[EM] Combined elections
Juho
juho4880 at yahoo.co.uk
Thu Dec 3 16:12:45 PST 2009
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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