Voting: Issues, Problems, and Systems, Continued

Transcription

1 Voting: Issues, Problems, and Systems, Continued 7 March 2014 Voting III 7 March /27

2 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. 1 Florida called for Gore 2 Gore declared the winner in Pennsylvania 3 The vote in Florida is now too close to call Voting III 7 March /27

3 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 7 March /27

4 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 7 March /27

5 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 7 March /27

6 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 7 March /27

7 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 7 March /27

8 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 Sequential yes no no yes Hare no no yes no CWC = Condorcet Winner Condition IIA = Independence of Irrelevant Alternatives Mono = Monotonicity Condition Voting III 7 March /27

9 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 7 March /27

10 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 H V Voting III 7 March /27

11 We computed Coleman s totals by: = = 110. The other totals were computed similarly. Thus, with the Borda count, Coleman wins the election. Voting III 7 March /27

12 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 7 March /27

13 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 7 March /27

14 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 7 March /27

15 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 7 March /27

16 Borda count Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Perot Perot Second Bush Perot Clinton Perot Clinton Bush Third Perot Bush Perot Clinton Bush Clinton We assign 2 points for each first place vote and 1 point for each second place vote. To make the arithmetic easier, let s interpret the percentages as votes. Voting III 7 March /27

17 Some Clicker Questions Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Perot Perot Second Bush Perot Clinton Perot Clinton Bush Third Perot Bush Perot Clinton Bush Clinton Q How many points does Clinton get? A Clinton: 43 first place votes and 18 second place votes. Total points, = 104 Voting III 7 March /27

18 Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Perot Perot Second Bush Perot Clinton Perot Clinton Bush Third Perot Bush Perot Clinton Bush Clinton Q How many points does Bush get? A Bush: 37 first place votes and 43 second place votes. Total points, = 117 Voting III 7 March /27

19 Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Perot Perot Second Bush Perot Clinton Perot Clinton Bush Third Perot Bush Perot Clinton Bush Clinton Q How many points does Perot get? A Perot: 19 first place votes and 38 second place votes. Total points, = 76 Since Bush has the most points, he would be elected. Voting III 7 March /27

20 Pairwise Sequential Voting Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Perot Perot Second Bush Perot Clinton Perot Clinton Bush Third Perot Bush Perot Clinton Bush Clinton Let us first order the candidates Bush, Clinton, Perot. We compare Clinton and Bush head to head. Bush is preferred to Clinton by 51% of the voters. Therefore, Clinton is eliminated. Bush is then compared to Perot head to head, and Bush is preferred. Perot is then eliminated. Bush is the one remaining, so he is elected. Other orderings of the candidates turn out to produce the same result. Voting III 7 March /27

21 Hare System We first eliminate the candidate with the fewest first place votes. This is Perot, so he is eliminated. We next reinterpret each vote to rank only Clinton and Bush. This yields the following table. Original vote Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Perot Perot Second Bush Perot Clinton Perot Clinton Bush Third Perot Bush Perot Clinton Bush Clinton Reinterpreted vote Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Clinton Bush Second Bush Bush Clinton Clinton Clinton Bush Voting III 7 March /27

22 Reinterpreted vote Rank 30% 13% 12% 25% 6% 13% First Clinton Clinton Bush Bush Clinton Bush Second Bush Bush Clinton Clinton Clinton Bush In the reinterpreted vote, Clinton receives 49% of the first place votes and Bush 51%. Therefore, Clinton is eliminated. Thus, Bush is elected with the Hare system. Voting III 7 March /27

23 The 2000 Presidential Election Since Gore won the popular vote, he would have won with plurality voting. Gore would also have won with sequential pairwise voting, and with the Hare system. It is less clear to me who would have won with the Borda count. Voting III 7 March /27

24 So, what voting system should we use? Thinking about the various voting systems we have considered, and the fact that all of them violate a reasonable condition, leads one to ask the following question: Is there a voting system which satisfies all four conditions we have introduced? Such a system would be better than any we have discussed. Voting III 7 March /27

25 Arrow s Theorem Kenneth Arrow, who won the Nobel prize in Economics in 1972, proved in 1951 that all voting systems have flaws. More precisely, he proved that there does not and cannot exist a voting system which satisfies both the Condorcet Winner condition and the Independence of Irrelevant Alternatives. Arrow s theorem says, therefore, that no matter what voting system we use, there will be flaws with it. To determine what voting system to use in a given situation, we should then consider what advantages and disadvantages systems have, and what are our priorities. We can then try to pick a reasonable system that will reflect the priorities. In practice, this is a complicated political problem. Voting III 7 March /27

26 Homework #6, due next Friday Assignment 6 is on the course website. It is due by next Friday. Next week we will discuss Apportionment. More specifically, we will discuss how seats in the U.S. House of Representatives have been and are assigned, or apportioned, among the states. Voting III 7 March /27