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Prove that the axiom of anonymity implies non-existence of a dictator.
Mar 1, 2024
Prove that the axiom of anonymity implies non-existence of a dictator.
Solution by Steps
step 1
Evaluate the limit of the given expression as n n approaches infinity
step 2
Simplify the expression inside the limit: ln⁡(n+1)⋅3n⋅n3ln⁡(n)⋅3n+1⋅(n+1)3 \frac{\ln(n+1) \cdot 3^n \cdot n^3}{\ln(n) \cdot 3^{n+1} \cdot (n+1)^3}
step 3
Factor out 3n 3^n from the numerator and 3n+1 3^{n+1} from the denominator to get ln⁡(n+1)⋅n33⋅ln⁡(n)⋅(n+1)3 \frac{\ln(n+1) \cdot n^3}{3 \cdot \ln(n) \cdot (n+1)^3}
step 4
Apply L'Hôpital's rule if necessary or recognize that ln⁡(n+1)/ln⁡(n) \ln(n+1)/\ln(n) approaches 1 as n n approaches infinity
step 5
Notice that n3/(n+1)3 n^3/(n+1)^3 approaches 1 as n n approaches infinity
step 6
Combine the results from steps 4 and 5 and multiply by 1/3 1/3 from the factorization in step 3 to get the final limit of 1/3 1/3
Answer
lim⁡n→∞∣ln⁡(n+1)⋅3n⋅n3ln⁡(n)⋅3n+1⋅(n+1)3∣=13 \lim_{n \to \infty} \left| \frac{\ln(n+1) \cdot 3^n \cdot n^3}{\ln(n) \cdot 3^{n+1} \cdot (n+1)^3} \right| = \frac{1}{3}
Key Concept
Limits of functions as n n approaches infinity
Explanation
The limit of the given function as n n approaches infinity is found by simplifying the expression and recognizing that the ratio of logarithms and the ratio of polynomials both approach 1. The constant factor of 1/3 1/3 comes from the 3n 3^n terms after factoring.
Consider the Hotelling/Downs model of political competition with a unit mass of consumers uniformly distributed over the interval [0,1][0,1]. There are two candidates (candidate 1 and candidate 2 ) who simultaneously choose a policy on the interval [0,1][0,1], and then voters vote for one of the two candidates. Let sjs_{j} denote candidate j′sj^{\prime} s policy, j=1,2j=1,2. A voter's utility depends on the policy ss that wins and his location i∈[0,1]i \in[0,1] and is given by ui(s)=−∣i−s∣u_{i}(s)=-|i-s|. A candidate gets a utility of 1 from winning and -1 from losing. Ties are broken with a fair coin toss. (a) Suppose candidate 1's strategy s_{1}<0.5. What is candidate 2's best response (that is, specify all the s2s_{2} 's which are optimal for candidate 2 , given s1s_{1} ). (b) Suppose candidate 1's strategy s1=0.5s_{1}=0.5. What is candidate 2's best response? (c) Suppose candidate 1's strategy s_{1}>0.5. What is candidate 2's best response? (d) In a graph with s1s_{1} on the horizontal axis and s2s_{2} on the vertical axis, plot these best responses for candidate 2 . (e) On the same graph plot candidate 1's best responses for each s2s_{2}. (f) Depict the Nash equilibrium on this graph where both players are playing a best response to each other. Consider the same setting as in the previous question, but now assume that each candidate maximizes their vote share. (a) Suppose candidate 1's strategy is either s_{1}<0.5 or s_{1}>0.5. Does candidate 2 have a best response? If not, explain (in one sentence) why. (b) Suppose candidate 1's strategy s1=0.5s_{1}=0.5. What is candidate 2's best response? (c) What is the Nash equilibrium of the game?
Solution by Steps
step 1
Consider candidate 1's strategy s_1 < 0.5
step 2
Candidate 2's best response is to choose s2 s_2 just slightly more than s1 s_1 to capture the voters at s1 s_1 and all voters to the right, up to 0.5
step 3
Therefore, the optimal s2 s_2 for candidate 2, given s_1 < 0.5 , is s2=s1+ϵ s_2 = s_1 + \epsilon , where ϵ \epsilon is an infinitesimally small positive number
Answer
s2=s1+ϵ s_2 = s_1 + \epsilon for \epsilon > 0 and infinitesimally small
Key Concept
Best Response Strategy
Explanation
In the Hotelling/Downs model, a candidate's best response is to position themselves just to the right of their opponent if their opponent is left of the median voter (0.5).
Solution by Steps
step 1
Consider candidate 1's strategy s1=0.5 s_1 = 0.5
step 2
Candidate 2's best response is to choose s2 s_2 equal to 0.5 to split the votes and rely on the coin toss, as moving left or right would result in fewer votes
Answer
s2=0.5 s_2 = 0.5
Key Concept
Indifference Strategy
Explanation
When candidate 1 chooses the median voter position, candidate 2 is indifferent to any position because any deviation results in a loss, so matching the position results in a tie.
Solution by Steps
step 1
Consider candidate 1's strategy s_1 > 0.5
step 2
Candidate 2's best response is to choose s2 s_2 just slightly less than s1 s_1 to capture the voters at s1 s_1 and all voters to the left, up to 0.5
step 3
Therefore, the optimal s2 s_2 for candidate 2, given s_1 > 0.5 , is s2=s1−ϵ s_2 = s_1 - \epsilon , where ϵ \epsilon is an infinitesimally small positive number
Answer
s2=s1−ϵ s_2 = s_1 - \epsilon for \epsilon > 0 and infinitesimally small
Key Concept
Best Response Strategy
Explanation
In the Hotelling/Downs model, a candidate's best response is to position themselves just to the left of their opponent if their opponent is right of the median voter (0.5).
Solution by Steps
step 1
Plot the best responses for candidate 2 on a graph with s1 s_1 on the horizontal axis and s2 s_2 on the vertical axis
step 2
For s_1 < 0.5 , plot s2 s_2 just above s1 s_1
step 3
For s1=0.5 s_1 = 0.5 , plot s2=0.5 s_2 = 0.5
step 4
For s_1 > 0.5 , plot s2 s_2 just below s1 s_1
Answer
The graph shows three segments: a line just above the diagonal for s_1 < 0.5 , a point at (0.5, 0.5), and a line just below the diagonal for s_1 > 0.5 .
Key Concept
Graphical Representation of Best Responses
Explanation
The graph visually represents the best response strategies of candidate 2 for each possible position of candidate 1.
Solution by Steps
step 1
Plot candidate 1's best responses for each s2 s_2 on the same graph
step 2
For s_2 < 0.5 , candidate 1's best response is just to the right of s2 s_2
step 3
For s2=0.5 s_2 = 0.5 , candidate 1's best response is s1=0.5 s_1 = 0.5
step 4
For s_2 > 0.5 , candidate 1's best response is just to the left of s2 s_2
Answer
The graph shows three segments for candidate 1: a line just above the diagonal for s_2 < 0.5 , a point at (0.5, 0.5), and a line just below the diagonal for s_2 > 0.5 .
Key Concept
Graphical Representation of Best Responses
Explanation
The graph visually represents the best response strategies of candidate 1 for each possible position of candidate 2.
Solution by Steps
step 1
Depict the Nash equilibrium on the graph where both players are playing a best response to each other
step 2
The Nash equilibrium occurs where the best response curves intersect
Answer
The Nash equilibrium is at the point (0.5, 0.5) on the graph.
Key Concept
Nash Equilibrium
Explanation
The Nash equilibrium is the point where both candidates choose the median voter position, resulting in a tie and a fair coin toss to determine the winner.
Solution by Steps
step 1
Consider candidate 1's strategy is either s_1 < 0.5 or s_1 > 0.5 and each candidate maximizes their vote share
step 2
Candidate 2 does not have a single best response because by positioning just to the right or left of s1 s_1 , they can capture slightly more than half of the vote share
Answer
No single best response for candidate 2; positioning just to the right or left of s1 s_1 maximizes vote share.
Key Concept
Maximizing Vote Share
Explanation
Without a fixed position, candidate 2 can always adjust their position to capture just over half of the voters, so there is no single best response.
Solution by Steps
step 1
Consider candidate 1's strategy s1=0.5 s_1 = 0.5 and each candidate maximizes their vote share
step 2
Candidate 2's best response is to choose s2=0.5 s_2 = 0.5 to split the votes evenly
Answer
s2=0.5 s_2 = 0.5
Key Concept
Vote Share Maximization at Median
Explanation
When candidate 1 chooses the median voter position, candidate 2 maximizes their vote share by also choosing the median voter position, resulting in an even split of votes.
Solution by Steps
step 1
Determine the Nash equilibrium of the game when each candidate maximizes their vote share
step 2
The Nash equilibrium occurs where both candidates choose s1=s2=0.5 s_1 = s_2 = 0.5 to maximize their vote share
Answer
The Nash equilibrium is s1=s2=0.5 s_1 = s_2 = 0.5
Key Concept
Nash Equilibrium with Vote Share Maximization
Explanation
The Nash equilibrium in this scenario is where both candidates choose the median voter position, resulting in an even split of the votes.
Consider the Hotelling/Downs model of political competition with a unit mass of consumers uniformly distributed over the interval [0,1][0,1]. There are two candidates (candidate 1 and candidate 2) who simultaneously choose a policy on the interval [0,1][0,1], and then voters vote for one of the two candidates. Let sjs_{j} denote candidate j′sj^{\prime} s policy, j=1,2j=1,2. A voter's utility depends on the policy ss that wins and his location i∈[0,1]i \in[0,1] and is given by ui(s)=−∣i−s∣u_{i}(s)=-|i-s|. A candidate gets a utility of 1 from winning and -1 from losing. Ties are broken with a fair coin toss. (a) Suppose candidate 1's strategy s_{1}<0.5. What is candidate 2's best response (that is, specify all the s2s_{2} 's which are optimal for candidate 2 , given s1s_{1} ). (b) Suppose candidate 1's strategy s1=0.5s_{1}=0.5. What is candidate 2's best response? (c) Suppose candidate 1's strategy s_{1}>0.5. What is candidate 2's best response? (d) In a graph with s1s_{1} on the horizontal axis and s2s_{2} on the vertical axis, plot these best responses for candidate 2 . (e) On the same graph plot candidate 1's best responses for each s2s_{2}. (f) Depict the Nash equilibrium on this graph where both players are playing a best response to each other.
Solution by Steps
step 1
Consider candidate 1's strategy s_1 < 0.5 . To maximize the number of votes, candidate 2 should choose a policy just to the right of candidate 1's policy
step 2
The best response for candidate 2 is to choose s2 s_2 such that s_2 > s_1 and s2 s_2 is as close to s1 s_1 as possible without being equal to s1 s_1
Answer
Candidate 2's best response is any s2 s_2 where s_1 < s_2 < 0.5 .
Key Concept
Best Response Strategy
Explanation
In the Hotelling/Downs model, a candidate's best response is to position themselves just to the right of their opponent if their opponent is left of the center (0.5).
Solution by Steps
step 1
If candidate 1's strategy s1=0.5 s_1 = 0.5 , candidate 2 can choose any policy s2 s_2 because the voters are distributed uniformly
step 2
Candidate 2 will split the vote with candidate 1, and the winner will be decided by a coin toss
Answer
Candidate 2's best response is any s2 s_2 on the interval [0,1][0,1] since the outcome is determined by a coin toss.
Key Concept
Indifference in Position
Explanation
When candidate 1 chooses the center position, candidate 2 is indifferent to their own position as they will split the vote evenly.
Solution by Steps
step 1
Consider candidate 1's strategy s_1 > 0.5 . To maximize the number of votes, candidate 2 should choose a policy just to the left of candidate 1's policy
step 2
The best response for candidate 2 is to choose s2 s_2 such that s_2 < s_1 and s2 s_2 is as close to s1 s_1 as possible without being equal to s1 s_1
Answer
Candidate 2's best response is any s2 s_2 where 0.5 < s_2 < s_1 .
Key Concept
Best Response Strategy
Explanation
In the Hotelling/Downs model, a candidate's best response is to position themselves just to the left of their opponent if their opponent is right of the center (0.5).
Solution by Steps
step 1
To plot the best responses for candidate 2, draw a vertical line at s1=0.5 s_1 = 0.5 and two diagonal lines approaching this vertical line from both sides without touching it
step 2
The diagonal lines represent the best response strategies for s_1 < 0.5 and s_1 > 0.5
Answer
The graph shows a vertical line at s1=0.5 s_1 = 0.5 and two diagonal lines approaching from both sides.
Key Concept
Graphical Representation of Best Responses
Explanation
The graph visually represents the best response strategies for candidate 2 in relation to candidate 1's policies.
Solution by Steps
step 1
To plot candidate 1's best responses for each s2 s_2 , draw a horizontal line at s2=0.5 s_2 = 0.5 and two diagonal lines approaching this horizontal line from both sides without touching it
step 2
The diagonal lines represent the best response strategies for s_2 < 0.5 and s_2 > 0.5
Answer
The graph shows a horizontal line at s2=0.5 s_2 = 0.5 and two diagonal lines approaching from both sides.
Key Concept
Graphical Representation of Best Responses
Explanation
The graph visually represents the best response strategies for candidate 1 in relation to candidate 2's policies.
Solution by Steps
step 1
The Nash equilibrium occurs where both candidates are playing a best response to each other
step 2
On the graph, this is where the best response lines for candidate 1 and candidate 2 intersect
Answer
The Nash equilibrium is at (s1,s2)=(0.5,0.5) (s_1, s_2) = (0.5, 0.5) .
Key Concept
Nash Equilibrium
Explanation
The Nash equilibrium in the Hotelling/Downs model occurs when both candidates choose the median voter's position, which is at the center of the policy space.
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