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Role of mathematics

The former two weblog texts discussed British and Scottish incomprehension of democracy. Our discussion used numbers though. A picture says more than one thousand words. Lorenz curves are a nice way to display inequality. The data, calculations and following charts are in this excel sheet: 2015-2017-UK-Holland-Lorenz-Gini-SainteLague-majority

The UK General Elections in 2015 and 2017

The following two charts show the results in the UK General Elections of 2015 and 2017.

The horizontal axis gives the cumulative percentage of the popular vote. The vertical axis gives the cumulative percentage of the seats. If there is proportional allocation of seats, then the blue line of the seats would cover the pink diagonal.

The parties in the line-up have been ordered by mismatch. Conforming to the Sainte Laguë / Webster criterion (see this discussion by Alan Renwick), the mismatch is determined by ((the ratio of % seats to the % of votes) minus 1). For example in 2015, UKIP with 12.6% of the votes got only 0.2% of the seats, namely 1 seat for Nigel Farage himself. Their mismatch is 0.2 / 12.6 – 1 = 0.012 – 1. LibDem got 7.9% of the votes but only 1.2% of the seats, a mismatch of 0.156 – 1 (with rounding). Another mismatch are the parties that got no seats: the “Others” still got 3.9% of the votes, which means a mismatch of 0 / 3.9 – 1 = -1.

One might argue that the wasted votes should be omitted from the graph and Gini, yet, they rather stand out as a sore spot in current representation. A proportional representation of the wasted vote w in total n is possible by leaving seats empty or by filling the seats and taking a qualified majority f = 1/2 / (1 – w / n). In 2017 the wasted vote was 3.5% and then f = 50% / 96.5% = 51.8%. A representative majority in a full House of 650 seats then requires 337 seats, and not 325. See here.

The situation in 2017 has improved mainly because UKIP no longer really participated. The LibDem still got 7.3% of the vote and 1.8% of the seats, which is a marginal improvement.

The Gini coefficients are 30.1% in 2015 and 15.6% in 2017.

The graphs and coefficients are inaccurate because of strategic or tactical voting. A voter who favours a Conservative candidate but sees a loss against a Labour candidate might vote for the LibDem, reflecting a {Conservative, LibDem, Labour} preference ordering. There will be some averaging out, but the official votes will likely not reflect the true proportions of the first choices.

The nice thing of these graphs also is that one can recognise some of the parties. In both graphs the Conservatives are on the right hand side, with 36.8% of the votes and 50.8% of the seats in 2015, and 42.2% of the votes and 48.8% of the seats in 2017.

(These percentage take the wasted votes including the invalid votes as part of the denominator. Other sources may report that e.g. the Conservatives got 43.8 of the vote, looking only at the parties that got seats.)

The Dutch General Elections of 2017

The PR Gini for the UK shows that it is strikingly disproportional. Holland provides a useful point of reference. Holland had general elections in 2017 too, and its PR Gini is 3.5%. The major impact in Holland are the 2% of voters for small parties who got no seats. The Dutch qualified majority is f = 50% / 98% = 51%. In a PR system there will be strategic voting too, for example w.r.t. the coalition government. This however is no good reason to adjust the PR Gini coefficient, since such considerations are not quite those of proportionality, see also the discussion in the Appendix.

Conclusion

Conclusions are:

  1. The UK is at an alarming distance from proportional representation. This is detrimental for: (a) The ability to work together, compromise, form coalitions, and respect the opposition, (b) The possibility for smaller parties to partake in government and responsibility. (c) The entry and exit of new parties. (d) The notion among the electorate that they are represented.
  2. The Lorenz graph is a useful tool to show proportionality. The graph and Gini coefficient are not difficult to make. The ordering via the Sainte Laguë / Webster criterion gives (slightly) higher Gini coefficients (less proportionality) than ordering by the difference between % seats and % votes.

Technical appendix

The calculation of the Gini is straightforward. Each step, from one party to the next one, generates a small trapezium, with the area h (a + b) / 2. The height h is in this case the horizontal distance, given by the vote share of the next party. The sides a and b are the differences (on the left and right) between the diagonal and the cumulated seat curve. Summation of all these areas gives a total A. The Gini is equal to 2A, since the whole area of the square is 1. (The formulas for the Gini in wikipedia are more complex than needed for this piecewise linear application.)

Once I had decided to use the Lorenz graph, a google generated some predecessors.

Orit Kedar, Liran Harsgor and Raz A. Sheinerman (2013) refer on page 5 to Taagepera and Shugart (1989) Seats and Votes. Let me reproduce the quote from the first authors quoting the second authors:

“They note that ‘an alternative [to the measure of deviation from PR which they use] is the Gini index of inequality, which has theoretical advantages but is more complex to calculate’ (p. 204). They add that ‘the Gini index is the most widespread index of inequality, and it does satisfy Dalton’s principle [of transfers]. The Gini index is useful for many purposes other than electoral studies (where it has been little used)’ (p.263).”

The calculation above has been straightforward. It must be mentioned that Kedar et al. have a more complex analysis with districts though.

Anish Tailor and Nicolas Veron (2014) look at inequality in the European Parliament. Their problem is that Germany has 700,000 votes per seat while Malta has 70,000 votes per seat. They find a Gini of the UK of 6.3%, but this thus concerns another research question. If one would look at representation by parties then the EU Parliament might be less disproportional.

Kestelman (2005) also considers measures of apportionment and proportionality, and also refers to Taagepera and Shugart for the Gini (p14). He states that the Gini would be complicated to explain and calculate, while it is rather simple, see above Excel sheet. Thus, curiously:

“Fortunately highly correlated with LHI [Loosmore-Hanby Index], the Gini Disproportionality Index (GnI) is rather complicated to explain and calculate (virtually necessitating computerisation).” (p16)

Kestelman also suggests that STV would be a proportional method, but then he might neglect that an application to districts causes disproportionality over the whole nation (see this example).

Alexander Karpov (2008) gives a more analytical overview of the various measures.  He sums over the ratio of shares (%seats / %votes) but I do not see the rationale for this yet. The calculation above uses the ratio only for ordering, and uses levels with their intuitive interpretation.

Karpov’s article got a comment from Michela Chessa (2012) who points to indices that look at power. Indeed, if a party has 49.9% of the vote and the mere technique of apportionment would generate a majority of 50.1% in the seats, then one might wonder whether this is merely technique, and not a major decision on content.

Update: June 23 2017: (1) Originally I sorted the parties on the difference between the % seats and the % votes, but the ratio is better, and indeed gives slightly higher Gini coefficients. One can easily check this by sorting differently in the excel sheet. (2) Alan Renwick repeats the useful distinction between the measure of proportionality and the measure on impact (like power). (3) If seats are allocated using one particular criterion (like the Sainte Laguë criterion), then it doesn’t seem so much useful to see what it means in terms of another criterion, for, if the other criterion really is better, then use this to assign the seats. Thus the issue is intellectually rather dead in continental Europe, that already applies proportionality. The issue only comes up here because of the situation in the UK.

Apparently I cannot find a picture of Max Lorenz (1876-1959) but there are some of Corrado Gini (1884-1965).

Corrado Gini (1884-1965) (Source: SIS, Instituto Centrale di Statistica)

If you don’t have proportional representation (PR) then some voters get representatives they did not vote for. Thus it isn’t very democratic not to have PR.

The last weblog criticised the UK Electoral Reform Society (ERS) for erroneously claiming that Single Transferable Vote (STV) was PR.

ERS namely adopts districts, which causes STV to lose the limited PR properties that it has.

A persons affiliated with ERS answered to this criticism:

“We are well aware of the tension between the desire for (overall) proportionality and the desire for guaranteed local representation.  This tension is apparent among British electors when opinion polls have asked relevant questions about the outcomes of voting systems.  British electors want both overall proportionality (of parties) AND the local representation provided by exclusively single-member districts.  That is just not possible, so we aim for a compromise between local representation and overall proportionality through appropriately-sized multi-member districts. (…).” (Personal communication)

This is an unsatisfactory answer since there simply is no such “compromise”. When one must choose between a square and a circle then the answer is not some other graphic with some measure of deviation. If there is no PR then there is no PR, and then ERS should not claim that they have PR. To express their “compromise”, ERS speaks about “STV-PR” but this is like speaking about square-circles, and comes with the grating sound from nails across a blackboard.

If n is the number of voters, s the number of seats, then q = n / s is the threshold or quota, of voters per member. A candidate can be elected when he or she meets the quota. When the district size is 2q, then the district representative must get 50%+1 of the vote to attain the quota. At best s / 2 seats can be filled in this manner, since s / 2 * 2q = n again. All unfilled seats can be allocated using overall PR. This shows that districts are not a key design feature, while PR is. (These formulas can be adjusted for turnout, when district size is defined in terms of the electorate and not actual voters. See here.)

By focusing on districts, ERS loses track of the key design feature, and it lets its logic be occluded by a less relevant issue.

Wikipedia follows ERS

Apparently the editors at wikipedia follow ERS rather uncritically. The wikipedia statement in red is what ERS claims falsely and what is adopted by wikipedia too. The statement in green is true. Since the statement in green is true, the statement in red can only be true by chance.

Proportional representation (PR) characterizes electoral systems by which divisions in an electorate are reflected proportionately in the elected body. If n% of the electorate support a particular political party, then roughly n% of seats will be won by that party. The essence of such systems is that all votes contribute to the result: not just a plurality, or a bare majority, of them. Proportional representation requires the use of multiple-member voting districts (also called super-districts); it is not possible using single-member districts alone.[1][2][3] In fact, the most proportional representation is achieved when just one super-district is used.

The two most widely used families of PR electoral systems are party list PR and single transferable vote (STV).[4][5] Mixed member proportional representation (MMP), also known as the Additional Member System, is a hybrid Mixed Electoral System that uses party list PR as its proportional component. MMP has the potential to be proportional or semi-proportional depending on a number of factors such as the ratio of first past the post (FPTP) seats to PR seats, the existence or nonexistence of compensatory seats to make up for overhang seats, and election thresholds.[6][7][8][9]   (Source: Wikipedia on PR)

ERS thus is confusing the world including wikipedia. My advice for the editors of wikipedia (and the ERS) is:

  • Maintain conceptual integrity.
  • Restrict PR to the notion that p% of the votes translates into p% of the seats.
  • For PR the first preferences are relevant and not what is done with the subsequent preferences. Thus do not label STV as a PR-system but as “potentially PR”, or as STV-PPR.
  • For PR it suffices when the electorate selects parties. A single candidate is a party with a single candidate.
  • The professionals in parliament can use more complex systems like STV. The use of STV (there) must be compared to other systems, like Borda Fixed Point.
  • Get rid of the hangup on district representation.

Unfortunately, the person affiliated with ERS writes to me, with an unrelenting hangup about districts, and neglecting that PR should hold nation-wide:

“Neither the ERS nor I would be prepared to label STV-PT as “potentially PR” or anything similar.  I have seen some academics describe STV-PR as “a semi-proportional” system.  That is just nonsense.  For the same district magnitude, STV-PR and party-list PR both deliver the same degree of proportionality.  The fact that some electorates are prepared to accept electoral districts that cover the whole country for party-PR but don’t like the idea of “large” electoral districts for STV-PR is completely irrelevant.  It is the district magnitude that is the determining factor, not the voting system.”  (Personal communication)

Scotland is an example

Scotland has four electoral systems, and I copy from Wikipedia:

Does this mean that Scotland comprehends democracy or that they don’t ?

The Party List System as used for the EU Parliament generates proportional representation (PR), and this would be the criterion for representative democracy.

(Obviously, for the election of a local council, the norm for PR are the local votes, and not nationwide PR. Once the issue here is reduced to apportionment, then STV is one of the options and a choice depends upon one’s criteria.)

Let us look at the Scottish implementation Additional Member System (AMS) a.k.a. Supplementary Member System a.k.a. Mixed Member System (MMS). I would prefer the latter term, since there is nothing “additional” about an elected MP. Sometimes the term “Mixed Member Proportional” (MMP) is used but this is only warranted when there really is overall PR.

The current Scottish system

Scotland has 73 constituencies, in which the candidate is selected by FPTP. There are 8 regions with 7 seats per region, to a total of 56 regional seats. These “additional seats” are used to make the outcome more proportional. Brief explanations of the current Scottish system are by the Parliament itself and The Scotsman. The Scottish Parliament elections of May 5 2016 have these full data. The turnout was 55.6%.

Scotland like the UK has a hangup on the distinction between the local candidate and the party. It is claimed: “In the second vote the voter votes for a party rather than a candidate.” Indeed, when the first vote has a FPTP selection, then voters may be forced to vote strategically for a candidate of reduced preference, in trying to prevent that a worst candidate is selected. Thus the explanation about local representation may be a misrepresentation about what might really motivate voters.

When we compare the votes for the constituencies (districts) and the regions, then we don’t see much of a difference, except for the Greens and Others. (This are totals though, and there might be differences over districts.)

District

Region

Party

Votes

Votes

Con

501,844

524,222

Green

13,172

150,426

Lab

514,261

435,919

LD

178,238

119,284

SNP

1,059,898

953,587

Others

11,741

102,314

Total

2,279,154

2,285,752

We take the summed region vote as determining what the proportions for the parties should be. The additional 56 seats and their restriction to regions are not enough for correction of the error in the local vote. The SNP got 7% more seats than warranted under PR.

District

Region

 All
Party

Seats

Seats

Seats

%Seats

%Votes

%S-%V

Con

7

24

31

24.0

22.9

1.1

Green

0

6

6

4.7

6.6

-1.9

Lab

3

21

24

18.6

19.1

-0.5

LD

4

1

5

3.9

5.2

-1.3

SNP

59

4

63

48.8

41.7

7.1

Others

0

0

0

0.0

4.5

-4.5

Total

73

56

129

100.0

100.0

An alternative for Scotland

Let us consider a rough alternative for Scotland:

  • A local winner must get at least 50% of the vote of a district (constituency).
  • All 129 seats are allocated in proportion to the summed region vote.

The data file allows us to determine which candidates are elected now. This generates a quite different result. In the local vote, only 29 candidates manage to get at least 50% of their district (constituency). 95 candidates are selected via the Party List, which puts the ERS argument for locality into perspective. In this rough alternative, there are 5 seats that cannot be allocated due to rounding errors. But having 4% empty seats is not unfair given that 4.5% of the votes are wasted on the small parties.

District

Region

 All
Party

Seats

Seats

Seats

%Seats

%Votes

%S-%V

Con

1

29

30

23.3

22.9

0.3

Green

0

8

8

6.2

6.6

-0.4

Lab

0

25

25

19.4

19.1

0.3

LD

2

5

7

5.4

5.2

0.2

SNP

26

28

54

41.9

41.7

0.1

Others  0  0  0

0.0

4.5

-4.5

Total

29

95

124

96.1

100.0

A general observation

The quota is q = 2,285,752 / 129 = 17720. Above criterion of 50% of the local vote might be too lax. With 73 districts, the number or electors per district might be too small. If the number of districts is 129 / 2 ≈ 65, then the average district has size 2q, and the criterion of at least 50% of the votes would fit the overall condition of winning a seat via satisfying q.

A google showed this page by Andrew Ducker who also wondered about PR in Scotland. He mentions: (1) The region votes must be summed for nationwide PR indeed. (2) A 50%:50% distinction between local and national seats would be helpful indeed. In reply to this: why still allow FPTP when it may cause that a minority winner would become the “representative” ? It is better to require at least q and/or at least 50% of the district.

The UK Electoral Reform Society (ERS)

The UK ERS falsely claims that STV applied to districts would be PR while it is not. The ERS also criticises the Scottish system, but perhaps for the wrong reasons.

In 2011, the current Scottish system was already in place, and the ERS advised a change. See the Guardian or the BCC:

One of the authors of the report, Prof John Curtice of Strathclyde University, said: “The widespread expectation that the Scottish Parliament would be a multi-party parliament, in which no party would ever have an overall majority, has been dashed. “In truth, although the electoral system bequeathed to the Scottish Parliament by Labour was far more proportional than first-past-the-post, it was never one that was best fitted to the realisation of that original expectation. “It still favours larger parties over smaller ones, who, indeed, are actually being discouraged from standing in the constituency contests.” (BBC 2011-11-15)

A similar critique is given w.r.t. the 2016 outcome. Some changes like an “open party” list (i.e. the ability to vote for individual party candidates on the list) and the allocation of Sainte-Laguë may indeed be mentioned.

But this is small beer compared to the major critique on the Scottish system, that it still allows for the hangup on district representation.

While the ERS should warn voters and legislators about this hangup, the ERS suffers from this hangup itself too, and propounds STV for districts, which destroys PR.

The person affiliated with ERS writes to me:

I am not in favour of electing MPs (or other representatives) in two different ways. In Scotland we have experience of the Scottish Parliament where MSPs are elected by the Additional Member System (AMS = a regionalised version of MMP). Some of the worst problems of electing MSPs in two different ways (Constituency and Region) have abated over the years, but the tension remains and surfaces from time to time. It would have been much better if all the MSPs had been elected by STV-PR, but AMS was a political compromise as one the major parties (Labour) would just not accept STV-PR at any price. (Personal communication)

Again, this person at ERS suggests that STV would be PR, even calling it STV-PR, while the very application of STV to districts destroys the PR.

Missing Scottish voters

ERS Scotland director Willie Sullivan wrote a book about the structurally low turnout for Scottish elections: “The Missing Scotland: Why over a million Scots choose not to vote and what it means for our democracy” (publisher).

In an article, Sullivan summarises:

“If the working people wanted democracy, why do so many now not vote? Surely these are the people that should be most eager to flex their democratic muscle? In research for my book, Missing Scotland, I tried to find out why more than a million Scots choose not to vote. What I found is worrying. Most important of all, people don’t think voting will make anything better. They have tried voting, and they have tried not voting, and there is no difference. They think politicians are all the same, don’t understand their lives and they make promises they never keep. This is not a question of not caring. The people I spoke to care a lot about their families and communities. They are worried about losing their homes or their jobs. They even like the idea of democracy, they just don’t think we have it. Not voting is often a deliberate act.”

I haven’t read this book, but only find it relevant enough to mention its existence. My guess is that Sullivan hasn’t mentioned two elements:

  1. When the Scottish electoral system was changed, they didn’t adopt the PR system like in Holland, but kept their hangup on districts, and ERS itself was an agent in this. Potentially Sullivan might feel ashamed that he doesn’t quite comprehend what democracy is.
  2. Scotland isn’t aware of the failure of the Trias Politica model and the need for an Economic Supreme Court.

Conclusion

My finding is that major political distortions in the UK, France, USA and India arise because of lack of PR.  A lobby for STV for districts doesn’t resolve this, and it is falsely claimed to be PR. Thus I would tend to advice electoral reform in this order of priority:

  • first PR, like the system in Holland or the EU Parliament (Open Party List)
  • if this is up and running, secondly allow for an element of locality for half of the seats (s / 2, district size 2q, and the district representative is elected with at least 50% of the district vote, potentially corrected for turnout)
  • if this is up and running, improve the system by allowing voters freedom on how they vote
  • compare STV and Borda Fixed Point and other methods for the selection of the local representative.

The UK Electoral Reform Society hinders clarity on electoral reform since they show a hangup on districts. They better focus on establishing Proportional Representation (PR), while regarding the issue of districts as of secondary importance.

Given overall PR, one might even let voters determine on the ballot how to deal with the district representation, for the s / 2 seats available for district representation.

  1. Some voters might vote for a party, and be done with that. Seats are allocated to the party in proportion to the total number of votes. (Closed Party List) Some voters might wish to select a party but also a particular person in that party, so that the party order takes over if the person would not be elected. (Open Party List) These approaches can be combined (as in Holland) when the Closed List voters vote for the party leader.
  2. Some voters might indicate where their vote would go, if their party of choice isn’t elected. (Remember that a single candidate is a party with a single candidate.)
  3. Other voters might wish to vote for particular candidates across parties, and then might want to indicate how votes would have to be transferred if the candidate doesn’t get elected. (Otherwise it is apportioned automatically.) There is still the comparison between STV and e.g. (repeated) application of the Borda Fixed Point method. STV runs the risk of eliminating a compromis candidate, who receives few votes in the initial stage, but who can collect support because of secondary preferences. This might not be relevant for the party proportion but be quite relevant for voters and the candidates themselves. This would not be an issue of PR but of Quality Representation (QR).

PS. Dan Hodges (Telegraph June 1 2015) has a very entertaining article “No, Britain does not want proportional representation“. The weak spot in his argument is that the 2011 referendum on AV was misrepresented as a referendum on PR while it actually was a referendum on AV. The strong point is that ERS cannot be convincing if its arguments are confused. There still is a case for sound arguments and good education.

PPS. The subtle relation between proportional representation (PR) and district representation (DR) shows also in the existence of a Senate or House of Lords, in which districts / States might be represented by 2 senators per State like in the USA. For a Senate the DR might be acceptable since the Senate has the role of guardian for the nation itself. The House would be sensitive to the preferences of the electorate, and in that case PR would be logical.

Given Kenneth Arrow’s impossibility theorem, it is a fair question to ask what voting system he himself would advise. There is a 2012 interview with him, with a phone recording and transcript, by Aaron Hamlin of the Center for Election Science. Arrow’s advice is:

  • Not plurality and no US Electoral College, with its winner-take-all selection of the US President
  • Not approval voting, since this uses too little information
  • A system that uses more information:

“Dr. Arrow: Well, I’m a little inclined to think that score systems [range voting] where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best. (…) In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.”

His statement about strategic voting – or manipulation:

“Dr. Arrow: There’s only one problem that bothers me about that. And that’s something my theorem really doesn’t cover. In my theorem I was assuming people vote sincerely. The trouble with methods where you have three or four classes, I think if people vote sincerely they may well be very satisfactory. The problem is the incentive to misrepresent your vote may be high. In other words, a classic view is that there’s a candidate I really like, but I know is hopeless. I may put him down at the bottom and vote for the next candidate simply because I feel there’s a chance. Now, if you have a very large electorate you might say no individual has much of an incentive to misrepresent. But I’m not sure. You probably need experience rather than theory.”

Observe that Arrow cautiously states “a little inclined to think (…) probably the best”. His advice to have more empirical research can be supported. The interview touches on some points that call for a closer discussion, also in the light of this earlier weblog text.

Definitions

In plurality, voters only can vote for their best candidate. In a district, often the one with the highest score wins, which is the “first past the post” (FPTP) system. If there are only two candidates, then the winner will also have more than 50%. If there are more candidates, the winner may have less than 50%. There may be ways to assure that a final vote only concerns two candidates. A Putin hack that eliminates a particular candidate will not quickly be accepted, yet, voting theorists still wonder what method would be reasonable. A current example is that Donald Trump got elected with 46% of the popular vote, while Hillary Clinton got 48%. With a turnout of 60% Trump has only 28% support in the electorate, while the House of Congress depends upon district results too. A prime minister who is elected by a coalition in a parliament that has proportional representation (PR) generally has more than 50% support in parliament, and by representation also in the electorate.

In approval voting, voters mention which candidates they approve. The candidate with the highest total approvement is selected.

  • In economics this links up with satisficing (Herbert Simon).
  • Strategic voters will tend not to approve of candidates that might harm their best candidate (even the second best), so that this system devolves into plurality. Steven Brams claims that such fears are overrated but are they ? Brams declines to look into non-satisficing alternatives like the Borda Fixed Point method.

In Borda ranking, each voter puts the candidates in order of preference, and assigns rank numbers.

  • In economics this reflects the notion of ordinal utility.
  • Strategic voters will give a low score to candidates that harm their best candidate (even the second best), which means that “dark horses” (of mediocre approval) might win. See the discussion below.

In range voting, the voters grade the candidates like on a report card, and the candidate with the highest grade point average (GPA) wins. There is the tantalizing but empirically perhaps small complexity of the distinction between a 0 grade (included in the GPA) and a blank vote (not included in the GPA).

  • In economics this reflects the notion of cardinal utility (with voters restricted to the same range).
  • Strategic voters will give a low score to candidates that harm their best candidate (even the second best), which means that the system devolves into plurality. (The use of ordinal preferences and Borda explicitly intends to resolve this again.)

(See also the distinction in levels of measurement.)

Beware of the distinction between cardinal and ordinal preferences

Arrow’s impossibility theorem is about aggregating individual rank orders into a collective rank order. The theorem uses rank orders, or ordinal preferences. Arrow does more than only use rankings. He also defends the “axiom of pairwise decision making” (APDM) a.k.a. the “axiom of independence of irrelevant alternatives” (AIIA) as reasonable and morally desirable (Palgrave Dictionary of Economics).

Range voting allows more information than just ordinal preferences, and it is similar to cardinal preferences (but limiting people to the same range). Cardinal preferences imply ordinal preferences. Yet rank voting doesn’t satisfy the requirements of Arrow’s impossibility theorem, for cardinality violates APDM or AIIA.

One might say that Arrow’s theorem is not about voting systems in general, since it only looks at ordinal and not at cardinal preferences. Instead, Arrow’s position is that he looks at voting theory in general and only proposes axioms that are “reasonable” and “morally desirable”  When cardinality and range voting are excluded from his axioms, then it is because they would be unreasonable or morally undesirable.

These distinctions are discussed – and Arrow’s notions are debunked – in my “Voting theory for democracy” (VTFD). (See especially chapter 9.2 on page 239.)

Arrow’s theorem is only about those voting systems that satisfy his axioms. Since his axioms cause an inconsistency, there is actually no system that matches his conditions. Something that doesn’t exist cannot be reasonable and morally desirable. Arrow’s theorem confuses voting results with decisions, see this earlier weblog discussion.

However, there still remains an issue for voting theory. Range voting allows more scope for strategic voting or manipulation. The reason to restrict votes to rank orders is to reduce the scope for strategic voting.

Gerry Mackie’s “Democracy Defended

A reader alerted me to Gerry Mackie’s thesis with Jon Elster, now commercially available as “Democracy Defended“. I haven’t read this but the blurb seems to confirm what I have been arguing since 1990 on Arrow (but not on Riker).

“Is there a public good? A prevalent view in political science is that democracy is unavoidably chaotic, arbitrary, meaningless, and impossible. Such scepticism began with Condorcet in the eighteenth century, and continued most notably with Arrow and Riker in the twentieth century. In this powerful book, Gerry Mackie confronts and subdues these long-standing doubts about democratic governance. Problems of cycling, agenda control, strategic voting, and dimensional manipulation are not sufficiently harmful, frequent, or irremediable, he argues, to be of normative concern. Mackie also examines every serious empirical illustration of cycling and instability, including Riker’s famous argument that the US Civil War was due to arbitrary dimensional manipulation. Almost every empirical claim is erroneous, and none is normatively troubling, Mackie says. This spirited defence of democratic institutions should prove both provocative and influential.” (Cover text of “Democracy Defended“)

My point however would be that issues of cycling are of concern, like we see with the Brexit referendum question. The concern causes support for representative democracy with proportional representation, rather than populism with referenda.

The key context is switching to parliaments with PR

Discussions about voting theory best be seen in the context of the switch towards parliaments that are elected with PR and that select the prime minister. The president may have a cerimonial role and be elected by parliament too (like in Germany).

It is most democratic when there is proportional representation (PR) of the electorate in the elected body. The more complex voting methods can then be used by the professionals in the elected body itself only. A prime minister is best elected by a parliament with PR, instead of a president by direct elections.

The interview with Arrow contains a criticism on plurality and FPTP compared to PR.

“Dr. Arrow: Yes. I think definitely. I think there’s no question about that. The Plurality system chokes off free entry. In other words, in the economic world we’re accustomed to the virtues of free entry. We don’t want a small number of corporations to be dominate. We favor the idea of new firms entering in order to compete to bring in new ideas, to bring in new products. Well, the same way in the political field. We should be encouraging free entry, I think, in order to have new political ideas come in. And they may flourish. They may fade. That’s what you want, them to be available. So I’m inclined that the Plurality system will choke off by encouraging, the two-party system will choke off new entry. So I’m really inclined to feel that we don’t want Plurality as a voting system. It’s likely to be very stifling.”

“(…) proportional representation [PR] plays very little role in The United States, but they do play a role in a number of countries. And the question of whether single-member districts are appropriate or not. The Germans, for example, have some kind of compromise between single-member and broader districts. (…)”

See my comparison between the Dutch PR and the UK district system.

Proposals that assume that the voters themselves would use the complexer voting systems – perhaps an enlightened form of populism – are complicating election reform, because these methods put too high demands upon the voters and the electoral process.

In the interview, Arrow referred to the proportional systems, but still expressed the idea that voters themselves would use the three or four categories. In this manner Arrow contributed to this confusion on context.

“CES: If you could, just sort of dictatorially, change something about the way that we do voting in the US, something that would make the biggest impact in your mind, what do you think you would do?

Dr. Arrow: The first thing that I’d certainly do is go to a system where people ranked all the candidates, or as many as they wish, and not just two. And that these data are used in some form or another to choose the candidate, say by eliminating the lowest, or some method of that kind. I’d be interested in experimenting with the idea of categorization and creating interpersonal comparisons by that. And those are the things that I would argue for, and certainly the abolition of the Electoral College. It goes without saying.”

In my experience Arrow is often more confused than one would expect. (1) His original theorem confused voting outcomes and decisions. (2) If he really assumed that people would vote sincerely, then he might as well have assumed cardinality, but he didn’t, for then he wouldn’t have had a theorem. (3) He made a theorem on ordinal preferences but now is inclined to cardinality, even though he defends his theorem that cardinality would be unreasonable and morally undesirable since it doesn’t satisfy APDM a.k.a. AIIA. (4) He now mentions PR but doesn’t draw the conclusion of the selection of the prime minister by parliament, and apparently still thinks in terms of a direct election of the president.

Arrow’s contributions to economics derive from the application of mathematics to economics in the 1950s, and not because he was exceptionally smart in economics itself. Paul Samuelson expressed this idea about himself once too, as a physicist entering into economics. If Arrow had been real smart then he also would have had the common sense to see that his theorem confuses voting results and decisions, and that it amounts to intellectual fraud to pretend that it is more than that.

A major issue is that abstract thinking mathematicians can get lost about reality. In VTFD I show that Amartya Sen is confused about his theorem about a Paretian liberal. Sen’s article with Eric Maskin in the NY Book Review about electoral reform also neglects the switch to a parliamentarian system with PR. A major problem in society is that many intellectuals have insufficient background in mathematics and follow such lost mathematicians without sufficient criticism, even when common sense would warn them.

Warren Smith’s parable of the bees

Warren Smith suggests that bees also use range voting to select the next location for their hive. My problem is that bees aren’t known for strategic voting. My VTFD already suggested – as Jan Tinbergen – that aggregation of cardinal utility would be best indeed. Thus I don’t feel the need to check how bees are doing it.

The problem in voting theory is that humans can vote strategically, also guarded by secrecy in the ballot box. Potentially this strategic vote might be less of a problem when votes for the prime minister in parliament are made public, so that people can wonder why a party has a particular vote. But transparency of the vote might not be the key issue.

Smith on Bayesian regret

Smith has a notion of Bayesian regret, as a more objective criterion to judge voting systems. I am amazed by the existence of such a notion for social optimality and haven’t looked into this yet.

Smith is too enthousiastic about Arrow’s support

Smith interpretes Arrow’s “a little inclined to think” as an endorsement for range voting.  Smith provides full quotes properly – and I must thank him for directing me to this interview with Arrow. But I would advise Smith to be more critical. Arrow mainly indicates an inclination, he is also confused and doesn’t repeal his interpretation of his theorem. Also Smith is advised to grow aware and alert readers of his website that the real improvement in democracy lies not in range voting but in a switch to a prime minister selected by a PR parliament. It is another issue how voting mechanisms operate in other situations, like the Eurovision Song Contest.

Smith’s discussion of the dark horse and the war of the clones

To reduce the options for strategic voting, the voters can be restricted to the use of rankings, and then we get systems like Borda, Condorcet, or my suggestion of the Borda Fixed Point method (BordaFP). The latter wasn’t designed to be a compromise between Borda and Condorcet but still can be seen as one. For example, in the 2010 general elections in the UK, with David Cameron, Gordon Brown and Nick Clegg, it appears that Clegg would be a Borda Choice, but Cameron would still be the BordaFP choice because he would beat Clegg in a pairwise contest.

The reader would enjoy Smith’s discussion of the dark horse and the war of the clones, in his criticism of the Borda method. There is no need for me to repeat his short statement, and I simply refer to here. While you are reading, there is also a picture of Frisian horse Fokke of 2013, and we continue the discussion below it. This discussion is not in VTFD since I mainly pointed to strategic voting but didn’t develop the argument, and thus I thank Smith for his succinct criticism.

 

Frisian Fokke 2013

War of the clones

This assumes the Borda system. Smith (point 8) compares the election between Mush (51%) and Bore (49%) with the election between Mush and some clones Bore1, Bore2, Bore3 (leaving unclear who the real Bore is). Supposedly it is publically known that Mush selects Bore1 in second place, so that the Bores can collect all their votes on Bore1 too. Now Mush loses. This criticism is accurate.  With Condorcet’s rule, Mush would beat all Bores, but the idea of Borda is to mitigate Condorcet. With enough Bores, the BordFP method is not immune to this either.

In above key context, the method would not be applied by the whole electorate but only by parliament. The number of parties would be limited, and each party would only mention one candidate. In the current Dutch parliament there are 13 parties, see Bloomberg with a graphical display of the political spectrum and my analysis on an application of BordaFP. Here the problem doesn’t really arise.

In general people might feel that parties and their candidates differ. If not, then this would require attention. For applications of Borda or BordaFP to smaller committees, it would be sensible to be aware of this. Committees might devise rules about when candidates are too much alike, bunch their votes as if they were one (and rerank), and only call for a decision vote between the clones when they would actually be chosen.

The dark horse

Smith (point 2) considers candidates A, B, C and various nonentities. Kenneth Arrow used the more polite term “irrelevant alternatives”. Let me settle for Dark Horse D. Let me also distinguish truthful voting and strategic voting. In a truthful vote there is no difference between the true preference and the ranking submitted to the ballot box. In a strategic vote there is the strategy provided by the truth and the tactic vote submitted to the box. (Potentially one might design a voting system in which a voter submits those two rank orders simultaneously, but then we must relabel between truth and those two submissions.)

A member of parliament (MP) faces a dilemma. If the MP prefers A > B > C > D then giving the ranks 4, 3, 2, 1 will give 3 points to B, which might cause that B is chosen instead of A. This MP has the incentive to shift points to the Dark Horse, as in 4, 1, 2, 3, hoping that nobody else will vote for this dark horse anyway. If all MPs think in this manner, then the Dark Horse will be elected with an impressive score.

Smith provides an anecdote how such an event happened in the selection of a job application, where there was disagreement about an excellent macro-economist and an excellent micro-economist, whereupon a mediocre candidate got the job.

This is the prisoners’ dilemma. (1) If everyone votes truthfully then they all benefit from the true selection. (2) If everyone votes strategically then they all suffer the worst outcome. (3) Each has an incentive to deflect from the true vote.

The BordaFP method is sturdier than Borda but is not immune to this situation.

A prime answer to Smith is that in parliament the rankings for the selection of the prime minister might be public, so that voters and the press can question party tactics. A party that gives so much points to a Dark Horse might be criticised for not appreciating a better candidate.

Looking for balance

For now, I find Smith’s discussion a bit unbalanced. He emphasizes the disadvantages of Borda, but these have the answers above, for the proper context, while the disadvantages of range voting don’t get as much attention. Range voting stimulates the strategy of giving zero points to alternative candidates, whence it reduces to plurality with all its drawbacks. A candidate with 51% of the vote in plurality might not be better, since more extremist, than a candidate with a higher Borda score who is more moderate. The main point remains that the key issue is that countries with district voting like the USA, UK and France better switch to PR.

By way of conclusion

It remains true that Borda has the risk of a Dark Horse, and that the search for better algorithms is open. How can we elicit information from voters about their true preferences ? In the ballot box we might numb their brains so that they vote like bees (perhaps also with the dance) ?

An idea that I already mentioned at another place: MPs might submit two inputs, one with the strategy (supposed to be true) and one with the intended tactic. (One would design a test whether these better be rankings or ranges.) The intermediate result would be based upon the tactics. A random selection of the true preferences then is used to revise the tactics to improve the results for those MPs who have the luck to be selected. This prospect encourages MPs to be truthful about the strategy.

Another possibility for such double submissions: One might first determine the outcome according to the submitted strategies (supposedly true) and then use a random selection to use the allowed tactics, and only uses these if they indeed cause an improvement in the eyes of the MP. This sanctions a moderate degree of unavoidable strategic voting, but reduces the chaos when all do it without information about others.

Such calculations are simple for a partial outcome for a single MP. The problem lies in the aggregation of all MPs. Perhaps money helps in solving this too. Voters in the electorate aren’t allowed to sell their vote directly, with the obvious horror stories, also involving the distribution of income. But in parliament there is coalition bargaining which involves money, i.e. budget allocations. Potentially this helps in designing better algorithms. Perhaps the Bayesian Regret comes into play here, but I haven’t checked this. In Holland there is professor Frans Stokman who studies coalition bargaining with his “Decide” model.

Thus the search for better voting schemes hasn’t ended. Yet the main step for the USA, UK and France would be to accept the choice of a prime minister by parliament selected by PR.

The Theresa May government has adopted Brexit as its policy aim and has received support from the Commons. Yet, economic theory assumes rational agents, and even governments might be open for rational reconsideration, even at the last moment.

Scientifically unwarranted referendum question

Based upon voting theory, the Brexit referendum question can be rejected as scientifically unwarranted. My suggestion is that the UK government annuls the outcome based upon this insight from science, and upon this insight alone. Let me invite (economic) scientists to study the argument and voting theory itself, so that the scientific community can confirm this analysis. This study best be done all over Europe, so that also the EU Commission might adopt it. Britons might be wary when their government or the EU Commission would listen to science, but then they might check the finding themselves too. A major worry is why the UK procedures didn’t produce a sound referendum choice in the first place.

Renwick et al. (2016) in an opinion in The Telegraph June 14 protested:

“A referendum result is democratically legitimate only if voters can make an informed decision. Yet the level of misinformation in the current campaign is so great that democratic legitimacy is called into question.”

Curiously, however, their letter doesn’t make the point that the referendum neglects voting theory, since the very question itself is misleading w.r.t. the complexity of the issue under decision. Quite unsettling is the Grassegger & Krogerus (2017) report about voter manipulation by Big Data, originally on Brexit and later for the election of Donald Trump. But the key point here concerns the referendum question itself.

The problem with the question

The question assumes a binary choice – Remain or Leave the EU – while voting theory warns that allowing only two options can be a misleading representation. When the true situation is more complex, then it may be political manipulation to reduce this to a binary one. As a result of the present process, we actually don’t know how people would have voted when they had been offered the true options.

Compare the question:

“Do you still beat your mother ?”

When you are allowed only a Yes or No answer, then you are blocked from answering:

“I will not answer that question because if I say No then it suggests that I agree that I have beaten her in the past.”

In the case of Brexit, the hidden complexity concerned:

  • Leave as EFTA or WTO ?
  • Leave, while the UK remains intact or while it splits up ?
  • Remain, in what manner ?

Voting theory generally suggests that representative democracy – Parliament – is better than relying on referenda, since the representatives can bargain about the complex choices involved.

Deadlocks can lurk in hiding

When there are only two options then everyone knows about the possibility of a stalemate. This means a collective indifference. There are various ways to break the deadlock: voting again, the chairperson decides, flip a coin, using the alphabet, and so on. There is a crucial distinction between voting (vote results) and deciding. When there are three options or more there can be a deadlock as well. It is lesser known that there can also be cycles. It is even lesser known that such cycles actually are a disguised form of a deadlock.

Take for example three candidates A, B and C and a particular distribution of preferences. When the vote is between A and B then A wins. We denote this as A > B. When the vote is between B and C then B wins, or B > C. When the vote is between C and A then C wins or C > A. Collectively A > B > C > A. Collectively, there is indifference. It is a key notion in voting theory that there can be distributions of preferences, such that a collective binary choice seems to result into a clear decision, while in reality there is a deadlock in hiding.

Kenneth Arrow (1921-2017) who passed away on February 21 used these cycles to create his 1951 “impossibility theorem”. Indeed, if you interprete a cycle as a decision then this causes an inconsistency or an “impossibility” w.r.t. the required transitivity of a (collective) preference ordering. However, reality is consistent and people do really make choices collectively, and thus the proper interpretation is an “indifference” or deadlock. It was and is a major confusion in voting theory that Arrow’s mathematics are correct but that his own verbal interpretation was incorrect, see my VTFD Ch. 9.2.

Representative government is better than referenda

Obviously a deadlock must be broken. Again, it may be manipulation to reduce the choice from three options A, B and C to only two. Who selects those two might take the pair that fits his or her interests. A selection in rounds like in France is no solution. There are ample horror scenarios when bad election designs cause minority winners. Decisions are made preferably via discussion in Parliament. Parliamentarian choice of the Prime Minister is better than direct election like for the US President.

Voting theory is not well understood in general. The UK referendum in 2011 on Proportional Representation (PR) presented a design that was far too complex. Best is that Parliament is chosen in proportional manner as in Holland, rather than in districts as in the UK or the USA. It suffices when people can vote for the party of their choice (with the national threshold of a seat), and that the professionals in Parliament use the more complexer voting mechanisms (like bargaining or the Borda Fixed Point method). It is also crucial to be aware that the Trias Politica model for democracy fails and that more checks and balances are required, notably with an Economic Supreme Court.

The UK Electoral Commission goofed too

The UK Electoral Commission might be abstractly aware of this issue in voting theory, but they didn’t protest, and they only checked that the Brexit referendum question could be “understood”. The latter is an ambiguous notion. People might “understand” quite a lot but they might not truly understand the hidden complexity and the pitfalls of voting theory. Even Nobel Prize winner Kenneth Arrow gave a problematic interpretation of his theorem.The Electoral Commission is to be praised for the effort to remove bias, where the chosen words “Remain” and “Leave” are neutral, and where both statements were included and not only one. (Some people don’t want to say No. Some don’t want to say Yes.) Still, the Commission gives an interpretation of the “intelligibility” of the question that doesn’t square with voting theory and that doesn’t protect the electorate from a voting disaster.

A test on this issue is asking yourself: Given the referendum outcome, do you really think that the UK population is clear in its position, whatever the issues of how to Leave or risk of a UK breakup ? If you have doubts on the latter, then you agree that something is amiss. The outcome of the referendum really doesn’t give me a clue as to what UK voters really want. Scotland wants to remain in the EU and then break up ? This is okay for the others who want to Leave ? (And how ?) The issue can be seen as a statistical enquiry into what views people have, and the referendum question is biased and cannot be used for sound statistics.

In an email to me 2016-07-11:

“The Electoral Commission’s role is to evaluate the intelligibility of referendum questions in line with the intent of Parliament; it is not to re-evaluate the premise of the question. Other than that, I don’t believe there is anything I can usefully add to our previously published statements on this matter.”

Apparently the Commission knows the “intent of Parliament”, while Parliament itself might not do so. Is the Commission only a facilitator of deception, and they don’t have the mission to put voters first ? At best the Commission holds that Whitehall and Parliament fully understood voting theory therefor deliberatedly presented the UK population with a biased choice, so that voters would be seduced to neglect complexities of how to Leave or the risks of a UK breakup. Obviously the assumption that Whitehall and Parliament fully grasp voting theory is dubious. The better response by the Commission would have been to explain the pitfalls of voting theory and the misleading character of the referendum question, rather than facilitate the voting disaster.

Any recognition that something is (very) wrong here, should also imply the annulment of the Brexit referendum outcome. Subsequently, to protect voters from such manipulation by Whitehall, one may think of a law that gives the Commission the right to veto a biased Yes / No selection, which veto might be overruled by a 2/3 majority in Parliament. Best is not to have referenda at all, unless you are really sure that a coin can only fall either way, and not land on its side.

Addendum March 31

  • The UK might repeal the letter on article 50 – see this BBC reality check. Thus science might have this time window to clarify to the general public how the referendum question doesn’t comply with voting theory.
  • The recent general elections in Holland provide another nice example for the importance of voting theory and for the meaning of Arrow’s Impossibility Theorem, see here.
Literature

BBC (2017), “Article 50: May signs letter that will trigger Brexit“, March 29

Carrell, S. (2017), “Scottish parliament votes for second independence referendum“, The Guardian, March 28

Colignatus (2001, 2004, 2011, 2014), “Voting theory for democracy” (VTFD), pdf online, https://zenodo.org/record/291985

Colignatus (2010, “Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands”, May 19 2010, MPRA 22782, http://mpra.ub.uni-muenchen.de/22782

Colignatus (2011a), “The referendum on PR“, Mathematics Teaching 222, January 5 2011, also on my website

Colignatus (2011b), “Arrow’s Impossibility Theorem and the distinction between Voting and Deciding”, https://mpra.ub.uni-muenchen.de/34919

Colignatus (2014), “An Economic Supreme Court”, RES Newsletter issue no. 167, October 2014, pp.20-21, http://www.res.org.uk/view/art7Oct14Features.html

Colignatus (2016), “Brexit: advice for young UK (age < 50 years), and scientific outrage for neglect of voting theory“, weblog text June 29

Colignatus (2017), “The performance of four possible rules for selecting the Prime Minister after the Dutch Parliamentary elections of March 2017″, March 17, MPRA 77616

Grassegger, H. and M. Krogerus (2017), “The Data That Turned the World Upside Down”, https://motherboard.vice.com/en_us/article/how-our-likes-helped-trump-win

Renwick, A. e.a. (2016), “Letters: Both Remain and Leave are propagating falsehoods at public expense“, The Telegraph, Opinion, June 14

From the BBC website

[ This is the same text as the former weblog (here), but now we follow Van Hiele’s argument for the abolition of fractions. The key property is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there. Thus we replace (y / x) with y xH with H = -1. ]

Robert Siegler participates in the “Center for Improved Learning of Fractions” (CILF) and was chair of the IES 2010 research group “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” (report) (video).

IES 2010 key advice number 3 is:

“Help students understand why procedures for computations with fractions make sense.”

The first example of this helping to understand is:

“A common mistake students make when faced with fractions that have unlike denominators is to add both numerators and denominators. [ref 88] Certain representa­tions can provide visual cues to help students see the need for common denominators.” (Siegler et al. (2010:32), refering to Cramer, K., & Wyberg, T. (2009))

For a bH “and” c dH kids are supposed to find (a d + b c) (b d)H instead of (a + c) (b + d)H.

Obviously this is a matter of definition. For “plus” we define: a bH + c dH = (a d + b c) (b d)H.

But we can also define “superplus”: a bHc dH = (a + c) (b + d)H.

The crux lies in “and” that might not always be “plus”.

When (a + c) (b + d)H makes sense

There are cases where (a + c) (b + d)H makes eminent sense. For example, when a bH is the batting average in the Fall-Winter season and c dH the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) (b + d)H. Kids would calculate correctly, and Siegler et al. (2010) are suggesting that the kids would make a wrong calculation ?

The “superplus” outcome is called the “mediant“. See a Wolfram Demonstrations project case with batting scores.

Adding up fractions of the same pizza thus differs from averaging over more pizzas.

We thus observe:

  • Kids live in a world in which (a + c) (b + d)H makes eminent sense.
  • Telling them that this is “a mistaken calculation” is actually quite confusing for them.
  • Thus it is better teaching practice to explain to them when it makes sense.

There is no alternative but to explain Simpson’s paradox also in elementary school. See the discussion about the paradox in the former weblog entry. The issue for today is how to translate this to elementary school.

[ Some readers may not be at home in statistics. Let the weight of b be w = b (b + d)H. Then the weight of d is 1 – w. The weighted average is (a bH) w + (c dH) (1 – w) = (a + c) (b + d)H. ]

Cats and Dogs

Many examples of Simpson’s paradox have larger numbers, but the Kleinbaum et al. (2003:277) “ActivEpi” example has small numbers (see also here). I add one more to make the case less symmetrical. Kady Schneiter rightly remarked that an example with cats and dogs will be more appealing to students. She uses animal size (small or large pets) as a factor, but let me stick to the idea of gender as a confounder. Thus the kids in class can be presented with the following case.

  • There are 17 cats and 16 dogs.
  • There are 17 pets kept in the house and 16 kept outside.
  • There are 17 female pets and 16 male pets (perhaps “helped”).

There is the phenomenon – though kids might be oblivious why this might be “paradoxical”:

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male pets, the proportion of cats in the house is larger than the proportion for dogs.
  3. For all pets combined, the proportion of cats in the house is smaller than the proportion for dogs.
The paradoxical data

The paradoxical data are given as follows. Observe that kids must calculate:

  • For the cats: 6 7H = 0.86, 2 10H = 0.20 and (6 + 2) (7 + 10)H = 0.47.
  • For the dogs: 8 10H = 0.80, 1 6H = 0.17 and (8 + 1) (10 + 6)H = 0.56.

A discussion about what this means

Perhaps the major didactic challenge is to explain to kids that the outcome must be seen as “paradoxical”. When kids might not have developed “quantitative intuitions” then those might not be challenged. It might be wise to keep it that way. When data are seen as statistics only, then there might be less scope for false interpretations.

Obviously, though, one would discuss the various views that kids generate, so that they are actively engaged in trying to understand the situation.

The next step is to call attention to the sum totals that haven’t been shown above.

It is straightforward to observe that the F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. This causes the following calculations about what pets would be kept at the house. We keep the observed proportions intact and raise the numbers proportionally.

  • For the cats: 0.86 * 10 ∼ 9, and (9 + 2) (10 + 10) H = 0.55.
  • For the dogs: 0.17 * 10 ∼ 2, and (8 + 2) (10 + 10) H = 0.50.

And now we find: Also for all pets combined, the proportion of cats in the house is larger than the proportion for dogs. Adding up the subtables into the grand total doesn’t generate a different conclusion on the proportions.

Closure on causality

Perhaps kids at elementary school should not bothered with discussions on causality, certainly not on a flimsy case as this. But perhaps some kids require closure on this, or perhaps the teacher does. In that case the story might be that the kind of pet is the cause, and that the location where the pet is kept is the effect. When people have a cat then they tend to keep it at home. When people have a dog then are a bit more inclined to keep it outside. The location has no effect on gender. The gender of the pet doesn’t change by keeping it inside or outside of the house.

Vectors in elementary school

Pierre van Hiele (1909-2010) explained for most of his professional life that kids at elementary school can understand vectors. Thus, they should be able to enjoy this vector graphic by Alexander Bogomolny.

Van Hiele also proposed to abolish fractions as we know them, by replacing y / x by y x^(-1). The latter might be confusing because kids might think that they have to subtract something. But the mathematical constant H = -1 makes perfect sense, namely, check the unit circle and the complex number i. Thus we get y / x = y xH. The latter would be the better format. See A child wants nice and no mean numbers(2015).

Conclusions

Some conclusions are:

  • What Siegler & IES 2010 call a “common mistake” is the proper approach in serious statistics.
  • Teaching can improve by explaining to kids what method applies when. Adding fractions of the same pizza is different from calculating a statistical average. (PM. Don’t use round pizza’s. This makes for less insightful parts.)
  • Kids live in a world in which statistics are relevant too.
  • Simpson’s paradox can be adapted such that it may be tested whether it can be discussed in elementary school too.
  • The discussion corroborates Van Hiele’s arguments for vectors in elementary school and the abolition of fractions as we know them (y / x) and the use of y xH with H = -1. The key thing to learn is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there.

PM. The excel sheet for this case is: 2017-03-03-data-from-kleinbaum-2003-adapted

Hans Rosling (1948-2017) was a professor of public health and at the Swedish Academy of Sciences. I hadn’t heard about him but his death caused newsmedia to report about his mission to better inform people by the innovative presentation of statistics. I looked at some of his presentations, and found them both informative and innovative indeed.

I applaud this chart in which he tabulates not only causes and effects but rather means and goals. (Clicking on the picture will bring you to the TED talk 2007, and at the end the audience may applaud for another reason, namely when he swallows a sword to illustrate that the “impossible is possible”.)

Hans Rosling 1948-2017

Hans Rosling 1948-2017

Continue the discussion

My impression is that we best honour Rosling by continuing the discussion about his work. Thus, my comments are as follows.

First of all, my book Definition & Reality in the General Theory of Political Economy shows that the Trias Politica model of democracy fails, because it allows politicians still too much room to manipulate information and to meddle in scientific advice on policy making. Thus, governance is much more important than Rosling suggested. Because of his analysis, Rosling in some of his simulations only used economic growth as the decisive causal factor to explain the development of countries. However, the key causal factor is governance. The statistical reporting on this is not well developed yet. Thus, I move one + from economic growth to governance.

Secondly, my draft book The Tinbergen & Hueting Approach in the Economics of Ecological Survival discusses that the environment has become a dominant risk for the world as we know it. It is not a mathematical certainty that there will be ecological collapse, but the very nature of ecological collapse is that it comes suddenly, when you don’t expect it. The ecology is so complex and we simply don’t have enough information to manage it properly. It is like standing at the edge of a ravine. With superb control you might risk to edge one millimeter closer, but if you are not certain that the ground will hold and that there will not be a sudden rush of wind, then you better back up. The table given by Rosling doesn’t reflect this key point. Thus, I move one + from economic growth to the environment.

In sum, we get the following adapted table.

Adapted from Hans Rosling

I have contemplated for the means whether I would want to shift another + from economic growth to either human rights (property rights) or education (I am also a teacher). However, my current objective is to highlight the main analytical difference only.

In the continued discussion we should take care of proper definitions.

What does “economic growth” mean ?

The term “economic growth” is confusing. There is a distinction between level and annual growth of income, and there is a distinction w.r.t. categories within. Economic welfare consists of both material products (production and services) and immaterial elements (conditions and services). If the term “economic growth” includes both then this would be okay. In that case, however, the whole table would already be included in the notion of welfare and economic growth. Apparently, Hans Rosling intended the term “economic growth” for the material products. I would suggest to replace his “economic growth” by “income level”, and thus focus on both income and level rather than annual change of a confusingly named statistic. Obviously, it is a policy target that all people would have a decent standard of living, but it is useful to remain aware that income is only a means to a higher purpose, namely to live a good life.

PM. This causes a discussion about the income distribution, and how the poor and the rich refer to each other, so that the notion of poverty is relative to the general standard of society. In the 1980s the computer was a luxury item and nowadays a cell-phone with larger capacity is a necessity. These are relevant aspects but a discussion would lead too far here now.

What does “environment” mean ?

In the adapted table, the environment gets ++ as both means and goal. There is slight change of meaning for these separate angles.

  • The environment as a goal means that we want to preserve nature for our descendants. Our kids and grandchildren should also have tigers and whales in their natural habitat, and not as photographs only.
  • The environment as means causes some flip-flop thinking.
    (1) In economic thought, everything that exists either already existed or mankind has crafted it from what was given. Thus we only have (i) the environment, (ii) human labour. There are no other means available. From this perspective the environment deserves +++.
    (2) For most of its existence (some 60,000 years), mankind took the environment for granted. Clear air and water where available, and if some got polluted it was easy to move to a next clean spot. The economic price of the environment was zero. (Or close to it: the cost of moving was not quite a burden or seen as an economic cost.) Thus, as a means, the environment didn’t figure, and from this viewpoint it deserves a 0. There are still many people who think in this manner. It might be an engrained cultural habit, but a rather dangerous one.
    (3) Perhaps around the middle of the past century, the 1950s, the environment has become scarce. As Lionel Robbins explained: the environment has become an economic good. The environment provides functions for human existence and survival, and those functions now get a price. Even more, the Tinbergen & Hueting approach acknowledges that the ecology has become risky for human survival. The USA and Europe might think that they can outsource most environmental pollution to the poorer regions of the world, but when the rain forests turn into deserts and when the CO2 turns the oceans into an acid soup that eats away the bones of fish, then the USA and Europe will suffer the consequences too. In that perspective, the environment deserves +++.
    (4) How can we make sure that the environment gets proper place in the framework of all issues ? Eventually, nature is stronger than mankind, and there might arise some natural correction. However, there is also governance. If we get our stuff together, then mankind might manage the world economy, save the environment at some cost, but still achieve the other goals. Thus governance is +++ and the environment is relative at ++. Thus we arrive at above adapted table.
Dynamic simulation

As a teacher of mathematics I emphasize the combined presentation of text, formula, numeric table, and graph. By looking at these different angles, there is greater scope for integrated understanding. Some students are better at single aspects, but by presenting the four angles you cover the various types of students, and all students get an opportunity to develop the aspects that they are weaker in.

Obviously, dynamic simulation is a fifth aspect. See for example the Wolfram Demonstrations project. Many have been making applets in Java and embedding this in html5, yet the use of Mathematica would allow for more exchangeable and editable code and embedding within educational contexts in which the manipulation of text, formula, numeric table, and graph would also be standard.

Obviously, role playing and simulation games are a sixth aspect. This adds human interaction and social psychology to the learning experience. Dennis Meadows has been using this to allow people to grow aware of the risk on the environment, see e.g. “Stratagem” or MIT-Sloan.

The economic crisis of 2007+

What I particularly like about Rosling’s table is his emphasis on culture as a goal. Artists and other people in the world of culture will already be convinced of this – see also Roefie Hueting on the jazz stage – yet others may not be aware that mankind exists by culture.

There is also an important economic angle on culture as a means. In recessions and depressions, the government can stimulate cultural activity, such that money starts flowing again with much less risk for competitive conditions. That is, if the government would support the automobile industry or steel and do specific investments, then this might favour some industries or services at the cost of others, and it might affect competitive conditions overall, and even insert imbalances into the economy in some structural manner. Yet stimulating cultural activity might be much more neutral and still generate an economic stimulus.

For example, Germany around 1920 got into economic problems and the government responded by printing more money, and this caused the hyperinflation. This experience got ingrained in the German attitude towards monetary issues. In the Eurozone Germany follows the hard line that inflation should be prevented at all costs. Thus the eurozone now has fiat money that still functions as a gold standard because of the strict rules. (See my paper on this.) By comparison, when the USA around 1930 got into economic problems and the central bank was hesitant to print money (no doubt looking at the German example), this eventually caused the Great Depression. Thus monetary policy has the Scylla and Charybdis character, with the risks of either too little or too much. Potentially, the option to organise cultural activity would be a welcome addition to the instruments to avoid such risks and smooth the path towards recovery.

I am not quite suggesting that the ECB should print money to pay the unemployed in Greece, Italy, Spain and Portugal to make music and dance in the streets, yet, when the EU would invest in musea and restorations and other cultural services so that Northern Europe can better enjoy their vacations in Southern Europe, then this likely would be more acceptable than when such funds would be invested directly in factories that start to compete with the North. The current situation that Southern Europe has both unemployment and less funds to maintain the cultural heritage is obviously less optimal.

The point is also made in my book Common Sense: Boycott Holland. Just to be sure: this notion w.r.t. culture is not the main point of CSBH. It is just a notion that is worthy of mentioning.

PM. Imagine a dynamic simulation of restoring the Colosseum. Or is it culturally more valuable as a ruin than fully restored ?

By Jaakko Luttinen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=22495158

By Jaakko Luttinen – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=22495158

Robert Siegler participates in the “Center for Improved Learning of Fractions” (CILF) and was chair of the IES 2010 research group “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” (report) (video).

IES 2010 key advice number 3 is:

“Help students understand why procedures for computations with fractions make sense.”

The first example of this helping to understand is:

“A common mistake students make when faced with fractions that have unlike denominators is to add both numerators and denominators. [ref 88] Certain representa­tions can provide visual cues to help students see the need for common denominators.” (Siegler et al. (2010:32), refering to Cramer, K., & Wyberg, T. (2009))

For a / b “and” c / d kids are supposed to find (ad + bc) / (bd) instead of (a + c) / (b + d).

Obviously this is a matter of definition. For “plus” we define: a / b + c / d = (ad + bc) / (bd).

But we can also define “superplus”: a / c / d =  (a + c) / (b + d).

The crux lies in “and” that might not always be “plus”.

When (a + c) / (b + d) makes sense

There are cases where (a + c) / (b + d) makes eminent sense. For example, when a / b is the batting average in the Fall-Winter season and c / d the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) / (b + d). Kids would calculate correctly, and Siegler et al. (2010) are suggesting that the kids would make a wrong calculation ?

The “superplus” outcome is called the “mediant“. See a Wolfram Demonstrations project case with batting scores.

Adding up fractions of the same pizza thus differs from averaging over more pizzas.

We thus observe:

  • Kids live in a world in which (a + c) / (b + d) makes eminent sense.
  • Telling them that this is “a mistaken calculation” is actually quite confusing for them.
  • Thus it is better teaching practice to explain to them when it makes sense.

There is no alternative but to explain Simpson’s paradox also in elementary school. See the discussion about the paradox in the former weblog entry. The issue for today is how to translate this to elementary school.

Cats and Dogs

Many examples of Simpson’s paradox have larger numbers, but the Kleinbaum et al. (2003:277) “ActivEpi” example has small numbers (see also here). I add one more to make the case less symmetrical. Kady Schneiter rightly remarked that an example with cats and dogs will be more appealing to students. She uses size (small or large pets) as a factor, but let me stick to the idea of gender as a confounder. Thus the kids in class can be presented with the following case.

  • There are 17 cats and 16 dogs.
  • There are 17 pets kept in the house and 16 kept outside.
  • There are 17 female pets and 16 male pets (perhaps “helped”).

There is the phenomenon – though kids might be oblivious why this might be “paradoxical”:

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male pets, the proportion of cats in the house is larger than the proportion for dogs.
  3. For all pets combined, the proportion of cats in the house is smaller than the proportion for dogs.
The paradoxical data

The paradoxical data are given as follows. Observe that kids must calculate:

  • For the cats: 6 / 7 = 0.86, 2 / 10 = 0.20 and (6 + 2) / (7 + 10) = 0.47.
  • For the dogs: 8 / 10 = 0.80, 1 / 6 = 0.17 and (8 + 1) / (10 + 6) = 0.56.

A discussion about what this means

Perhaps the major didactic challenge is to explain to kids that the outcome must be seen as “paradoxical”. When kids might not have developed “quantitative intuitions” then those might not be challenged. It might be wise to keep it that way. When data are seen as statistics only, then there might be less scope for false interpretations.

Obviously, though, one would discuss the various views that kids generate, so that they are actively engaged in trying to understand the situation.

The next step is to call attention to the sum totals that haven’t been shown above.

It is straightforward to observe that the F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. This causes the following calculations about what pets would be kept at the house. We keep the observed proportions intact and raise the numbers proportionally.

  • For the cats: 0.86 * 10 ∼ 9, and (9 + 2) / (10 + 10) = 0.55.
  • For the dogs: 0.17 * 10 ∼ 2, and (8 + 2) / (10 + 10) = 0.50.

And now we find: Also for all pets combined, the proportion of cats in the house is larger than the proportion for dogs. Adding up the subtables into the grand total doesn’t generate a different conclusion on the proportions.

Closure on causality

Perhaps kids at elementary school should not bothered with discussions on causality, certainly not on a flimsy case as this. But perhaps some kids require closure on this, or perhaps the teacher does. In that case the story might be that the kind of pet is the cause, and that the location where the pet is kept is the effect. When people have a cat then they tend to keep it at home. When people have a dog then are a bit more inclined to keep it outside. The location has no effect on gender. The gender of the pet doesn’t change by keeping it inside or outside of the house.

Vectors in elementary school

Pierre van Hiele (1909-2010) explained for most of his professional life that kids at elementary school can understand vectors. Thus, they should be able to enjoy this vector graphic by Alexander Bogomolny.

Van Hiele also proposed to abolish fractions as we know them, by replacing y / x by y x^(-1). The latter might be confusing because kids might think that they have to subtract something. But the mathematical constant H = -1 makes perfect sense, namely, check the unit circle and the complex number i. Thus we get y / x = y xH. The latter would be the better format. See A child wants nice and no mean numbers(2015).

Conclusions

Some conclusions are:

  • What Siegler & IES 2010 call a “common mistake” is the proper approach in serious statistics.
  • Teaching can improve by explaining to kids what method applies when. Adding fractions of the same pizza is different from calculating a statistical average. (PM. Don’t use round pizza’s. This makes for less insightful parts.)
  • Kids live in a world in which statistics are relevant too.
  • Simpson’s paradox can be adapted such that it may be tested whether it can be discussed in elementary school too.
  • The discussion corroborates Van Hiele’s arguments for vectors in elementary school and the abolition of fractions as we know them (y / x) and the use of y xH with H = -1. The key thing to learn is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there.

PM. The excel sheet for this case is: 2017-01-30-data-from-kleinbaum-2003-adapted.