## Math 405 Practice Exam 2

• [1] Answer these questions about the preference schedule for candidates A, B, C, and D given below:
# of voters  7   4   2
Choice 1     A   B   D
Choice 2     B   D   A
Choice 3     C   C   C
Choice 4     D   A   B

• (a) Is there a candidate with a majority of first place votes?
• (b) Is there a Condorcet candidate; if so, which one is it?
• (c) Find the Borda count winner.
• (d) Find the Borda count winner, assuming that C drops out of the race.
• (e) Which of the four fairness criteria are violated in this election under the Borda count?

• [2] Consider the weighted voting system [q: 4, 3, 1], where 4 < q <=8.
• (a) For what values of q, if any, is there a dummy?
• (b) For what values of q, if any, does player 2 have veto power?
• (c) For what values of q, if any, is there a dictator?
• (d) Find the Banzhaf power index for each player if q = 5.
• (e) Find the Shapley-Shubik power index for each player if q = 5.

• [3] In a weighted voting system, explain why the Banzhaf power index of a player with veto power is never less than that of any other player. Is this also true for the Shapley-Shubik power index? Why or why not?

• [4] Ms. Brown, Ms. Green and Ms. White share an apartment with Ms. Black, who is moving out, but doesn't want to take her stereo, bicycle, microwave or VCR. The other three decide to use the method of sealed bids to divide these items between them. Based on the sealed bids given below, determine who gets which items, and determine how much cash each receives or pays.
Item     B     G     W

b       150    50   110
s       300   280   320
m       200   100   150
v       100   140   110

• [5] A pizza is one third each of pepperoni, anchovy, and mushroom. Ted and Alice like mushroom but hate pepperoni and anchovy, but Carol likes all three equally. Assume none of the players knows the preferences of the others. Also assume Carol is the first player and Ted is the second player. Decide who ends up with what and who, if anyone, is envious, in each of the following cases:
• (a) The lone divider method is used, and Carol divides the pizza into three pieces, separating the pepperoni, anchovy, and mushroom.
• (b) The lone chooser method is used, and Carol divides the pizza into two pieces, with the pepperoni in one, the mushroom in the other, and the anchovy divided equally between the two pieces.
• (c) The last diminisher method is used, and Carol cuts off the mushroom to start.
• (d) The Selfridge-Conway method is used, and Carol cuts the cake as in (a).