How Not To Waste Your Vote: A Mathematical Analysis

A look at mathematical probabilities suggests you should vote third party if you want your vote to matter.

During this especially contested election, a lot of people are talking about people “wasting” or “throwing away” votes. However, many people who say this do not have a complete grasp of the full mathematical picture – or worse, they are only mentioning the part that supports their position. First let’s define what a “wasted” vote is.

Mathematical Definition of Wasted Votes

A wasted vote is a vote that provides no determination or effect on the final outcome of the election. According to Wikipedia: “Wasted votes are votes cast for losing candidates or votes cast for winning candidates in excess of the number required for victory. For example, in the UK general election of 2005, 52% of votes were cast for losing candidates and 18% were excess votes – a total of 70% wasted votes.”

There are two kinds of wasted votes that mathematically have no effect on the final election:

  1. Votes cast for candidates who did not win
  2. Excess votes cast for winning candidates

Clearly, neither of these kinds of votes statistically affect the election. However, many arguments only mention the first type without mentioning the second. Mathematically and logically, both categories are ineffectual votes.

First Past the Post

The value of your vote is what you give it. Should you spend it on a candidate you don’t believe in?The United States, along with several other nations, uses the First Past the Post (FPTP) or “winner take all” election. This method is defined as “the candidate who receives more votes than any other candidate wins.”

This is one of the reasons that many people mention wasted votes – our system creates that result. Sociologically speaking, the FPTP system tends to favor a two-party system. The French sociologist Maurice Duverger created “Duverger’s Law” which says just that.

The Electoral College

For U.S. Presidential elections, a state-by-state system is used called the Electoral College. Each state gets a proportional amount of electoral votes which are then used to find a majority for president. Interestingly, what happens in each separate state is a smaller FPTP election, followed by a counting of electoral votes.

The Electoral College is slightly different from a pure FPTP system because it requires an actual number threshold (currently 270 electoral votes) for a candidate to win instead of a simple majority of the votes. 

We can sum things up as follows:

  1. States hold “winner take all” FPTP elections for electoral votes
  2. Electoral votes are counted
  3. The winner must have 270 electoral votes
  4. If there is no candidate that reaches it, the House of Representatives chooses the president

These distinctions are important, because they can change the math and the concept of the “wasted” vote phenomenon.

Wasted Votes in Presidential Elections

The general concept that is proposed by many is that you must always vote for a Republican or a Democrat because you must stop the worst candidate from winning. In a sense, you are voting a negative vote – against someone – rather than for a candidate of your choice. However, this actually depends on the scenario of the vote. Let’s look at some examples.

Bush vs. Gore: 2000

People voting out of fear of the worst candidate is a self-perpetuating cycle. Let’s examine a common example used in this discussion.

Following the extremely close 2000 U.S. presidential election, some supporters of Democratic candidate Al Gore believe that one reason he lost the election to Republican George W. Bush is because a portion of the electorate (2.7%) voted for Ralph Nader of the Green Party, and exit polls indicated that more of these voters would have preferred Gore (45%) to Bush (27%), with the rest not voting in Nader’s absence.

The argument for this case is even more pronounced because the election was ultimately decided on the basis of the election results in Florida where Bush prevailed over Gore by a margin of only 537 votes (0.009%), which was far exceeded by the number of votes, 97,488 (0.293%), that Nader received. (Wikipedia)

At first, this may look like a clear example of the need to vote for a major party. However, let’s break this situation down mathematically. In every single state election, Bush or Gore won. There were millions of mathematically wasted votes in this election of both types.

In California, Gore won by 1,293,774 votes. Mathematically speaking, there were over one million wasted votes for Gore in this state alone. None of these excess votes could have helped Gore, since he had already mathematically won the state. The California votes didn’t matter in Florida. In fact, the votes in Florida have much more relevance than any other state. 

Conclusions: Sometimes a vote for a major party winner is wasted anyway. Sometimes everything will come down to one state. However, there is no way to predict in advance which votes will be this important. If the parties knew that Florida would have been the deal breaker, then they would have acted differently. However, we simply don’t know the future well enough to predict that.

We do know that battleground states are generally more important than “safe” states for each candidate, but it is hard to know exactly which state might matter. (There are plenty of scenarios you can research online about possibly electoral outcomes, I encourage you to do so.) This leads us into our next example.

Clinton vs. Trump 2016

Let’s do some math about the state of California and our current presidential election. The average RCP poll has Hillary Clinton ahead by 22.2 percent. The registered voters in California add up to 17.7 million. Not all of them will vote, but we can use the 2012 presidential election as a predictor, where 13.2 million people voted.

Out of those 13.2 million, according to current predictions, 52.6% will vote for Clinton. However, Clinton only needs about 31% to beat Trump. The other 21% of excess votes for Clinton will be wasted. This means that approximately 3 million votes for Clinton in California will be wasted. Now, this is only a mathematical model, but we have several reasons to believe in it.

  1. California has a history of being a heavily Democratic state
  2. Polls usually swing within a single digit margin of error
  3. 21% is quite a large margin of leeway

Even if the polling changes significantly, we are still looking at millions of wasted Clinton votes in California.

Now let’s throw Jill Stein into the math. As part of the Green Party, she is to the left politically of Hillary, so we will assume that votes for her will be taken from Clinton’s pool. (Though this isn’t always a true assumption, as we will see later.) Right now she is polling at around 4%, but we could even give her 5%. If you take away 5% from Hillary’s margin of 22.2%, that leaves a huge margin of 17.2%: still millions of votes. The takeaway from this: you can safely vote for Jill Stein in California without fear of changing the state election results. Therefore, it will not affect the national vote either.

Since we have the Electoral College, your votes will have no influence beyond the state to change other vote counts. Those who prefer Jill Stein can with a clear conscience vote for her, since it will make no difference mathematically. Later we will look at the ethics of voting as it relates to this math.

Mathematical Importance of a Single Vote

There are a few theories on voting power calculations; we will look at two of them here. John F. Banzhaf III created a probabilistic system for determining individual voting power in a block voting system, such as the Electoral College. According to his calculations, because of differences in each state, it gives different voters different amounts of “voting power.”

A computer science researcher at UNC ran the Banzhaf power numbers for the 1990 U.S. Presidential election and determined that the state of California had the voters with the highest power index: 3.3. This index is measured as a multiple of the weakest voting state, which was Montana (1.0 voting power).

A newer method of measuring voting power was created by a research team from Columbia University using a more empirical (based on existing data) and less randomized model. They concluded that the smaller states had more mathematical voting power due to the fact that they received 2 votes minimum as a starting point. This model tends to generate smaller multipliers for voting power but more accurately matches empirical data from past elections.

Using these power ratings as a guide, we can estimate an estimated maximum voting power for each vote. We will be making some assumptions for this calculation.

  1. The minimum voting power multiplier is 1
  2. The highest multiplier from both models will be used as a maximum

Starting numbers

In the United States there are currently 218,959,000 eligible voters with 146,311,000 actual registered voters. In the 2012 Presidential election, 126,144,000 people actually voted. This is our voting pool.

Each vote, legally speaking, has the same weight. So if we start from that assumption, taking into account a probable amount of voters (126 million), the power of your vote is:

1
_____

126 million

This is: 0.0000000079 or 0.00000079%. That is the weight of your vote mathematically. Now we can multiply it by the highest power index to show the highest potential of your vote. Our California historical data from 1990 shows a 3.3 index, but to be conservative we will raise it to 4. So now the power is: 0.00000317%

Using probabilistic equations and analysis, this is the result. This is how powerful your vote is in the U.S. Presidential election is if you end up in the most heavily weighted state. 

Addressing Weighted Vote Fallacies

As we have seen, many people argue that we should not “waste” votes, yet many millions of votes for the winner are wasted every year. It is difficult to predict whether a vote will end up in either wasted category. We’ve also seen past and possible scenarios where voting third party or major party can have no influence on the final election.

Fallacy 1: Treating Single Voters as One Block

A false assumption that people make about voting is treating a single vote as a block. For instance, let’s use our current election again as an example.

Someone insists that if you do not vote for Hillary, then you are helping Trump to be elected. (The reverse of this can also apply here.) You claim that you wish to vote for Gary Johnson, the Libertarian candidate. You’re then told that the current national poll with all parties shows that Johnson is polling at 7%, which is less than the difference between Clinton (39%) and Trump (40%). Therefore, you must vote for Clinton to make up that difference.

There are several problems with this proposal. It does not take each state into consideration. It assumes all Gary Johnson supporters have Clinton as their second choice. And it treats your single vote as the entire 7%.

As we have seen, the current picture in California shows that Clinton has a huge margin. If this voter lived in California, a vote for Gary Johnson would not help Trump and also would not hurt Hillary, even if the entire 7% voted for Johnson. Anyone who says it is your duty to vote negative in this scenario does not know the math of this state.

This also assumes that all Johnson votes would choose Hillary as the second choice, but given that Libertarians take some platform elements from both the Left and the Right, this assumption would be highly unlikely. The same would go for Trump.

When people look at the 7% and tell you that you must vote a certain way, it is assuming you will somehow influence the entire 7%. However, we have seen that you are just one voter, and that your voting power is a very tiny number by itself. You cannot be entirely responsible for a candidate winning or losing with your single vote. In theory, it’s mathematically possible for one vote to decide an election, but given there are an exponential number of possible scenarios with millions of voters (imagine raising a few million to an exponent), it’s astronomically unlikely, especially if you live in a non-battleground state.

It’s also astronomically unlikely that all 7% (8,820,000 people) would vote for who they polled for. Even if you gave each voter a 99% chance of voting for who they polled for, the chance that all of them would vote the way they polled is (0.99) to the power of 8,820,000, which is less than 0.000000000000000000000000000000000000000000000000001%

Individuals are not entire blocks of voters, and it’s problematic to treat them as such.

Fallacy 2: Third Party Votes Have No Value

If enough people vote their conscience and vote for what they believe in, things can change.On the surface, this might appear to be true. A third party candidate for President has never won an election. We also have Duverger’s law that states our FPTP favors two party systems. However, it is mathematically possible for a third party to win, and there are also other measurable gains for voting for a third party.

Part of this fallacy is the “winner take all” perspective. In other words, if you don’t win the presidency, you’ve wasted your time.

However, there are many benefits of voting third party, even for president. It makes a political statement to the majority parties. It helps local politicians of that party in elections. It can help change platforms to include third-party elements. And it provides recognition for the party among voters as a viable alternative.

Third party candidates can and have won local and state elections in the past. This is a fact.

In 1968, George Wallace ran as a third party option for President. He received nine million votes and 45 electoral votes. Though he did not expect to win the popular vote, one of his aims was to force the House of Representatives to choose the President by denying either candidate the 270 electoral votes needed to win – and he nearly succeeded. Since our system is not a true First Past the Post, but a hybrid, this kind of situation is possible. In fact, calculations have been done showing that Gary Johnson could in fact force that situation this year. It is very unlikely, but it is possible.

Regardless of his loss, the impact of the Wallace campaign was substantial. He was able to affect the dialogue and events of that election significantly. (This is meant in no way as an endorsement of George Wallace’s political positions.) If his supporters had mostly voted for a majority party, his impact would have been less significant.

In most scenarios given by the “wasted” vote crowd, all of the votes that are considered are ones from the current voting electorate. Yet we have seen from figures previously mentioned that over 50 million eligible voters are not registered. Even among registered voters, almost 20 million didn’t vote in the last election. These potential votes are never placed into the scenario.

The simple truth is, there are millions of uninterested voters out there, yet candidates are not inspiring them to vote. If candidate X or Y were truly worthy of votes, would not some of these voters decide to register? And another question, would it be easier to get a third party voter to choose a majority candidate, or a non-voter? These are not mathematical questions, but they are logical. The fact is, with this many votes at stake, if these non-voters could be encouraged to register, they would undoubtedly change the election as they make up one-third of total eligible voters.

Ethics and Math

It has been demonstrated that the potential individual power of a vote is mathematically very small. It also has been shown that wasted votes can be cast for the winner of an election as well as the losers, as well as demonstrating that it is sometimes hard to predict exactly which vote will be wasted. Given this information, where do we derive the value of a vote?

It’s hard to get it purely from the math or practicality. In fact, it would seem our single vote is of very little import at all. Therefore, we must find meaning and value for our votes outside of the math.

Certainly, the Founders never envisioned an endless cycle of US citizens voting for the "lesser of two evils."Certainly, the Founders never envisioned an endless cycle of United States citizens voting for the “lesser of two evils,” as the argument is often presented. The idea was for free and open elections where the people’s voice would be heard. It was simple: the candidate who best represented your interests earned your vote.

Your vote is, therefore, an expression of yourself and your beliefs. Your vote has power as a statement. People voting out of fear of the worst candidate is a self-perpetuating cycle. If no one ever has the courage to vote outside of the two main parties, it will never be broken. However, if enough people vote and it shows in the total election count, it will give cause for us to reconsider and embolden even more to vote outside of the two parties.

Yes, our current electoral system has some serious mathematical flaws. It simply does not encourage people to vote for their conscience – but we have seen that things are not as bad as we would be led to believe by some. The true value of a vote is in the people.

The Value of Your Vote

The value of your vote is what you give it. Should you spend it on a candidate you don’t believe in? Should it be an exercise in fear? It’s up to you. It is my hope that these mathematical calculations will bring you freedom from the idea that only majority party votes matter. A vote is a statement, a vote is personal, a vote is an expression of your citizenship in this country. If enough people vote their conscience and vote for what they believe in, things can change.

If you are already a staunch supporter of a major party, then you should vote that way. This paper is not against the major parties at all – but rather against the concept that votes somehow “belong” to only Democrats or Republicans. Votes belong to the voter. There has never been a more important time to vote your conscience.

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