Radical By Choice

Voting Power: Packing A Punch

Photo by Heather Mount on Unsplash

Radicals pop up in unusual places. In a previous post, we saw how they emerge to maximize Utility. In this post, let’s explore Voting Power.

Voting Power

A cornerstone of a thriving democracy is the enfranchisement of its body politic. Qualified elected-representatives act on its behalf. In flexing voting power, citizens affect lasting changes to society. Its core idea is rooted in Game Theory and entails radicals!

The vote is the most powerful nonviolent tool we have — John Lewis

Power Vs Right

Voting power differs from voting right. The former measures the punch a single vote packs, whereas the latter determines if a vote counts in deciding the matter at hand. They are related insofar as representation, fairness, in/out-group dynamics, and other socioeconomic factors affecting both. While voting-right justifiably features in our zeitgeist, little is understood of voting power unless one is in the throes of jurisprudence, political science, or has a passing interest.

To Choose

A binary choice (yea or nay) is a poor substitute for the real-world-choices most of us face. Most issues provoke a continuum of views based on one’s intuitions, morality, and worldview. All told, an election ends up picking a winner and a loser. It approximately tracks majority opinion, however slim. And its outcome hinges upon a decisive block of votes that holds sway. These are the so-called swing votes in an otherwise stable electorate. A tie-breaking mechanism is required to break deadlocks.

Swing Vote

Photo by Markus Spiske on Unsplash

Defining voting power involves translating an abstract idea into a concrete mathematical model. Let us pose a question that could hint at one.

What is the chance a single vote influences the outcome?

Combinatorics Answer: A subset of all combinations wherein a single vote swings the outcome.

This fraction is a measure of its influence. Lionel Penrose developed it in 1946. John Banzhaf independently obtained an identical result in 1968. The combinatorics approach relies only on simple counting and has several underlying assumptions that we will explore shortly.

One can argue a modern data-based approach like stochastic modeling is more suitable. No doubt, it captures the election-complexity more realistically. But, it also requires the knowledge of probability distributions and empirical data to model accurately.

For a vote to break a tie, the remainder must cause it!

A lone swing vote holds the most influence when it tips the balance breaking the impasse! Recently, Ms. Harris broke crucial ties by flexing this power in an otherwise deadlocked U.S Senate. Let’s illustrate the combinatorics approach with an example.

Three Convicts


Pete: Wait a minute. Who elected you leader of this outfit?

Everett: Well Pete, I figured it should be the one with the capacity for abstract thought. But if that ain’t the consensus view, then hell, let’s put it to a vote.

Pete: Suits me. I’m voting for yours truly.

Everett: Well I’m voting for yours truly too.

Delmar: Okay… I’m with you fellas.

O Brother, Where Art Thou, 2000

Everett (E), Pete (P), and Delmar (D) are scheming to get to the buried treasure. Everett and Pete, both harboring an independent streak, have their heels dug in. Delmar is intent on avoiding conflict. But first, they must elect a leader before embarking on their plan.

The size (N) of the outfit is three. In general, the population-size could be any N > 3. One can model the real-world-complexity by suitably altering the following prototypical assumptions.

  1. Voters are rational and act independently.
  2. They have equal voting rights.
  3. All votes have equal weight (w = 1)
  4. Ties get settled in a mutually agreed-upon way (e.g., coin-flip).

The chance a lone-swing-vote tips the outcome is denoted by the Greek letter ψ (psi). The situation applies equally to any swing-voter, but we just happened to pick Delmar.

Q: How Can We Quantify Voting Power?

A: We need the sample-space (denominator, v) and a subset where a swing-vote gets cast (numerator, u). Their ratio (ψ = u/v) is the chance a vote swings the outcome — or the swing-vote-probability.

The effective-voting-power is swing-vote-probability (ψ) scaled by the population size (N).

P = ψ · N

Q1: In how many different ways can three votes get cast? A: 8

Delmar has two voting choices. He may vote either yea (✔︎) or nay (✘). For each of his ways, Pete can independently vote in one of two ways for a total of 2·2 = 2² = 4. And for each of their four ways, Everett can independently vote in two ways for a total of 2·2·2 = 2³ = 8 (columns #, D, E, P in the table).

Q2: In how many cases will Delmar’s swing-vote affect the outcome? A: 4

Fig 1: When Delmar’s vote flips, it affects 4/8 = 50% of the cases (shown in green and red). Image by author.

Delmar’s flipped-vote is shown in the column labeled !D. He holds sway in a subset of four (cases #2, #3 favorably, and cases #6, #7 unfavorably, as shown in the column ☯). In each of these cases, it’s noteworthy that Everett’s and Pete’s votes remain equally split. Delmar’s power lies in his ability to tip the scale either way by casting his critical vote.


Our intuition may lead us to conclude that any single voter has absolute control (ψ =100%) of their vote alone (N=1). If so, their voting power is:

Pᵢ = 100% · 1 = 100% = 1.0

Before trusting our naive intuitions, a closer inspection reveals something else. Delmar holds sway in (ψ = 4/8 = ) 50% of cases. The same applies to Pete or Everett. The outfit is a bloc (or coalition) of three votes (N=3) that acts as a single unit. Delmar’s effective voting power is:

Pᵦ = 50% · 3 = 150% = 1.5 > Pᵢ

Let’s pause for a moment and let that sink in. This is not intuitive to say the least! How can an individual’s power exceed 100%?

The answer lies in unit-vote or a bloc-vote. An individual’s power within a bloc (Pᵦ) seemingly (and actually) exceeds that of his acting in self-interest alone (Pᵢ). As the saying goes — sticks in a bundle are unbreakable!

Q: Does the voting power depend on the bloc-size? A: Yes!

The voting power of a bloc is directly proportional to the square root of the population.

To see why that is true, let’s look at the general case of N voters. Although this generalization works for both odd and even N, we choose odd-N for the sake of convenience.

N=2k+1 (k = 1, 2, …) (N ≥ 3)

The total number of combinations (v) for the general case is:

Equation 1: All possible vote combinations, denominator (v)

A swing vote tips the balance. Of the remaining 2k voters, they remain 50:50 split — half of them (i.e., k=(N-1)/2 ) end up voting yea (✔︎) and the remaining half (k) vote nay (✘). How many such equally-split combinations exist? 2k-choose-k, with the formula:

Equation 2: Swing vote combinations, numerator (u)

An approximation (called Stirling’s Formula) can simplify the expression for ψ to obtain the voting power when the population size (N) grows.

Equation.3: Stirling’s Formula for k!

We have the ingredients necessary to discover the relationship between voting power and the total population. The two answers, again, are not very intuitive but are both couched in radicals!

  • Individual’s power (ψ) proportionally diminishes as the square root of the bloc-size (ψ ∝ 1/√N).
  • Voting power (P) grows as the square root its bloc-size (P ∝ √N).
Equations 4, 5: Voting power of (an individual) bloc is (inversely) proportional to square-root of its population

So What?

Elections have consequences. Voting power underlies the ability of an individual to affect society at large. Voting blocs hold enormous sway compared to individuals fending for themselves — like out in some wild west. The larger the coalition-size, the greater its power. All else being equal, elections come down to the swing states that end up picking winners. The Electoral College is another prime example of the voting power in action. And radicals emerged in defining voting power!


  1. The mathematics and statistics of voting power, Andrew Gelman, Jonathan N. Katz and Francis Tuerlinckx, Statistical Science, 2002
  2. One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College, John Banzhaff III, Law Review, 1968
  3. Alpha Proportionality and Penrose Square Root Law, Piotr Dniestrzański, European Proceedings of Social and Behavioral Sciences, 2016