# Game of Elections: election rules restrict the outcome???!!!

Welcome back to the channel of the Political Academy. In this video we will talk about the “Game of Elections”. Elections are subject to rules that restrict the outcome. Politics has hardly anything to do with that. My name is Joost Smits. First I will show how many parties can be expected to have at least one seat. We will look at election results to prove the value of such a calculation. We will speculate on why the “game of elections” works. And then we will take it one step further. How large will parties be? Can we even predict elections using this tool? This is the first video of this kind that I am making. We have a board we can draw on … … and we have an arts and crafts table with books and stuff to play with. Let’s get started. The Netherlands and Belgium use proportional representation … … to fill the national parliament, provincial assemblies and the municipal councils. In a future video I will treat this electoral system … … and compare it to other systems that are used in the world. This video uses proportional representation as framework, … … but the mechanism applies everywhere. Let’s start with an example. Here we have a town council with 9 seats. If we know nothing about this town, we can assume … … there can be a maximum of 9 parties filling the benches. And a minimum of 1. If elections are random, … … we can expect 1 plus 9 divided by 2 parties … … is 5 parties on average to have at least one seat. If elections are random … … a party will have at least 1 and at the most 9 seats in the council. So, on average 1 plus 9 divided by 2 is 5 seats. Five parties on average times five seats on average equals 25 seats. But we only had 9 seats to fill? So, apparently there exists some other mechanism … … why some parties are big and others are small. §1. Literature Here I have some books about the subject. One is by Arend Lijphart, a Dutch-American professor … … of political science, currently at the University of California, San Diego. This is one of the books in which he treats electoral systems all over the world. Lijphart is also honorary member of the Dutch Political Science Association. Lijphart pays attention to Rein Taagepera, … … who is an Estonian former politician and now researcher. Currently at the University of California, Irvine. With colleagues he did a lot of research on … … the average number of parties that get at least one seat in an election. Taagepera found that in general the number of parties that get at least one seat … … is the square root of the number of available seats. At least that is the formula in countries like the Netherlands and Belgium. In other electoral systems the formula is just a little bit more complicated. I will not go into that now. So, the formula is p=square root of M Where p is the number of parties … … and M the number of available seats. Which can also be written like this. How to apply this? The Dutch and Belgian parliaments have 150 seats, … … therefore 12.2 parties are expected. This value of 0.5 is commonly referred to as Taagepera’s Index. It is not a true norm, it is an average. Taagepera describes that in some countries or other periods … … the fragmentation factor may vary. Since later in this video we will play with different values … I will refer to it as “the fragmentation factor”. Some of you may have a problem with this approach. It is very strange that there is no political content. Aren’t elections about debate? About choices to make for the country, for the province, for the city? Isn’t it true that voters only make up their mind in the last weeks before elections? What about Facebook campaigns, social media, fake news? Steven Reed commented on this in 1996. “Political scientists are traditionally less comfortable … … with the assumption of mathematical elegance than are physicists. We tend to be more comfortable … with the presumption that ‘things are much more complicated than that’. More importantly, … … we expect some behavioral model to underlie our theories. There is no particular behavioral basis to Taagepera’s theory. It does not depend on rational voters or strategic political parties. It is less a ‘political’ theory than a mechanical one. Taagepera apologised that he can not help it. “To paraphrase Winston Churchill’s dictum about democracy… … ‘p=M to the power ½’ is the worst possible prediction … … except for all others. This prediction is pulled almost completely from thin air, … … but all the others would be completely so, in the absence of further information.” In 1999 he called it “ignorance-based quantative modelling”. Let’s go back to the board to see some examples of how this works out in practice. §2. The Number of Parties Here you see the development of the number of parties … … in Dutch national parliament since the early seventies. The blue line is Taagepera’s 0.5 value. You see that Dutch parliament is quite “normal” … … with 13 parties securing at least one seat in March 2017. Three months before, in December 2016, 81 political parties said … … they wanted to take part in the March 2017 elections. Only 28 passed all formalities to wind up on the ballot. You see the strong influence of the “game”. Then we look at the Belgian national parliament timeline. In Belgium it is very easy to start a political party. But, they have Flemish-speaking parties who only compete in the northern part of the country … … and French-speaking parties who only compete in the southern part, … … and party families of both language groups. But still, the international average is met. 13 Parties in the 150 seat parliament since 2014, just like the Netherlands. In provincial and municipal elections, however, the picture is different. The “fragmentation factor” is on the rise in the Netherlands. Ever since the mid nineteen eighties, which was a period of economic prosperity. Often researchers say that fragmentation rises … … under conditions of economic downturn. How to read this? Municipal councils have different sizes … … depending on the size of the population. Amsterdam has 45 seats. The yellow line points to 0.647. We raise 45 to 0.647 and get 11.7. The city council indeed has 12 parties since 2018. Of course, the number in the chart is an average and it will not always fit perfectly. On 20 March we will again have Dutch Provincial elections … … and we can expect the line to rise again. Belgium has been experimenting with electoral law since the mid nineteen nineties … … which influences their development of the number of parties. It is rising gradually, … … but it is not at the high level of the Dutch. In a future video I will go more into that. So, even though voters are free to vote for any party they like, … … people are free to create new parties, … … it is very easy to take part in elections (more in Belgium than in the Netherlands), … … there is a clear influence that keeps the number of parties with seats … … low for national elections. Why is this? I think there are four main reasons. §3. Four main reasons The first reason is about electoral law … … and the way in which votes are transferred into seats. Something for a future video. Belgium uses a different calculation to turn votes into seats, … … and small parties are disadvantaged by that system. Taagepera says it is wrong to just assume … … that votes are turned into seats just like that. Votes indeed lead to seats, but the electoral system … … has influence on how many seats a party gets. The second reason is psychological and cultural influences. Will voters vote for small parties or only for potential winners? That also addresses so-called “strategic voting”. Strategic voting means that voters will not vote for the party they really prefer, … … but for some other party that has more chances of winning … … and deliver the preferred policies. I will get back to strategic voting later in this video. An example of cultural influence is that Belgian local elections produce much less parties … … with seats because of the so-called “List of the mayor”. Something also for future videos. The third main reason is the choice of groups. Parties can only get votes if there are groups … … in society that support their program. Or, a new party or political candidate can find out … … if there is support in society for issues … … that are not yet addressed by existing parties or candidates. In 1948 Duncan Black combined economic theory with elections. Let’s go to the arts-and-crafts table so I can show you. In 1948 Duncan Black combined economic theory with elections. He found out that when all voters’ preferences are arranged as single peaked, … … then the group’s preference can be found in the highest peak. An example of a single peaked line is for example this. If all preferences of voters … … are like this. Now there is one peak on the right. This is apparently the preference of this group. Another possibility may be … … is this. Also now there is one peak. For the neutral position. If one would have … … an arrangement like this … … there are two peaks. So this is not a valid conclusion. Later, in 1988 Andrew Caplin and Barry Nalebuff showed that if 64% … … of voters are aligned in their preferences, … … this is an unbeatable proposal winning every election. It is quite obvious that although in a society there may be a zillion issues, … … only a limited number have enough broad support to allow for electoral success. And the fourth and last reason is simple statistics of distributions. If you measure the length of a group of North-western European men or women … … some may be short and some may be tall. But the average is more or less predictable by the knowledge we have … … of the length of North-western European men or women from earlier measurements. Beforehand we can know more or less how many in a group of 100 … … will be 5 centimetres taller than the average. Same goes I think for groups of issues. This is not the end of it. Taagepera even shows he can predict the size of parties. §4. The Size of Parties In 1993 Rein Taagepera and Matthew Shugart came up with the following formula. The largest party has a share equal to the number of parties to the power of -½. Let’s get another marker. From this follows that the second party has a size of … This is s1, the first party. (1 minus s1) divided by (N0 – 1) So the share of the first party is equal to the number of parties to the power -½ … … or equal to 1 divided by the number of parties to the power ½. From this follows that the second party has a size of (1 minus the share of the first party) … … divided by (the number of parties from the beginning minus 1). Because the first party already had a share. And the third party … … (1-s1-s2) divided by (N0 – 1 – 1) … … to the power ½ We can do this example with 9 seats from the beginning. We will expect square root of 9 is 3 parties to have at least one seat. The share of party 1 is 0.57. Which means, times 9 seats … … 5.2 seats. The second party … … (1 minus 0.5774 ) … … divided by (3 – 1) to the power ½ … … is 0.2989 times 9 … … is 2.7 seats. And the third party … … (1- 0,5774 – 0,2989) … … divided by (3 – 1 – 1) to the power ½ is … … this is basically 1, so the share … … is 0.1237 Times 9 is … 1.1 If we go back to the beginning … … now we know the distribution of this town council of 9 seats. We now know, party 1 … … has 5 seats. Party 2 … … has 3 seats. And party 3 … … has only 1 seat. This means that an election is now downgraded to … … a contest of parties for a number of predefined slots. In this case there are three slots. There is a slot of 5 seats, a slot of 3 seats and a slot of 1 seat. In this case the orange party opts for the slot of 5 seats … … green got the slot of 3 seats, … … en purple only had the slot for 1 seat. These slots exist by definition. Before the election. The parties can only try to gain one of these slots. Taagepera’s sequence does not compute well in practice. First it predicts 12-13 parties for 150 seats, … … and then the smallest parties wind up with a seat below 1. So they do not have a seat. The formula also needs to be extended if we implement strategic voting. In 2006 Rein Taagepera and Mirjam Allik found that because of strategic voting … … the two largest parties normally get more votes than the formula predicts. Even 50% extra of their probabilistic share, at the expense of the smaller parties. Here is a diagram of a town council with 25 seats, according to the original formula. However, extending it with strategic voting … … we would need to add … … more seats to the two largest parties … … which would come at the expense of the smaller parties. They would get less. §5. Prediction If we want to apply Taagepera’s formula we need to modify it a bit … … and also choose whether we expect strategic voting or not. In December 2016 I expected 13 parties to win at least one seat … … in Dutch parliament, … … and because of our coalition governments in which even small parties can play an important role … … note, this is a psychological influence … … I left out strategic voting. Using a slightly modified form of Taagepera’s formula I got the following sequence: 31 25 21 17 13 11 9 7 5 4 3 2 1 and 1. Back to the main board. In March, a week before the election, together with political enthusiasts … … from Rotterdam (VVD Studio), we added party names to the sequence and posted it on Facebook. Later we could compare our prediction with those of major Dutch polling organisations. The prediction of slots was as good or better than most pollsters. The result-prediction, with added party names,, was a bit less lucky … … because we switched two parties. This was slots only, and this was the prediction by VVD Studio. Still our prediction did not cost any money, was still very close, … … was only based on an “ignorance-based model”, and we had fun. And guess what… The Belgian parliament also has 150 seats and 13 parties. If we apply the same December 2016 sequence to the … … Belgian national elections of 2014… we get even closer. If in the upcoming elections for the Belgian parliament only 13 parties … … are likely to get at least 1 seat, … … this prediction of 2016 still holds for that election. Thank you for watching this video. Your comments and questions are welcome below. Do not forget to subscribe, and please ring that bell … … so you will be notified for the next video.