MTH 110 Week 1 hapter 1 Worksheet NME The Mathematics of Elections It s not the voting that s democracy; it s the counting. Tom Stoppard We have elections because we don t all think alike. Since we cannot all have things our way, we vote. ut voting is only half the story. Voting theory consists methods of answering the following questions: How do we shift through the many choices of individual voters to find the collective choice of the group? How well does the process work? Is the process always fair? Wait, voting theory?! Why do we need a fancy theory to figure out how to count the votes? Isn t it pretty simple? We have an election; we count the ballots and based on that count we decide an outcome of the election in a consistent and fair manner. What issues might we need to account for in a voting theory? 1.1 asic Elements of an Election candidates: voters: ballots: outcome: voting method: 1
Preference allots and Preference Schedules onsider the following preference ballot. What does this ballot tell us about the voter s preferences? Things seems to get a little messier when we start considering multiple preference ballots. Some voters will have identical ballots but not all of them. One way to keep track of this information is a preference schedule. Example: Math lub Election The following are preference ballots from a math club election where the candidates are : lice, : en, : armen, and : arius. We can make a preference schedule by filling in the following chart: Number of Voters: Still we have not answered: Who is the winner of the election? efore answering this question, we will briefly discuss one other potential element of an election. Ties The most common method for breaking a tie is with a runoff election, but runoff elections are expensive and time consuming so there are other methods. In general, tie-breaking procedures add a new level of complexity to an already rich subject so we will not delve into them. Now, let us turn to methods of determining a winner. 2
1.2 The Plurality Method The plurality method is one of the most commonly used and simplest methods for determining the outcome of an election. With this method, all that matters is how many first place votes a candidate gets. In a winner-only election, the candidate with the most first place votes is the winner. In a ranked election, the candidate with the most first place votes is winner, the candidate with the second most votes is second, and so on. What happens in the Math lub Election example? The main appeal of this method is its simplicity. an you think of an potential problems with this method? s an example of when one particular candidate may be preferred by voters over all other candidates yet not win, consider the following preference schedule. 1.3 The orda ount Method Number of Voters: 50 49 1 The orda count method is the second most commonly used method for determining the winner of an election. In this method, each ballot is assigned points. In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points. The points are tallied for each candidate separately, the candidate with the most points wins the candidate with the second most points gets second place, and so on. Use the orda method to determine the outcome of the Math lub Election example. Number of Voters: 4 2 2 1 1 There are many variations of the orda count method. For instance, sometimes getting voted first place is weighted heavier. 3
1.4 The Plurality-with-Elimination Method In the US most municipal and local elections have a majority requirement a candidate needs a majority of the votes to get elected. With two candidates this is rarely a problem but with more than two it can easily occur that no candidate has the majority. In these cases, as opposed to doing a run-off, we use the plurality-with-elimination method: Round 1: ount the 1st place votes for each candidate, just as youu would in the plurality method. If a candidate has a majority of 2st place votes, then that candidate is the winner. eliminate the candidate(s) with the fewest 1st place votes. Otherwise, Round 2: ross out the names of the candidates eliminated from the preference schedule and transfer those votes to the next eligible candidates on those ballots. Recount the votes. If a candidate has a majority then declare that candidate the winner. candidate with the fewest votes. Otherwise eliminate the Round 3, 4,...: Repeat the process, each time eliminating the candidate with the fewest votes and transferring those votes to the next eligible candidate. ontinue until there is a candidate with the majority. That candidate is the winner of the election. In a ranked election, the candidates should be ranked in reverse order of elimination. Example: ity ounsel Election Use the plurality-with-elimination method to determine the outcome of the following preference schedule. Number of Voters: 14 10 8 4 1 gain, there are variations on this method that are outlined in the chapter. One such example is instant runoff voting that is used for mayoral elections in Minneapolis and St Paul (see p. 17 and Example 1.15 of the text for more information). 1.5 The Method of Pairwise omparisons One of the most useful features of the preference schedule is that it allows us to find the winner of an pairwise comparison between candidates. Specifically, given any two candidates, we can count how many voters votes one above the other and vice versa, the one with the highest number wins the pairwise comparison. This is the basis for the method of pairwise comparisons. For each possible pairwise comparison between candidates, give 1 point to the winner and 0 points to the loser (if a tie, give each 1/2 a point). The candidate with the most points is the winner. 4
Use the plurality-with-elimination method to determine the outcome of the ity ounsel Election example. What do you notice about the result here versus the result found using the plurality-with-elimination method? What might be one disadvantage of this method? This method is not used in real life elections. However, it does have one interesting application, the NFL draft! (see the bottom of p. 20 and Example 1.18 of the text) 1.6 Fairness riteria and rrow s Impossibility Theorem Of all of the methods discussed above which one is the best? In the late 1940 s, merican economist Kenneth rrow attempted to answer the question: What would it take for a voting method to at least be a fair voting method? To answer this questions, he set a minimum set of requirements that a mothod should have, these are called rrow s fairness criteria: Majority riterion: ondorcet riterion: Monotonicity riterion: Independence-if-irrelevant-alternatives (II) riterion: 5
This list does not exhaust all fairness criteria but is a set of basic principles we expect a democratic election to satisfy and can be used as a benchmark by which we can measure any voting method. If a method violates any one of these criteria then there is potential for unfair results under this method. It turns out that every voting method is flawed. rrow demonstrated that, in an election involving three or more candidates, it is mathematically impossible to satisfy all four of these fairness criteria. rrow s Impossibility Theorem For elections involving three or more candidate, a method for determining election results that is always fair is mathematically impossible. Note though that this does not mean that every election is unfair or that every voting method is equally bad, nor does it mean that we should stop trying to improve the quality of our voting experience. 6