Voting Flashcards
Explain what we mean by social choice theory
Social choice theory is a framework for analysis of combining individual opinions, preferences, interests or welfare to reach a collective decision or social welfare in some sense.
How to find and decide what choices to make depending on agents preferences.
- A social choice function - finding the one best choice overall
- A social welfare function - finding a ranking of the best choices overall
Explain Plurality voting, its strengths and weaknesses
Every voter submits their preference order and the winner is the outcome
ranked first most times. It is a very simple voting procedure, but is open for strategic manipulation.
Can also have a condorcet paradox.
- No matter the winner the majority of the voters will be unhappy with the outcome
Explain Simple majority voting, its strengths and weaknesses
Simple majority voting is plurality voting with only two possible outcomes. It is not easy to manipulate, but in reality there are often more than 2 possible outcomes.
It is straightforward but limited to systems that can implement it.
Explain Sequential majority election, its strengths and weaknesses
Sequential majority elections are a series of plurality elections in order to determine a winner. Typically there are pairwise elections (simple majority voting).
The outcome is sensitive to order and strategic manipulation from voters.
Explain the Borda count
Aggregate through for each preference list for all agents and give it some number based on its rank. The one with highest Borda count is the social outcome.
-Takes into account more information in the voter preference list than top rank.
a) Explain the majority graph
b) What is a possible winner?
c) What is a Condorcet winner?
a)
Majority graph is a directed graph from voter preferences. Nodes represent outcomes and edges represent pairwise winners.
Must have:
- Completeness - Either outcome A or outcome B has to win
- Asymmetry - if A defeats B, then B can’t defeat A
- Irreflexivity - an outcome cannot defeat itself
b)
Every node that has an edge arrow pointing out is a possible winner
c)
The Condorcet winner is the outcome that is the overall winner for all possible election agendas. In a majority graph the Condercet winner is the one node that points out to all other nodes or connected in some way to all other nodes.
Explain the Slater rule
The slater rule is to choose the social ordering that minimizes the disagreement between the majority graph and the social choice, i.e. the order with the lowest Slater ranking.
How many edges arrows do we need to flip to construct a social preference list.
Go through all preference lists and pick the one with lowest Slater count.
Explain dictatorship
The outcome is only dependent on one agents preference list - the dictator.
What does Arrow’s theorem say?
Assuming voters have 3 or more distinct alternatives there exist no ranked voting system that can convert the ranked preferences of individuals into as social preference order while at the same also meet a set of specific desirable criteria
1 Unrestricted domain
2 Pareto efficiency
3 Independence of irrelevant alternatives
4 non dictatorship
- Unrestricted domain conditions that all preferences of all voters are allowed.
- Pareto efficiency states that there are no other outcomes that make one voter
better off without making any other voter worse off.
- IIA depends on the individual preference between w w for the social preference w w - only Dictatorship satisfies this criteria.
- If the non dictatorship condition is dropped as a criterion then the Dictatorship is the only system that satisfies Arrow’s theorem
What is Condorcet’s paradox?
There can exist a winner that the majority of all voters are unhappy with.
What do we mean by tactical voting and strategic manipulation?
It is a way of changing your preference list when voting in order to push the results of the voting more closely related to your actual preference list.
Give an interpretation of the Gibbard-Satterthwaite theorem
Assuming voters have 3 or more distinct alternatives, according to the Gibbard-Satterthwaite theorem in general there exists no voting system except for dictatorship that is non-manipulable.
However:
- Strategic manipulation can be made hard to compute (Second order Copeland)
- Uncertainties could make manipulation strategies more difficult to obtain.
Gibbard-Satterthwaite is in a sense related to Arrow’s theorem Exist no voting system except dictatorship that is non-manipulative
Why could the Second-order Copeland be an advantageous election scheme in terms of strategic manipulation?
Because it makes it very hard to compute ways for strategic manipulation
