Start the Lab Math Alive Welcome Page

Problem Set. Voting and Social Choice, Part 2.

You can answer by filling in the blank spaces. If there is not enough space attach other sheets.

In this lab we will work on two things: weighted voting systems, and fair division. This first section will deal with weighted voting systems.

In this part of the lab, you will explore how much power different voters hold in systems where they have an unequal number of votes, such as shareholders' meetings, or the Electoral College, or many kinds of boards.

The first screen in the unit Weighted Voting Systems on the web explains and illustrates the basic principles.

Note that there are a few different definitions around:

• in class (and in the handouts), we computed the ratio

• In this lab (and in this problem set), we just count, for each voter, the number of coalitions where that voter holds a swing vote, and we do not divide by anything. In any case, it is only the relative power of all the different players that really matters; voter A has twice as much power as B if the number of coalitions in which A has a swing vote is double the number of coalitions in which B has a swing vote. You can check this ratio (2 if A has twice as much power as B) by comparing either the straight numbers of coalitions, or also by using the "normalized" ratios above (the denominators cancel out).

• in Banzhaf's original definition (also the one used in "Rational Politics" by Steven Brams), yet a different denominator is used. There one computes the sum N, over all the voters, of the number of coalitions where that voter has a swing vote, and uses this for the denominator; the index is then given by

In this case, the sum of the power indices of all the voters is always 1.

In the first example on the web it is explained how to compute the relative powers of a system with 3 voters, one of which holds two votes, while the other two have both just one. A measure can pass only if at least 3 votes are cast in favor of it.

This system is denoted

[3:2,1,1]

the first 3 denotes the majority necessary; the numbers after the colon give the number of votes held by the different players.

After this is explained in detail, you will be asked to figure out for yourself the indices for other systems. In this problem set, you will calculate the indices for different systems.

Problem 1. Calculating Banzhaf Power Indices.

The first one is [51: 48,45,7] : a system with three voters again, with relative weights 48, 45 and 7, and where a 51 majority is needed to pass a measure.

To find the power indices, we start by making a table as in the [3:2,1,1] example:

Complete table

A B C Number of Votes Pass/Fail
48 45   7
y y y 100 pass
y(s) y(s) n 93 pass
y(s) n y(s) 55 pass
n y y
y n n
n y n
n n y
n n n

Again, you compute on every line how many votes have been cast in favor of the measure (yes votes), and whether the measure passes or fails (that is, whether it got 51 votes or more). Then you check for each voter whether the status (pass or fail) of the measure would change if that voter had a mind change. You could put a little (s) next to each swing vote (see the first three lines in the example) to make sure you don't forget any, but this is optional.

After you have done all the lines, count the number of (s) marks for each voter.

[51: 48,45,7] =

Problem 2. Same example with a new quorum.

The next example is [60: 48,45,7] . The three voters have the same weights as in the previous question, but the majority rule has changed. Now 60 votes are necessary to pass a measure.

Complete the following table again:

A B C Number of Votes Pass/Fail
48 45   7
y y y
y y n
y n y
n y y
y n n
n y n
n n y
n n n

[60:48,45,7] =

Problem 3. Another example of calculating BPI's.

Next: [51: 35,35,30]

Complete the following table again:

A B C Number of Votes Pass/Fail
35 35 30
y y y
y y n
y n y
n y y
y n n
n y n
n n y
n n n

[51: 35,35,30] =

Compare this with the first example, [51: 48,45,7] that you worked out? In particular, compare the amount of power each voter has versus the percentage of the votes they have in the two cases.

Problem 4. An example with four voters.

This example has four voters: [60: 31,31,31,28].The table has now 16 lines:

Complete the table

A B C D Number of Votes Pass/Fail
31 31 31 28
y y y y
y y y n
y y n y
y n y y
n y y y
y y n n
y n n y
n n y y
n y n y
y n y n
n y y n
y n n n
n y n n
n n y n
n n n y
n n n n

[60: 31,31,31,28] =

Any comments? In particular, is the distribution of power what you would expect given the percentages of the votes each voter has?

Problem 5. Maximum BPI.

Assuming that the quorum is greater than or equal to the simple majority, if one of the voters has the maximum Banzhaf Power Index (as determined by the number of voters), explain why the other voters have zero power.

Next you can experiment with some more complicated systems, with up to 10 voters on the webpage Create Your Own Example of Weighted Voting System. You should enter the different weights yourself in the lab. If there are fewer than 10 voters, enter the weights (from left to right) in the boxes and leave the unused boxes empty (except the first box, which must be filled in) - the system will ignore the empty boxes (as it will voters with zero weight). The "Banzhaf power index" (BPI) calculated by the software is the total number of coalitions that a voter can swing.

If you are interested in simple majority rule, then you can click the button "Simple majority", and the required majority quota will appear; if you want to work with another quota, then you have to enter it by hand. When you then click "Power Index", you will see the answer appear. To work with a slightly changed example, you don't need to click "Clear Table" and enter everything again; you can just go back to the boxes with entries that will change, and change them there; if you then click "Power Index" again, you get the new result.

Problem 6. Blockvotia Example.

In the Scientific American article that was handed out in class, different weights were given to six districts in fictional Blockvotia.

a) At first, the distribution of weights was

 Sheepshire 10 9 7 3 1 1

What are the power indices of the six districts under simple majority rule?

b) The two smallest districts complained, and lobbied hard to get each an extra vote, so that the distribution of weights would be

 Sheepshire 10 9 7 3 2 2

What would the power indices now be under simple majority rule?

c) But this solution turned out to be infeasible. However, if the biggest district Sheepshire is given two extra votes, so that the distribution is

 Sheepshire 12 9 7 3 1 1

then the following power indices are obtained (again under simple majority rule):

Can you compare this with the previous situations?

Problem 7. Power in the European Community.

Next, let us experiment with the European Community numbers illustrated in the class handout.

a) Originally there were 6 countries:

 Weight 4 4 4 2 2 1

The quorum needed to pass a measure was 12 (out of 17, which is much larger than simple majority!). What are the corresponding power indices?

b) Experiment with raising and lowering the quorum. How does that affect Luxembourg's power? And that of the other countries? (Give the numerical results!)

c) Next, three new members were added, England, Denmark and Ireland, with a redistribution of the weights, so that we get

 Weight 10 10 10 5 5 2 10 3 3

and the new quorum is 41. How does the power of Luxembourg fare now? Compare it with that of the other countries.

d) Later, Greece was added, and the new list (with weights) was

 Weight 10 10 10 5 5 2 10 3 3 5

with quorum 45. What is the power distribution now? Compare Luxembourg's power with that of other countries.

This second part of the lab deals with Fair Division. For this part of the lab, you won't need anything from the web.

Problem 8. Fair 4-way division of an inheritance.

We are going to work through an example of fair division in an inheritance, involving four people, Janice, Scott, Andrea and Eric.

The fair division procedure concerns three big items, the house in the small town in which they all grew up, a very nice cabin in the mountains where they spent many vacations when they were children, and a boat in which they all remember going on fishing trips.

Each of the four is asked to assign their own subjective value to the house. All of the different items need some work, and they are probably worth more to them, effectively, than the market value if they just simply got them appraised.

They all have different preferences as well; Eric has remained in the same area of the country and he has a lot of affection for the house in which he grew up. His own family is growing, and they need more space. He really would like to move into his parents' house. Janice doesn't care much about the house - she certainly doesn't want to live in it, and she knows it would need a lot of fixing up before they could get a good price for it. On the other hand, she has fond memories of the mountain cabin, and if she could get it, she would use it often for hiking get-aways from her city job. Scott wouldn't mind taking over the house, but he does not feel about it very strongly; he certainly doesn't intend to take vacations in the mountain cabin - his job doesn't leave him enough time to go there often for a few days, and he prefers to spend longer vacations elsewhere. But he is very much interested in the boat, with which he could go on day-outings during the weekend. Finally, Andrea is not so determined to take over the boat or the house or the cabin, and she just tries to gauge what she thinks would be a reasonable price for them.

When they are asked to put a money value on each item, taking into account not only what they think it is worth, but also what it is worth to THEM, they come up with:

 house cabin boat 200,000 40,000 16,000 100,000 80,000 20,000 180,000 60,000 32,000 140,000 60,000 24,000

They have to make these evaluations separately and simultaneously - no jockeying for position or lying about one's own interest after having had a peek at what the others said.

a) Try now to divide up this estate according to the Knaster procedure that we saw in class.

The main steps are :

(i) assign each item to the person who wanted it most

• who gets the house?

• who gets the cabin?

• who gets the boat?

(ii) imagine that each of them now pays into a common "pot" the amount corresponding to what they got "over" their share. For instance, Eric gets the house, worth in his estimate 200,000, of which his share would normally have been 25%; so he has to pay the excess, 75% of 200,000, into the pot.

• amount Eric pays into the pot:

• amount Janice pays into the pot:

• amount Scott pays into the pot:

• amount Andrea pays into the pot:

(iii) next, each person withdraws from the pot the money equivalent, according to their own evaluation, of the shares in the items that they didn't get.

• amount Eric withdraws from pot at this stage:

• amount Janice withdraws from pot at this stage:

• amount Scott withdraws from pot at this stage:

• amount Andrea withdraws from pot at this stage:

(iv) what remains in the pot after this stage gets divided up evenly between all the heirs.

amount each of them gets at this step:

• Eric:

• Janice:

• Scott:

• Andrea:

(v) compute for everyone what they got, in items (if they got any), and in money (include in this the monetary value of any item a person got. At least one of them will have to pay the others money. You can use negative amounts to represent money owed or paid.):

• Eric:

• Janice:

• Scott:

• Andrea:

For each of them, compare the value of what they got (according to their own evaluation again) with what they thought the whole estate was worth.

• Eric:

• Janice:

• Scott:

• Andrea:

b) Imagine now a slightly different situation. We still have the same three items, and the same evaluations by each of the four heirs. But, because Eric spent much more time in the final years of their parents' life taking care of many things for them, all the children have agreed that this entitles him to 40% of the estate, while the other three get 20% each. Please work through the whole division again with these unequal weights.

Challenge Questions

Problem C1. Combinations of BPI's.

For three voters, give all possible combinations of Banzhaf power indices that can occur. Explain first why all the power indices are even numbers.

Problem C2. Strategic evaluations in "fair" division.

In Problem 8. a) above, on the fair division of the inherited estate equally among the four siblings, consider what Eric could do to maximise his gain if he had secretly seen the evaluations of each of his other siblings before making his own evaluations.