Spatial Modelling Flashcards
(24 cards)
What is one way to overcome the issue that Arrow runs into of incoherent group choices (dictators)?
One way to overcome this problem is to change the universal domain (condition U) in such a way that an individual’s preferences reflect some sort of a consensus
Who invented the idea of domain restriction?
Duncan Black
What is the effect of preference orderings being limited in the real world (domain restriction)?
- Some preferences do not occur
- Some preference orderings are more likely to occur than others
Duncan Black facts
- Economist (1908-1991)
- Never came to england (stayed in Scotland)
- Social choice theorist (another founding father)
- Working on a solution for Condorcet paradox
- Black rediscovered ideas that had been advanced earlier by Borda and Condorcet, which were lost, rediscovered by Dodgson, then lost again “[so] he must be viewed as the true founder of social choice”
- Invented the famous Universal Domain restriction: Single Peakedness
What do you put on the x and y axis of a single peakedness graph?
- X-axis: Alternatives (fixed scale)
- Y-axis: Ordinal rank
What are the requirements for a group of agents to be said to have single-peaked preferences?
- Each agent has an ideal choice in the set
- For each agent, outcomes that are further from his ideal choice are preferred less
Characteristics of single-peakedness
- All alternatives to be decided can be linearly ordered, left to right
- All voters agree on the left to right ordering; they disagree on their choices
- Everyone has a favorite point; but the favorite point differs among voters
- For each voter, as we move to the left of her favorite alternative, her utility goes down; as we move to the right of her favorite alternative her utility goes down
What types of plot can be considered single-peaked?
“We will refer to a curve which is either always upward-sloping, or always downward-sloping, or which is upward-sloping to a particular point and downward-sloping beyond that point, as being a single-peaked curve. On this definition a single-peaked curve is one which changes its direction at most once, from up to down”
Relation between Condorcet paradox/winner and single-peakedness
- Condorcet cycles cannot occur when preferences are single-peaked
- Single-peakedness as a domain restriction, removes the occurrence of Condorcet cycles
Black’s theorem
- If there is an odd number of voters that display single-peaked preferences, then a Condorcet winner exists
– ie. if preferences are single-peaked, then majority voting on pairwise alternatives yields transitive group decisions - In other words: by restricting condition U(D), we can prove that the Condorcet paradox will never arise
What is one possible reason that voter’s preferences are single-peaked?
Deliberation
Assumptions of single-peakedness
- Number of voters is odd
- We can order alternatives on 1 single dimension
What does the maximum of a single-peaked curve identify?
The maximum of the ordering is the bliss point of the median voter - the voter whose bliss point has, on the common ordering, as many voters’ bliss points to one side as it has to the other
- If all voters have single-peaked curves as preferences, then the median motion will be adopted by the committee
What does Black’s median voter theorem posit?
Black’s median voter theorem states that, when preferences are single-peaked, majority rule preferences are transitive and the feasible alternative which lies highest on the preferences of the median voter is a majority winner
- When preferences are single-peaked, the ideal point of the median voter is the majority preference
What is Black’s median voter theorem a formalization of?
Hotelling’s model (1929)(example of ice cream sellers moving along beach)
- Also inspired the work of Anthony Downs
How did Black formalize Hotelling’s model?
If (1) we can order alternatives along a single dimension and (2) preference orderings are single-peaked along this dimension, then:
- There is an identifiable majority preference
- This is the preference of the median voter
- The ideal point of the median voter has an empty winset
– The ideal point of the median voter cannot be defeated by a majority
– The ideal point of the median voter is the majority preference
When graphed what do empty and non-empty winsets look like?
- In an empty winset the vertical line will hit the peak of the curve (the single peak)
- In a non-empty winset the vertical line will hit somewhere other than the peak of the curve creating a triangle
What does a winset mean?
The winset of alternative A is the set of alternative that can defeat A by a majority vote
What does it mean if A’s winset it empty?
If A’s winset is empty, then A cannot be defeated by another alternative
- If so, we have identified the majority preference: a Condorcet winner
- A majority winset of y, is the set of alternatives that is preferred by a majority over y
Winset in connection with Black
The ideal point of the median voter has an empty winset
Winset in connection with Arrow
- It is not always the case that an empty winset exists
- Condition U: all preferences (if complete and transitive) may be held (even cyclic ones)
- In case of cyclic majority; the winset is not empty
Median voter theorem and politics
- This finding is very important for the proper functioning of democratic systems of government, which rest upon choices made by groups
- If preferences are organized along a single dimension, we can identify a group choice that is preferred by a majority
Assumptions of median voter theorem
- Full participation is assumed ie. all members of the group have to vote
- They participate sincerely ie. they do not misrepresent their preferences and reveal them honestly
- The median voter theory assumes that politician’s lack ideological convictions, which could lead them to position themselves away from the median voter
- The median voter model assumes perfect information along three dimensions
– Voter knowledge of the issues
– Politician knowledge of the issues
– Politician knowledge of voter preferences - There is only one single dimension that plays a role
What types of voting behavior would violate the assumptions of median voter theorem (and potentially render its predictions invalid?
- Abstention; many voters do not vote
- Some voters may vote strategically; voting for a certain candidate not because they prefer that candidate, but to avoid other candidates (eg. independent party voters in the US may vote for democrats)
