Mathematics in politics
Introduction
Since around 500 B.C., the ancient Greek civilization introduced one of the earliest forms of democracy. Back then, voters would write their choices on broken pieces of pottery called ‘ostraka’. Imagine a world where the outcome of a major election isn’t just about the number of people voting, but about the mathematical weight of each vote. Surprisingly, this is often the case even today. Behind every election or vote, mathematics plays a significant—sometimes hidden—role in determining who truly holds power. This article explores the importance of voting theory and power indices in modern elections.
Voting Theory
Voting theory studies how groups make collective decisions. While a simple majority (>50%) might seem like the fairest way to decide, real-world elections are often much more complex.
Factors such as the number of candidates, voters’ preferences, and the weighting of votes allinfluence the outcome. Voting theory seeks to design systems that are as fair andrepresentative as possible.
For example, consider an election with three people—Person A, Person B, and Person C—voting among three candidates: 1, 2, and 3. Suppose their preferences are:
𝑃𝑒𝑟𝑠𝑜𝑛 𝐴: 1 > 2 > 3
𝑃𝑒𝑟𝑠𝑜𝑛 𝐵: 2 > 3 > 1
𝑃𝑒𝑟𝑠𝑜𝑛 𝐶: 2 > 1 > 3
If we look only at first-choice votes (plurality), candidate 2 is the most popular, receiving two out of three first-choice votes. However, in other scenarios, plurality can result in a tie or fail to represent the voters’ true preferences. This is why mathematicians use more sophisticated methods to measure representation and fairness.
Proportional Representation
Some countries, such as Norway and Sweden, use proportional representation in their elections. In this system, the number of seats a party wins is proportional to the percentage of votes it receives.Mathematically, the proportionality ratio is defined as:
\[proportionality = \dfrac{seats \, won}{vote \, share}\]
For example, if a party receives 30% of the votes and wins 30% of the seats, the ratio is:
\[\frac{0.30}{0.30} = 1.0\]
This is perfectly proportional. However, if a party receives 20% of the votes but wins 40% of the seats, the ratio is:
\[\frac{0.40}{0.20} = 2.0\]
This indicates the party is over-represented, revealing a potential flaw in the system. The proportionality ratio helps us mathematically assess the fairness of an election.
Why Is There No Perfect Voting System?
In 1951, American economist and mathematician Kenneth Arrow proved the so-called “impossibility theorem”. Arrow showed that no voting system can be completely fair when there are three or more candidates and voters rank their preferences. Every system must violate at least one criterion of fairness, such as respecting all voters’ preferences equally, avoiding strategic manipulation, or always reflecting the majority’s wishes. As a result, all voting methods involve trade-offs—some may be more representative, while others are more resistant to manipulation.
The Banzhaf Power Index
Introduced by John Banzhaf in 1946, the Banzhaf Power Index is a mathematical tool for measuring the power of each voter in a decision-making body. A voter is considered “critical” if removing their vote from a winning coalition causes it to lose. The index calculates how often each voter is critical in all possible winning coalitions.
For example, suppose three countries (A, B, and C) are voting on a policy, with the following weights:
𝐴: 4 votes
𝐵: 3 votes
𝐶: 1 votes
A majority of 5 votes is needed to pass the policy. The possible coalitions and their total votes
are:
𝐴 + 𝐵 = 7 → winning
𝐴 + 𝐶 = 5 → winning
𝐵 + 𝐶 = 4 → not enough
𝐴 + 𝐵 + 𝐶 = 8 → winning
Now, let’s identify the critical voters in each winning coalition:
𝐴 + 𝐵:
Remove 𝐴 (𝐵 = 3) → not enough ⇒ 𝐴 is critical
Remove 𝐵 (𝐴 = 4) → not enough ⇒ 𝐵 𝑖𝑠 critical
𝐴 + 𝐶:
Remove 𝐴 (𝐶 = 1) → not enough ⇒ 𝐴 is critical
Remove 𝐶 (𝐴 = 4) → not enough ⇒ 𝐶 is critical
𝐴 + 𝐵 + 𝐶:
Remove 𝐴 (𝐵 + 𝐶 = 4) → not enough ⇒ 𝐴 is critical
Remove 𝐵 (𝐴 + 𝐶 = 5) → still enough ⇒ 𝐵 is NOT critical
Remove 𝐶 (𝐴 + 𝐵 = 7) → still enough ⇒ 𝐶 is NOT critical
Counting the critical appearances:
𝐴: 3 times
𝐵: 1 time
𝐶: 1 time
The total number of critical votes is 5. Therefore, the Banzhaf power of each voter is:
\[A= \frac{3}{5} = 60\%\]
\[B = \frac{1}{5} = 20\%\]
\[C = \frac{1}{5} = 20\%\]
This example shows that even though A holds only 4 out of 8 total votes, A has 60% of the power, highlighting how voting weights and coalition dynamics can create large disparities in influence.
Conclusion
In times of political instability, concerns about unfair elections become especially pressing. For instance, in the European Parliament, Germany has 29 votes while Estonia has only 4, illustrating significant disparities in voting power. Mathematics provides tools to analyse and expose these imbalances, helping us strive for fairer and more transparent institutions.
Ultimately, understanding the mathematics behind political systems empowers us to recognise both the strengths and weaknesses of the ways we are governed.
