[EM] Combined elections

[Juho]

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.