[EM] Amalgamation details

Jameson Quinn jameson.quinn at gmail.com
Mon Aug 1 21:42:39 PDT 2011

```OK, that's what I thought. So, candidate hijacking does not work for any
amalgamated "ballot blind" method, that is, a method which forgets which
rating came from which ballot. However, on a non-ballot-blind system,
including the ranking-based DSC which was the next step in your
SODA-inspired "sequential play" method, it can work. Basically, it involves
finding a faction a bit smaller than yours, and ranking its favorite
candidate first. Since your faction is larger, you will be able to set the
ranking of the remaining candidates, and you will gain the ballot weight of
the smaller faction. Of course, you must be sure that the "false flag"
candidate does not win. This is similar to Wodall free riding in PR.

JQ

2011/8/1 <fsimmons at pcc.edu>

> To amalgamate factions so that there is at most one faction per candidate X
> (in the context of range
> style ballots) take a weighted average of all of the ballots that give X
> top rating, where each ballot has
> weight equal to one over the number of candidates rated equal top on that
> ballot.  The total weight of the
> resulting "faction rating vector" for candidate X is the sum of the weights
> that that were used for the
> weighted average.
>
> Note that these faction rating vectors are efficiently summable.  A running
> sum (together with its weight)
> is kept for each candidate.  Any single ballot is incorporated by taking a
> weighted average of the running
> sum and the ballot, where the respective weights are those mentioned above.
>  For the running sum it is
> the running sum weight.  For the ballot it is zero if the candidate is not
> rated top, and 1/k if it is rated top
> with (k-1) other candidates..
>
> To combine two running sums for the same candidate take a weighted average
> of the two using the
> running sum weights, and then add these weights together to get the
> combined running sum weight.
>
> If you multiply each faction rating vector by its weight and add up all
> such products, you get the vector of
> range totals.
>
> Of course Range as a method is summable more efficiently without
> amalgamating factions, but other
> non-summable methods, when willing to accept amalgamated factions, thereby
> become summable.
>
> So, for example, we can make a summable form of Dodgson:
>
> (1) Use ratings instead of rankings.
>
> (2) amalgamate the factions.
>
> (3) let each candidate (with help from advisors) propose a modification of
> the ballots that will created a
> Condorcet Winner.
>
> (4) the CW that is created with the least total modification is the winner.
>
> Modifications are measured by how much they change the ratings on how many
> ballots.
>
> For example if you change X's rating by .27 on 10 of the 537 ballots of one
> faction, and by .32 on 15
> ballots from another faction, then the total modification is  2.7 + 4.8 =
> 7.5
>
> The reason for the competition is that otherwise the method would be
> NP-complete.
>
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