I got a comment that the component election descriptions were too
vague. Here's a more detailed description of what would have happened
if there had been two separate elections.
If there were two separate elections then they would be
- Election 1: A vs. B
- Election 2: C vs. D
The votes (using rankings only, ignoring the ratings) would be
Election 1:
27: A>B
26: B>A
25: A>B
22: B>A
and
Election 2:
27: D>C
26: C>D
25: C>D
22: D>C
- A would win B (52 - 48) (supported by the first and third group of
voters)
- C would win D (51 - 49) (supported by the second and third group of
voters)
But if we combine these two elections and use ratings then we would
get those combined votes (with ratings) that are listed below. I thus
just derived the votes of the component elections (what they would
have been) from the combined ratings.
Juho
On Dec 4, 2009, at 2:12 AM, Juho wrote:
Here's one more method in the series of how to collect sincere
ratings.
The point is to combine several elections into one. I'll give one
example. Let's take two Condorcet elections where the ballots are
ratings based. The first election is between A and B. The second
election is between C and D. The preferences (ratings) are as follows.
27: A=1 B=0 C=0 D=2
26: A=0 B=2 C=1 D=0
25: A=2 B=0 C=1 D=0
22: A=0 B=1 C=0 D=2
A would win the first Condorcet election (or Plurality or whatever
common single-winner method). C would win the second Condorcet
election.
Let's then combine these elections into one election in which the
outcome alternatives (sets of winners of the two component
elections) will be AC, AD, BC and BD. We can sum up the preferences
so that each voter is considered to prefer outcome x to outcome y if
the sum of his/her ratings of the candidates is higher in outcome x
than in y. The first 27 voters are thus considered to prefer outcome
AD (1+2 points) to BD (0+2) and AC (1+0) and BD (0+0).
27: AC=1 AD=3 BC=0 BD=2
26: AC=1 AD=0 BC=3 BD=2
25: AC=3 AD=2 BC=1 BD=0
22: AC=0 AD=2 BC=1 BD=3
Based on the resulting preference orders we will then use some
Condorcet method (=some good single winner method) to determine the
winning outcome.
With these votes the winner is outcome AD. The combined election
thus doesn't elect both Condorcet winners of the component elections
but changes winner from C to D in the second component election. The
combined election collects some additional information when compared
to having two independent elections, and that additional information
leads in this case to different results (although we still use
Condorcet to pick the winner).
It is possible to allow the voters to use whatever means to indicate
their preferences between different (combined) outcomes. Typically
the number of different possible outcomes is however high, so it is
not feasible to evaluate and rate all possible outcomes. Some more
compact approach is needed. Sum of ratings is a quite natural way to
derive the required preferences from a small(ish) amount of input
(often the opinions are quite well "summable" in this sense).
The input values (in the ballots) could be anything, e.g. from -
infinite to +infinite. Some agreed fixed points could be named (e.g.
1="acceptance threshold" 2="excellence threshold") to make the votes
of different voters comparable (for other uses like statistics) (or
one could normalize them if one wants all votes to have "equal
weight").
Also the set of outcomes can be determined quite freely. It is for
example possible that candidates are allowed to take part in
multiple component elections but only outcomes in which all the
winners (of different component elections) are different are
acceptable (i.e. nobody can get two jobs). Or one might agree that
party x must win y elections, each gender to get at least 40% of
some set of seats etc.
This method does not avoid the typical Condorcet related problems
and strategic incentives. In many cases the strategic problems may
however be slightly smaller due to the added complexity (more
difficult to master). In what aspects would this type of combined
election be worse than Condorcet or would fail to collect sincere
ratings (at about the same level as Condorcet collects sincere
rankings)?
Juho
P.S. One could add still more complexity by covering also multi-
winner elections (a la CPO-STV, or why not also list/tree based).
Proportionality can be seen as an absolute requirement on what
outcomes are acceptable or as one target that will be evaluated
numerically (and this result then will have an impact on what
outcomes the society is considered to like). The formula that
determines the winning outcome can be flexible (just like the voter
preferences and allowed outcomes). One could still have similar
rules for required supermajorities etc. Things may get complex, so a
good approach is to just determine the conditions and preferences
(at user and society level) and then use generic optimization
algorithms in some agreed way to seek (and hopefully find or
approximate) the best possible outcome.
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