Voting Research - Voting Theory

Paul Cuff - Sanjeev Kulkarni - Mark Wang - John Sturm


At times a group of people, with differing and inconsistent opinions, must come together to make a decision between several options. This happens in business, it happens in social organizations, and it happens in politics.

When there are only two options, the rules of democracy are straightforward. The compromise goes the way of the majority. But when there are more than two options, there isn't just one obvious system for determining a democratic best choice. There are many different “voting system,” and a great deal of controversy surrounds the discussion of which voting system is most fair.

In our opinion, the Condorcet method of voting is the appropriate mechanism for democracy. Here we give two compelling reasons. First, it is the only voting system that is not affected by other candidates entering or leaving the race (unless they win, of course). Second, the Condorcet method does not reward exaggerated voting, making the voter's job simpler.

Those of you who are familiar with voting theory might be throwing your arms in the air and saying that our claims are not true. Indeed, these properties are the Holy Grail of voting theory, sought after rigorously by mathematicians and economists for at least 60 years. The voting method we're promoting is not new and is very intuitive to invent. What we present here is a practical argument for how the Condorcet method effectively satisfies these golden properties.

Does the Voting System really Matter?

Elections and campaigns will always be tumultuous, especially in politics and government, but the voting mechanism doesn't have to be. Typical voting systems used by society have flaws that pervade the election process. These flaws play a big part in the development of campaign strategy, primary elections, and party politics. The right voting system does not stress party relationships, allowing multiple candidates from a single party to run in the same race without hurting each other's chances, and rewards honest, straight-forward voting by the voters.

Let us be clear that this discussion is not about the Electoral College or things of that nature. That is for another discussion. We also do not address proportional representation or systems that elect more than one candidate.

First things first — let's decide how to select a single candidate out of several, treating every voter and candidate equally.

What is the Problem?

The most common and familiar voting system is called plurality voting. Each voter chooses one candidate to support. The candidate with the most supporters wins.

Why should we not be satisfied with this system?

Although most people have not thought explicitly about the mathematics of voting, most people have noticed the pitfalls of plurality voting. It is possible, and even common, for a “spoiler” to throw off the race. In US presidential elections, Ralph Nader may have cost Al Gore the race in 2000, and Ross Perot may have done the same thing to both Bob Dole in 1996 and George H. W. Bush in 1992. Furthermore, in the primary elections, this effect may be even more prevalent. The problem is that each voter has to make a judgment call and pick only one candidate to support. Similar candidates can end up splitting votes and losing to a less popular alternative.

Preferential Voting

One simple modification to voting is to allow voters to support more than one candidate by providing a list of who they support, in the order or their preference. Let's look at a few different voting systems that can use this information.


Imagine, for the sake of illustration, an unrealistic example where the entire population fits into three different likeminded groups. The table below shows the “preference profile” of this population when choosing between three candidates.

Group 1 (37% of pop.) Group 2 (30% of pop.) Group 3 (33% of pop.)
First Choice Ann Betty Carl
Second Choice Betty Carl Betty
Third Choice Carl Ann Ann

Plurality vote

The preferential voting ballot tells us much more than we need to know for plurality voting. In plurality voting, each voter can only put their support behind one candidate. If we assume that each voter will put their support behind their top choice, then Ann wins the race with 37% of the votes.

In reality, the voters may not decide to support their top choice. For example, voters in Group 2 may decide to abandon Betty and throw their support behind Carl. If enough support Carl then he will win. This highlights the two main concerns of this discussion. We see that voters can be rewarded for exaggerating their vote (Group 2 voting for Carl instead of Betty, for example). We also see that Carl would beat Ann if they were the only two candidates in the race, yet Ann might win the plurality vote because Betty is in the race.

Instant run-off

A better method, and one which has a wave of political support, is instant run-off. It is currently used in a number of local government and non-political elections. The way it works is by running several rounds of a plurality election, removing the loser in each round, and moving the voters who supported them to the next choice on their ballot. Voters don't actually need to cast a ballot more than once. They just supply their preference list once.

In the simple example in our table there are only two rounds. In the first round we assume voters support their first choice on their ballot. Since Betty only gets 30% of the support, she is eliminated first. In the second round, all of Betty's supporters move to their second choice. Usually they won't all have the same second choice, but in this example they do, and they all vote for Carl in the second round. Carl wins the instant run-off election by getting 63% of the vote in the last round.

Borda count

In the instant run-off election, Betty was unfortunate to be eliminated in the first round. We see that she has broad support — she is the first or second choice of every voter. The Borda count is a system that takes that into account. In this system, each position on the ballot is given a score. With only three candidates it would be like this: first choice gets 2 points; second choice gets 1 point; third choice gets 0 points.

The average score determines the winner. Ann has an average score of 0.74, Betty has an average score of 1.3, and Carl has an average score of 0.96. Betty wins.

Resolving the Controversy

We see that there are many different choices of voting system that can give different answers. Is there a correct system to use? Let us now try to design one.

Basic Assumptions

First, let us make some obvious assumptions:

  • The voting system treats each voter equally

  • The voting system treats each candidate equally

  • If there are only two candidates, the voting system chooses the majority choice.

Two Golden Properties

Now let's be a little more ambitious about the properties we want out of a voting system.

  1. Robust to Candidates: The voting result is not affected by candidates entering or leaving the race (unless they win).

  2. Robust to Voters: The voters are not rewarded for exaggerating their vote.

Either of these properties is crucial in a voting system. What we will show in the remainder of this page is that each of these properties alone points to a particular method of voting, the Condorcet method. But first a word on why these properties are so important.

The first property assures that we avoid the spoiler effect and splitting of votes. For many who have contemplated this topic, this satisfies a fundamental sense of fairness. It also has very practical consequences. Without this property, populations tend to apply after-market fixes to the system. They form primary elections. They attempt to persuade weaker candidates to leave the race. In short, populations have a natural motivation to fight against the spoiler effect.

The second property assures that voters can vote honestly on their ballots. No need to artificially move the closest rival of your first choice to the bottom of your list. No need to list your second choice first simply because your first choice has “no chance of winning.” No need to strategize or hear the latest polls before casting your vote.

Unfortunately, neither of these two properties can be satisfied in all circumstances, and that is the controversy!

Robust to Candidates (IIA)

In the research literature, what we are calling robust to candidates is referred to as independence of irrelevant alternatives, or IIA for short. Since that is a mouthful and sounds a bit technical, we simply call it robust.

A famous foundational result in voting theory is Arrow's Impossibility Theorem. The significant contribution of Arrow's analysis is the observation and proof that no voting system can be robust in all circumstances. We can construct examples where the preferences of the population are so bizarre that the winner of the election will always be affected by which candidates are in the race, no matter which system is used (assuming our basic assumptions).

The good news is that our survey results presented on this website, along with a great amount of other data, corroborate our hypothesis that these circumstances are very rare. We can design a voting system that is almost always robust.

The Condorcet Method

In order for a voting system to be robust to candidates, it must elect the same winner even if any of the non-winners were not in the election. This means, among other things, that the winner must beat every other individual candidate if they were the only two candidates in the race. And we already know, by assumption, how to decide between two candidates. We use a simple majority vote. Thus, a voting system is only robust if it picks a winner who would beat every other candidate in a head-to-head majority vote. This candidate is called the Condorcet winner.

Thus, the Condorcet winner can rightfully be called the robust winner or the IIA winner.

It should be quite obvious that there cannot be more than one Condorcet winner. What is less obvious is that there may not be any. It is possible that a cycle occurs among the top choices, where A beats B beats C beats A. When this happens, no voting system can be robust. This is precisely the situation that Arrow's Impossibility Theorem is identifying.

A Condorcet method is actually any voting system that selects the Condorcet winner when one exists. Different Condorcet methods differ in how they break ties (cycles).

Calculate the Condorcet winner

Fortunately, we don't actually need to hold an election between every pair of candidates to identify the Condorcet winner, as long as we use preferential voting ballots (each voter submits a ranked list) and make one very believable assumption. For each voter, if a candidate is removed from the race, we assume that the rest of the candidates would stay in the same order on the ballot. With this modest assumption, a head-to-head match between two candidates can be calculated by separating the voters according to who ranked which candidate higher, regardless of how high or how low they are on the list.

Consider the example of a preference profile found in the table earlier on this page. Notice that Betty beats Ann with 63% of the vote because Groups 2 and 3 vote for Betty. Also, Carl beats Ann by the same margin. Finally, Betty beats Carl with 67% of the population because Groups 1 and 2 vote for Betty. So, Betty is the Condorcet winner. The Borda count was the only other system that happened to get the right answer in this case, but it won't in general.

A Condorcet winner always exists

Well, not really “always.” We know that we can easily construct examples where there is no Condorcet winner. But we would like to have some sort of understanding of how likely those situations are. It is hard to approach this with probability theory because the results will be entirely dependent on the distribution used to model the population. Some research literature analyzes what happens when voters cast completely random votes. This doesn't seem to give us the type of understanding we're looking for.

The good news is that real data shows that a Condorcet winner exists except in very rare cases. Thus the Condorcet method is almost always robust.

A glance at the content of this website (using the menu at the top left) will confirm that a Condorcet winner exists when making important decisions, even though opinions may be diverse. In our four data sets, we see different demographics and different winners, but in all cases there is a Condorcet winner. In fact, even when we divide the data sets by political party or by date, giving many more chances to find a cycle that breaks the otherwise robust Condorcet method, we still always have a Condorcet winner. Even more incredible is that we almost always have a complete Condorcet order with no cycles among any candidates. This means we still would have had a Condorcet winner had we conducted this survey with any subset of the candidates. To top it off, if we combine all data from all four polls, even though the demographics and voting results disagree so dramatically, we still find a complete Condorcet order.

Find many other sources of voting data through links from We find that at least 95% of the data sets we investigated have Condorcet winners.

Some other methods claim to be robust

There are a handful of other voting methods (which use a different type of ballot — not a ranked list) that are purported to satisfy the independence-of-irrelevant-alternatives (IIA) property. Although there is some mathematical accuracy to these claims, they miss the intent of the IIA property.

Among these methods, range voting has found enthusiastic support. Range voting is very simple to explain. Each voter can give a score to each candidate within a range. The candidate with the highest average score wins. In fact, our survey looks a lot like a range voting ballot. We asked participants to score candidates between 0 and 100.

In range voting, if a non-winning candidate is removed from the system, and no scores are changed on the ballots, then the winner does not change.

Unfortunately, it is not plausible that scores would not be changed when a candidate is removed. In the least, voters tend to spread out their scores to use the entire range. If a voter's favorite candidate had not been part of the race, it's likely that they would have given another candidate the highest score allowable. If there are only two candidates, for example, it doesn't make sense for a voter to do anything other than give the highest possible score to the preferred candidate and the lowest score to the other candidate. Anything else would be equivalent to only casting a fraction of a vote.

In summary, it is not realistic to assume that voters will behave the way that they need to in order to make range voting a robust system (or other similar methods such as approval voting). In our survey data, we find situations where there exists a Condorcet winner but range voting selects someone else. See, for example, the Mercer County Poll.

On a technical note, these voting systems do not satisfy our third assumption, which is that elections between two candidates are determined by a majority vote.

Robust to Voters (Strategic voting)

The research literature addresses what is known as strategic voting or tactical voting, which we refer to here as exaggerated voting. You may not realize it, but there is a good chance you have cast a strategic vote instead of an honest vote at some point in your life. In plurality voting, what happens most often is that a voter abandons their favorite choice to cast their vote for a candidate that has a stronger chance of winning.

Plurality voting is not the only system that rewards exaggerated voting. In fact, the Gibbard-Satterthwaite Theorem asserts that all voting systems with our basic assumptions will reward exaggerated voting in some situations. When a voting system rewards this behavior, concerned voters have a right, perhaps even a duty, to use the system the way that is more likely to benefit them. Voters have to think strategically about how to vote.

However, for the Condorcet method, the opportunities for exaggerated voting to be effective are fundamentally different and less frequent.

Our research papers (in preparation) show two fundamental observations. First, for all voting systems that we inspected that are not Condorcet methods, in all voting situations (preference profiles) there is some portion of the population that has an incentive to cast an exaggerated vote. No rest for the weary voter. In contrast, the Condorcet method has many situations where no voters in the entire population have an incentive to exaggerate. In fact, any time that the vote ultimately identifies a Condorcet winner, no voter had an incentive to exaggerate their vote. This is not a coincidence. There is a geometric reason why the Condorcet method, based on pairwise comparisons, does not reward exaggerated voting.

The second observation is actually quite fascinating. Suppose that you do not accept our philosophical argument for using the Condorcet method, and you decide to use another voting system, such as the Borda count. Consider that an election process may not happen in an instant. As a campaign progresses, polls and other incomplete feedback help inform the voters of the best strategy for casting their vote. Under a very simple model that allows individuals in a population to dynamically adjust their strategy, and their vote, as they receive feedback about the rest of the population, a remarkable thing happens. The population unwittingly adjusts their votes until they elect the Condorcet winner anyway. This doesn't happen for all voting systems, but it does for the Borda count, and perhaps others.

Agree or Disagree

On this page we've attempted to lay out a case for using the Condorcet method in all democratic decisions. It's practical; it makes life easier for voters, who don't have to put strategy into their vote; and it satisfies a crucial fairness property that the public naturally struggles for — candidates who don't win do not affect the race.

Some other preferential voting systems have a great deal of momentum and advocacy. It's important for us to know how often the choice of voting system will make a difference in an election. From what we've seen, the plurality voting system elects the wrong candidate a significant portion of the time (>10%). Of the other systems we looked at, instant run-off elected the Condorcet winner most often. But feel free to take a look at the survey results on this website to form your own opinion.