# [EM] D2MAC can be much more efficient than Range Voting (corrected)

Warren Smith wds at math.temple.edu
Wed Mar 7 12:01:08 PST 2007

```>Jobst Heitzig:
A typical voting situation is

55%:  A>C>B
45%:  B>C>A

with C being considered a good compromise by all voters
(in the sense that all voters would definitely prefer C strongly
to tossing a coin between A and B).

--WDS: to be concrete, let us consider utility(A)=10, utility(C)=9, utility(C)=0
for the first group of voters, and utility(A)=0, utility(C)=9, utility(C)=10
for the second group.

>Jobst continues:
At first, it seems that this is exactly the situation where
methods which claim to maximize "social utility" or "social benefit"

If all voters were sincere, indeed both Approval Voting and
Range Voting will elect C, since the ballots would then
look like this:

Approval Voting:
55%: A,C
45%: B,C
Winner: C

Range Voting:
55%: A 100, C 50+whatever, B 0
45%: B 100, C 50+whatever, A 0
Winner: C

--WDS: social average utility = 9.

The problem is, rational voters just *won't* vote sincere in this
situation. And since Approval and Range Voting are *majoritarian*
methods, the real outcome will rather look like this:

Approval Voting:
55%: A
45%: B,C
Winner: A

Range Voting:
55%: A 100, C 0, B 0
45%: B 100, C 99, A 0
Winner: A

--WDS: social average utility = 5.5.

So, both these methods *fail* to do just what they were apparently
constructed to do!

Now let us look how D2MAC performs in this typical situation,
a democratic, non-majoritarian method:

Recall that in D2MAC you specify a favourite and as many "also approved"
options as you want. Then two ballots are drawn and the winner is the
most approved option amoung those that are approved on both ballots
(if such an option exists), or else the favourite option of the first
ballot.

If voters are sincere, the result will be this:
55%: favourite A, also approved C
45%: favourite B, also approved C
Winner: C

--WDS correction: actually, it appears to me that the winner is
A with probability 30.25%
B with probability 20.25%
C with probability 49.50%
with social average utility = 7.03

>Jobst Heitzig continues:
Can the A-faction improve their result upon this by voting differently?

If they switch to
55%: favourite A, none also approved
then A will win whenever an A-ballot is drawn first, i.e., with 55%
probability. However, B will win in the remaining cases, i.e., with 45%
probability.

--WDS agreement: that the winner is
A with probability 55.00%
B with probability 45.00%
with social average utility = 5.05

For the A-supporters, this "almost coin tossing" is not
preferable to C, so the strategy won't help them.

--WDS:
true if C's utility is sufficiently near to A's, since 5.05 < 7.03.

Thus, the "obvious" A-strategy cannot destroy the compromise under D2MAC!

>Jobst continues:
The only way the A-voters could have to strategically improve the result
is to switch to something like this:
55-x%: favourite A, none also approved
x%:    favourite A, also approved C

--WDS correction of Jobst wrong probabilities: in that case, winner is
A with probability 55 - 0.45*x %
B with probability 45 - 0.45*x %
C with probability 0.9*x %

The utility for the A voters is then a linear function of x and
their best strategy is therefore either x=0 or x=55.

>Jobst continues:
This lottery may or may not be preferable to C for the A-voters.

--WDS correction:
First, Jobst is acting falsely like comparing versus (C wins) is the right comparison to
make.  Second this lottery is NEVER preferable to the x=55 endpoint.

--WDS conclusions after correcting JObst errors:
Jobst is correct that D2MAC incentivizes voters to act differently than approval,
in this case.  Further, Jobst is correct that in certain situations, this
can result in higher social utility than range voting - in this example,
utility=7.03 for D2MAC with honest=strategic voters, versus
utility=5.50 for approval=range with strategic voters, versus
utility=9.00 with honest (or zero info strategic) approval voters,  versus
utility=9.00 with honest range voters.

However, my computer study of 1D politics [paper 95 here
http://www.math.temple.edu/~wds/homepage/works.html]  showed
EVERY voting system studied there  (even "random winner")
does better (in terms of social utility) than range voting
in SOME situations.  Indeed every voting syste there does better than every
other in some situations.  But on average over all situations, range is
the clear winner.  I believe includin D2MAC in that study would
not alter this conclusion.  Indeed I spsuect D2MAC is quite poor
on average.

Warren D Smith
http://rangevoting.org

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