ECON3106 Politics and Economics

Exercises 1

1 .

Definition 1. A preference ranking > over a set of alternatives A is transitive if, for any three alternatives, A, B, C ∈ A, if A > B and B > C , then A > C.

There is a society with 3 individuals:  i, j, k (Irma, Jakie and Kelly).  Their preferences are represented as:

Irma: A >i  B >i  C

Jakie: B >j  A >j  C

Kelly: C >k  B >k  A

Irma proposes a system where each individual associates 3 points to his or her favourite alternative,  2 to the second  and  1 to the third.   The  sum  of each individual points will constitute the social ranking.

1.1 Show the social ranking resulting from Irma’s method.

1.2    Show that Irma’s method always gives a transitive so- cial ranking (hint: notice that in the natural numbers, i.e. 1; 2; 3; . . . , “greater than” is transitive)

2 .

There is a society with 3 individuals: i, j, k (Irma, Jakie, Kelly, Louise). Their preferences are represented as:

Irma: A >i  C >i  B

Jakie: B >j  A >j  C

Kelly: B >k  C >k  A

Louise: A >l  C >l  B

2.1    Find the set of Pareto efficient alternatives

The  society  needs  to  choose  one  of  the  three  alternatives.    As  Louise  and Kelly are the youngest ones, the society wishes to give their preferences extra- consideration. Consider the following social choice method:

Round 1—select between B and C: each individual “votes” for the alter- native she prefers the most between the two.  The alternative with most votes is selected for Round 2.  In case of a tie, Louise’s preferences will determine the selected alternative.

Round 2—choose between selected alternative and A: each individual“votes” for the alternative she prefers the most between the two. The alternative with most votes is chosen. In case of a tie, Kelly’s preferences will deter- mine the selected alternative.

2.2    Which alternative would be chosen if the society was to use this method?

Now consider the following social choice method:

Round 1—select 2 alternatives: each individual “votes” for the alternative she prefers the most between the three.  If an alternative gets the most votes, then it is chosen.  Otherwise, the two top alternatives are selected for Round 2.

Round 2—choose between the 2 selected alternatives: each individual“votes” for the alternative she prefers the most between the two. The alternative with most votes is chosen.  In case of a tie, Louise’s preferences will deter- mine the selected alternative.

2.3    Which alternative would be chosen if the society was to use this method?

3 Condorcet Method (Open Agenda)

A society is composed of 3 individuals named 2, 6, and 10.  There are three alternatives, whether to have one, three, or five parties.  We label these three alternatives, respectively, 1, 3, and 5.  For any individual i (where i is a name like 2, etc), her utility if alternative A is chosen is given by

ui (A) = - (i - 2A)2 .

For example, if the alternative chosen is A = 5, individual i = 6 receives utility equal to

u6 (5)   =    - (6 - 2 × 5)2

            =    - (6 - 10)2

=    -16.

3.1    What is the most favourite alternative for each indi- vidual?

For the remaining of this exercise, assume voters vote sincerely.  I also invite you to think about whether sincere voting would be a Nash equilibrium of the voting game.

3.2    Consider a majority vote between alternatives 1 and

3. Which alternative would win?

3.3    Consider a majority vote between alternatives 3 and

5. Which alternative would win?

3.4    What can we conclude about the alternative 3?

4 Arrow’s Impossibility Theorem

A friend of yours proposes a system to choose between different alternatives and proves to you that this is not a dictatorship.  Using Arrow’s impossibility theorem, what must you conclude?

5    Strategic Voting in Plurality Elections

There is a plurality election with three candidates, {X, Y, Z}. You are a voter with preferences X > Y > Z. You read in an accurate poll that there are three types of voters:

1.  circa 49% will surely vote for Z;

2.  circa 48% will surely vote for Y ;

3.  circa 3% have the same preferences you have, but have not yet decided for whom to vote.

5.1 What is a plurality election?

5.2 If all voters like you (group 3) vote sincerely, which candidate would you expect to win?

5.3 If all voters like you (group 3) vote strategically, which candidate would you expect to win?

6 Strategic Voting and the Swing Voter’s Curse

Assume that you are a member of a jury voting by simple majority rule between two alternatives: A or B.  In case of a tie, the jury will toss a fair coin to choose between the alternatives.  There are other 99 jurors.  You have been told the following: if A is the correct alternative, then 50 of the other voters will vote for A and 49 will vote for B ; If B is correct, then 50 of the remaining voters vote for B and 49 vote for A.  That is, in each possible state, a majority of 50vs49 voters are guessing correctly. This means that you are the so called swing voter and the result of the ballot depends on you.  You think that  A is the correct alternative with probability 80%.

6.1 If you vote for A and your vote is pivotal (i.e. deci- sive), which alternative must be the correct one?

6.2 Is voting A a good idea for you?

6.3 If you vote for B and your vote is pivotal (i.e. deci- sive), which alternative must be the correct one?

6.4 Is voting for B a good idea for you?

6.5    If you were given the alternative between voting A, B , or abstaining, what would you prefer to do?

If you have answered correctly, then you have shown an example of a result known as the swing voter’s curse. Congratulations!

7    "Majority voting aggregates information dis-persed among the voters." Comment in no more than two paragraphs. Make sure to refer clearly to major results in voting theory.

8 .

There is a society with 71 individuals.  Each individual’s name is a (natural) number between 1 and 71.  Call i the name  (number) of each individual.  Her preferences are given by the utility function

ui  = - jA - ij

where A is an alternative and  jxj  is the  absolute value of x.   The set of the alternatives is A ≡ {1, 40, 81}.

8.1 For any individual i ∈ {1; . . . ; 71}, find her bliss point.

(Hint: if you make a list of 71 bliss points, you are not being very efficient).

8.2 Are the preferences of these individuals single-peaked?

You might notice that there is more than one median voter.

8.3 What is their bliss point?

8.4 Suppose that the bliss point of the median voter(s) is put to vote against another alternative of your choice. What would be the result of the vote?   (How many votes for each alternative?)

8.5    According to the median voter theorem, what is the unique equilibrium outcome of an open agenda method in this society?

9 .

There is a society with three voters, I = {a, b, c}.  Voters have preferences over three possible alternatives {0, 1, 3} as follows:

0 >a  1 >a  3;

1 >b  0 >b  3;

3 >c  1 >c  0.

9.1 Do the voters exhibit single-peaked preferences?  (Pro- vide a justification for your answer)

9.2 Is there a Condorcet winner in this society? If so, which theorem guarantees its existence and why?

9.3    Which alternative is the Condorcet winner, and why? 10 .

In an election there are two candidates, L and R.  Both candidates only care about winning the election, i.e. they are office-motivated.  There is a continuum of voters of total mass 1.  A generic voter has name i.  A fraction γ ∈ (1/2, 1) of the voters have income yi   = yl.  The remaining fraction  (1 — γ) of voters have income yi  = yh  > yl.  The set of possible alternatives is all the tax rates

τ between 0 and 1.  The tax is purely redistributive:  if y- is the mean income, consumption for voter i is equal to

ci  = yi + τ (y- - yi ) .

Each voter wants to maximize her own consumption.

Before the election, candidates L and R choose platforms τL and τR , respec- tively. That is, each chooses a tax rate.  Each voter then observes the platforms and votes for the candidate whose platform she prefers.

10.1 Express the mean income as a function of the income of the two groups?

10.2 What is the median income? How does it compare with the mean income?

10.3    What is the tax rate most preferred by a voter with income yl?

10.4 Use the theorems seen in class to predict the plat- forms of the two candidates and the policy that will be implemented by this society.