions of RRVC (Representative Range Voting with
Compensation) all had a flaw which I saw only yesterday night. Although
they did achieve efficiency and strategy-freeness, they did not achieve
my other goal: that voters who like the winner more than the random
ballot lottery compensate voters who liked the random ballot lottery
more than the winner. In short, the flaw was to use the three randomly
drawn voter groups for only one task each, either for the benchmark, or
the compensation, or the decision.
I spare you the details and just give a new version which I think may
finally achieve all three goals: efficiency, strategy-freeness, and
voter compensation.
The basic idea is still the same: Partition the voters randomly into
three groups, let one group decide via Range Voting, and use each group
to benchmark another group and to compensate still another group.
To make an analysis more easy, I write it down more formally this time
and assume the number of voters is a multiple of 3.
DEFINITION OF METHOD RRVC (Version 3)
=
Notation:
-
X,Y,Z are variables for options
i,j,k are variables for voters
f,g,h are variables for groups of voters
Input:
--
All voters give ratings and mark a favourit. Put...
R(X,i) := the rating voter i gave option X
F(i) := the option marked favourite on ballot of voter i
A(i) := balance on voter i's voting account before the decision
Tally:
--
Randomly partition the N voters into three groups of equal size.
The winner is the range voting winner of group 1.
The voting accounts are adjusted as follows. Put...
S := N/3
Q := (S-1)/S
G(i) := group in which voter i landed
T(X,f) := total rating group f gave option X
= sum { R(X,i) : i in group f }
W(g) := range voting winner of group g
= that W with T(W,g)T(X,g) for all X other than W
P(X,h) := proportion of group h favouring X
= probability of X in group h's random ballot lottery
= # { i in group h : F(i)=X } / S
D(f,g,i) := rating difference on voter i's ballot
between the range voting winner of group f
and the random ballot lottery of group g
= R(W(f),i) - sum { P(X,g)*R(X,i) : X }
E(f,g,h) := total rating difference in group h
between the range voting winner of group f
and the random ballot lottery of group g
= sum { D(f,g,i) : i in group g }
For each voter i, add the following amount to her voting account C(i):
If i is in group 1:
deltaC(i) := E(1,2,1)-D(1,2,i) - E(2,2,2) - Q*E(3,3,2) + E(3,3,3)
If i is in group 2:
deltaC(i) := E(3,3,2)-D(3,3,i) - E(3,3,3) - Q*E(1,1,3) + E(1,1,1)
If i is in group 3:
deltaC(i) := E(1,1,3)-D(1,1,i) - E(1,1,1) - Q*E(1,2,1) + E(2,2,2)
(Remark: E(1,2,1) and D(1,2,1) are not typos!)
(END OF METHOD RRVC)
Analysis:
-
1.
The sum of all C(i) remains constant, so voting money retains its
value. To see this, note that
sum { E(1,2,1)-D(1,2,i) - E(2,2,2) : i in group 1 }
= S*E(1,2,1) - E(1,2,1) - S*E(2,2,2) )
= S*( Q*E(1,2,1) - E(2,2,2) )
= sum { Q*E(1,2,1) - E(2,2,2) : i in group 3 }
and analogous for the other terms in the above sums.
2.
Note that the terms E(1,2,1)-D(1,2,i), E(3,3,2)-D(3,3,i), and
E(1,1,3)-D(1,1,i) in the above sums do not depend on voter i's ratings!
Hence the only way in which the ballot of voter i can affect her own
voting account is trough the dependency of W(1) on her ratings, and
this is only the case for voters in group 1, the deciding group.
So, as only voters in group 1 can influence their outcome, an analysis
of individual voting strategy is only required these voters. For such a
voter i the net outcome, up to some constant which is independent of
i's behaviour, is this:
O(i) := sum { R(W(1),j) : j other than i } + U(W(1),i)
where
U(X,i) := true value of X for i.
If voter i is honest and puts R(X,i)=U(X,i), this simply adds up to
O(i) = T(W(1),1) (if i is honest).
Now assume this honest voter i thinks about changing the winner from
W(1) to some other option Y by voting dishonestly. The net outcome for
i after this manipulation would be
O'(i) = sum { R(Y,j) : j other than i } + U(Y,i)
= T(Y,1)-R(Y,i) + U(Y,i)
= T(Y,1)
T(W(1),1) = O(i).
So after all, i has no incentive to manipulate the outcome because she
would have to pay more than she gains from this.
3.
Now consider a large electorate of honest voters, and think about what a
voter can expect, before the random process of drawing the three groups
is applied, of how much her voting account will be adjusted. If I got
it right this time, this expected value of deltaC(i) should be, up to
some constant term which is equal for all voters, just
the rating difference on
