1What is the concept of Condorcet winner?
A Condorcet winner is the unique equilibrium policy (stable) and coincides with the median-ranks bliss point
2How is the Median Voter Theorem proven?
By ordering individuals based on their bliss point
3What does the Median Voter Theorem state?
The median voter's policy choice is rational, Paretian, IIA and non-dictatorial (no UD)
4How can the Arrow's Impossibility Theorem be 'solved'?
By putting structure on preferences - assuming single-peaked preferences.
5What does the Condorcet winner correspond to?
The Median voter's policy choice
6What is the significance of the Median voter in voting?
The Median voter cuts in half the distribution of bliss points.
7How does the Median voter theorem relate to majority voting?
It shows that majority voting is rational, Paretian, IIA and non-dictatorial
8What is the Median Voter Theorem
The theory states that in direct democracy, the median voter is the one with exactly half of the voters preferring policies to their left and half preferring policies to their right
9What is the Arrow Impossibility Theorem?
The Arrow Impossibility Theorem states that it is impossible to have a decision rule that is Paretian, Rational, Nondictatorial, based on Universal Domain, and Independent of Irrelevant Alternatives.
10The Condorcet paradox occurs when...
Rational individuals in a majority voting system lead to an irrational social preference.
11 Single-peaked preferences in pairwise majority voting lead to...
a Condorcet winner where each voter has a unique bliss point
12Who is the median voter in a society?
The median voter is the individual in the middle of the distribution of bliss points
- $\frac{N-1}{2}$ to the left and $\frac{N+1}{2}$
- voter denoted by a$_m$ and his bliss point by p$_m$
13What are the conditions for the Median Voter Theorem?
The conditions for the Median Voter Theorem include direct democracy via majoritarian voting, sincere voting (vot according to highest utility U(p*(a$_i$)), open agenda, and all voters having single-peaked preferences.
- if these are satisfied, there is always a Condorcet winner, unique equilibrium policy that coincides with the median voter's bliss point
14When does cycling occur?
Occurs when rational individuals lead to an irrational social preference ie when majority voting does not deliver a consistent aggregation of individual preferences and there is no equilibrium of the voting process that selects a unique policy choice
15What is cycling?
Keep voting/cycling through possibilities when there is no stable electoral outcome, without a Condorcet winner, the sequence of voting matters a lot
16⪰$_S$ =
F(⪰$_1$, ⪰$_2$, ⪰$_2$, …, ⪰$_3$ )
17Write the definition of single peaked unidimensional preferences mathematically
- a finite set of $\mathcal{P}\ ⊂ ℝ
- let p*(a$_i$) ∈ $\mathcal{P}\ be individual i's unique bliss point over $\mathcal{P}\
- policy preferences of citizen i are single peaked if and only if:
/*For all p'', p' ∈ $\mathcal{P}\ such that p''<p'≤ p*(a$_i$) or p’’>p’≥ p*(a$_i$), Ui(p’’)< Ui(p’)≤ Ui(p*(a$_i$) )
- strict concavity of U$_i$ is sufficient
18Prove the Median Voter Theorem
- Order individuals according their bliss point p*(a$_i$)
- N is an odd number so p$_m$ is uniquely defined (but there can be several individuals at the median
- Suppose there is a vote between p$_m$ and other policy p'' < p$_m$
- Apply the definition of single peaked preferences
- For every individual i with p*(a$_i$) > p$_m$, we have p*(a$_i$) > p$_m$>p'' so U$_i$(p$_m$) > U$_i$(p'')
- Individuals will vote sincerely and thus favour p$_m$
- The same conclusion goes for p'' > p$_m$