**Voting Systems** Andrew Glassner @AndrewGlassner The Imaginary Institute (This is supplementary material from my upcoming machine learning book) What's This All About? === >" TK Reviewer's note: Underlined passages > are responses to comments in the first version > I shared. Note that this is now a supplement > to my book, rather than a part of it." I wrote this document as part of a chapter in a new book I'm writing on deep learning, and machine learning in general. In this field, we sometimes create collections of algorithms and let them vote on a final answer to a problem. So I wrote this discussion of different ways to vote and tally the results. On reflection, it's really more of a side subject than core material. So I've removed it from the book, taken out the material specific to machine learning, and am making it available now as a supplement. Anyone can be wrong from time to time. One way we work around that problem, both in programming and in society, is to allow multiple, independent voters to present their opinions on ballots. Then we combine the ballots in some way, hoping to produce a winner that best reflects the desires of the voters. Perhaps the most popular way of deciding elections today is to enable each voter to name one candidate, and then say that the winner is the the candidate with the largest number of votes. This is called **plurality voting**. It's simple, fast, and easy to understand and program. But it also suffers from some serious problems that can produce unfair results, and push voters into voting for candidates other than their first choice. In the discussion below we'll look at what it means for an election to be "fair," and then we'll look at a variety of voting systems, including plurality voting, that attempt to produce satisfying results. This document is all about voting, and not at all about machine learning. If you'd like to know more about my book on deep learning and machine learning, see Section [End of Supplementary Material]. Voting ==== Many people are used to elections where voters each get to vote for one candidate, and the candidate with the most votes wins. But this is only one of many different ways to run an election, and despite its popularity, it's widely considered to be among the worst of all the voting systems in popular use [#FairVote16]. This approach, called "plurality voting", suffers from a wide range of commonly occurring problems in which the declared winner can easily violate some of our common ideas of fairness. We'll look at this voting method, and some better alternatives, more closely below. To evaluate a voting method, there are two key issues to look at: the **ballot** which identifies someone's choices, and a **voting method** (or **tallying method**) for determining a result. In the following, we will assume that each ballot contains a list of candidates. In most methods, voters can choose more than one candidate if they want to, and present them in a list. The order of the candidates in this list will matter in some tallying methods, and not in others. There are many ways to organize the broad spectrum of voting methods. Our first dividing line will be whether a system produces one winner, or a set of them. Voting systems that produce only one winner are called **single-winner methods**. Our discussion here will be focused exclusively on single-winner methods. Single-winner methods can be broken down into two broad categories. The first category is **preferential voting methods**, in which we list candidates on our ballot in order of preference. Note that if we're only allowed to vote for one candidate, this is still a preferential system, in which only our most preferred candidate is listed. The second category is **non-preferential voting methods**, where we can still list multiple candidates, but the order in which we choose them doesn't make any difference. We'll cover voting systems from both categories. We'll make frequent use of the concepts of a **majority** (more than half the ballots cast), and a **plurality** (the largest number of ballots cast for a single candidate). For example, if there are 3 candidates and 100 voters, the votes might come in as 33-33-34. In this case, the third candidate has a plurality, but nobody has a majority. If the results are 60-30-10, then the first candidate has both a majority and plurality. The word "majority" has also come to sometimes mean merely the largest group of votes [#Merriam-Webster16]. That is, "majority" sometimes is used to mean the same thing as "plurality". This is unfortunate, since it makes the word "majority" ambiguous. To add to the confusion, some authors use "most" to mean a majority, while others use it to mean a plurality. These ambiguities make it hard to talk about these concepts simply, since we cannot tell what is intended by phrases like "the candidate with the majority wins," or "the candidate with the most votes wins." One way to test what an author means by "majority" is to ask if the vote they're describing must always produce a winner (or multiple winners if there's a tie). If there is always a winner, then they're necessarily talking about a plurality. Majority voting in the sense of "more than half the votes" *produces no winner* unless one candidate gets more than half the votes cast. If that condition isn't met, then that election simply doesn't have a winner. To eliminate this ambiguity and keep our discussions concise, in this book "majority" means *more than half*, and "plurality" means *the largest share.* When we speak of "majority voting," then only if one candidate has more than half the votes is there a winner. When we speak of "plurality voting," there is always a winner, and it is the candidate who has more votes than any other. In both situations, ties can produce multiple winners. Note that a candidate with a majority will necessarily also have a plurality, but not vice-versa. Fairness ==== Central to any voting system is the concept of *fairness*. In some objective sense, we want the winner to be the candidate that can be credibly considered the most popular or desired, or the most acceptable the most number of voters. To choose a "good" voting system we have to be able to first describe what kind of results we want from it. This is not a question we can answer logically, because it depends on human judgements of fairness. For instance, if we have an election where 70% of the people are satisfied with a choice, and 30% are deeply outraged, is the 70% candidate the "fair" winner? Or would it be more fair to select someone who is merely acceptable to everyone, with nobody too happy or too upset? These are questions of ethics, not logic or mathematics [#Malkevitch16]. A popular concept in real-world voting is that an election should channel "the will of the people" into a preferred candidate. The choice of voting system defines how that "will" is extracted from the ballots and then acted upon. Trying to formalize "the will of the people" in a rigorous way that everyone agrees on has proven to be a formidable task. Still, we need some kind of framework in which to create and judge different voting systems. To that end, researchers have put forth a wide variety of "fairness criteria," and many have been studied closely [#VotingSystem16]. From these proposals, there has emerged a set of four fairness criteria that many people find acceptable, and those have become one of the commonly-accepted yardsticks for comparing voting systems. Those four criteria are the ones we'll focus on here [#Bowen99]. For clarity in the discussion we'll describe our fairness criteria in the context of a political election using ballots that can contain multiple candidates. We'll present each criterion in terms of its effects, describing the kind of results that it is intended to achieve, rather than in terms of the internal mechanisms it might use to bring about those results. We're motivated to do this for three reasons. First, these ideas can involve complex logic and mathematics, which we want to avoid here. Second, many discussions of these ideas use very different interpretations of the math, so that reading one verbal definition can make others seem surprisingly unrelated. Finally, the internals of these ideas can be subtle, and can drag us into deep and narrow analyses. We avoid these problems by presenting only the outcomes of the criteria, rather than their mechanics, To illustrate our criteria, we'll use a consistent set of ranked ballots shown in Figure [fig-voting-start]. There are four candidates, named Green, Blue, Orange, and Yellow, and 11 voters. Each voter has chosen to rank each of the candidates in preferred order, with the most desired candidate on top, then the second-most, and so on. ![Figure [fig-voting-start]: The ballots in an imaginary election involving 4 candidates and 11 voters. Each voter creates a ballot that ranks the choices of candidate with the most desirable at the top, and least desirable at the bottom. In this election, every voter has ranked every candidate. Below the ballots we've tallied the number of votes received by each candidate in each position of the balloting.](Images/voting-start.jpg width="700px") Fairness Criterion 1: The Majority Criterion ---- The **majority criterion** says that if a candidate receives a majority of first-place votes, that candidate should win. This criterion uses the word "majority" as we do in this chapter. In other words, if a candidate is preferred by more than half of the voters, that candidate is the winner. Figure [fig-majority] shows this graphically. ![Figure [fig-majority]: In majority voting, the candidate with more than half the cast ballots wins. In our example, a winning candidate would need at least 6 of the 11 voters to place that candidate in the first-choice position. Looking at the first-choice results in Figure [fig-voting-start], reproduced here as (a), we see that none of the candidates have that many, so this election has no winner. By comparison, if all the voters had swapped their first and second choices, then we'd have the result in (b), and Orange would have a majority and win.](Images/majority.jpg width="400px") Fairness Criterion 2: The Condorcet Criterion ---- The **Condorcet criterion** says that in a fair election, the winner should be the candidate who is preferred by the most people when compared one-on-one to each of the other candidates. For example, with our four candidates there are six one-on-one pairings, shown in Figure [fig-Condorcet]. The candidate who wins more of these matchups than any other should be the winner. Intuitively, the winner is the candidate who is most frequently preferred when compared with each of the other candidates one-on-one. ![Figure [fig-Condorcet]: The Condorcet criterion involves making a comparison between each pair of candidates. The winner receives 1 point. Ties award a half-point to both candidates. We see that Green wins 3 comparisons, Blue wins 1 and ties 1 (for 1.5 points), Orange wins none, and Yellow wins 1 and ties 1. Thus Green is declared the winner. ](Images/Condorcet.jpg width="700px") Fairness Criterion 3: The Monotonicity Criterion ---- The **monotonicity criterion** is a little different than the previous two. It doesn't tell us how to pick a winner, but it tells us that if there was a mistake in the balloting, and as a result the named winner becomes only more popular than before, that candidate should remain the winner. This might seem self-evident, which is what makes it an attractive fairness criterion. After all, if the ballots change so that the winning candidate becomes even more popular than before, but then that candidate loses, something will have gone very wrong. To re-phrase the idea, if any of the ballots change, and in each changed ballot the winning candidate is more popular than before, that candidate should remain the winner. In our example election of Figure [fig-voting-start], let's assume that the candidate with a plurality is the winner. In this case, it's Green. But now suppose that after the election, we discover that some of the ballots in Figure [fig-voting-start] are incorrect, because we scanned them in upside-down. So we remove those ballots from the tallies, read them back in right side up, and then add the corrections. Figure [fig-monotonicity] shows this idea. The monotonicity criterion says that this election is a fair one if Green, after *moving upwards on every changed ballot*, remains the winner. ![Figure [fig-monotonicity]: The criterion of monotonicity is about making sure the winner remains the winner if ballots are changed to prefer that candidate even more. In our original ballots (top row) Green is declared the winner by plurality, but then we discover that votes 2, 5, and 10 were read in upside-down. Notice that Green is in the bottom half of each ballot. When we correct them by flipping them over (lower row), Green's ranking increases in each corrected ballot. The monotonicity criterion tells us that election is fair if, in the corrected tally, Green is still the winner. In this case, the condition is satisfied. ](Images/monotonicity.jpg width="700px") Fairness Criterion 4: The Independence of Irrelevant Alternatives ---- The **independence of irrelevant alternatives criterion** is a mouthful. Like the monotonicity criterion, it refers to what we want from our elections after a winner has been declared, but then something changes. This criterion is subtle, and has many implications. Perhaps the most important involves what happens if a candidate is introduced after the election, perhaps because they were accidentally left off the ballots. We decide to allow voters update their votes, but they cannot change their ballots in any way *except* to insert the new candidate somewhere in their rankings. When we re-tally the ballots, then this fairness criterion says that only the old winner or the new candidate can be the new winner [#Electorama12]. In other words, if a new candidate enters the race and is inserted into the ballots, then the only possible winners are the previous winner or the new candidate. The losing candidates (the "irrelevant alternatives") don't become more popular, and therefore cannot move up in the results. Figure [fig-iia] shows this idea. We start with our original balloting in which Orange wins with a plurality. After the election, we discover that Green had been accidentally left off. After voters insert Green and we re-tally the votes, only the old winner, Orange, or the new candidate, Green, should be able to win. In this case Green has the plurality and wins. ![Figure [fig-iia]: The criterion of the independence of irrelevant alternatives addresses what should happen after we declare a winner and then add a new candidate. We decide to allow voters to insert the new candidate where they please, but that's the only change allowed. In row (a) we show a set of ballots where Green was accidentally left off. In this election Orange wins by plurality. In row (b) voters are allowed to insert Green where they wish. In row (c) we tally the new results. This criterion tells us that only Orange or Green should be the new winner. In this case, it's Green. ](Images/iia.jpg width="700px") Let's recap the four fairness criteria we've considered: - **Majority**: If one candidates wins a majority, that candidate wins the election. - **Condorcet**: If one candidate is preferred more than any other in head-to-head comparisons, that candidate wins the election. - **Monotonicity**: If a winner is declared, but the ballots are changed so that the winner becomes only more popular, that candidate should remain the winner. - **Independence of Irrelevant Alternatives**: If a winner is declared, and a new candidate is allowed to be inserted, only the old winner or the new candidate should be the new winner. Tallying Methods ===== Now that we've considered different measurements of fairness, let's look at several popular methods of tallying the ballots. Plurality Voting --- Perhaps the best-know single-winner method is **plurality voting** (also called **winner-take-all**, **first-past-the-post**, and **relative majority**). This is a preferential voting system in which each voter lists their candidates in order of preference. Whoever gets the most first-choice votes is declared the winner. The name "first-past-the-post" invites comparison to a foot race, where the winner is the participant with the fastest time, regardless of the time taken by the other runners. Figure [fig-plurality] shows the idea. ![Figure [fig-plurality]: In plurality voting, the candidate with the most votes is the winner. In our election, Green has the most first-choice votes and is thus the winner by plurality, even though Green does not have a majority.](Images/plurality.jpg width="200px") This system can also be used when voters are allowed to select only one candidate in the race. Since we only look at the first choice, we can interpret the candidate selected by a voter to be their first choice. Plurality-With-Elimination --- The method of **plurality-with-elimination** (also called **instant runoff voting** or **sequential runoff voting**) is another preferential voting system. We begin by tallying up the first-choice candidate for every voter. If there are only two candidates, then the technique is just like plurality voting, because the candidate with the larger number of votes is the winner. But if there are more than two candidates, we identify the candidate with the fewest number of first-place votes, and we *remove that candidate from the election*. In effect, we cross that candidate out on every ballot. Now we re-tabulate the ballots, again counting up how many times each candidate was named the first choice on these potentially modified ballots. Figure [fig-plurality-with-elimination] shows plurality with elimination at work with our ballots from Figure [fig-voting-start]. Notice that although Orange was only one voter's first choice, Orange was by far the most popular second choice candidate. Nevertheless, Orange is eliminated on the very first round, and by the time we get down to two candidates, Green is the winner. ![Figure [fig-plurality-with-elimination]: Plurality with elimination in action. Row (a) shows our starting ballots, and on the right the number of first-choice votes for each candidate. Orange received the fewest votes, so it is immediately eliminated. Row (b) shows the ballots with Orange removed, and row (c) shows those ballots tightened up vertically. At the right of row (c) we see the new voting record for first choices. Notice that Yellow has picked up the single first-choice vote that Orange had, in vote 5, since Yellow was the second choice in that vote. Since Blue now has the fewest first-place votes, it's eliminated in row (d). In row (e) we've tightened things up visually again. At the right we can see that Green has picked up two votes that previously went to Blue, and Yellow picked up one. Since Green now has more first-choice votes than Yellow, Green is the winner of the election. ](Images/plurality-with-elimination.jpg width="700px") In this example, Green led or was tied for the lead at every round, but that's irrelevant. The only thing that matters is which candidate has the most votes in the final round between the last two candidates. It's interesting to note that Orange's overwhelming support in the second place of the ballots did that candidate no good. This shows that predicting the results of plurality with elimination just by looking at the ballots can be difficult. The interplay of which candidate is eliminated on each round, and how those votes are distributed, can make it hard to predict the final results before actually running the process. Borda Count --- The **Borda Count** method is a preferential voting system based on assigning a numerical score to each candidate. The idea is that voters give more points to the candidates they like the most, and fewer points to each candidate working down their list. The candidate with the biggest score is declared the winner. To start the counting, each first-choice vote gets as many points as there are candidates. Each second-choice vote gets one less point, and each level going down continues to be worth one less point than the level above. In our ballots from Figure [fig-voting-start], we have 4 candidates. So each first-choice vote is worth 4 points, each second-choice vote is worth 3 points, each third-choice is worth 2 points, and finally each fourth-place vote is worth 1 point. Figure [fig-Borda-count] shows the idea. ![Figure [fig-Borda-count]: In the Borda Count, each vote in the first level gets points equal to the number of candidates running. Votes in each subsequent level are worth one less than the level above. In our ballots we have 4 candidates, so each first-choice vote is worth 4 points. Because Green has 4 first-choice votes, and each first-choice vote is worth 4 points, Green gets 16 points from these votes. Blue and Yellow each have 3 first-choice votes, so they each get 12 points. Orange has only one first-choice vote, so it gets 4 points. Now we look at the second choices, where each vote is worth 3 points. Orange has 8 votes at this level, for a total of 24 points. When the points are all tallied, Orange's strong showing on the second level puts it over the top, and Orange is the winner. ](Images/Borda-count.jpg width="700px") Pairwise Comparisons --- The method of **pairwise comparisons** is another preferential voting method that is score-based. It is closely related to the Condorcet fairness criterion. After all of the votes have been tallied, we find every pairing of two candidates and compare their totals. If one candidate has more votes than the other, the more popular candidate gets 1 point and the other gets 0 points. If their vote counts are tied, each gets a half point. Whoever has the most points at the end is the winner. Figure [fig-pairwise] shows the idea. ![Figure [fig-pairwise]: The pairwise process is the Condorcet criterion turned into a tallying algorithm. We compare the first-place scores of each pair of candidates head to head. The winner of each pairing gets 1 point, while ties score a half-point to both candidates. In our balloting of Figure [fig-voting-start], Green is the winner with a total of 3 points.](Images/pairwise.jpg width="700px") Approval Voting --- The only non-preferential voting method we'll look at is **approval voting.** Here, each voter simply lists all the candidates that would be acceptable, in no particular order. When we look at a ballot, each candidate that appears gets 1 point. Whoever gets the most points is the winner. The intention is to keep everything as simple as possible. Voters merely identify each candidate that they would find acceptable. That candidate who is acceptable to the greatest number of voters is the winner. In our example balloting of Figure [fig-voting-start], every one of our 11 voters listed every candidate, relying on ranking to express preference. If we used approval voting to score these ballots, then because every voter approved of every candidate, we'd have a four-way tie, with each candidate getting 11 points. So to illustrate this form of tallying, Figure [fig-approval] shows a modified set of our original ballots. Each ballot now only lists the candidates that voter approved of. For continuity, we retained the same naming order of candidates in each vote, but that order is irrelevant in approval voting. ![Figure [fig-approval]: In approval voting, every voter simply lists all the candidates they approve of. The order in which they're named is irrelevant. Every candidate gets 1 point for each appearance. In this election, Orange wins with a total of 9 votes.](Images/approval.jpg width="700px") Weighted Voting === In all of the discussions above, each ballot was treated independently. This implies that each voter is considered equal. But in many situations, voters do not carry equal weight. Suppose we have a meeting between four people who jointly own a company that has been divided up into 100 ownership shares. One person owns 40 shares, and the other three own 20 shares each. It's common in this kind of situation for each person to "vote their stock," meaning that they get 1 vote for each share they own. So the largest shareholder gets 40 votes. Let's say that it's against the rules for anyone to split their vote. Then we can rephrase the situation by saying that the shareholder with 40 shares gets 1 vote that carries a **weight** of 40. The other owners would each get 1 vote with a weight of 20. To find the outcome, each vote is multiplied by its weight before being added to the final tallies. Suppose that a particular yes-no decision by this group is to be decided by plurality voting. If the largest shareholder is opposed by the others, they will win, because their 3 votes each carry a combined weight of 60, compared to a single vote of weight 40. But if just one of the smaller shareholders votes with the largest owner, their combined total of 60 would be an unbeatable majority. We can add weighting to any of the systems above, but doing so makes it much more difficult to analyze and determine if the system meets our fairness criteria. Discussion === Let's look a bit more closely at some of the implications of the discussions above. Fairness by System --- One might judge a voting system to be fair only if it satisfies all four of our fairness criteria. Figure [fig-voting-chart] summarizes which criteria are always satisfied by our five example voting systems. ![Figure [fig-voting-chart]: Each of our example voting systems either always obeys a fairness criterion (green box) or it doesn't (red box) [#VotingSystem16]. Note that simply obeying most of these criteria doesn't guarantee the system will always produce predictable results. As noted in the text, variations in the details of each voting system can bring about different results, so this chart is only a guideline. ](Images/voting-chart.jpg width="600px") We won't discuss the results in this chart because that would require a great deal of space and detailed analysis (those analyses can be found in the references, such as [#VotingSystem16]). But it's important to note that small variations in the details of how these voting methods are carried out can change these results, so they should be taken as only a starting point [#Vaughen15]. Evaluating a specific system with respect to each of these fairness criteria, and perhaps others as well, requires a careful analysis of every detail of its design, and exactly how it's implemented. Since none of these systems satisfy all the criteria, it's natural to ask if we can design an alternative that does. Unfortunately, no voting system can satisfy these four rules of fairness, even in theory. Arrow's Impossibility Theorem --- One can prove that no system can be fair in all the ways we want. The broadly-named **Arrow's Impossibility Theorem** tells us that no voting system can be designed to satisfy all four fairness criteria simultaneously [#Arrow50]. Specifically, it says that if there are at least 2 voters and 3 candidates, then there cannot exist a voting method which satisfies all four of the fairness criteria above. There is always a way that a winner could be declared in violation at least one of these seemingly straightforward rules. Note that merely saying that no voting system can be guaranteed to always be perfectly fair does not mean that every election is unfair. In practice, many elections will satisfy the fairness criteria for many voting systems. We just can't guarantee any of them. Thus the art of choosing a voting system for a particular situation is to select one that will usually produce fair results in that context. But we always have to keep in mind that if the votes are cast in just the right way (or wrong way), unfair results are always possible. Discussion of Plurality Voting ---- It's worth a moment to look more carefully at plurality voting, because it's probably the most popular voting method used both in public elections. Plurality voting suffers from a number of problems that can offend our sense of fairness. The problems are important enough and frequent enough that despite its popularity, many voting researchers consider plurality voting to be the very worst system in use today [#Erdmann11]. Let's look at just three of these problems. First, we have the problem of **majority opposition**. Because a winner only needs a plurality, the winning candidate might be opposed by a majority of the voters, as shown in Figure [fig-plurality-majority]. ![Figure [fig-plurality-majority]: In our balloting of Figure [fig-voting-start], Green has about 36% of the vote. That means Green is opposed by a majority of the voters (the other three candidates also share this problem). Nevertheless, in plurality voting, this majority opposition is irrelevant and Green is declared the winner.](Images/plurality-majority.jpg width="200px") This flaw of plurality voting can be especially pernicious when several similar candidates are running against a single opposition candidate. For example, suppose that our election of four candidates represents choices for a group dinner for 6 adults and their 4 children. Green, Blue, and Orange represent three different restaurants that all of the adults have been to, and would be happy to return to. Yellow was included by the kids, who want ice cream for dinner. In our election, suppose that the three restaurants get 2 votes each, while all 4 children vote for ice cream, as shown in Figure [fig-plurality-pines]. ![Figure [fig-plurality-pines]: Green, Blue, and Orange are different restaurants for dinner. Each gets 2 votes. Yellow represents ice cream, preferred by the 4 children. In this election, Yellow wins, even though all 6 adults would have been happy with any restaurant. In plurality voting we have no way to know that any of the restaurants would have won if compared directly to ice cream. ](Images/plurality-pines.jpg width="200px") In a plurality voting system, ice cream is the winner, even though a majority would have been happy with any of the restaurants. It's important to note that this phenomenon is not an inherent or inevitable part of elections. It is a result of the specific way that plurality voting determines a winner. The second flaw of plurality voting that we'll consider is the **spoiler effect**. Suppose that there are three candidates in an election. We represent a 3-vote block of voters who prefer Orange, even though Orange has only a small base of support. We don't like Blue very much, but we definitely prefer either candidate to Green, who we oppose. Suppose the election has 100 voters, and the balloting so far is shown in the first row of Figure [fig-spoiler]. Here, Green has nearly a majority win with 48 votes, Blue trails by only 1 vote, and Orange is far behind with a mere 2 votes. If we cast our 3 votes for our preferred candidate of Orange, then Green will win. To prevent that, we can vote for our second choice of Blue. This will deprive Green of victory and elect Blue. This is the spoiler effect, where the influence of a candidate with very small support can determine the election between other candidates with large, even near-majority, support. ![Figure [fig-spoiler]: The spoiler effect encourages defensive voting, or the lesser of two evils effect. In this election, we strongly oppose Green. Our first choice is Orange, with Blue a distant second preference. We'll suppose that we have control of a small block of 3 votes. The voting is shown in row (a) when we arrive. Our 3-vote block is shown in gray. If we vote for Orange, the candidate we prefer, then the race between Blue and Green is unaffected and Green wins, as in row (b). But if we abandon Orange and vote defensively for Blue, as in row (c), we deprive Green of the victory. By settling for Blue we've deprived our favored candidate Orange of our support. Orange is the spoiler here, whose small body of support can decide the election of two other candidates with relatively massive support.](Images/spoiler.jpg width="700px") This effect also highlights a third problem with plurality voting: **defensive voting**. We want to vote for Orange, but to block the election of Green we vote for Blue instead. We might not want Blue very much, but consider Green even worse. This is also called the **lesser of two evils effect**, and it seems to violate one of our basic notions of what voting is all about. We should be able to vote affirmatively, *for* the candidate we want. But plurality voting can encourage us to instead vote *against* a candidate we don't want, settling for a choice we consider only less bad, rather than good. Though we demonstrated these effects with a ballot of 3 candidates, they hold true for larger numbers of competitors. Discussion of Plurality with Elimination (Instant Runoff) Voting ---- Let's now turn our attention to instant runoff voting, or plurality with elimination. This is popular for some public elections, but it too has problems. Suppose that out of four candidates, Green has 49% of the vote. This is practically a majority, promising nearly certain victory under many voting schemes. Suppose the other candidates split the rest of the votes with 16%, 17%, and 18% each, as shown in Figure [fig-irv-surprise]. In plurality with elimination, Orange can win the election, even though Orange started with only 17% of the vote and Green was only 2% away from a majority. ![Figure [fig-irv-surprise]: Plurality with elimination, or instant runoff voting, can give surprising results. We start with the initial vote in row (a). Note that Green has 49% of the vote, nearly a majority. The other three candidates roughly split the remaining vote. Since Blue has the smallest vote, it's dropped from the next round. Let's suppose that in the ballots that listed Blue as their first choice, 10% listed Orange as their second choice, and the other 6% listed Yellow. Adding these votes to those candidates, as in row (b), gives us the new totals shown in row (c). Of the three remaining candidates, Yellow has the fewest votes, so in this round we eliminate Yellow. Let's suppose that all of the votes that now have Yellow as the first choice list Orange as their second choice. When Yellow is eliminated, Orange gets all of those votes, as in row (d). Row (e) shows the final election result. Orange wins, despite the fact that almost half the voters preferred Green as their first choice, and only 17% selected Orange as their first choice. ](Images/irv-surprise.jpg width="700px") This phenomenon can reach surprising extremes with a larger slate of candidates. Suppose we have an election with 9 candidates, in which one has a near-majority of 48% of the vote, and the rest is split among the other 8 candidates, where one has 3%, the next 4%, then 5%, and so on, up to 10%. Following the same logic as in Figure [fig-irv-surprise], the candidate who starts with only 5% of the first-choice vote can amass 52% by the final round, and defeat the very popular 48% candidate. The key thing to note is that although instant runoff voting allows voters to express more opinions than single-choice plurality-based voting, it can produce a winning candidate that is widely considered to be far down on the list of preferences. Discussion Summary --- Candidates in popular elections are often affiliated with parties. Party affiliations, whether there are 2 parties or many, can definitely affect electoral campaigns. But note that the voting systems we've discussed have nothing to do with parties. All of their issues, pro and con, are inherent in the design of the voting systems themselves. The plurality voting method is probably today's most common choice in popular use because it's familiar, simple, efficient, and produces reasonable results when the election is not close. On the other hand, it is prone to some very unpleasant types of failures, and it can even push voters into undesirable behaviors, like voting against a candidate they oppose rather than for one they support. Instant runoff voting, despite being a popular alternative, can also produce surprising results. Given that Arrow's Impossibility Theorem tells us that a completely fair system is impossible to achieve, the responsibility passes to us to choose which voting system's problem scenarios we're willing to live with. Given all the tradeoffs above, I have become a big fan of Approval Voting. It's easy for voters to understand, and easy for election officials to tally. Although it can produce unwelcome results in very particular situations (as all voting algorithms do), it avoids the problems of defensive voting and the spoiler effect inherent in plurality voting. Approval Voting also encourages the participation of candidates and parties with initially small bodies of support. It lets them build up visibility and influence over time by earning the active support of voters, without the risk of "throwing away one's vote" inherent in two-party plurality voting. It allows voters to vote affirmatively, voting *for* the candidates they prefer the most, whether they're front-runners or small independents. This can open up the election to candidates with new ideas. Ultimately, the art of choosing the right voting technique for any task depends on our ability to anticipate the types of election results that are likely to come up. Then we can pick an algorithm that will usually give reasonable results in those situations. Such predictions may be difficult, but it's worth making the effort. The payoff is an election that is fair, satisfying, and invites positive participation. End of Supplementary Material === This material is a supplement to my upcoming book on deep learning, and machine learning in general. Like the material above, the book covers the essential topics in machine learning without the usual mathematics and algorithmic complexities. The idea is to convey the techniques deeply enough so that readers will be confident and knowledgeable in designing machine learning systems using any of the popular libraries. Readers will be able to accomplish both serious, real work in big data and data analysis, and carry out fun, personal investigations involving creative and original uses of machine learning. By understanding the fundamentals of what is happening inside machine-learning libraries (rather than treating them as "black boxes"), we can design better systems, explore new and creative approaches, pinpoint any problems that come up and correct them, and ultimately create better-functioning systems in less time. Note that this material is a preview of work in progress, and might differ from the final publication. To keep up to date on my book as it continues to take shape, know when it's released, and read other excerpts or supplements as I post them, follow me on Twitter at [@AndrewGlassner](https://twitter.com/andrewglassner?lang=en). References ==== To make the references as easy to obtain as possible, both the book and this supplement use documents that are available online whenever possible. [#Arrow50]: Kenneth J. Arrow, "A Difficulty in the Concept of Social Welfare", Journal of Political Economy. 58 (4), pages 328–346, 1950. https://www.stat.uchicago.edu/~lekheng/meetings/mathofranking/ref/arrow.pdf [#Bowen99]: Larry Bowen, "The Mathematics of Voting", The University of Alabama, 1999. http://www.ctl.ua.edu/math103/voting/mathemat.htm#The%20Mathematics%20of%20Voting [#Electorama12]: Electorama, "Independence of Irrelevant Alternatives", 2012. http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives [#Erdmann11]: Eric Erdmann, "Strengths and Drawbacks of Voting Methods for Political Elections", University of Minnesota Duluth, 2011. http://www.d.umn.edu/math/Technical%20Reports/Technical%20Reports%202007-/TR%202011/TR_2011_4.pdf [#FairVote16]: The Center for Voting and Democracy, "Comparing RCV With Plurality Voting," 2016. http://archive3.fairvote.org/reforms/instant-runoff-voting/irv-and-the-status-quo/comparing-irv-with-plurality-voting/ [#Malkevitch16]: Joseph Malkovitch, "Voting and Elections: Evaluating Election Systems", American Mathematical Society, 2016. http://www.ams.org/samplings/feature-column/fcarc-voting-evaluate [#Merriam-Webster16]: Merriam-Webster Editorial Board, "Majority", in "The Merriam-Webster Dictionary of the English Language", 2016. http://www.merriam-webster.com/dictionary/majority [#Vaughen15]: Christopher S. Vaughen, "The Plurality Method Fails the Independence of Irrelevant Alternatives Criterion", Monterey County Community College Department of Mathematics, 2015. http://faculty.mc3.edu/cvaughen/mgf1107/voting/Plurality_fails_IIA.ppt [#VotingSystem16]: Wikipedia, "Voting System", 2016. https://en.wikipedia.org/wiki/Voting_system All contents copyright (c) 2016 Andrew S. Glassner