Elections and Mathematics, or Divide and Conquer
The problem of choice has always been before us. Primitive man also faced a dilemma: in what light to live? On the other hand, the election of tribal leaders was simpler: the one who killed the competitor ruled. Today is more difficult. It's also good.
The Latin sentence used in the title of the article means "divide and conquer". It has always been used. Cause a quarrel in a nation and it will be easier for you to win it. The Spanish conquistadors of the 1990th and XNUMXth centuries skillfully turned some Indian tribes against others. At the end of the XNUMXth century, the Russian ambassador Repnin achieved a lot: he managed to create unrest in the last years of independent Poland. So did the British in their former empire, and the XNUMX Yugoslav war began with Serbs pitting against Croats and vice versa.
We know examples of deliberate incitement of conflicts within one country. Fortunately, this is not the case in today's Poland. The ruling party is an example of softness, restraint and common sense, filled with respect for the opposition, respecting the law, the Constitution and the will of a simple man. At the international forum we win, often with zero (a memorable victory 27:0). In sports, we are doing well: we remember the dramatic hockey match with Cameroon. There are no scandals, the politicians are crystal clear. Where do they have their own pockets in their heads! The party is in the lead. We will help!
Stop, stop. We are not a journalistic magazine. Let's see how you can bend the decision-making process in the grandeur of mathematics and ... logic. A full description would be a big job, more journalistic than scientific.
The following options are possible.
First, manipulating the division of the country into districts.
Secondly, the choice of the method of converting votes into parliamentary seats or (for example, in the case of presidential elections) into electoral seats.
Third: interpreting when the voice is important and when it is not.
I'm not mentioning explicit abuses here, such as manipulation of voter ignorance (for the Polish People's Republic, empty voting meant voting for candidates listed at the top of the list), fraud in the counting of votes, and sending the data above.
I will begin . What is this strange term? I explain in a slightly roundabout way.
Your readers probably know the score in tennis. We get points, games and sets. To win the game, you need to win at least four balls (points), but at least two more than your opponent. The exception is the tie-break game - it is played up to seven winning points (balls), also with a two-ball advantage rule. The won balls are numbered strangely: 15, 30, 40, then we use only the terms “advantage – balance”.
1. Left classic gerrymandering. The global balance turns into a victory for the blue. That's right: in each district of the northern district, the blues have only 25% support, in the rest they still - but they do not mind.
Gems are collected in sets. To win a set, you must have at least six games and at least two more than your opponent. When the score is 6:6, a tie-break is usually played. Matches are played with two or three sets won. "Up to two wins" means that the one who wins two sets wins. Thus, the result can be 2:0 or 2:1 (and symmetrically 0:2, 1:2). These rules mean that you don't need to win more balls (points) to win the game. Simply put, you have to win the more important ones. An extreme example is where player A wins the first set 6-0 and the other two lose 4-6. Loses a match despite winning 14 games and his opponent 12.
I will refer to what I wrote a moment ago. There are more and less important moments in tennis. A good tennis player focuses on what matters most.
The fate of millions in the paws of the salamander
Let's move on to political elections. More generally, to elections that are decided by thousands or millions.
You must first have a country for constituencies. Because? Doesn't matter how? Oh no! The first to figure out how to do this in order to increase the chances of his own party was Elbridge Jerry, an American politician of two hundred years ago. One of the circles he proposed was in the shape of ... a salamander, and the combination of his name with this tailed amphibian led to the term. It works quite well with single-member districts, so it is not directly applicable to Poland. With a multi-member office, the situation is quite different. You can get burned from time to time. And an interesting thing.
2. Master of fraud. Left: 40% of global support turned into a 4-2 win. Right: Geometry does a great job of turning 32% support into a 4:3 global win.
So, let's imagine a country, densely populated and with very regular borders: a perfect square with small field towns inside it. The city and the mayoral election is the best analogy, but mathematically it doesn't matter. The ruling party in blue has support in the sectors marked in blue on fig. 1. Greens lead in green squares. Since we are talking about single-member districts, it does not matter what the advantage is. We are nationally connected, as many blue squares as there are green ones. But the blues rule and divide the country into regions. There are eight constituencies (1). What are the voting results? Unexpected! Blue players win in A, C, E, F, G, that is, in five out of eight circles. In the case of single member constituencies, they have a 5:3 advantage across the country (possibly cities if it is a mayoral election).
electoral geography this has an important advantage for a party where scandals are common. Let's imagine that a scandal erupted in constituency B - the mayor embezzled budget money and said that everything was in order. Many voters turned their backs on him. If earlier the votes were distributed almost equally (51:49 in favor of one or another party), now in district B in each small district, green received 75%, and blue only 25. However, on a national scale, this did not hurt at all (Table 1). To use the tennis analogy, they only lost an empty point.
| constituency | Blue | Greens | Who is winning |
| A | 251 | 249 | Blue |
| B | 100 | 300 | Greens |
| C | 251 | 249 | Blue |
| D | 198 | 202 | Greens |
| E | 251 | 249 | Blue |
| F | 251 | 249 | Blue |
| G | 251 | 249 | Blue |
| H | 149 | 151 | Greens |
| Total votes | 1702 | 1898 | 5 to 3 for blue |
Table 1. Number of votes 1898: 1702 in favor of the greens, but 5: 3 seats in parliament for the blue! In the US presidential election, it happens that the winner receives fewer votes.
The single system has its advantages and disadvantages. It came from the English parliamentary tradition. A variety of mathematical formulas have been proposed to slightly reduce the principle of "winner takes all". The most common rule was "largest fractional part". Let's assume that four parties A, B, C and D compete in Grodzisko Nadmorsky region. There are seven places to win. In the elections, these parties received respectively 9934 5765, 4031 1999, 21 729 and XNUMX XNUMX votes; total XNUMX XNUMX. We expect:
7∙9934/21729= 3,20
7∙5765/21729= 1,86
7∙4031/21729= 1,30
7∙1999/21729= 0,64
Clear; if the Commonwealth were, as Prince Radziwiłł says in The Flood, a red cloth, the parties would tear it apart in the proportion of 320:186:130:64. But there are only seven places to share. Lots A deserve three places (because the quotient is greater than 3), lots B, C deserve one place each. How can I select the other two? The following solution is proposed: to give it to those parties that “least lack a full vote”, i.e., those with the largest fractional part. Therefore, they fall into parts B, D. Let's represent the result in a clear graph on fig. 3.
fig.3 The method of "greatest fractional part". Coalition B + C + D defeats Party A
What will the so-called. d'Hondt's rule? I discuss this a little further. I recommend it as an exercise. Result on fig. 4.
fig.4 Results of the d'Hondt method. Party A rules on its own.
For the next easy exercise, I recommend that readers do something like this: imagine parties B, C, and D agree and go to the polls in one bloc—call it E. Then, as d'Hondt's rule suggests, they take away one party A has a mandate, i.e. the result of A:E is 3:4. The conclusion has been known for many years as a proverb: Consent creates, disagreement destroys.
Fortunately, the examples I give here are fictitious and any resemblance to known countries is purely coincidental.
D'Ond
How does the mentioned d'Hondt method work? An example is best suited for this. Suppose a particular constituency voted in an episcopal election, as shown. Table 2.
| Party name | Voices, N. | H/2 | H/3 | H/4 | H/5 |
| Full Prosperity Party | 10 000 | 5000 | 3333 | 2500 | 2000 |
| party of abundance | 6600 | 3300 | 2200 | 1650 | 1320 |
| The locomotive of progress | 4800 | 2400 | 1600 | 1200 | 960 |
| Fraudsters and scammers | 3600 | 1800 | 1200 | 900 | 720 |
Table 2. Voting results in the Klapucko Male constituency in the elections in Klapadocsy.
It turned out that the party of swindlers and gochstaplers had succeeded so well only in Klaputsky Maly. Globally, they did not score 5%, so their results are not taken into account. We place the rest in turn, not forgetting which party they are from:
10 (PTD), 000 (SO), 6600 (PTD), 5000 (LP), 4800 (PTD), 3333 (SO), 3300 (PTD), 2500 (LP), 2400 (SO), etc. We assign tickets in the specified order. The result largely depends on the number of tickets available.
| Xnumx places | PTD 2, SO 1, LP 0 |
| Xnumx places | PTD 2, SO 1, LP 1 |
| 5 Places | PTD 3, SO 1, LP 1 |
| 6 Places | PTD 3, SO 2, LP 1 |
| 7 Places | PTD 4, SO 2, LP 1 |
| 8 Places | PTD 4, SO 2, LP 2 |
| 9 Places | PTD 4, SO 3, LP 2 |
Table 3. Distribution of seats depending on their number.
It is said that such a system smoothes the results - reduces the possible dominance of one party. However, the matter is more complicated. It all depends on the specific data. I have no room for longer discussions, I will note only two interesting facts:
1. If the scammers and fraudsters had reached the national electoral threshold, the results could have been different. They would not change if three or four seats were to be won, but if five people from the constituency entered the parliament, the result would be: PTD 2, SO 1, PL 1, JG 1. The PTD party would lose its absolute right. majority. It works the other way around: if a small faction breaks out of the party, everyone loses, including those who disagree.
2. If SO and LP got along and went to the polls together, then they would be no worse in any scenario, but usually better.
Let us also see how the d'Hondt method treats the situation with fig. 2when there are two or three empty seats in the ward. Let me remind you that in the case of single-member districts, this gave a strong victory to the Blues. In the case of doubles, there is a total defeat, but in the case of triples, he wins again.
| constituency | Blue | Greens | Method d'Hondt |
| A | 251 | 249 | Gear ratios: 251/249; schedule 1-1 |
| B | 100 | 300 | 300/100; 0-2 |
| C | 251 | 249 | 251/249; 1-1 |
| D | 198 | 202 | 202/198; 1-1 |
| E | 251 | 249 | 251/249; 1-1 |
| F | 251 | 249 | 251/249; 1-1 |
| G | 251 | 249 | 251/249; 1-1 |
| H | 149 | 151 | 151/149; 1-1 |
| Total votes | 1702 | 1898 | Blue 7 - Green 9 |
Table 4. Situation with fig. 2, but with dual-member constituencies. The failure of blue 7:9.
| constituency | Blue | Greens | Method d'Hondt |
| A | 251 | 249 | Gear ratios: 251/249/125,5; graph 2-1 |
| B | 100 | 300 | 300/150/100; 0,5-2,5 |
| C | 251 | 249 | 251/249/125,5; 2-1 |
| D | 198 | 202 | 202/198/101; 1-2 |
| E | 251 | 249 | 251/249/125,5; 2-1 |
| F | 251 | 249 | 251/249/125,5; 2-1 |
| G | 251 | 249 | 251/249/125,5; 2-1 |
| H | 149 | 151 | 151/149/75,5; 1-2 |
| Total votes | 1702 | 1898 | Blue 12,5 - Green 11,5 |
Table 5. Situation with fig. 2, but with three-member constituencies.
Among some features, I include "geometry" in qualifying votes as important or unimportant. In many countries, the sign of approval is a “tick”, that is, a v, and sometimes a Y. We have an x, which is more associated with a strikethrough (and therefore a rejection). The legislator wanted to clarify this and gave a quasi-mathematical definition - “two intersecting lines”, interpreting that the two lines of the letter v do not intersect.
First, in mathematics, "intersecting" means "having a common point" - this should be especially associated with younger people (under fifty), because that's how school is now. However, if someone does not believe in mathematics, then he may remember that a U-turn on the road is also a crossroads.
It is better to leave an inaccurate definition: any sign that unambiguously indicates the election of a candidate to a position that was once honorary, but now has only a pejorative association.
