### Election Analysis with 7 Voters and 4 Candidates**Prompt:** Consider the following election with 7 voters and 4 candidates. Determine the winner using the methods listed below.#### Preference Table| Number of Voters | 2 | 2 | 1 | 1 | 1 ||------------------|---|---|---|---|---|| **First** | A | B | A | C | D || **Second** | D | D | B | B | B || **Third** | C | A | D | D | A || **Fourth** | B | C | C | A | C |#### Methods to Determine the Winner- **e) Plurality Runoff**- **f) Borda Count**- **g) Condorcet’s Method**- **h) Sequential Pairwise with Agenda: B, D, C, A**### Explanation of Methods- **Plurality Runoff:** This method involves first identifying the top two candidates with the most first-place votes and then conducting a runoff election only between these two.- **Borda Count:** Candidates are scored based on their position in each voter's ranking. The candidate with the highest total score wins.- **Condorcet’s Method:** A candidate is the winner if they can beat every other candidate in a head-to-head comparison.- **Sequential Pairwise:** Using a predefined agenda (B, D, C, A in this case), each candidate is paired one-on-one in a sequence. The winner of each matchup moves on to the next round.This setup showcases various voting systems used to determine outcomes in elections, analyzing voter preferences for a fair decision-making process.

As per the guidelines I am solving 3 subparts only. Plurality Method: In this method, we count the first-choice votes and the choice with the most first-choice votes is declared the winner. In case of ties, we use run-off method. Borda Count: If there are 4 choices, then assign first-choice with 4 points, second-choice with 3 points, third-choice with 2 points and fourth-point with 1 point and sum all the points. The one who gets the highest point becomes the winner. Condorcet's Method: There occurs a one-one comparison. Plurality Method: VOTES--> 2 2 1 1 1 First A B A C D Second D D B B B Third C A D D A Fourth B C C A C Count the first-choice votes: A gets 2+1=3 votes B gets 2 votes. C and D get 1 vote each. Thus, A is the winner in Plurarity method.Borda Count: VOTES--> 2 2 1 1 1 First (4points) A B A C D Second (3points) D D B B B Third (2points) C A D D A Fourth (1point) B C C A C Count the total points VOTES--> 2 2 1 1 1 First (4points) A 2*4=8 B 2*4=8 A 1*4=4 C 1*4=4 D 1*4=4 Second (3points) D 2*3=6 D 2*3=6 B 1*3=3 B 1*3=3 B 1*3=3 Third (2points) C 2*2=4 A 2*2=4 D 1*2=2 D 1*2=2 A 1*2=2 Fourth (1point) B 2*1=2 C 2*1=2 C 1*1=1 A 1*1=1 C 1*1=1 Total points of A = 8+4+4+2+1=19 Total points of B = 8+3+3+3+2=19 Total points of C = 4+4+2+1+1=12 Total points of D = 4+6+6+2+2=20 Therefore by Borda count method D is the winnerCondorcets Method: VOTES--> 2 2 1 1 1 First A B A C D Second D D B B B Third C A D D A Fourth B C C A C A vs B : VOTES--> 2 2 1 1 1 First A B A Second B B B Third A A Fourth B A Here we see A gets 2+1=3 votes and B gets 2+1+1=4 votes. So betwwen A and B, B is the winner. A vs C : VOTES--> 2 2 1 1 1 First A A C Second Third C A A Fourth C C A C Here A gets 2+2+1+1=6 votes and C gets 1 vote, hence A is the winner. A vs D : VOTES--> 2 2 1 1 1 First A A D Second D D Third A D D A Fourth A Here, A gets 2+1=3 votes and D gets 2+1+1=4 votes . So D is the winner in this case. B vs C: VOTES--> 2 2 1 1 1 First B C Second B B B Third C Fourth B C C C Here B gets 2+1+1=4 votes and C gets 2+1=3 votes, hence B is the winner. B vs D: VOTES--> 2 2 1 1 1 First B D Second D D B B B Third A D D Fourth B Here, B gets 2+1+1=4 votes and D gets 2+1=3 votes. So B is the winner. C vs D: VOTES--> 2 2 1 1 1 First C D Second D D Third C D D Fourth C C C Here C gets 1 vote and D gets 2+2+1+1=6 votes. So D is the winner Comparison table: one-one A vs B A vs C A vs D B vs C B vs D C vs D Winner B A D B B D B has won thrice in one-one comparisons. Therefore, B is the Condorcet winner.