MA 103Q-002

Topics in Contemporary Mathematics: Mathematics and Politics

Mathematics

Hints and Answers

Jan. 12 assignment

2a. Dollars spent are like population, pearls are like seats in a legislature. Standard divisor is 413.8889. Quotas are 14.255, 18.362, 3.383. Hamilton apportionment is 14, 18, 4.
4. Enrollment is like population, sections are like seats in a legislature. Standard divisor is 19.4. Quotas are 2.680, 1.701, 0.619. Hamilton apportionment is 3, 2, 0.

Jan. 14 assignment

8. Standard divisor and quotas are as in problem 4. Tentative apportionment given by rounding is 3, 2, 1, so we must increase divisor to remove a seat. Critical divisors are 20.8, 21.333, 24. Smallest is 20.8, so the Webster apportionment is 2, 2, 1.

Supplementary problem on aportionment
1. 500
2. 2.47, 6.49, 41.46, 49.58
3. 2, 7, 41, 50
4. 2, 6, 42, 50
5. 2, 6, 42, 50

Jan. 16 assignment

Additional problem: The Jefferson apportionment is 14, 14, 16, 56.

Jan. 21 assignment

14. Standard divisor is 18.2. Quotas are 3.077, 1.538, .385. The geometric means used in rounding are 3.464, 1.414, and 0. The tentative Huntington-Hill apportionment is therefore 3, 2, 1, which makes 6 classes: too many. The critical divisors are 56/2.449=22.866, 28/1.414=19.801, and 7/0=infinity. (Important: In order to reduce the number of algebra classes from 3 to 2, we must reduce algebra's quota to 56/(square root of 2 times 3)=56/2.449.) The critical divisor that is closest to the original divisor is 19.801, so geometry loses a class. The final Huntington-Hill apportionment is 3, 1, 1. (Hamilton and Webster give 3,2,0.)

Jan. 23 assignment

22(a) California 640,204; Utah 745,571. (b) 105,367; 16.46%. (c) New district sizes are California 652,515; Utah 559,179. Absolute difference is 93,336, relative difference is 16.69% (d) Huntington-Hill makes relative difference in district size as small as possible, so uses the first apportionment (the real one). In this case Huntington-Hill did not make the absolute difference in district sizes as small as possible.
24. To do this problem, figure out the representative share (seats/people) for the given apportionment AND for the apportionment that gives Massachusetts 11 seats and Oklahoma 5. Then use the fact that Webster's method makes the absolute difference in representative shares as small as possible. Given apportionment: representative shares are Massachusetts .0000016586, Oklahoma .0000019074, difference is .0000002489. New apportionment: representative shares are Massachusetts .0000018245, Oklahoma .0000015895, difference is .0000003350. Webster would pick the given apportionment. (Question: Then why did Massachusetts sue?)
26(a) The quotas add up to the house size. If you round all the quotas up, the new, bigger numbers will add up to more than the house size. (b) California would not like this. Its delegation would go from 53 to 54, but the house size would go from 435 to 485 (since all 50 states would gain a seat). Californians would be a smaller percentage of the new house than they were originally.


Feb. 2, 4 and 6 assignments

Skills Check 2 c, 3 a, 4 a, 5 a.
Exercise 16a or b: A wins, and B is a loser. Then the last voter changes his preferences, but NOT in a way that moves B from below A to above A. Nevertheless, if the election is held again, B becomes a winner.
20: D is eliminated in the first round, then B and C are eliminated in the second round, so A wins. Check out what happens if the last voter changes his preferences to ADBC.

Game theory problems assigned Feb. 25

1(a) 1/2 fastballs, 1/2 curves (b) .300 (c) 2/3 fastballs, 1/3 curves
2(a) 3/4 run, 1/4 pass (b) .575 (c) 5/8 run, 3/8 pass

Game theory problems assigned Mar. 5

"Game Theory and Strategy," p. 43, 5e. Only one of Colin's strategies is not dominated: R/LR. So Rose uses LL or RL, Colin uses R/LR, outcome is RLR, payoff to Rose is -1.
"Game Theory and Strategy," p. 72, 2a. No dominated stategies, (3,4) and (4,3) are Nash equilibria, they are also Pareto optimal.
"Game Theory and Strategy," p. 72, 2b. Rose A dominates Rose B, (2,2) is Nash equilibrium, it is not Pareto optimal.
Final game. Rose A dominates Rose B, Colin A dominates Colin B, (2,4) is Nash equilibrium, it is Pareto optimal.


Game theory problems assigned Mar. 19

1(a) Yes, A. (b) Yes, A. (c) No. Commitment to B results in (2,2), which is worse for Rose.
2(a) No. (b) Yes, A. (c) A, (2,4). (d) No. Commitment to B results in (1,2), which is worse for Rose.
3(a) Yes, A. (b) No. (c) A, (2,2). (d) Colin: no. if Colin commits to B, the outcome will be (4,1), which is worse for Colin. Rose: yes. If Rose commits to B, the outcome will be (3,4), which is better for Rose.
4(a) No. (b) Yes, A. (c) Rose should use A, outcome is (3,2). (d) Colin gets a better outcome by committing to B.
5. No.
6. Answer in text.

Game theory problems assigned Mar. 19

1. Rose can threaten, "If you do A, I'll do B."
2. Rose must both threaten, "If you do A, I'll do B," and promise, "If you do B, I'll do A." The threat or promise alone won't help.
3. Rose can promise, "If you do B, I'll do B." Colin can make the same promise.
4. Rose can promise, "If you do B, I'll do B." Colin can make the same promise.
5. Each can make a promise (as we saw with the Israelis and Palestinians).
6. Answer in text.

Game theory problems assigned Mar. 26

1. There is just one Nash equilibrium, ABC 2, NBC 1, CBS 1.
2. (a) There is just one Nash equilibrium, AAA. (b) Yes. (c) Yes. (d) Yes.
3. Of the eight outcomes, six have two companies in one mall and one company in the other mall. All six of those outcomes are Nash equilibria.

April 14 assignment

Skills Check 1. (b)
Skills Check 2. (c)
Skills Check 3. (c)
Skills Check 4. (b)
Exercise 4. {A,B}, {A,C}, {A,D}, {A,B,C}, {A,B,D}, {A,C,D}, {B,C,D}, {A,B,C,D}.
Exercise 6. Since B, C, and D together have 70 votes, the quota should be increased to 71.

April 19 assignment

10. Partial answer: For part a, the winning coalitions are {A,B}, {A,C}, {A,B,C}, {A,B,D}, {A,C,D}, {B,C,D}, {A,B,C,D}. The extra votes are 3, 2, 27, 24, 23, 18, 48.
18. (a) [4:2,1,1,1]. (b) [6:2,2,1,1,1].
33. (a) [8:6,1,1,1,1,1,1,1,1]

April 23 assignment

24. (1/2, 1/6, 1/6, 1/6)

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