Lecture 4: Political Economy

Question 1

Majority voting with n=2 alternatives

Answer 1

Majority voting over two alternatives is one of the most common mechanisms of preference aggregation in democratic systems
However it does not necessarily lead to a Pareto-improvement in social welfare
Simple majority (50%+1) does not necessarily minimize the cost/benefit ratio
In reality there are almost always more than two alternatives

Question 2

Cost and benefits of majority voting

Answer 2

External costs: Costs arising from the imposition of the majority choice on the minority (max if there is only person deciding, 0 if unanimity)
Internal costs: costs arising from the time and resources needed to reach a consensus (increasing with the approval quorum)

Question 3

Majority Voting in the presence of n>2

Answer 3

Majority voting on pairs of alternatives
3 alternatives (A, B, C) and 4 types of voters (Alfa, Beta, Gemma, Delta)
Voters have to choose among the alternatives by voting sequentially each pair
The winning alternative (The one which wins wrt alll the others in the pair comparison) is the Condorcet winner

Question 4

The median Voter Theorem

Answer 4

If there is an odd number of voters and the preferences of all voters are “single peak”, there will always exist a condorcet winner and it corresponds to the median of the distribution of the option preferred by the voters
“Single peak” or unimodal preferences: When ordering alternatives accoridng to a specific criterion, the utility of any individual is always increasing when approaching her preferred option and always decreasing when moving away from it

Question 5

Median Voter Theorem does not apply when individual has double-peaked preferences

Answer 5

With double-peaked preferences, the majority voting on pairs of alternatives is not able to produce a social order of preferences which is transitive -> No Condorcet winner
A vs B -> B wins

A vs C -> A wins

Hence, by the transitive property -> B preferred to C

Instead: C vs B -> C wins

Cycle: B>A>C>B

Question 6

Bimodal Preferences

Answer 6

Example: Public health care system that is financed by taxation
Rich people may prefer a low public health coverage, their second best option would be extensive coverage and high taxation. The worst case intermediate coverage and intermediate taxation
A: High > Low > Middle
B: Low > High > Middle

Question 7

Arrows impossibility Theorem

Answer 7

Arrow: Isit posible to find a mechanism of public choice that is able to order different social alternatives while satisfying at the same time reasonable and ethical norms?
Arrow’s impossibility theorem: No

Question 8

Arrow’s Axioms

Answer 8

Interdependence of irrelevant alternatives: Adding new options should not affect the initial ranking of the old options: So the collective ranking over the old options should be unchanged
Non dictatorship: The collective preference should not be determined by the preferences of one individual
Pareto criterion: If everybody agrees on the ranking of all the possible options, so should the group; the collective ranking should coincide with the common individual ranking
Unrestricted domain: The collective choice of method should accommodate any possible individual ranking of options (Jede Politik muss gewählt werden können)
Transivity: If the group prefers A to B and B to C; then this group cannot prefer C to A. The Condoorcet Paradox shows that majority voting fails to meet this condition and can lead to cycles in collective preference
When choosing among more than two options, there exists no collective decision-making process that satisfies all the above conditions

Question 9

Agenda Setting/Manipulation

Answer 9

Suppose that agents apply majority voting on pairs of alternatives with elimination of losing alternative
A vs B -> B wins, A is eliminated
B vs C -> C wins
However:
A vs C -> A wins, C is eliminated
A vs B -> B wins
The final outcome will depend on the order of voting. This opens the door to the issue of “Agenda manipulation”

Violation of the 2nd Axiom

Question 10

Simple majority and (relevance) of irrelevant alternatives

Answer 10

Suppose that there is a simple majority voting over the set of these three alternatives
A wins with five votes
However, if C did not exist: B would win with seven votes
Incentive to introduce “Irrelevant alternatives” to modify outcome
Violation of the 1st Axiom

Question 11

Borda Voting System

Answer 11

N alternatives. Each of the voter may attribute a score N to their preferred alternative, N-1 to their second and so on

Question 12

Borda Voting System

Answer 12

N alternatives. Each of the voter may attribute a score N to their preferred alternative, N-1 to their second and so on
Votes for D = 3x9+2x7+2x2+2x1 = 47
Votes for R = 2x9+3x7+2x1+2x2 = 45
Votes for E = 1x9+1x7+3x2+3x2 = 28
D wins
Suppose now that Beta voters gather and think about how they can increase their utility
So now they place E as their second choice: now R wins with 45 (D: 40, E: 35)

Question 13

Problems with the Borda System

Answer 13

Strategic voting which does not satisfy the second Axiom
Clearly in the presence of a large number of voters, the influence of strategic voting by a single individual is virtually 0

Question 14

Income Distribution

Answer 14

The income distribution in the population is usually such that the income of the median voter is lower than the mean income
If the MRS between the public good and private good is higher for individuals with lower income: MRS median voter > Mean of MRS of voters -> Overproduction of the public good
If the MRS between the public good and private good is lower for individuals with lower income: MRS median voter < Mean of MRS of voters -> Underproduction of the public good

Question 15

The Voting Paradox

Answer 15

When is it rational to vote?
Electoral participation comes with some costs:
Physical cost of going to the voting booth
Time spent in going to vote (opportunity cost)
Other implicit costs (Information cost etc.)
Benefit: Utility from electing a party that will set the production of a given public good at a level closer to the one preferred by the voter
This benefit should be “weighted” for the probality that a single vote may actually contribute to elect a given party/candidate (I.e. probality of being pivotal)
If citizens were rational and maxmized their utility, they would vote only if their expected benefit from voting is above the cost
If the voter were to know for certain that their party will win, they will choose not to vote. If they would go to vote, they would not have any “extra gain” from her vote and would still have to pay the cost C
At the same time, if the voter were to know for sure that the other party would win, they would also choose not to vote
-> For the voter, it is rational to vote only if their vote may affect the electoral outcome

Question 16

A voter may influence the voting outcome in two cases

Answer 16

If, in the absence of their vote, the voting outcome will be a draw. In this case, their vote would decide the voting outcome in favour of one of the parties
If the party preferred by the voter would get one vote less than the other one in the absence of their vote. That is: The vote of the voter would allow their preferred party to obtain a draw

Question 17

Voting Paradox Equation

Answer 17

X1=votes received by party 1 if voter n decides not to vote
X2= votes received by party 2 if voter n decides not to vote

Let’s assume that in the case of a draw, there will be a coin-tossing to decide the outcome of the elections (probability ½)
Hence, the probability of influencing the final voting outcome is P such that
P = 1/2Pr((X1=X2)+1/2Pr(X1=X2-1)
Note: if X1=X2, with probality ½ party 1 would win even in the absence of the vote by voter 1
Given P, the expected benefit of voting for the voter is PB
Hence the rational voter would find it optimal to vote if and only if PB>C

Question 18

Why people vote anyway

Answer 18

Ethical concerns
Non-correct expectations of being pivotal
Voting as a consumption good