Social Choice Quiz Results
Dominik Peters, February 2026
Note: The models were given all 25 questions in a single prompt and asked to answer all questions. Thus, this also tests stamina and not getting confused between questions. Where possible, I deactivated web access. Grok ran many web searches.
| Model | Score |
|---|
| gemini-3-1-pro-preview.txt | 20/25 |
| gemini-3-flash-attempt2.txt | 17/25 |
| gemini-3-flash.txt | 16/25 |
| claude-sonnet-4-6.txt | 12/25 |
| gemini-3-pro-preview.txt | 12/25 |
| gpt-5-1-thinking-extended.txt | 12/25 |
| gpt-5-2-thinking-extended.txt | 12/25 |
| grok.txt | 8/25 |
| o3.txt | 8/25 |
| mistral.txt | 6/25 |
| claude-haiku-4-5.txt | 5/25 |
Question 1: Which of the following properties does the Borda count satisfy?
- A) Majority criterion
- B) Condorcet criterion
- C) Monotonicity
- D) Independence of Pareto-dominated alternatives
| A | B | C | D |
|---|
| Correct | | | ✓ | |
| claude-haiku-4-5.txt | | | ✓ | |
| claude-sonnet-4-6.txt | | | ✓ | |
| gemini-3-1-pro-preview.txt | | | ✓ | |
| gemini-3-flash-attempt2.txt | | | ✓ | |
| gemini-3-flash.txt | | | ✓ | |
| gemini-3-pro-preview.txt | | | ✓ | |
| gpt-5-1-thinking-extended.txt | | | ✓ | |
| gpt-5-2-thinking-extended.txt | | | ✓ | |
| grok.txt | | | ✓ | ✓ |
| mistral.txt | | | ✓ | ✓ |
| o3.txt | | | ✓ | |
Question 2: Suppose I only have access to voter preferences via an oracle that tells me, given a voter and a subset of alternatives, which alternative in the subset is the most preferred by the voter. To find a winner according to Install-runoff voting, how many queries to the oracle do I need in the worst case?
- A) O(n)
- B) O(m)
- C) O(nm)
- D) O(nm log m)
- E) O(n log m)
| A | B | C | D | E |
|---|
| Correct | | | | | ✓ |
| claude-haiku-4-5.txt | | | ✓ | | |
| claude-sonnet-4-6.txt | | | ✓ | | |
| gemini-3-1-pro-preview.txt | | | ✓ | | |
| gemini-3-flash-attempt2.txt | | | ✓ | | |
| gemini-3-flash.txt | | | | | ✓ |
| gemini-3-pro-preview.txt | | | ✓ | | |
| gpt-5-1-thinking-extended.txt | | | ✓ | | |
| gpt-5-2-thinking-extended.txt | | | ✓ | | |
| grok.txt | | | ✓ | | |
| mistral.txt | | | ✓ | | |
| o3.txt | | | ✓ | | |
Question 3: Which of the following properties does the Method of Equal Shares satisfy in the context of approval-based committee selection?
- A) Pareto-optimality
- B) Core
- C) If everybody approves candidate x, then x is part of every winning committee
- D) Extended justified representation (EJR)
- E) Polynomial-time computability
| A | B | C | D | E |
|---|
| Correct | | | | ✓ | ✓ |
| claude-haiku-4-5.txt | ✓ | | ✓ | ✓ | ✓ |
| claude-sonnet-4-6.txt | | | ✓ | ✓ | ✓ |
| gemini-3-1-pro-preview.txt | | | | ✓ | ✓ |
| gemini-3-flash-attempt2.txt | | | | ✓ | ✓ |
| gemini-3-flash.txt | | | | ✓ | ✓ |
| gemini-3-pro-preview.txt | | | ✓ | ✓ | ✓ |
| gpt-5-1-thinking-extended.txt | ✓ | | | | ✓ |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | | ✓ | ✓ |
| grok.txt | | | | ✓ | ✓ |
| mistral.txt | ✓ | ✓ | | ✓ | ✓ |
| o3.txt | ✓ | | ✓ | ✓ | ✓ |
Question 4: Which of the following properties is compatible with Condorcet consistency, in the context of resolute social choice functions?
- A) Independence of clones
- B) Monotonicity
- C) Strategyproofness
- D) Participation
- E) Borda efficiency: whenever there is a unique Borda winner, then this candidate is the unique winner of the social choice function.
| A | B | C | D | E |
|---|
| Correct | ✓ | ✓ | | | |
| claude-haiku-4-5.txt | ✓ | ✓ | | ✓ | ✓ |
| claude-sonnet-4-6.txt | ✓ | ✓ | | | |
| gemini-3-1-pro-preview.txt | ✓ | ✓ | | | |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | | |
| gemini-3-flash.txt | ✓ | ✓ | | | |
| gemini-3-pro-preview.txt | ✓ | ✓ | | | |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | | | |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | | | |
| grok.txt | ✓ | ✓ | | | |
| mistral.txt | ✓ | ✓ | | ✓ | ✓ |
| o3.txt | ✓ | | | | |
Question 5: Which of the following voting rules satisfies independence of clones?
- A) Plurality
- B) Borda count
- C) Instant-runoff voting
- D) Copeland's method
- E) Plurality with runoff
- F) Schulze method
| A | B | C | D | E | F |
|---|
| Correct | | | ✓ | | | ✓ |
| claude-haiku-4-5.txt | | | | ✓ | | ✓ |
| claude-sonnet-4-6.txt | | | ✓ | | | ✓ |
| gemini-3-1-pro-preview.txt | | | ✓ | | | ✓ |
| gemini-3-flash-attempt2.txt | | | ✓ | | | ✓ |
| gemini-3-flash.txt | | | ✓ | | | ✓ |
| gemini-3-pro-preview.txt | | | ✓ | | | ✓ |
| gpt-5-1-thinking-extended.txt | | | ✓ | | | ✓ |
| gpt-5-2-thinking-extended.txt | | | | | | ✓ |
| grok.txt | | | ✓ | | | ✓ |
| mistral.txt | | ✓ | | ✓ | | ✓ |
| o3.txt | | | | | | ✓ |
Question 6: Which of the following voting rules satisfies participation?
- A) Plurality
- B) Borda count
- C) Instant-runoff voting
- D) Copeland's method
- E) Plurality with runoff
- F) Schulze method
| A | B | C | D | E | F |
|---|
| Correct | ✓ | ✓ | | | | |
| claude-haiku-4-5.txt | ✓ | ✓ | | | | |
| claude-sonnet-4-6.txt | ✓ | ✓ | | | | |
| gemini-3-1-pro-preview.txt | ✓ | ✓ | | | | |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | | | |
| gemini-3-flash.txt | ✓ | ✓ | | | | |
| gemini-3-pro-preview.txt | ✓ | ✓ | | | | |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | ✓ | | ✓ | |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | | | | |
| grok.txt | ✓ | ✓ | | | | |
| mistral.txt | | ✓ | ✓ | ✓ | ✓ | ✓ |
| o3.txt | ✓ | | | | | |
Question 7: Which of the following properties does Proportional Approval Voting (PAV) satisfy in the context of approval-based committee selection?
- A) Pareto-optimality
- B) Core
- C) Extended justified representation (EJR)
- D) If x is the approval winner, then x is part of at least one winning committee
- E) Committee monotonicity
| A | B | C | D | E |
|---|
| Correct | ✓ | | ✓ | | |
| claude-haiku-4-5.txt | ✓ | | ✓ | | |
| claude-sonnet-4-6.txt | ✓ | | ✓ | ✓ | |
| gemini-3-1-pro-preview.txt | ✓ | | ✓ | ✓ | |
| gemini-3-flash-attempt2.txt | ✓ | | ✓ | | |
| gemini-3-flash.txt | ✓ | | ✓ | | |
| gemini-3-pro-preview.txt | ✓ | | ✓ | | |
| gpt-5-1-thinking-extended.txt | ✓ | | ✓ | | |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | ✓ | | |
| grok.txt | ✓ | | ✓ | ✓ | |
| mistral.txt | ✓ | ✓ | ✓ | ✓ | ✓ |
| o3.txt | ✓ | | ✓ | | |
Question 8: Which of the following properties does the Max Nash Welfare (MNW) rule satisfy in the context of allocation of indivisible goods with additive utilities?
- A) Pareto-optimality
- B) Envy-freeness up to one good (EF1)
- C) Envy-freeness up to any good (EFX)
- D) Strategyproofness
- E) Maximizes utilitarian social welfare
- F) Non-bossiness
| A | B | C | D | E | F |
|---|
| Correct | ✓ | ✓ | | | | ✓ |
| claude-haiku-4-5.txt | ✓ | ✓ | | | | |
| claude-sonnet-4-6.txt | ✓ | ✓ | | | | |
| gemini-3-1-pro-preview.txt | ✓ | ✓ | | | | |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | | | |
| gemini-3-flash.txt | ✓ | ✓ | | | | |
| gemini-3-pro-preview.txt | ✓ | ✓ | | | | |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | | | | |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | | | | |
| grok.txt | ✓ | ✓ | | | | |
| mistral.txt | ✓ | ✓ | | | | ✓ |
| o3.txt | ✓ | ✓ | | | | |
Question 9: Which of the following properties does the leximin rule satisfy in the context of allocation of indivisible goods with additive utilities?
- A) Pareto-optimality
- B) Envy-freeness up to one good (EF1)
- C) Envy-freeness up to any good (EFX)
- D) Strategyproofness
- E) Maximizes utilitarian social welfare
- F) Proportionality whenever it is satisfiable
| A | B | C | D | E | F |
|---|
| Correct | ✓ | | | | | ✓ |
| claude-haiku-4-5.txt | ✓ | | | | | ✓ |
| claude-sonnet-4-6.txt | ✓ | ✓ | | | | |
| gemini-3-1-pro-preview.txt | ✓ | | | | | |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | | | ✓ |
| gemini-3-flash.txt | ✓ | ✓ | | | | ✓ |
| gemini-3-pro-preview.txt | ✓ | ✓ | | | | |
| gpt-5-1-thinking-extended.txt | ✓ | | | | | |
| gpt-5-2-thinking-extended.txt | ✓ | | | | | |
| grok.txt | ✓ | ✓ | ✓ | | | |
| mistral.txt | ✓ | ✓ | ✓ | | | ✓ |
| o3.txt | ✓ | ✓ | | | | |
Question 10: Does the Borda count satisfy the majority criterion?
- A) Yes
- B) Yes for two candidates, but not for three or more candidates
- C) Yes for two or three candidates, but not for four or more candidates
- D) No
| A | B | C | D |
|---|
| Correct | | ✓ | | |
| claude-haiku-4-5.txt | | | | ✓ |
| claude-sonnet-4-6.txt | | | | ✓ |
| gemini-3-1-pro-preview.txt | | ✓ | | |
| gemini-3-flash-attempt2.txt | | ✓ | | |
| gemini-3-flash.txt | | ✓ | | |
| gemini-3-pro-preview.txt | | ✓ | | |
| gpt-5-1-thinking-extended.txt | | ✓ | | |
| gpt-5-2-thinking-extended.txt | | ✓ | | |
| grok.txt | | ✓ | | |
| mistral.txt | | ✓ | | |
| o3.txt | | ✓ | | |
Question 11: Consider the domain of single-peaked preferences with an odd number of voters. Which of the following statements is true?
- A) The Borda count satisfies the majority criterion on this domain.
- B) A Condorcet winner always exists on this domain.
- C) A Condorcet loser always exists on this domain.
- D) The Schulze method selects the same alternatives as Copeland's method on this domain.
- E) If voter 1 ranks x bottom-most and voter 2 ranks y bottom-most, then voter 3 ranks either x or y bottom-most.
| A | B | C | D | E |
|---|
| Correct | | ✓ | ✓ | ✓ | ✓ |
| claude-haiku-4-5.txt | | ✓ | | ✓ | |
| claude-sonnet-4-6.txt | | ✓ | | ✓ | ✓ |
| gemini-3-1-pro-preview.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-flash-attempt2.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-flash.txt | | ✓ | | ✓ | |
| gemini-3-pro-preview.txt | | ✓ | | ✓ | |
| gpt-5-1-thinking-extended.txt | | ✓ | ✓ | ✓ | |
| gpt-5-2-thinking-extended.txt | | ✓ | ✓ | ✓ | ✓ |
| grok.txt | | ✓ | | ✓ | |
| mistral.txt | | ✓ | | | |
| o3.txt | | ✓ | ✓ | ✓ | |
Question 12: Suppose there are exactly three candidates and voters have linear order preferences. Which of the following statements is true?
- A) The Borda count selects the same winners as the plurality rule on this domain.
- B) The Schulze method selects the same winners as the minimax method on this domain.
- C) The Gibbard-Satterthwaite theorem applies to this domain.
- D) A Condorcet winner always exists on this domain.
| A | B | C | D |
|---|
| Correct | | ✓ | ✓ | |
| claude-haiku-4-5.txt | | | ✓ | |
| claude-sonnet-4-6.txt | | ✓ | ✓ | |
| gemini-3-1-pro-preview.txt | | ✓ | ✓ | |
| gemini-3-flash-attempt2.txt | | ✓ | ✓ | |
| gemini-3-flash.txt | | ✓ | ✓ | |
| gemini-3-pro-preview.txt | | | ✓ | |
| gpt-5-1-thinking-extended.txt | | ✓ | | |
| gpt-5-2-thinking-extended.txt | | ✓ | ✓ | |
| grok.txt | | | ✓ | |
| mistral.txt | | ✓ | ✓ | |
| o3.txt | | ✓ | ✓ | |
Question 13: Suppose there are exactly two candidates, voters have linear order preferences, and the number of voters is odd. Which of the following statements is true?
- A) The Borda count selects the same winners as the plurality rule on this domain.
- B) The Schulze method selects the same winners as the minimax method on this domain.
- C) The Gibbard-Satterthwaite theorem applies to this domain.
- D) A Condorcet winner always exists on this domain.
- E) The number of different voting rules that satisfy the anonymity, neutrality, and positive responsiveness is odd.
| A | B | C | D | E |
|---|
| Correct | ✓ | ✓ | | ✓ | ✓ |
| claude-haiku-4-5.txt | ✓ | ✓ | | ✓ | |
| claude-sonnet-4-6.txt | ✓ | ✓ | | ✓ | ✓ |
| gemini-3-1-pro-preview.txt | ✓ | ✓ | | ✓ | ✓ |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | ✓ | ✓ |
| gemini-3-flash.txt | ✓ | ✓ | | ✓ | ✓ |
| gemini-3-pro-preview.txt | ✓ | ✓ | | ✓ | ✓ |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | | ✓ | ✓ |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | | ✓ | ✓ |
| grok.txt | ✓ | ✓ | | ✓ | ✓ |
| mistral.txt | ✓ | ✓ | ✓ | ✓ | |
| o3.txt | ✓ | ✓ | | ✓ | ✓ |
Question 14: Which of the following rules satisfy both Justified Representation (JR) and committee monotonicity in the context of approval-based committee selection?
- A) Proportional Approval Voting (PAV)
- B) Method of Equal Shares (MES)
- C) Sequential Proportional Approval Voting (SPAV)
- D) Sequential Phragmén's method
- E) Greedy Chamberlin-Courant (Greedy CC)
| A | B | C | D | E |
|---|
| Correct | | | | ✓ | ✓ |
| claude-haiku-4-5.txt | | ✓ | | ✓ | |
| claude-sonnet-4-6.txt | | | ✓ | ✓ | |
| gemini-3-1-pro-preview.txt | | | | ✓ | ✓ |
| gemini-3-flash-attempt2.txt | | | | ✓ | ✓ |
| gemini-3-flash.txt | | | | ✓ | ✓ |
| gemini-3-pro-preview.txt | | | ✓ | ✓ | ✓ |
| gpt-5-1-thinking-extended.txt | | ✓ | ✓ | ✓ | ✓ |
| gpt-5-2-thinking-extended.txt | | | ✓ | ✓ | ✓ |
| grok.txt | | | ✓ | ✓ | |
| mistral.txt | ✓ | ✓ | ✓ | ✓ | |
| o3.txt | | ✓ | | ✓ | |
Question 15: Which of the following rules satisfy the Condorcet loser criterion?
- A) Plurality
- B) Borda count
- C) Instant-runoff voting
- D) Minimax Condorcet
- E) Schulze method
| A | B | C | D | E |
|---|
| Correct | | ✓ | ✓ | | ✓ |
| claude-haiku-4-5.txt | | | ✓ | ✓ | ✓ |
| claude-sonnet-4-6.txt | | ✓ | ✓ | | ✓ |
| gemini-3-1-pro-preview.txt | | ✓ | ✓ | | ✓ |
| gemini-3-flash-attempt2.txt | | ✓ | ✓ | | ✓ |
| gemini-3-flash.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-pro-preview.txt | | ✓ | ✓ | | ✓ |
| gpt-5-1-thinking-extended.txt | | ✓ | ✓ | | ✓ |
| gpt-5-2-thinking-extended.txt | | ✓ | ✓ | | ✓ |
| grok.txt | | ✓ | ✓ | ✓ | ✓ |
| mistral.txt | | ✓ | ✓ | ✓ | ✓ |
| o3.txt | | | | ✓ | ✓ |
Question 16: Which of the following properties of a social choice function are preserved under refinement, i.e., if a social choice function f satisfies a property P, then every refinement of f also satisfies P?
- A) Condorcet consistency
- B) Participation
- C) Neutrality
- D) Pareto-optimality
- E) Majority loser criterion
| A | B | C | D | E |
|---|
| Correct | ✓ | | | ✓ | ✓ |
| claude-haiku-4-5.txt | ✓ | ✓ | | ✓ | ✓ |
| claude-sonnet-4-6.txt | ✓ | | | ✓ | ✓ |
| gemini-3-1-pro-preview.txt | ✓ | | | ✓ | ✓ |
| gemini-3-flash-attempt2.txt | ✓ | | | ✓ | ✓ |
| gemini-3-flash.txt | ✓ | | | ✓ | ✓ |
| gemini-3-pro-preview.txt | ✓ | | | ✓ | ✓ |
| gpt-5-1-thinking-extended.txt | ✓ | | | ✓ | ✓ |
| gpt-5-2-thinking-extended.txt | | | | ✓ | ✓ |
| grok.txt | | | | ✓ | ✓ |
| mistral.txt | ✓ | ✓ | ✓ | ✓ | ✓ |
| o3.txt | ✓ | | | ✓ | ✓ |
Question 17: Consider social choice functions with linear order preferences. Suppose that the number of voters is strictly smaller than the number of candidates. Which of the following statements is true in this case?
- A) The Borda count satisfies the majority criterion on this domain.
- B) A Condorcet winner always exists on this domain.
- C) The plurality rule is strategyproof on this domain.
- D) At least one alternative is Pareto-dominated on this domain.
- E) None of the above statements is true.
| A | B | C | D | E |
|---|
| Correct | | | | | ✓ |
| claude-haiku-4-5.txt | | | | ✓ | |
| claude-sonnet-4-6.txt | | | | | ✓ |
| gemini-3-1-pro-preview.txt | | | | | ✓ |
| gemini-3-flash-attempt2.txt | | | | | ✓ |
| gemini-3-flash.txt | | | | | ✓ |
| gemini-3-pro-preview.txt | | | | | ✓ |
| gpt-5-1-thinking-extended.txt | | | | | ✓ |
| gpt-5-2-thinking-extended.txt | | | | | ✓ |
| grok.txt | | | | | ✓ |
| mistral.txt | | | | | ✓ |
| o3.txt | | | | | ✓ |
Question 18: Consider the allocation of indivisible goods with submodular utilities. Which of the following methods satisfy envy-freeness up to one good (EF1)?
- A) Max Nash Welfare (MNW)
- B) Leximin
- C) Round-robin
- D) Envy-cycle elimination
- E) Every method maximizing a strictly increasing and concave welfare function
| A | B | C | D | E |
|---|
| Correct | | | | ✓ | |
| claude-haiku-4-5.txt | ✓ | ✓ | | | ✓ |
| claude-sonnet-4-6.txt | | | ✓ | ✓ | |
| gemini-3-1-pro-preview.txt | | | | ✓ | |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | ✓ | ✓ | |
| gemini-3-flash.txt | ✓ | ✓ | ✓ | ✓ | |
| gemini-3-pro-preview.txt | ✓ | | ✓ | ✓ | |
| gpt-5-1-thinking-extended.txt | | | | ✓ | |
| gpt-5-2-thinking-extended.txt | | | | ✓ | |
| grok.txt | ✓ | ✓ | ✓ | ✓ | ✓ |
| mistral.txt | ✓ | ✓ | ✓ | ✓ | |
| o3.txt | ✓ | ✓ | ✓ | ✓ | |
Question 19: Consider the allocation of indivisible goods with additive utilities. Which of the following statements is true?
- A) The round robin method satisfies envy-freeness up to one good (EF1).
- B) The Max Nash Welfare (MNW) rule satisfies fractional Pareto-optimality (fPO).
- C) Envy-freeness up to any good (EFX) is equivalent to envy-freeness up to one good (EF1) in this setting.
- D) Every Pareto-optimal allocation gives at least one item to each agent.
| A | B | C | D |
|---|
| Correct | ✓ | | | |
| claude-haiku-4-5.txt | ✓ | ✓ | | |
| claude-sonnet-4-6.txt | ✓ | ✓ | | |
| gemini-3-1-pro-preview.txt | ✓ | | | |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | |
| gemini-3-flash.txt | ✓ | ✓ | | |
| gemini-3-pro-preview.txt | ✓ | ✓ | | |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | | |
| gpt-5-2-thinking-extended.txt | ✓ | ✓ | | |
| grok.txt | ✓ | ✓ | | |
| mistral.txt | ✓ | ✓ | | |
| o3.txt | ✓ | ✓ | | |
Question 20: Which of the following properties does the Borda count satisfy?
- A) Independence of clones
- B) Never selects a Condorcet loser
- C) When x is the unique winner in profile P, and x is among the winners in profile P', then x is the unique winner in profile P'
- D) If every voter prefers x to y, then x is the unique winner
| A | B | C | D |
|---|
| Correct | | ✓ | ✓ | |
| claude-haiku-4-5.txt | | | ✓ | |
| claude-sonnet-4-6.txt | | ✓ | ✓ | |
| gemini-3-1-pro-preview.txt | | ✓ | ✓ | |
| gemini-3-flash-attempt2.txt | | ✓ | | |
| gemini-3-flash.txt | | ✓ | | |
| gemini-3-pro-preview.txt | | ✓ | | |
| gpt-5-1-thinking-extended.txt | | ✓ | | |
| gpt-5-2-thinking-extended.txt | | ✓ | | |
| grok.txt | | ✓ | | |
| mistral.txt | ✓ | ✓ | | ✓ |
| o3.txt | | | | |
Question 21: Suppose there are m alternatives that are ordered on a left-to-right axis. How many different linear orders are single-peaked with respect to this axis?
- A) m!
- B) 2^m
- C) 2^(m-1)
- D) \binom{m}{2} + 1
| A | B | C | D |
|---|
| Correct | | | ✓ | |
| claude-haiku-4-5.txt | | | ✓ | |
| claude-sonnet-4-6.txt | | | ✓ | |
| gemini-3-1-pro-preview.txt | | | ✓ | |
| gemini-3-flash-attempt2.txt | | | ✓ | |
| gemini-3-flash.txt | | | ✓ | |
| gemini-3-pro-preview.txt | | | ✓ | |
| gpt-5-1-thinking-extended.txt | | | ✓ | |
| gpt-5-2-thinking-extended.txt | | | ✓ | |
| grok.txt | | | ✓ | |
| mistral.txt | | | ✓ | |
| o3.txt | | | ✓ | |
Question 22: Suppose there are exactly 4 alternatives. What is the minimum number of voters with linear order preferences such that there exists a profile with no Condorcet winner?
| A | B | C | D |
|---|
| Correct | | ✓ | | |
| claude-haiku-4-5.txt | | | ✓ | |
| claude-sonnet-4-6.txt | ✓ | | | |
| gemini-3-1-pro-preview.txt | | ✓ | | |
| gemini-3-flash-attempt2.txt | ✓ | | | |
| gemini-3-flash.txt | ✓ | | | |
| gemini-3-pro-preview.txt | ✓ | | | |
| gpt-5-1-thinking-extended.txt | ✓ | | | |
| gpt-5-2-thinking-extended.txt | ✓ | | | |
| grok.txt | ✓ | | | |
| mistral.txt | | ✓ | | |
| o3.txt | ✓ | | | |
Question 23: Which of the following apportionment methods satisfies lower quota?
- A) Hamilton's method
- B) D'Hondt's method
- C) Sainte-Laguë's method
- D) Adams' method
- E) Webster's method
- F) Balinski-Young's quota method
| A | B | C | D | E | F |
|---|
| Correct | ✓ | ✓ | | | | ✓ |
| claude-haiku-4-5.txt | ✓ | | | | | |
| claude-sonnet-4-6.txt | ✓ | ✓ | | | | ✓ |
| gemini-3-1-pro-preview.txt | ✓ | ✓ | | | | ✓ |
| gemini-3-flash-attempt2.txt | ✓ | ✓ | | | | ✓ |
| gemini-3-flash.txt | ✓ | ✓ | | | | ✓ |
| gemini-3-pro-preview.txt | ✓ | | | | | ✓ |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | | | | ✓ |
| gpt-5-2-thinking-extended.txt | ✓ | | | | | ✓ |
| grok.txt | ✓ | | | ✓ | | ✓ |
| mistral.txt | ✓ | | | ✓ | | ✓ |
| o3.txt | ✓ | | | | | ✓ |
Question 24: Which of the following apportionment methods satisfies house monotonicity?
- A) Hamilton's method
- B) D'Hondt's method
- C) Sainte-Laguë's method
- D) Adams' method
- F) Balinski-Young's quota method
| A | B | C | D | F |
|---|
| Correct | | ✓ | ✓ | ✓ | |
| claude-haiku-4-5.txt | | ✓ | ✓ | ✓ | |
| claude-sonnet-4-6.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-1-pro-preview.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-flash-attempt2.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-flash.txt | | ✓ | ✓ | ✓ | ✓ |
| gemini-3-pro-preview.txt | | ✓ | ✓ | ✓ | |
| gpt-5-1-thinking-extended.txt | | ✓ | ✓ | ✓ | |
| gpt-5-2-thinking-extended.txt | | ✓ | ✓ | ✓ | ✓ |
| grok.txt | | ✓ | ✓ | ✓ | ✓ |
| mistral.txt | | ✓ | ✓ | ✓ | |
| o3.txt | | ✓ | ✓ | ✓ | |
Question 25: Which of the following statements about apportionment methods is true?
- A) Hamilton's method can violate house monotonicity, but it always satisfies lower quota.
- B) D'Hondt's method never gives more seats to a larger party than to a smaller party.
- C) Every population monotone apportionment method satisfies upper quota.
- D) When there are only two parties, all divisor methods give the same apportionment.
- E) Under the Sainte-Laguë method, if we multiply the vote count of every party by an odd number, then the apportionment does not change.
| A | B | C | D | E |
|---|
| Correct | ✓ | | | | ✓ |
| claude-haiku-4-5.txt | ✓ | | | | |
| claude-sonnet-4-6.txt | ✓ | | | ✓ | ✓ |
| gemini-3-1-pro-preview.txt | ✓ | | | | ✓ |
| gemini-3-flash-attempt2.txt | ✓ | | | | ✓ |
| gemini-3-flash.txt | ✓ | | | | ✓ |
| gemini-3-pro-preview.txt | ✓ | | | | ✓ |
| gpt-5-1-thinking-extended.txt | ✓ | ✓ | | | ✓ |
| gpt-5-2-thinking-extended.txt | ✓ | | | | ✓ |
| grok.txt | ✓ | | | ✓ | ✓ |
| mistral.txt | ✓ | | | ✓ | |
| o3.txt | ✓ | ✓ | | ✓ | ✓ |