Proportional Representation is the goal of a variety of voting methods, as described e.g. in a 2017 presentation Proportional Representation for the League of Women Voters, Boulder County (LWVBC) and in a 2019 presentation on Multi-Winner Approval Voting, available at Voting Methods Team - League of Women Voters of Boulder County.
The pr_voting_methods project provides code to implement a variety of proportional representation voting methods, and some data on how individual ballots have been cast in some multi-winner contests, and analyzes how different election methods would handle the same votes.
Proportional representation is perhaps easiest to implement in the US when working with voting methods which use the same ballots used for plurality voting contests, and for approval voting.
The Plurality Block Voting method is commonly used in council and board of election contests in the US. In an election with N winners, it allows voters to vote for up to N candidates. So, with the important exception that we don't see how the voters would vote if they could vote for more than N candidates, we can tabulate them using voting methods that use an approval voting ballot, and get at least some insight into methods like Proportional Approval Voting.
You can click 
pav.ipynb notebook) online.
On a notebook on Binder, you can use File/Open/Upload to upload your own Cast Vote Records in csv format (like the canned example in Byers_SD_32J_Adams.csv and analyze them in the same way.
- pav.py: Proportional Approval Voting
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City of Westminster Colorado City Councilor (Vote for 3)
- Cast Vote Record data on each ballot from 2015, when there were 10 candidates
- Westminster_Adams.csv
- Westminster results for Jefferson county: should be available soon....
- Cast Vote Record data on each ballot from 2015, when there were 10 candidates
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Byers School District (Vote for 4)
- Cast Vote Record data on each ballot from 2015, when there were 8 candidates
- Byers_SD_32J_Adams.csv
- We have no Cast Vote Records for the Arapahoe County portion of this contest for 2015
- Cast Vote Record data on each ballot from 2015, when there were 8 candidates
-
Official Results by county in 2015
See the PAV notebook for PAV results, so far with just the Adams County results in the 2015 Westminster Colorado City Council contest and Byers SD contest.
To run tests on the pav module (where no output means it is working):
python3 pav.py --test
Work relating to audits of PAV contests is in progress, and suggests that a risk-limiting audit method would not be difficult to implement. That is also true for Satisfaction Approval Voting (SAV).
Sequential Proportional Approval Voting (SPAV) is likely to be trickier because it proceeds in multiple rounds. See related work at https://people.eng.unimelb.edu.au/michelleb/IRV-auditing.pdf and https://www.ece.rutgers.edu/~asarwate/pdfs/SarwateCS13irv.pdf .
Note that any voting method can be scientifically audited via Bayesian methods.
Since there are more possible outcomes to distinguish, margins in general will be tighter for multi-winner methods than for single-winner methods.
But if changing a small number of ballots would alter just one winner in a 5-winner contest, it might not affect the balance of power or the legislative decision-making much. So the impact might be considered far less than with a single-winner contest in a highly-polarized electorate.
It would also be interesting to look at the outcomes in other ways, e.g. the risk that each individual candidate is or is not actually a winner, or the risk that the "utility" of the outcome for the electorate (as defined e.g. by the PAV score) is off by some fraction.
Add more statistics. E.g. for a PAV score, show it also as an average score per ballot, or per non-blank ballot, or show as a fraction of the maximum possible score.
This project is licensed under the terms of the MIT license.
Warren D. Smith has an extensive page on Proportional Approval Voting. RangeVoting.org - Optimal proportional representation
Fast C language code for the algorithms discussed in that paper is available at CleanOptPRVote.c and runs about 100 times faster than this Python code. Besides being written in C, it also uses Knuth's clever "revolving-door" algorithm R. as described on page 9 of his pre-fascicle 3A, generating all combinations. It has the advangage of changing only one element of the winner set at each step, so we can track and update the number of winners for the next ballot in constant time. See also the "MC" algorithm at Combinations.
The first risk-limiting audit result for proportional representation is published at Verifiable European Elections: Risk-limiting Audits for D’Hondt and its relatives by Philip B. Stark and Vanessa Teague, March 26, 2015
The code to both allocate seats, and audit the allocation, is at DKDHondt14: IPython notebook for Risk-Limiting Audit of 2014 Danish portion of the EU Parliamentary elections
It is demonstrated on Denmark's 2014 European Union Parliamentary election which uses an open list proportional representation voting method, with seats allocated via the highest averages method.
Note that in Denmark, parties can form coalitions, in which case first seats are allocated across the coalitions, and then from the the seats for each coalition, the parties within the coalition are allocated seats.