Question

exercises - voting theory score: 0/20 answered: 0/10 question 7 number of voters 15 7 10 1st choice b c a 2nd choice c b c 3rd choice a a b find the condorcet candidate, if there is one. if there is not a condorcet candidate, enter none condorcet candidate =

Explanation:

Step1: Calculate total voters

Total voters = \( 15 + 7 + 10 = 32 \). A majority is more than \( \frac{32}{2}=16 \).

Step2: Compare A vs B

- Voters preferring A over B: In the third group (10 voters, A 1st, B 3rd) and second group (7 voters, B 2nd, A 3rd? Wait, no: For A vs B, look at rankings where A and B are compared.
- Group 1: B 1st, A 3rd → prefer B over A (15 voters)
- Group 2: C 1st, B 2nd, A 3rd → prefer B over A (7 voters)
- Group 3: A 1st, B 3rd → prefer A over B (10 voters)
- So A vs B: A gets 10, B gets \( 15 + 7 = 22 \). B wins over A.

Step3: Compare A vs C

- Voters preferring A over C: Group 3 (A 1st, C 2nd) → 10 voters; Group 1 (B 1st, C 2nd, A 3rd → prefer C over A? Wait, A vs C:
- Group 1: B 1st, C 2nd, A 3rd → prefer C over A (15)
- Group 2: C 1st, B 2nd, A 3rd → prefer C over A (7)
- Group 3: A 1st, C 2nd → prefer A over C (10)
- A vs C: A gets 10, C gets \( 15 + 7 = 22 \). C wins over A.

Step4: Compare B vs C

- Voters preferring B over C: Group 1 (B 1st, C 2nd) → 15; Group 2 (C 1st, B 2nd) → prefer C over B (7); Group 3 (A 1st, C 2nd, B 3rd → prefer C over B? Wait, B vs C:
- Group 1: B 1st, C 2nd → B over C (15)
- Group 2: C 1st, B 2nd → C over B (7)
- Group 3: A 1st, C 2nd, B 3rd → C over B (10)
- Wait, no: For B vs C, in group 3, 1st is A, 2nd is C, 3rd is B → so they prefer C over B (10 voters).
- So B vs C: B gets 15, C gets \( 7 + 10 = 17 \). C wins over B (17 > 15, which is more than 16? Wait 17 > 16 (majority). Wait, but let's recalculate:

Wait, total voters 32. For B vs C:
- Voters who rank B higher than C: Group 1 (B 1st, C 2nd) → 15; Group 2 (C 1st, B 2nd → no, B is 2nd, C is 1st → C higher. Group 3: A 1st, C 2nd, B 3rd → C higher than B. So B vs C: B has 15, C has \( 7 + 10 = 17 \). So C beats B (17 > 15, which is a majority? Wait 17 is more than 16 (half of 32). So C beats B.

Wait, earlier:
- C beats A (22 - 10, 22 > 16)
- C beats B (17 > 15, 17 > 16? Wait 17 is more than 16 (since 32/2=16). So C wins both A and B? Wait no, wait in B vs C: 15 voters for B, 17 for C. 17 > 16, so C beats B. And C beats A (22 > 16). Wait, but let's check again.

Wait A vs B: B gets 15 + 7 = 22, A gets 10. 22 > 16, so B beats A.

B vs C: B gets 15, C gets 7 + 10 = 17. 17 > 16, so C beats B.

A vs C: A gets 10, C gets 15 + 7 = 22. 22 > 16, so C beats A.

Wait, so C beats both A and B? Wait no, wait B beats A, C beats B, C beats A. Wait, but a Condorcet candidate is one who beats every other candidate in a head-to-head. So C beats A (22-10) and C beats B (17-15). Wait 17 is more than 16? Wait 32 voters, so majority is 17 (since 32/2=16, so more than 16 is 17 or more). So C beats B (17 > 15) and C beats A (22 > 10). Wait, but let's check the counts again.

Wait for A vs C:

- Group 1: B, C, A → so when comparing A and C, these voters have C above A (since C is 2nd, A is 3rd) → 15 voters for C.

- Group 2: C, B, A → C above A (C is 1st, A is 3rd) → 7 voters for C.

- Group 3: A, C, B → A above C (A is 1st, C is 2nd) → 10 voters for A.

So A vs C: C has 15 + 7 = 22, A has 10. 22 > 16, so C beats A.

For B vs C:

- Group 1: B, C, A → B above C (B is 1st, C is 2nd) → 15 voters for B.

- Group 2: C, B, A → C above B (C is 1st, B is 2nd) → 7 voters for C.

- Group 3: A, C, B → C above B (C is 2nd, B is 3rd) → 10 voters for C.

So B vs C: C has 7 + 10 = 17, B has 15. 17 > 16, so C beats B.

For A vs B:

- Group 1: B, C, A → B above A (B is 1st, A is 3rd) → 15 voters for B.

- Group 2: C, B, A → B above A (B is 2nd, A is 3rd) → 7 voters for B.

- Group 3: A, C, B → A above B (A is 1st, B is 3rd) → 10 voters for A.

So A vs B: B has 15 + 7 = 22, A has 10. 22 > 16, so B beats A.

Now, check if any candidate beats all others:

- C beats A and C beats B → so C beats both A and B. Wait, but B beats A, but C beats B. So C is the Condorcet candidate? Wait, let's confirm:

A Condorcet candidate must win head-to-head against every other candidate. So C vs A: C wins (22-10). C vs B: C wins (17-15). So C wins both, so C is the Condorcet candidate? Wait, but wait 17 is more than 16 (majority), so yes. Wait, but let's check the numbers again.

Wait total voters: 15 + 7 + 10 = 32. So majority is 17 (since 32/2 = 16, so need more than 16, i.e., 17 or more).

C vs B: C has 17 (7 + 10), B has 15. 17 > 16, so C wins.

C vs A: C has 22 (15 + 7), A has 10. 22 > 16, so C wins.

So C beats both A and B, so C is the Condorcet candidate.

Wait, but earlier I thought maybe I made a mistake, but let's re-express the head-to-heads:

1. A vs B:
- Voters who prefer A over B: Only group 3 (A 1st, B 3rd) → 10 voters.
- Voters who prefer B over A: Groups 1 (B 1st, A 3rd) and 2 (B 2nd, A 3rd) → 15 + 7 = 22 voters.
- Since 22 > 16, B beats A.

2. A vs C:
- Voters who prefer A over C: Only group 3 (A 1st, C 2nd) → 10 voters.
- Voters who prefer C over A: Groups 1 (C 2nd, A 3rd) and 2 (C 1st, A 3rd) → 15 + 7 = 22 voters.
- Since 22 > 16, C beats A.

3. B vs C:
- Voters who prefer B over C: Only group 1 (B 1st, C 2nd) → 15 voters.
- Voters who prefer C over B: Groups 2 (C 1st, B 2nd) and 3 (C 2nd, B 3rd) → 7 + 10 = 17 voters.
- Since 17 > 16, C beats B.

Now, check each candidate:

- A: Lost to B and C → not Condorcet.
- B: Lost to C → not Condorcet.
- C: Beat A and B → Condorcet candidate.

Answer:

C