Voting: Issues, Problems, and Systems, Continued 9 March 2012 Voting III 9 March 2012 1/1
Last Time We’ve discussed several voting systems and conditions which may or may not be satisfied by a system. We’ll review them. But first, let’s look at some clips from reporting of the 2000 presidential election. Florida called for Gore 1 Gore declared the winner in Pennsylvania 2 The vote in Florida is now too close to call 3 Voting III 9 March 2012 2/1
Plurality Voting The method we use in the U.S. to decide most elections is called Plurality Voting. In this method, a voter chooses one candidate, and the candidate with the most votes wins. It is nearly the most simple method of deciding elections. Voting III 9 March 2012 3/1
The Borda Count With the Borda count voters rank order the candidates. If there are n candidates, each first place vote is worth n − 1 points, each second place vote is worth n − 2 points, and so on. The person who received the most points wins the election. Voting III 9 March 2012 4/1
Sequential Pairwise Voting In this voting system, voters rank candidates as in the Borda count. The winner is determined by comparing pairs of candidates. The loser between a comparison is eliminated and the winner is then compared to the next candidate. The last person remaining is then the winner of the election. Voting III 9 March 2012 5/1
The Hare System In the Hare system, candidates are ranked. The candidate (or candidates) with the fewest number of first place votes is eliminated. This process continues until one candidate remains, and is then elected. Once a candidate is elected, we remove that candidate from each ballot. For example, if B is eliminated, then a ballot which lists C > B > A would then be reinterpreted to have C > A . That is, C would be listed first and A second. Voting III 9 March 2012 6/1
The Four Conditions Condorcet winner criterion: The Condorcet winner (preferred over all others head to head), if there is one, always wins the election. Voting III 9 March 2012 7/1
The Four Conditions Condorcet winner criterion: The Condorcet winner (preferred over all others head to head), if there is one, always wins the election. Independence of Irrelevant Alternatives: It is impossible for a non-winning candidate B to change to winner unless at least one voter reverses the order in which they listed B and the winner. Voting III 9 March 2012 7/1
The Four Conditions Condorcet winner criterion: The Condorcet winner (preferred over all others head to head), if there is one, always wins the election. Independence of Irrelevant Alternatives: It is impossible for a non-winning candidate B to change to winner unless at least one voter reverses the order in which they listed B and the winner. Pareto: If everybody prefers one candidate to another, then the latter is not elected. Voting III 9 March 2012 7/1
The Four Conditions Condorcet winner criterion: The Condorcet winner (preferred over all others head to head), if there is one, always wins the election. Independence of Irrelevant Alternatives: It is impossible for a non-winning candidate B to change to winner unless at least one voter reverses the order in which they listed B and the winner. Pareto: If everybody prefers one candidate to another, then the latter is not elected. Monotonicity: If no voter were to switch his/her preference between the winner and another, then the outcome of the election would be the same. Voting III 9 March 2012 7/1
The following table summarizes how each system behaves with respect to each of the conditions. A yes means the system satisfies the condition. CWC IIA Pareto Mono Plurality no no yes yes Borda no no yes yes yes no no yes Sequential no no yes no Hare CWC = Condorcet Winner Condition IIA = Independence of Irrelevant Alternatives Mono = Monotonicity Condition Voting III 9 March 2012 8/1
Back to the 1998 Minnesota Gubernatorial Election How would the 1998 Minnesota gubernatorial election come out if other voting systems were used? Voting III 9 March 2012 9/1
Back to the 1998 Minnesota Gubernatorial Election How would the 1998 Minnesota gubernatorial election come out if other voting systems were used? To answer this we’d need to have had candidates rank ordered. Let’s suppose the voters would rank the three candidates, Coleman, Humphrey, and Ventura, as follows. This table is not based on exit polls. I made up the table, trying to make a reasonable guess as to what would have happened. Voting III 9 March 2012 9/1
Back to the 1998 Minnesota Gubernatorial Election How would the 1998 Minnesota gubernatorial election come out if other voting systems were used? To answer this we’d need to have had candidates rank ordered. Let’s suppose the voters would rank the three candidates, Coleman, Humphrey, and Ventura, as follows. This table is not based on exit polls. I made up the table, trying to make a reasonable guess as to what would have happened. Recall that Ventura received 37% of the first place votes, while Coleman received 35% and Humphrey 28%. 20% 25% 8% 10% 20% 17% H C H C V V C H V V C H V V C H H C Voting III 9 March 2012 9/1
Using the Borda Count With the Borda count, since there are three candidates, we assign 2 points for first place votes, 1 point for second place votes, and 0 points for third place votes. To make it easier, we pretend there are 100 votes, so the percentage refers to the number of votes. Voting III 9 March 2012 10/1
Using the Borda Count With the Borda count, since there are three candidates, we assign 2 points for first place votes, 1 point for second place votes, and 0 points for third place votes. To make it easier, we pretend there are 100 votes, so the percentage refers to the number of votes. 20% 25% 8% 10% 20% 17% H C H C V V C H V V C H V V C H H C Voting III 9 March 2012 10/1
Using the Borda Count With the Borda count, since there are three candidates, we assign 2 points for first place votes, 1 point for second place votes, and 0 points for third place votes. To make it easier, we pretend there are 100 votes, so the percentage refers to the number of votes. 20% 25% 8% 10% 20% 17% H C H C V V C H V V C H V V C H H C first place votes 2nd 3rd total points C 35 40 25 110 H 28 42 30 98 V 37 18 45 92 Voting III 9 March 2012 10/1
We computed Coleman’s totals by: 35 · 2 + 40 · 1 + 25 · 0 = 70 + 40 = 110 . The other totals were computed similarly. Thus, with the Borda count, Coleman wins the election. Voting III 9 March 2012 11/1
Sequential Pairwise Voting 20% 25% 8% 10% 20% 17% H C H C V V C H V V C H V V C H H C Voting III 9 March 2012 12/1
Sequential Pairwise Voting 20% 25% 8% 10% 20% 17% H C H C V V C H V V C H V V C H H C If we use sequential pairwise voting, and order them C, H, V, then Coleman is preferred to Humphrey 55% to 45%, so Humphrey is eliminated. Since Coleman is also preferred 55% to 45% over Ventura, so Coleman is elected in this system. Voting III 9 March 2012 12/1
Sequential Pairwise Voting 20% 25% 8% 10% 20% 17% H C H C V V C H V V C H V V C H H C If we use sequential pairwise voting, and order them C, H, V, then Coleman is preferred to Humphrey 55% to 45%, so Humphrey is eliminated. Since Coleman is also preferred 55% to 45% over Ventura, so Coleman is elected in this system. It turns out that, in this election, the order in which we list the candidates does not affect the outcome Voting III 9 March 2012 12/1
Hare System If we use the Hare system, then Humphrey is eliminated since he received the fewest first place votes, 28%, compared to 35% and 37% for the other two candidates. We then reinterpret the ballot by eliminating Humphrey. This gives Voting III 9 March 2012 13/1
Hare System If we use the Hare system, then Humphrey is eliminated since he received the fewest first place votes, 28%, compared to 35% and 37% for the other two candidates. We then reinterpret the ballot by eliminating Humphrey. This gives 20% 25% 8% 10% 20% 17% C C V C V V V V C V C C Voting III 9 March 2012 13/1
After reinterpreting the ballots, Coleman then has 55% first place votes to Ventura’s 45%, so Coleman would win under the Hare system. Voting III 9 March 2012 14/1
After reinterpreting the ballots, Coleman then has 55% first place votes to Ventura’s 45%, so Coleman would win under the Hare system. So, in all of the systems, other than plurality voting, Coleman would win the election. Voting III 9 March 2012 14/1
The 1992 Presidential Election Revisited Let’s make an estimate of how the voting would have been conducted if voters ranked the three main candidates, and determine who would be elected in each of the four systems we have studied. Voting III 9 March 2012 15/1
The 1992 Presidential Election Revisited Let’s make an estimate of how the voting would have been conducted if voters ranked the three main candidates, and determine who would be elected in each of the four systems we have studied. I’ve made an attempt to predict how voters would rank the candidates, in order to come up with the following table. Voting III 9 March 2012 15/1
Recommend
More recommend
