# [EM] Combined elections

Juho juho4880 at yahoo.co.uk
Thu Dec 3 18:20:10 PST 2009

```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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