In most democratic countries, voting is an essential part in deciding how a country is led. For example, in the Netherlands there are local and nationwide elections in which you decide which political parties will run different aspects of governmental life. Naturally one is quick to assume that one vote does not matter much in the grand scheme of things, which is what often causes people not to vote. To limit the number of people that choose not to vote as a result of this, voting power is analysed. This entails an inquiry into how much each vote is “worth”. It is in democracy’s interest to maximise voting power, as this is seen to be more representative of how a public feels about its leadership.

Typically, countries are divided into constituencies or provinces, where local elections are held, and these are used to build up to national level. For example, in the UK, one votes for a local member of parliament, who, if elected, represents their constituency on a nationwide scale. In the US, presidents are elected in a state, and then this state will count as a certain number of votes nationally. The weighting of each constituency or state nationally depends on the population size – for example California weighs much more than Missouri for national elections, due to the dramatic difference in population between these two states. From this, the mathematical theory of voting power becomes clear, which is usually defined to be the amount of influence a voter or set of voters have on the outcome of an election. In mathematical terms, we can define voting power as the probability that a vote affects an election’s outcome.

With regard to the recent French election, let us focus on the concept of voting power with 2-party states, which is the case with many democratic countries, notably the presidential elections in the US. Voting power analysis is essential in evaluating the fairness of a democratic system, so that each individual’s vote leads to the most equally weighted outcome.

We may use the notation *i *for the “i-th voter”, and the outcome of a two-party vote being *v _{i} *= ±1, for each of the two parties. Then for a set of

*n*voters, their outcomes can be listed as a vector

*v*= (

*v*), and have that the outcome of the election be

_{1 }, v_{2 }, … ,_{ }v_{n}*R*= R(

*v*) = ±1. Of course R may be stochastic, in the event of a tie, but in most cases when looking at large population samples this will not be the case. From this, we can define the general formula for the voting power of an individual

*i*as: power

_{i }= P(R = -1 if

*v*= -1) – P(R = -1 if

_{i}*v*= -1). From this, it is clear that if one’s voting power is equal to 0, it does not matter which party an individual chooses to vote for, as each of these probabilities would be the same and therefore cancel out. Now let us consider the following scenarios: weighted voting and two-stage voting.

_{i}Weighted voting theorises that each of *n *voters has a certain weighting, defined by *w _{i}*. From this, the outcome is determined by the weighted average of each of the voter’s choices, mathematically calculated as setting R(v) equal to the signum function of the sum

*w*. This scenario represents perhaps the election system in the US, where each state is assigned a predetermined weight. To calculate the power of each voter with their weight, we define the total weighted vote of all voters excluding voter

_{i}v_{i}*i*to be V

_{-i }equal to the sum of

*w*for all k not equal to i, and the sum of the squares of all the weights to be W

_{i}v_{i}^{2}. Then, the power of voter

*i*is defined as: power

_{i}= P(|V

_{-i }| < w

_{i }) + 0.5*P(|V

_{-i }| = w

_{i}). With this, we must assume a random voting model, which implies that votes are decided by independent coin flips. This is of course not representative of an election process, as voters choose based on personal preferences, however, this is a simplified way to represent a population’s election. With this, any 2

^{n}combinations of votes are equally likely. This is of course not representative of an election process, as voters choose based on personal preferences, however, this is a simplified way to represent a population’s election. In addition, the random voting model implies a voting margin with mean 0 and standard deviation 0.001, however, in reality elections are usually won by a much larger margin.

Using this random voting model, we can derive that E(V_{-i }) = 0 and sd(V_{-i}) is the square root of W^{2} – w_{i}^{2}. From this, if we assume usual conditions for our elections (large population, fair counting, etc.), then we can assume a normal distribution for V_{-i }. Thus, we can approximate the power of voter *i *by using the cumulative normal distribution function:

We see that if w_{i}^{2} << W^{2}, voting power is linear, which is often the case with countries like the US, where the states have smaller weights, but have a large total. From this we can calculate the power of each state, or each voter, and hopefully see a linear relationship between weight and power.

In the case of two-stage voting, there are two steps to calculating voting power. First, we can calculate the voting power of each jurisdiction *j, *then the voting power of each *n _{j} *voters within each jurisdiction. Again assuming ordinary circumstances for an election, we can assume that the voting power of each jurisdiction is equal to

*w*under the random voting model. For the individual voter

_{j}*i*in jurisdiction

*j,*we set V

_{-i }equal to the sum of all

*n*-1 votes in said jurisdiction. Then, the probability that the voter

_{j}*i*has an effect within their jurisdiction is: P(V

_{−i}= 0) + 0.5*P(|V

_{−i}| = 1). With two-stage voting, the votes within a jurisdiction and outside a jurisdiction are independent, meaning we can approximate the overall power of voter

*i*to be w

_{j}/n

_{j}.

From these two different types of voting, we can see how different structures of voting change the effect of one individual vote. In terms of evaluating fairness of these structures, one needs to consider the circumstances of a country. This would include population spread, for example, countries with an uneven population spread would benefit more from a weighted voting system. On the other hand, countries with a splintered political climate with many candidates may benefit from two-stage voting. Overall, both theorems provide a mathematical explanation for the reason that almost every democratic country has a slightly different election system, and why these systems may all be considered valid. In addition, now one can see the logic behind two of the most common election systems in the Western world and that these systems do attempt to maximise the power of each voter. So, next time there is an election, avoid assuming that your vote is inconsequential and remember the power of your vote.