MA 111 Worksheet Sept. 9 Name:
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1 MA 111 Worksheet Sept. 9 Name: 1. List the four fairness criteria. In your own words, describe what each of these critieria say. Majority Criteria: If a candidate recieves more than half of the first place votes, that candidate should win the election. Condorcet Criteria: If a candidate beats all other candidates when put up head to head, then that candidate should win the election. Monotonicity Criteria: Suppose a candidate wins an election, then in a reelection the only changes to ballots rank that candidate even higher. This candidate should win the reelection. Independence of Irrelevant Alternatives (IIA) Criteria: Suppose that a candidate wins an election. Then a non-winning candidate withdraws from the election. The original candidate should win the recount as well. 2. List the four voting methods discussed in this chapter. For each voting method, describe at least one defect. Plurality Method: Fails Condorcet criteria Borda Count Method: Fails the Majority and Condorcet criteria Plurality with Elimination: Fails the Monotonicity Criteria Pairwise Comparisons: Fails the IIA Criteria, can be very time consuming if many candidates are involved. 3. Explain the difference between a plurality and a majority. A majority requires more than half of the first place votes. A plurality just requires the most number of first place votes. There is no difference in two candidate elections. The difference can emerge when three or more candidates are involved.
2 4. An election involves 10 candidates. (a) How many distinct ways can a first place candidate be picked? There are 10 candidates and therefore 10 ways to choose a first place candidate (b) How many distinct ways can a first and second place candidate be picked? There are 10 ways to choose the first place candidate. Once the first place candidate is choosen, there are 9 remaining choices for second place. Thus, there are 10 9 = 90 ways to pick first and second. (c) How many distinct ways can a first, second, and third place candidate be picked? From the previous, there are 10 9 ways to choose first and second. Now, the third place candidate could be any one of the remaining 8 candidates. Thus, there are = 720 ways to choose first, second, and third. (d) What is the largest number of columns that can appear in the preference schedule for an election with 10 candidates? Following the same reasoning in the above parts, there are = possible orderings of all ten candidates. (e) Suppose this election is decided using Pairwise Comparisons. How many comparisons will need to be made? Suppose the candidates are A, B, C, D, E, F, G, H, I, J. A has to face off against each of A, B, C, D,..., J. Thus, A is involved in 9 pairwise competitions. Likewise, for B, C,... J. Thus each of the ten candidates is involved in 9 pairwise competitions. If we re not careful, we may think there are 90 comparisons, but this counts each comparison twice. (A vs B is counted once from A s point of view and once from B s point of view.) Thus, there are 90/2 = 45 distinct pairwise comparisons. 5. An election involves 4 candidates (A, B, C, D) and 35 people vote. (a) How many first votes are needed for a Majority? Exactly half would be 35/2 = For a majoroty, we need more than half, (and there s no such thing as a partial vote) so a majority requires 18 votes. (b) Suppose the election is decided using Borda Count. What is the most number of points a candidate could receive? A candidate receives 4 points for each first place. The best case for a candidate is if all 35 place this candidate in first, in which case the candidate receives 35 4 = 140 (c) Suppose the election is decided using Borda Count. What is the least number of points a candidate could receive? The worst case if if this candidate is in last place on every ballot, in which case the candidate only receives one point per voter, so 35 1 = 35 points. (d) Suppose A recieves 10 first place votes, B receives 4 first place votes, C receives 6 first place votes, and D recieves 8 first place votes. 7 of the ballots have not been counted yet. How many more first place votes must D receive to guarantee victory, if the election is decided using Plurality? Notice that D COULD win with only 3 additional votes, for then D would have 11 votes, which beats any of the remaining candidates... except this only accounts for 3 of the 7
3 remaining votes. If some of the remaining votes go to A, then D still might not win. Thus, we need to look at the WORST CASE for D. Let x denote the number of additional votes that D recieves. D s strongest competitor is A, so the worst case for D would be if the remaining votes all go to A, in which case A receives 7 x additional votes. Thus, after the remaining 7 votes are accounted for, D would have 8 + x votes and A would have 10 + (7 x) = 17 x votes. In order for D to win, we need 8 + x > 17 x. Adding x to both sides gives 8 + 2x > 17, subtracting 8 from both sides gives 2x > 9 so x > 9/2 = 4.5 Since x must be a whole number, we see that D must recieve at least 5 of the remaining votes to guarentee a victory. 6. An election has 8 candidates. List all of the required comparisons, if the candidates are A, B, C, D, E, F, G, H. The easiest way is to list them in alphabetic order: 7. (a) How many people voted in this election? = 27 AB AC BC AD BD CD AE BE CE DE AF BF CF DF EF AG BG CG DG EG FG AH BH CH DH EH FH GH Number of voters First choice A C D B Second choice B B C A Third choice C A B C Fourth choice D D A D (b) What is the minimum number of votes needed for a majority? Is there a majority candidate? Half is 27/2 = 13.5 so a majority requires at least 14 first place votes. This election has no majority candidate (c) Determine the winner using the Plurality Method. How many first place votes does each candidate get? A receives 7 first place votes, B receives 5 first place votes, C receives 7 first place votes, D receives 8. Thus, D is the winner using plurality. (d) Determine the winner using the Borda Count Method. How many points does each candidate receive (assuming 4 points for each first place, 3 for each second place, etc) A: = 65
4 B: = 78 C: = 55 D: = 51 Thus, D wins by Borda count. (e) Determine the winner using the Plurality with Elimination. Who is eliminated in the first round? Who is eliminated in the second round? B is eleiminated in the first round. The preference schedule in the second round is then Number of voters First choice A C D Second choice C A C Third choice D D A C is eliminated in the second round. The preference schedule in the third round is then Number of voters 19 8 First choice A D Second choice D A Therefore, A wins using Plurality with Elimination. (f) Determine the winner using the Pairwise Comparisons Method. How many head to head matches does A win? B? C? D? A vs B: 7 against = 20, so B A vs C: = 12 against = 15, so C A vs D: = 19 against 8, so A B vs C: = 12 against = 15, so C B vs D: = 19 against 8, so B C vs D: = 19 against 8, so C C wins more pairwise comparisons than any other candidate, so C is the winner using Pairwise Comparisons (g) Is there a Condorcet candidate? Each candidate was involved in 3 pairwise comparisons. C won all three of its pairwise comparisons, so C is a Condorcet candidate.
5 8. Number of voters First choice A A C D D B Second choice B D E C C E Third choice C B D B B A Fourth choice D C A E A C Fifth Choice E E B A E D (a) Determine the winner using the Plurality Method. A receives 8 first place votes, which is more than any other candidate, so A wins by plurality. (b) Determine the winner using the Borda Count Method. A: = 60 B: = 64 C: = 72 D: = 65 E: = 48 C wins by Borda count (c) Determine the winner using the Plurality with Elimination. E has no first place votes, so E is eliminated in the first round. Eliminating E does not change the first place votes, so B will be eliminated in the second round. Number of voters First choice A A C D Second choice C D D C Third choice D C A A C and D EACH have the fewest first place votes in the next round, so they will both be eliminated, leaving only A. Thus A wins. (d) Determine the winner using the Pairwise Comparisons Method. A vs B: = 13 against = 8, so A A vs C: = 11 against = 10, so A A vs D: = 11 against = 10, so A A vs E: = 10 against = 11, so E B vs C: = 11 against = 10, so B B vs D: = 8 against = 13, so D B vs E: = 16 against 5, so B C vs D: = 13 against = 8, so C C vs E: = 18 against 3, so C D vs E: = 13 against = 8, so D
6 A wins more pairwise comparisons than any other candidate, so A is the winner using Pairwise Comparisons (e) Is there a majority candidate? 21 voters so 11 votes are needed a majority. No candidate is a majority candidate. (f) Is there a Condorcet candidtae? No. Each candidate loses at least one pairwise comparison.
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