By Jonny Lupsha, News Writer

The long wait for the 2020 US presidential election results isn’t the first of its kind, BuzzFeed reported. A photo gallery with pictures as old as 1940 shows number crunchers tallying votes, crowds gathered in Times Square, and more. Vote tallying isn’t always simple.

Election Night was Tuesday, but it took until late Saturday morning for most news outlets to even begin to project former Vice President Joseph R. Biden, Jr., as the winner. Georgia, which is heading for a recount due to its close race, has until Tuesday, November 17, to certify its results. However, in terms of anxiously awaiting results, this isn’t America’s first rodeo.

BuzzFeed’s gallery shows photos from over a dozen elections with candidates, private citizens, pollsters, and others. They pore over voting tallies, nervously watch televisions, gather in crowds around a big screen in Times Square. More than anything, it reminds us of the perceived stakes of Election Night and the tense days that follow.

The democratic process extrapolates meaning from overwhelming amounts of data. Part of the reason tensions run so high during this period of waiting is because it’s impossible to make everyone happy.

Arrow’s Impossibility Theorem

The quickest way to realize the hurdles a voting system must jump is by defining what that voting system is from a functional perspective.

“A voting system is a method by which you take the preferences of all the people in the voting population and, from them, assemble a winner or an ordered list of the societal preferences,” said Dr. Michael Starbird, Professor of Mathematics and University Distinguished Professor at The University of Texas at Austin. This sounds simple enough, so what makes it so tense? Arrow’s impossibility theorem addresses this issue.

“What Arrow’s impossibility theorem says is that there is no system for summarizing the opinions of people that satisfies all three of these conditions: “Go along with Consensus,” “Better Is Better,” and “Irrelevant Is Irrelevant,” Dr. Starbird said.

The first desirable condition, Go along with Consensus, is self-explanatory. If one candidate gets more votes than the other, the consensus is that they should be elected, and the system elects them. However, this becomes a problem when we use a sequential voting system. For example, if Candidate A wins one bracket against Candidate B, then B is discarded. If Candidate C is then put up against A and Candidate A loses to C, then Candidate C advances and A is discarded.

“We pair C to D, and look what happens,” Dr. Starbird said. “D is better than C here; C is better than D here; but D is better than C here. So, two of the people think that D is better than C and, therefore, D beats C. Well, there’s only one slight problem with this, and that is everyone likes B better than D.”

Now, due to the pair-based voting system, the people are stuck with a candidate who was never compared to several other candidates, nor are they everyone’s first choice.

Which Is Irrelevant and Which Is Better?

The “Better Is Better” desirable property of a voting system makes sense theoretically as well. “Better is better means, if you do better, if more people want you and vote for you, prefer you in a higher position, then that should help you in the outcome, not hurt you,” Dr. Starbird said. “But in the case of run-off elections, [they] fail this property.”

Pretend that in the first round of voting there are two candidates each in two major political parties. In each party, one candidate does better than the other and earns that party’s nomination. One candidate is Dave, in which the D is also for Democrat; the other candidate is Rick, where the R is also for Republican. Their performances and appeal to voters cause them to win out against their opponents within their respective parties.

But even if Rick or Dave perform the best overall, they may be running in an area in which the voters lean politically to their opponent’s side anyway, thus costing them the election. Therefore, the candidate who performed second-best wins and the candidate who performed the best loses.

The simplest way to explain the “Irrelevant Is Irrelevant” quality is if you have an election and candidates come in in first, second, and third place. If the second-place candidate is then found to have been illegible to run at all, they’re discarded and the third-place candidate becomes the second-place candidate even though they earned the least amount of votes. This may not matter when deciding a president, but if we imagine that the vice presidency were awarded to the second-place candidate, it changes things.

There’s only one way to avoid these problems in any voting system, and it’s pretty ironic.

“There is only one system that has all three of these [desirable qualities], and that is a dictatorship,” Dr. Starbird said. “If you just say there is one voter in the country, and whatever it is that that voter says goes, [then] that does, in fact, satisfy all three of these conditions.”

This article was proofread and copyedited by Angela Shoemaker, Proofreader and Copy Editor for The Great Courses Daily.

Dr. Michael Starbird contributed to this article. Dr. Starbird is Professor of Mathematics and University Distinguished Teaching Professor at The University of Texas at Austin, where he has been teaching since 1974. He received his BA from Pomona College in 1970 and his PhD in Mathematics from the University of Wisconsin–Madison in 1974.