Multiple Choice Voting systems

Here are my notes on voting systems. I used them to write https://davelevy.info/voting-systems-theory-and-stv/ on my blog. Yhis wa originally written in 2019.

So as the rerun of the referendum becomes more likely, some people are considering a multiple choice; the problem is how to weight and count the votes. It interests me that instant run off is people’s first port of call, and I have some links that refer to it. On first sight, the problem is that instant run off fails to weight people’s commitment to their 1st vs. 2nd choice. This problem is what Parliaments are designed to solve; the polity can negotiate 1st, 2nd and 3rd choices and negotiate the importance of each packages features, this may be impossible in an instant run-off election. On further reading, I came across the two theorems that state all ranked voting systems enable tactical voting and so I add, may not be able to win loser’s consent.

This article became a commentary on Voting Systems; some clever people have thought hard about this and it’s not simple. For much more see below/overleaf …

It seems that political scientists and mathematicians study this problem in some detail and have developed some principles to guide in voting systems design.

I have found wikipedia very useful, often, as a start point.

Wikipedia, on their article, Comparison of electoral systems , says,

Attitudes towards systems are highly influenced by the systems’ impact on groups that one supports or opposes, which can make the objective comparison of electoral systems difficult. … To compare methods fairly and independently of political ideologies, voting theorists use voting method criteria, which define potentially desirable properties of voting methods mathematically.

I originally found on Wikipedia, instant run off voting, where its voting method criteria section lists these criteria. Later reading led me to its article, “Comparison of Electoral Systems“, which lists and examines the various criteria. I am not sure if the criteria spoken about take into account motivational effects which are impossible to eliminate; how do we eliminate the motivational effect of Scottish independence i.e. where people vote leave because they expect it to accelerate Scottish independence.  Neither is are all systems immune to tactical voting strategies. Possibly most importantly, for AV/Instant Runoff, the Condorcet majority principle , together with participation and reverse symmetry criterion are not met.

The Condorcet candidate ( a.k.a. Condorcet winner) is the person who would win a two-candidate election against each of the other candidates in a plurality vote. … A voting system satisfies the Condorcet criterion (English: /kɒndɔːrˈseɪ/) if it always chooses the Condorcet winner when one exists.

The wikipedia page identifies 12 qualities of a voting system,and suggests that instant run off systems satisfy six and fail six. I am unsure why I only identified three failures, i.e, the Condorcet majority principle , and participation and reverse symmetry criteria. Each of these qualities has it’s own page.

My meanderings over wikipedia took me to this page, where wikipedia states,

In social choice theory, the Gibbard–Satterthwaite theorem is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner. It states that for every voting rule, one of the following three things must hold:

1. The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
2. The rule limits the possible outcomes to two alternatives only; or
3. The rule is susceptible to tactical voting: in certain conditions some voter’s sincere ballot may not defend their opinion best.

Since for a referendum as proposed, one and two above are untrue, it must be that three would be true.

Further reading means I want to highlight the Gibbard–Satterthwaite theorem which states, according to wikipedia, that voting systems, with more than two outcomes will always enable tactical voting. This built on Arrow’s impossibility theorem which makes similar claims but restricts the proposition to ranked voting systems.

I have found these links which are much lighter, as in not written by mathematicians.

1. Beyond the binary: what might a multiple-choice EU referendum have looked like?, from Democratic Audit, an examination of how the ballot paper might look but no discussion on how to mark or count the votes.
2. A paper to Labour’s NPF, An introduction to the idea of a three-option Brexit referendum by Nick Donovan, catalogues the 19 such referendums that have occurred.
3. A second Brexit referendum: the practical questions, at the Institute for Govt, written in July 2018, looks at the choice,the voting system, the victory threshold (in terms of devolved nation consent) and the franchise by Akash Paun.

These are a bit more scientific

1. Beyond instant runoff: A better way to conduct multi-candidate elections, by Christopher Borgers at the conversation.com. Examine’s instant run off, the Condorcet rulesand looks at the Borda count system of weighted votes, and describes results propensity of the two systems and games to be played.
2. Borda Count, at the ERS, a weighted preference vote, like that used in the Eurovision, mainly descriptive.
3. By Toby Piera, Proportional Approval Method using Squared loads, Approval removal and Coin-flip approval transformation (PAMSAC) – a new system of proportional representation using approval voting

Some further reading, from wikipedia;s sources,

1. Four Condorcet-Hare Hybrid Methods for Single-Winner Elections
2. Range Voting and the Mathematics of Democracy, includes a fascinating example where each voting system produces a different vote, written by Warren D Smith, founder of the centre for range voting who argues that range voting is best.
3. Point/Counterpoint: Terry Bouricious Attempts To Rip Professor Gierzynski A New One Over Instant Runoff Voting Controversy (Now With All New Gierzynski Update!), a serious row about the advantages and disadvantages of instant run-off.
4. This, Properties of Preferential Election Rules by Douglas Woodall, Reader in Pure Mathematics at Nottingham University argues that more attention should be paid to properties of electoral systems, and less to procedures. He lists many properties that a preferential election rule may or may not have, and discusses them with reference to STV which, in my opinion he finds wanting.

Late browsing of the wikipedia sources list led me to this,

1. Arrow’s impossibility theorem, which poses an impossible trilemma, based on trading the accuracy of the voting system and the representative nature.

A note on STV.

Single Transferable Vote is a ranked voting system, with multiple winners, each winner requiring a proportion of 1st votes or transfers. Surplus votes are transferred in proportion to ranked preferences and then the lowest scoring candidates are eliminated abd their votes transferred. The ERS describe it https://electoral-reform.org.uk/voting-systems/types-of-voting-system/single-transferable-vote/ on their web site.

This system is vulnerable to being gamed.

In the Republic of Ireland. they use STV with varying number of winners, between 3 & 5. There is a trade off between equality of the value of the vote and geographic coherence. This is discussed in Fixed-boundary constituencies and the principle of equal representation in Ireland by Coakley J. This is important because one quota’s worth of votes ( minus one per victor) does not contribute towards a winner, meaning that with three winners, 25% of the vote is lost and with five winners, just under 17% counts. Obviously, this remains better than FPTP, where a majority of voters may have voted against the single winning candidate. However, this is a potential means of gerrymandering.

See also the ERS on Does Ireland use proportional representation?

See also https://www.tcd.ie/Political_Science/about/people/michael_gallagher/IrishElectSys.php , by MIchael Gallagher and published at TCD.

And https://www.irishtimes.com/election2020/results-hub which suggests that the independents are under represented as are Sein Fein & Labour, while the right wing parties, and the Social Democrats are over represented. NB 2020.

ooOOOoo

Image Credit: SMARTBOY10 VIA GETTY IMAGES, found at Huffpo, cropped, and stored for the usual reasons