Do You Have Apoorva’s Approval?

Gems in STEM: A Quick Study of Ballots

Apoorva Panidapu
Geek Culture


It all started in fourth grade when I finished my debut novel The Magic Inside, a 10,000 word masterpiece for NaNoWriMo (National Novel Writing Month) about three friends who learn magic together and unite to defeat an evil sorcerer…definitely not a Harry Potter rip off. As I proudly typed my name “Apoorva Panidapu” to make sure that it would forever be known that I wrote this tour de force, I was shocked when the author suddenly changed to “approval pandas.” I tried again. And again. And again. But to no avail. I accepted my fate: I would forever be known as Approval Pandas, and I would never be recognized for my future bestseller.

This outrage continued throughout my school career until I figured out how to stop autocorrect, and instead had to endure ugly red lines constantly underlining my name in every document. But the turning point came when I actually responded to the word “approval” in a group chat as if it were my name…I had to put my foot down. I did not approve this message — enough was enough. I emailed and pled my case to make “Apoorva” an official word…or else. Of course, they wrote back because I’m very compelling, but said their hands were tied unless I could win the approval vote from higher-ups (which was a really mean way to put it considering the situation). They didn’t specify how or where or anything else really, so hence began my study of voting and elections — which is what we’re talking about today! Along the way, we’ll figure out how to get justice.

​​There are two pillars of an election: the candidates and the voters. Voters elect a winner by casting a ballot which records their preference for the candidates. Ballot design has evolved from a small ball into a range of designs across different voting systems. Based on which ballot system is used, the outcome of the election might change. From one of the earliest types of ballots, blackball ballots, to approval ballots, there are a variety of ballot designs with different ideas of fairness and representing the voters’ best interests.

There are three general properties that are initially used to see if a voting system is fair: anonymity, neutrality, and monotonicity. For simplicity, we’ll talk about their definitions with respect to a two-candidate election.

A voting system is anonymous if whenever any two voters exchange ballots, the result of the election remains unchanged. Basically, anonymity means that all voters are equal in an election.

A voting system is neutral if switching all the votes between the two candidates switches the outcome of the election accordingly. So, neutrality is just a property that says that the candidates are equal in an election.

Finally, a voting system is monotonic if a winner of an election cannot become a loser by gaining more votes. Similarly, a loser cannot become a winner by losing votes, all else being equal.

Now, these properties seem fairly obvious, but let’s see them in action!

Not really the typical ballot that first comes to mind, blackballing is a form of voting where the ballots are marbles, in which a black marble (or “ball”) is counted as a vote against a candidate and a white marble is a vote in support of a candidate. This form of voting is common when you want an election to be almost unanimous, meaning that only a couple votes in opposition, i.e. black marbles, should cost a candidate their win.

Blackballing has been around since the 17th century and is mostly used in gentlemen’s clubs, Masonic lodges, or similar organizations to appoint a new member. In this voting process, each vote is anonymous since no one can see which color ball they put into the ballot box. The outcome of the election is also revealed to all at the end after all votes have been cast, so it doesn’t reveal which balls belong to whom.

Blackballing is unlike the traditional “majority wins,” since just a few voters can change the outcome with black balls, and even if a majority of voters are in favor of a candidate, they may still be rejected. In this way, black ball votes (votes in opposition) hold more weight than votes in support. To see this, let’s say that the two “candidates” of this election that people are voting for are 1: “Appointing a new member” or 2: “Rejecting the member.” Then, this voting system is anonymous since if any two voters traded votes, the outcome of the election would remain the same because the number of white balls and black balls would remain the same, and that count is how the election is decided.

However, this voting system is not neutral because if all the white balls switched to black balls and all the black balls switched to white balls, the outcome of the election would not necessarily change accordingly. To see this, suppose there are two white balls with the rest of the votes as black balls, meaning the outcome is rejecting the potential member. Switching all the votes means there are now two black balls with the rest as white balls, but the number of black balls is enough opposition (in a typical blackball system) for the outcome to still be rejecting the potential member. Finally, the system is definitely monotonic because more white ball votes wouldn’t cause a member to be rejected and less black balls votes wouldn’t cause a member’s rejection.

Though this two-candidate system doesn’t fulfill all the fairness properties of anonymity, neutrality, and monotonicity­, it makes sense that the fraternal club sorts have adopted blackball ballots and voting. When they’re appointing a new member of such a brotherhood/community, it is probably preferable that everyone is glad for them to join (instead of just a majority), otherwise there might be friction amongst the members, which you don’t want for a social club.

Now, it’s finally time for the point of this article!

Another type of ballot is an approval ballot, where a voter can “approve of” multiple candidates, i.e., they can vote for more than one candidate. Approval voting was proposed by economist Robert J. Weber as a fairer voting system for elections with more than two candidates. Approval voting is in use across various organizations and societies, such as the Mathematical Association of America, the American Mathematical Society, the Society for Social Choice and Welfare, and more.

An argument for approval voting is that it simplifies a voter’s job–there may be difficulty ranking a large number of candidates, so it’s much easier to either “approve” or “disapprove” of each candidate. But, this is also a drawback since it provides less data on preferences between all the candidates a voter approves of and the candidates a voter disapproves of–even if a candidate slightly likes one candidate and completely loves another, they are still considered equal in their approval ballot.

An approval ballot can be thought of as a preference ballot with only two ranks, where one or more candidates tie for first place and all the remaining candidates are in a tie for last place. The interesting thing about this type of system is that it can be manipulated…dun, dun, DUN.

Approval ballots allow for something called bullet voting, which is when a voter approves of both candidate A and B, but only votes for Candidate A. They do this because voting for Candidate B could cause Candidate A to lose, and they prefer A slightly over B, but this means they are not voting according to their sincere preferences. Approval voting also allows for a type of voting manipulation called compromising, where a voter might vote for an extra candidate that they don’t actually like in order to stop a different candidate, who they consider even less desirable, from winning.

Okay, now that we know what approval voting actually is, it’s time to win the approval vote to make Apoorva a word so I don’t have to see an ugly red line under my name ever again! But actually, now that I think about it, that email was kind of suspicious…they didn’t even really use the term “approval vote” correctly. Maybe are not actually the autocorrect creators. :(

Fear not, I have a better idea.

Sign my petition at to make Apoorva an official word! All you have to do is write, “My name is ____, and I apoorva this message!” If the site cannot be reached, either it is overwhelmed by millions of people who are joining the global movement or the actual autocorrect creators have gotten to me and you must promise to carry on the mission without me.

Now, I have one last question for you:

Are you running for office? Because I want to candiDATE you. ❤

Until next time! If you found this interesting, make sure to follow me to stay updated on the next ones.

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As a reminder: this column, Gems in STEM, is a place to learn about various STEM topics that I find exciting, and that I hope will excite you too! It will always be written to be fairly accessible, so you don’t have to worry about not having background knowledge. However, it does occasionally get more advanced towards the end. Thanks for reading!



Apoorva Panidapu
Geek Culture

Math, CS, & PoliSci @ Stanford. Advocate for youth & gender minorities in STEAM. Winner of Strogatz Prize for Math Communication & Davidson Fellows Laureate.