University of Kansas, Fall 2007
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu

Problems for credit

rules (as of August 17):

  1. Answers are due in writing at the beginning of class unless otherwise specified. If you are going to miss class, you can drop them off at my office, but you must get them to me by the time I leave for class.
  2. Answers submitted by e-mail must be submitted by the time I leave for class, and will be subject to a penalty of 10 percentage points.

problems due Wednesday, August 22 (Eggleston, section 2):

  1. Give an example of a choice situation that would most appropriately be handled by utility theory. (It does not have to be different from the situation you thought of for class on Monday, August 20.)

[infinitely many correct answers—see me if you have questions about this one]

  1. Give an example of a choice situation that would most appropriately be handled by game theory. (It does not have to be different from the situation you thought of for class on Monday, August 20.)

[infinitely many correct answers—see me if you have questions about this one]

  1. Give an example of a choice situation that would most appropriately be handled by social choice theory. (It does not have to be different from the situation you thought of for class on Monday, August 20.)

[infinitely many correct answers—see me if you have questions about this one]

problems due Wednesday, August 29 (Eggleston, section 7):

  1. Suppose we have the options a, b, c, and d, and the following preferences:
  1. a P b
  2. c P d
  3. a P d

Do these preferences satisfy the completeness condition?

Answer: No. (You don’t know about preferences between a and c, or b and c, or b and d.)

  1. Consider the options and preferences given in problem 4. Do they satisfy the transitivity condition?

Answer: Yes. (They satisfy the transitivity condition as long as they don’t violate it. And no violation of the transitivity condition can be derived from the given preferences.)

  1. Let A be the following proposition: Between options f and m, Ralph has no preference. Clearly, if A is true, then Ralph’s preferences violate the completeness condition. State some preferences involving options f and m (and, if necessary, others that you make up) such that, if A is true and Ralph has those preferences, then Ralph’s preferences violate not only the completeness condition (as just observed), but also the transitivity condition.

Answer: There are multiple acceptable ways to go here. The most elegant approach is to impute to Ralph preference such as f P z and z P m. Then, if it remains true that Ralph has no preference between f and m, he is violating not only the completeness condition (in the say way as before), but also the transitivity condition, because the preferences we’ve newly imputed to him would imply (if his preferences did satisfy the transitivity condition) that f P m.

Another acceptable approach is to impute to Ralph some preferences that don’t connect with f and m in the way just described, but that result in a cycle of preferences on their own. For example, you could impute the following preferences to Ralph, and Ralph's preferences would then violate the transitivity condition: a P b, b P c, c P a. And Ralph’s preferences would still violate the completeness condition, because of his having no preference between f and m.

  1. Consider the following (valid or invalid) proof of the claim that cyclical preferences over four options violate the transitivity condition:
# claim justification
1 a P b given
2 b P c given
3 c P d given
4 d P a given
5 a P d 1–3, transitivity
6 contradiction 4 and 5, completeness

What is the largest number n that makes the following sentence true? “Lines 1 through n are all correct and correctly justified.” (If the proof is valid, give the number of the last line. If the proof has one or more mistakes, give the number of the line immediately preceding the one containing the first mistake.)

Answer: 4.

problems due Friday, September 7 (Eggleston, section 11):

  1. Suppose you are going to cater an outdoor event this weekend and are deciding whether to bring your assistant to help you. If you bring your assistant, you’ll be happy with your decision if there is no running water near the bar area or if it is raining. If neither of these things is the case, you’ll be mad that you wasted some of your payroll budget. If you do not bring your assistant, you’ll be happy with your decision as long as there is running water near the bar area and it’s not raining. But if either of those things fails to be the case, you’ll be very tired at the end of the day. Set up and fill in the matrix for this situation. Make sure that no two columns have identical outcomes in every row, even if this means combining what might originally appear to be states of the world that should be listed separately.
  rain or no running water running water and no rain
bring assistant happy with your decision mad that you wasted some of your payroll budget
not bring assistant tired at the end of the day happy with your decision
  1. Suppose you are working in an alchemy lab and are deciding whether to use proteins or radiation in your next procedure. Your leaves can be in state A, B, or C, and your roots can be in state P or Q. If you use proteins, you’ll end up with iron if your leaves are in state A and your roots are in state P, tin otherwise if you roots are in state P, and aluminum if your roots are in state Q. If you use radiation, you’ll end up with gold if your leaves are in state A and your roots are in state P, silver if your leaves are in state B and your roots are in state P, bronze if your leaves are in state C and your roots are in state P, and platinum if your roots are in state Q. Set up and fill in the matrix for this situation. Make sure that no two columns have identical outcomes in every row, even if this means combining what might originally appear to be states of the world that should be listed separately.
  leaves in state A and roots in state P leaves in state B and roots in state P leaves in state C and roots in state P roots in state Q
use proteins iron tin tin aluminum
use radiation gold silver bronze platinum

problems due Monday, September 10 (Eggleston, section 12):

Use the choice situation represented by the following matrix for problems 10–14:

  S1
(1/2)		 S2 S3
A1 5 19 18
A2 13 8 11
A3 18 7 13
A4 6 7 20
  1. dominance, maximin, and maximax:
  1. Would any option be selected by the dominance principle? If so, which one?

No.

  1. Which option(s) would be selected by the maximin rule?

A2.

  1. Which option(s) would be selected by the maximax rule?

A4.

  1. optimism/pessimism:
  1. Which option(s) would be selected by the optimism/pessimism rule with an optimism index of 1/4?

A3.

  1. Which option(s) would be selected by the optimism/pessimism rule with an optimism index of 2/3?

A4.

  1. Is there any optimism index that would make the optimism/pessimism rule select option A2? If so, is there more than one optimism index that would make the optimism/pessimism rule select option A2? If so, state two of them.

Yes, Yes, 0 and 1/10. (Any α of less than 1/6 will make the α-index of A2 less than the α-index of each of the other options.)

  1. Which option(s) would be selected by the minimax regret rule? What is the maximum regret associated with the option(s) selected by that rule?

A2, 11.

  1. Which option(s) would be selected by the method of maximizing expected utility using the principle of insufficient reason? What is the expected utility associated with the option(s) selected by that method?

A3, 14.

problems due Friday, September 14 (Eggleston, section 14):

(Note: Problems 15–18 are very similar to problems for credit nos. 16–19 that I assigned in my Fall 2006 section of this course. Those problems, and their answers, are at the web site that I set up for that course, at http://web.ku.edu/~utile/courses/rct2. Problems 19-20 are involve some slightly more complicated math than is found in nos. 15–18, but not additional conceptual complexities.)

  1. Suppose you prefer more money to less, and also prefer $400 to an option giving you a 30-percent chance at $1,500 and a 70-percent chance at $0. If u($400) = u($0) + x, and u($1,500) = u($400) + y, what constraint(s) on x and y (in addition to x > 0 and y > 0) imply utility assignments for the three dollar amounts that make the principle of maximizing expected utility agree with your preferences? (Show your derivation of whatever constraint(s) you state.)

EU($400) > EU(30-percent chance at $1,500 and 70-percent chance at $0)
   u($400) > (30/100)u($1,500) + (70/100)u($0)
100u($300) > 30u($1,500) + 70u($0)
100[u($0) + x] > 30[u($0) + x + y] + 70u($0)
100u($0) + 100x > 30u($0) + 30x + 30y + 70u($0)
100u($0) + 100x > 100u($0) + 30x + 30y
                 100x >                 30x + 30y
                  70x >                 30y
               (7/3)x >                    y
                      y < (7/3)x

  1. Suppose you have the preferences stated in problem 15. Give utility assignments for the three dollar amounts that make the principle of maximizing expected utility agree with your preferences. (You do not have to show that your stated utility assignments work.)

There are infinitely many correct answers. Here is one:

u($0) = 0
u($400) = 10
u($1,500) = 11

  1. Suppose you prefer more money to less, and also prefer an option giving you a 60-percent chance at $1,000, and a 40-percent chance at $0, to a simple $700. If u($700) = u($0) + x, and u($1,000) = u($700) + y, what constraint(s) on x and y (in addition to x > 0 and y > 0) imply utility assignments for the three dollar amounts that make the principle of maximizing expected utility agree with your preferences? (Show your derivation of whatever constraint(s) you state.)

EU(60-percent chance at $1,000 and 40-percent chance at $0) > EU($700)
(60/100)u($1,000) + (40/100)u($0) > u($700)
60u($1,000) + 40u($0) > 100u($700)
60[u($0) + x + y] + 40u($0) > 100[u($0) + x]
60u($0) + 60x + 60y + 40u($0) > 100u($0) + 100x
             100u($0) + 60x + 60y > 100u($0) + 100x
                             60x + 60y > 100x
                                      60y > 40x
                                            y > (2/3)x

  1. Suppose you have the preferences stated in problem 17. Give utility assignments for the three dollar amounts that make the principle of maximizing expected utility agree with your preferences. (You do not have to show that your stated utility assignments work.)

There are infinitely many correct answers. Here is one:

u($0) = 0
u($700) = 1
u($1,000) = 11

  1. Suppose you prefer more money to less, and also prefer an option giving you a 25-percent chance at $600 and a 75-percent chance at $500 to an option giving you a 60-percent chance at $1,000 and 40-percent chance at $0. If u($500) = u($0) + x and u($600) = u($500) + y and u($1,000) = u($600) + z, what constraint(s) on x and y and z (in addition to x > 0 and y > 0 and z > 0) imply utility assignments for the four dollar amounts that make the principle of maximizing expected utility agree with your preferences? (Show your derivation of whatever constraint(s) you state.)

EU(25-percent chance at $600 and 75-percent chance at $500) > EU(60-percent chance at $1,000 and 40-percent chance at $0)
(25/100)u($600) + (75/100)u($500) > (60/100)u($1,000) + (40/100)u($0)
                 25u($600) + 75u($500) > 60u($1,000) + 40u($0)
     25[u($0) + x + y] + 75[u($0) + x] > 60[u($0) + x + y + z] + 40u($0)
25u($0) + 25x + 25y + 75u($0) + 75x > 60u($0) + 60x + 60y + 60z + 40u($0)
                    100u($0) + 100x + 25y > 100u($0) + 60x + 60y + 60z
                                               40x > 35y + 60z
                                                 8x > 7y + 12z

  1. Suppose you have the preferences stated in problem 19. Give utility assignments for the four dollar amounts that make the principle of maximizing expected utility agree with your preferences. (You do not have to show that your stated utility assignments work.)

There are infinitely many correct answers. Here is one:

u($0) = 0
u($500) = 10
u($600) = 11
u($1,000) = 12

problems due Friday, October 5 (Resnik, sections 5–2 and 5–3):

Use dominance considerations to analyze the next two games (which you can assume to be zero-sum games). Circle all outcomes not ruled out by such considerations.

  1.  
  C1 C2 C3 C4
R1 8 5 3 3
R2 7 4 1 1
R3 6 2 7 6

Answer: Four outcomes are not ruled out:

  1. the 5 (R1, C2)
  2. the second 3 (R1, C4)
  3. the 2 (R3, C2)
  4. the last 6 (R3, C4)
  1.  
  C1 C2 C3 C4
R1 2 6 4 3
R2 7 8 7 8
R3 8 5 5 8
R4 4 6 6 9

Answer: One outcome is not ruled out: the second 7 (R2, C3).

Circle every equilibrium outcome in the next two games (which you can assume to be zero-sum games).

  1.  
  C1 C2 C3 C4
R1 3 5 4 3
R2 1 4 5 5
R3 3 3 4 8

Answer: the first 3 (R1, C1) and the third 3 (R3, C1)

  1.  
  C1 C2 C3 C4
R1 7 1 2 9
R2 7 8 5 7
R3 4 5 9 7
R4 7 9 8 9

Answer: the last 7 (R4, C1)

  1. Prove that if a zero-sum game has an equilibrium outcome in a dominated column, then it has at least two equilibrium outcomes whose values are equal to each other. At any point in your proof, you may (1) assert that the row player does not have an incentive to unilaterally deviate from whatever strategy pair you want to say that he does not have an incentive to unilaterally deviate from and (2) justify this claim with the word ‘shortcut’. (This is a shortcut to make the proof easier.)

Answer:

# claim justification
1 Suppose the outcome of (Ri, Cj) is an equilibrium outcome but that Cj is a dominated column. assumption for proving conditional
2 The outcome of (Ri, Cj) is an equilibrium outcome. 1
3 Neither player has an incentive to unilaterally deviate from the strategy pair (Ri, Cj). 2, definition of equilibrium
4 The column player does not have an incentive to unilaterally deviate from the strategy pair (Ri, Cj). 3
5 vij is at least as small as any other value in row i. 4
6 Cj is a dominated column. 1
7 There is some column Ck, with k ≠ j, that dominates Cj. 6
8 vik is at least as small as vij. 7
9 vik is at least as small as any other value in row i. 8, 5, transitivity of ≤
10 The column player does not have an incentive to unilaterally deviate from the strategy pair (Ri, Ck). 9
11 The row player does not have an incentive to unilaterally deviate from the strategy pair (Ri, Ck). shortcut
12 Neither player has an incentive to unilaterally deviate from the strategy pair (Ri, Ck). 10, 11
13 The outcome of (Ri, Ck) is an equilibrium outcome. 11, definition of equilibrium
14 This game has at least two equilibrium outcomes—the outcome of (Ri, Cj) and the outcome of (Ri, Ck). 2, 13
15 vij is at least as small as vik. 5
16 The values of the outcome of (Ri, Cj) and the outcome of (Ri, Ck) are equal to each other. 8, 15
17 The game has at least two equilibrium outcomes whose values are equal to each other. 14, 16
18 If a game has an equilibrium outcome in a dominated column, then it has at least two equilibrium outcomes whose values are equal to each other 1–17

problems due Monday, October 8 (Resnik, section 5–3a):

  1. Using the method described in parts A.1 and B.1 of handout 19, verify that ((4/9 R1, 5/9 R2), (1/3 C1, 2/3 C2)) is an equilibrium pair for the following game:
  C1 C2
R1 8 3
R2 2 6

Answer:

To verify that [(4/9 R1, 5/9 R2), (1/3 C1, 2/3 C2)] is an equilibrium pair, we need to verify that neither the row player nor the column player can do better by unilaterally changing his or her strategy.

First, consider the game from the row player’s point of view. If the column player is playing strategy (1/3 C1, 2/3 C2), then the expected utility of the game for the row player, with his strategy (p R1, (1 – p) R2), is (p)[(1/3)(8) + (2/3)(3)] + (1 – p)[(1/3)(2) + (2/3)(6)], or p(8/3 + 6/3) + (1 – p)(2/3 + 12/3), or p(14/3) + (1 – p)(14/3), or p(14/3) + 14/3 – p(14/3), or 14/3. Because this expected utility does not depend on p, in that the p drops out of the expression as we simplify it, the row player’s expected utility is unaffected by his own strategy, and thus he has no incentive to unilaterally deviate from whatever strategy he happens to be playing. This goes for (4/9 R1, 5/9 R2) as much as for any other strategy.

Second, consider the game from the column player’s point of view. If the row player is playing strategy (4/9 R1, 5/9 R2), then the expected utility of the game for the column player, with her strategy (q C1, (1 – q) C2), is (q)[(4/9)(8) + (5/9)(2)] + (1 – q)[(4/9)(3) + (5/9)(6)], or q(32/9 + 10/9) + (1 – q)(12/9 + 30/9), or q(42/9) + (1 – q)(42/9), or q(42/9) + 42/9 – q(42/9), or 42/9. Because this expected utility does not depend on q, in that the q drops out of the expression as we simplify it, the column player’s expected utility is unaffected by her own strategy, and thus she has no incentive to unilaterally deviate from whatever strategy she happens to be playing. This goes for (1/3 C1, 2/3 C2) as much as for any other strategy.

Since neither the row player nor the column player can do better by unilaterally changing his or her strategy, [(4/9 R1, 5/9 R2), (1/3 C1, 2/3 C2)] is an equilibrium pair.

  1. Compute the value of the following game, assuming that the players are using the strategy pair [(½ R1, ½ R2), (½ C1, ½ C2)]. Show your work.
  C1 C2
R1 a –b
R2 –b a

Answer:

To compute the value of the game, we find the expected utility of the playing of [(½ R1, ½ R2), (½ C1, ½ C2)], by computing the weighted average of the values of the four possible outcomes of the game: (½)(½)(a) + (½)(½)(–b) + (½)(½)(–b) + (½)(½)(a) = a/4 – b/4 – b/4 + a/4 = 2a/4 – 2b/4 = a/2 – b/2 = ½(a – b).

  1. If the row player’s strategy in the following game is (1/3 R1, 2/3 R2), what expression of the form xq + y represents the column player’s expected utility if she plays strategy (q C1, (1 – q) C2)?
  C1 C2
R1 8 3
R2 2 6

Answer:

EUcolumn(game)
= q[(1/3)(8) + (2/3)(2)] + (1 – q)[(1/3)(3) + (2/3)(6)]
= q(8/3 + 4/3) + (1 – q)(3/3 + 12/3)
= q(12/3) + (1 – q)(15/3)
= q(12/3) + 15/3 – q(15/3)
= q(–3/3) + 15/3
= –q + 5

  1. Suppose the answer to problem 28 were –(1/2)q + 3/2. What value of q would be best for the column player?

Answer:

The column player wants to minimize the score she gets upon the playing of the game. To minimize –(1/2)q + 3/2, the column player would make q as large as possible, by setting q (which must be between 0 and 1) equal to 1.

problems due Wednesday, October 10 (Resnik, section 5-3c):

  1. Derive an equilibrium pair of mixed strategies for the following game (in which you may assume that a, b, and c are all positive). You can use any method you want, but you must show your work.
  C1 (q) C2 (1 – q)
R1 (p) a –b
R2 (1 – p) –c a

Answer:

There are two fairly manageable techniques for this one. I’ll use one technique to derive p, and the other to derive q.

First, let's derive p by figuring out what value of p would make the column player’s expected utility for this game not depend on q:

EUColumn(game)
= q[(p)(a) + (1 – p)(–c)] + (1 – q)[(p)(–b) + (1 – p)(a)]
= q(ap – c + cp) + (1 – q)(–bp + a – ap)
= q(ap + cp – c) + (–ap – bp + a) – q(–ap – bp + a)
= q(ap + cp – c + ap + bp – a) + (–ap – bp + a)
= q(2ap + bp + cp – a – c) + (–ap – bp + a)
Now it’s time to figure out what value of p would make the coefficient of q equal to 0:
2ap + bp + cp – a – c = 0
p(2a + b + c) = a + c
p = (a + c)/(2a + b + c)

Second, let’s derive q by figuring out what value of q would make the row player’s expected utility for R1 equal to his expected utility for R2:

EU(R1) = EU(R2)
qa + (1 – q)(–b) = q(–c) + (1 – q)(a)
qa – b + qb = –qc + a – qa
qa + qb + qc + qa = a + b
2qa + qb + qc = a + b
q(2a + b + c) = a + b
q = (a + b)/(2a + b + c) 

  1. Consider the following game.
  C1 C2
R1 1 9
R2 6 5

For this game, the strategy pair [(1/9 R1, 8/9 R2), (5/9 C1, 4/9 C2)] is in equilibrium, and the value of the game is 49/9. If this game is played a large number of times using the specified strategy pair, what proportion of the time would you expect the outcome to be 49/9?

Answer:

None—even though 49/9 is the value of the game, it is not a possible outcome of the game.

problems due Monday, October 15 (Resnik, section 5-3c):

  1. Consider the following game, which you can assume to be a zero-sum game with no equilibrium strategy pair except one consisting of mixed strategies:
  C1 (q) C2 (1 – q)
R1 (p) a b
R2 (1 – p) c d

Show that solving for p by assuming EU(C1) = EU(C2) yields a formula for p that makes the coefficient of q (in the formula for the expected utility of the game from the column player’s point of view) equal to 0.

Answer:

The answer involves fractions that are hard to read unless the numerator can be written above the denominator, which does not seem to be an option in this format (i.e., I don’t know HTML code that allows for that), so here I’m just going to say that this answer is closely parallel to the demonstration (done in class) that solving for q by assuming EU(R1) = EU(R2) yields a formula for q that makes the coefficient of p (in the formula for the expected utility of the game from the row player’s point of view) equal to 0. Please see me if you need more explanation.

  1. What values of p and q make [(p R1, (1 – p) R2), (q C1, (1 – q) C2)] an equilibrium pair for the following game? (You do not have to show your work. An answer of the form ‘p = __, q = __’ can earn full credit.)
  C1 (q) C2 (1 – q)
R1 (p) 13 16
R2 (1 – p) 18 4

Answer:

p = 14/17, q = 12/17

  1. What values of p and q make [(p R1, (1 – p) R2), (q C1, (1 – q) C2)] an equilibrium pair for the following game? (You do not have to show your work. An answer of the form ‘p = __, q = __’ can earn full credit.)
  C1 (q) C2 (1 – q)
R1 (p) 19 1
R2 (1 – p) 5 9

Answer:

p = 4/22, q = 8/22

problems due Monday, October 22 (Resnik, section 5-4a–5-4c):

  1. Write the pure-strategy equilibrium pair(s) for the following game (or, if there aren’t any, write ‘no pure-strategy equilibrium pairs’). For each strategy pair you write, use the form (Rx, Cy), where x = 1 or 2 and y = 1 or 2. Is the game a coordination game (also known as a battle of wills), a prisoner’s dilemma, or neither?
  C1 C2
R1 5, 4 9, 6
R2 6, 8 3, 2

Answer:

(R1, C2) and (R2, C1) are the pure-strategy equilibrium pairs.
The game is a coordination game.

  1. Follow the instructions for the previous question, but for the following game:
  C1 C2
R1 6, 8 1, 1
R2 8, 3 3, 7

Answer:

(R2, C2) is the pure-strategy equilibrium pair.
The game is neither a coordination game nor a prisoner’s dilemma.

  1. Follow the instructions for the previous question, but for the following game:
  C1 C2
R1 2, 7 7, 5
R2 3, 2 9, 1

Answer:

(R2, C1) is the pure-strategy equilibrium pair.
This game is a prisoner’s dilemma.

problems due Wednesday, October 31:

  1. Suppose that, for the following two profiles, social welfare function M generates the indicated social orderings. Suppose that proposition S is the claim that M satisfies condition I. Which of the following is true?
    1. These profiles, along with the indicated social orderings generated by M, entail that S is true.
    2. These profiles, along with the indicated social orderings generated by M, entail that S is false.
    3. These profiles, along with the indicated social orderings generated by M, neither entail that S is true nor entail that S is false.
profile 1 society   profile 2 society
Othello Desdemona Cassio Iago Othello Desdemona Cassio Iago
c b d a b b b c a c
a a b d a a a d c d
d d a c d c d a d b
b c c b c d c b b a

Answer:

b (look at alternatives a and d)

  1. Follow the same instructions as for the previous question, but for the following two profiles:
profile 1 society   profile 2 society
Antony Cleopatra Antony Cleopatra
a b c b a c
b d b d d a
c a a c b b
d c d a c d

Answer:

c (no violation of condition I is evident, but we cannot be sure that M doesn’t violate condition I elsewhere)

  1. Consider the following profiles, with some unreadable entries (i.e., suppose they are smudged out, or something like that), and with the social preference ordering for the second profile not yet specified:
profile 1 society   profile 2 society
Hamlet Ophelia Horatio Hamlet Ophelia Horatio
c b a b [unreadable] [unreadable] a  
b [unreadable] [unreadable] c [unreadable] [unreadable] c  
a [unreadable] [unreadable] a b a b  

What social preference ordering for profile 2 would make it the case that, regardless of how the unreadable entries are filled in, it could not be concluded from these two profiles and their corresponding social orderings that the social welfare function used to generated those two social preference orderings violates condition I?

correction added October 31: ‘generated’ should be ‘generate’

Answer:

b; c; a (violations of condition I arise only when the social preference orderings for two profiles are different and certain other conditions are not satisfied—if the social preference orderings are the same, there is no violation of condition I)

problems due Monday, November 5:

  1. Suppose that, for the two profiles given in problem 38, social welfare function M generates the social orderings indicated there. Suppose that proposition S is the claim that M satisfies condition ND. Which of the following is true?
    1. These profiles, along with the indicated social orderings generated by M, entail that S is true.
    2. These profiles, along with the indicated social orderings generated by M, entail that S is false.
    3. These profiles, along with the indicated social orderings generated by M, neither entail that S is true nor entail that S is false.

Answer: a

  1. Suppose that, for following two profiles (which also appeared in problem 39), social welfare function M generates the indicated social orderings (which are different from those indicated in problem 39). Suppose that proposition S is the claim that M satisfies condition ND. Which of the following is true?
    1. These profiles, along with the indicated social orderings generated by M, entail that S is true.
    2. These profiles, along with the indicated social orderings generated by M, entail that S is false.
    3. These profiles, along with the indicated social orderings generated by M, neither entail that S is true nor entail that S is false.
profile 1 society   profile 2 society
Antony Cleopatra Antony Cleopatra
a b a b a b
b d b d d d
c a c c b c
d c d a c a

Answer: c

  1. Suppose that, for the two profiles given in problem 40, social welfare function M generates the social ordering indicated for profile 1. What is a social ordering that could be stipulated for profile 2 that would ensure that M satisfies condition ND?

Answer:

There are several correct answers. Based on profile 1, the only possible violation of condition ND would come from Ophelia’s being a dictator. So, as long as the social ordering corresponding to profile 2 did not match Ophelia’s ordering there, condition ND would be assured of being satisfied. We know that Ophelia’s ordering there has a in third place, so any ordering with a in first or second place would be fine.

  1. Suppose that, for the two profiles given in problem 38, social welfare function M generates the social orderings indicated there. Suppose that proposition S is the claim that M satisfies condition CS. Which of the following is true?
    1. These profiles, along with the indicated social orderings generated by M, entail that S is true.
    2. These profiles, along with the indicated social orderings generated by M, entail that S is false.
    3. These profiles, along with the indicated social orderings generated by M, neither entail that S is true nor entail that S is false.

Answer: c

  1. Suppose that, for following two profiles (which also appeared in problems 39 and 42), social welfare function M generates the indicated social orderings (which are different from those indicated in problems 39 and 42). Suppose that proposition S is the claim that M satisfies condition CS. Which of the following is true?
    1. These profiles, along with the indicated social orderings generated by M, entail that S is true.
    2. These profiles, along with the indicated social orderings generated by M, entail that S is false.
    3. These profiles, along with the indicated social orderings generated by M, neither entail that S is true nor entail that S is false.
profile 1 society   profile 2 society
Antony Cleopatra Antony Cleopatra
a b d b a d
b d b d d b
c a a c b a
d c c a c c

Answer: c

  1. Suppose that, for the two profiles given in problem 40, social welfare function M generates the social ordering indicated for profile 1. What is a social ordering that could be stipulated for profile 2 that would ensure that M satisfies condition CS?

Answer:

a; c; b (then the social orderings for profiles 1 and 2 would have each pair of alternatives ranked in each of their two possible orders)

problems due Monday, November 12:

special instructions: Type your answers. You can present these proofs in paragraphs rather than numbered lists of claims and accompanying justifications.

  1. State which of the following propositions is true, and then prove it.
    1. Pairwise minority rule satisfies condition U.
    2. Pairwise minority rule violates condition U.

Answer:

Proposition 2 is true, and can be proved as follows. Consider the following profile:

A B C
a c b
b a c
c b a

For this profile, pairwise minority rule would say that in the social ordering, b P a and a P c and c P b. Those three preferences, being cyclical, do not admit of being represented by a preference ordering. Thus, pairwise minority rule does not specify a preference ordering for every profile. (This is essentially the Concorcet paradox, in the case of pairwise minority rule rather than pairwise majority rule.)

  1. State which of the following propositions is true, and then prove it.
    1. If a social welfare function violates condition I, then it satisfies condition ND.
    2. If a social welfare function violates condition I, then it violates condition ND.

Answer:

Proposition 1 is true, and can be proved as follows. Suppose condition I is violated. Then there is some pair of alternatives a and b, and some pair of profiles—call them Profile 1 and Profile 2—such that every individual’s pairwise ranking of a and b is the same in Profile 2 as in Profile 1, but such that the social preference orderings corresponding to the two profiles contain different pairwise rankings of a and b. Since every individual’s pairwise ranking of a and b is unchanged from Profile 1 to Profile 2, but the social pairwise ranking of a and b does change from Profile 1 to Profile 2, there must not be any individual whose preference ordering is always matched by the social preference ordering. Thus, condition ND will be satisfied.

  1. State which of the following propositions is true, and then prove it.
    1. If a social welfare function satisfies condition U but violates condition ND, then it satisfies condition CS.
    2. If a social welfare function satisfies condition U but violates condition ND, then it violates condition CS.

Answer:

Proposition 1 is true, and can be proved as follows. Suppose condition ND is violated. Then there is some individual—the dictator—whose preference ordering is always matched by the social preference ordering. By condition U, every possible profile is included in the domain. This means that, for every pair of alternatives a and b, there is a profile in which the dictator prefers a to b. Since this individual is a dictator, this profile’s corresponding social preference ordering ranks a above b. So, for every pair of alternatives a and b, there is a social preference ordering in which a is ranked above b, and thus condition CS is satisfied.

  1. State which of the following propositions is true, and then prove it.
    1. If a social welfare function satisfies conditions U and P, then it satisfies conditions I, CS, and PA.
    2. If a social welfare function satisfies conditions U and P, then it does not necessarily satisfy conditions I, CS, and PA.

Answer:

Proposition 2 is true, and there are a couple of good ways to prove it. One is to point to a Paretian social welfare function that violates condition I, such as the Borda count. Another is to point to a Paretian social welfare function that violates condition PA, such as pairwise minority rule modified to respect unanimity and to deal with Condorcet-paradox cases. The one strategy that won’t work is to point to a Paretian social welfare function that violates condition CS, since there aren’t any (see handout 24, section 24.3.4).

problems due Monday, November 19:

  1. Consider the following two propositions.
    1. No social welfare function satisfies conditions U, I, ND, CS, and PA.
    2. No social welfare function satisfies conditions U, I, ND, CS, PA, and P.

    Which of the following claims is true?

    1. Proposition 1 implies proposition 2.
    2. Proposition 2 implies proposition 1.
    3. Each proposition implies the other.
    4. Neither proposition implies the other.

Answer: a

For problems 52–54, consider the following partially specified profile and corresponding social preference ordering, and let M (which may or may not satisfy condition I—don’t assume anything either way) be the social welfare function generating the indicated social preference ordering:

A B C society
p r   r
r p   p
q q   q
  1. Is there a preference ordering for person C such that the resulting profile and corresponding social preference ordering is (1) consistent with the claim that M makes the set consisting of persons A and B quasi-decisive for r over q but (2) not consistent with the claim that M makes the set consisting of persons A and B decisive for r over q? If so, what is one such preference ordering?

Answer:

No, there is no such preference ordering for person C. Any preference ordering for person C would make the resulting profile and corresponding social preference ordering (1) consistent with the claim that M makes the set consisting of persons A and B quasi-decisive for r over q, because with society already ranking r over q there is no way to counter that claim with this profile and corresponding social preference ordering. By similar reasoning, no preference ordering for person C would make the resulting profile and corresponding social preference ordering (2) not consistent with the claim that M makes the set consisting of persons A and B decisive for r over q. Therefore, there is no preference ordering for person C that has both characteristics.

  1. Is there a preference ordering for person C such that the resulting profile and corresponding social preference ordering is (1) consistent with the claim that M makes the set consisting of person A quasi-decisive for p over r but (2) not consistent with the claim that M makes the set consisting of person A decisive for p over r? If so, what is one such preference ordering?

Answer:

Yes, there is such a preference ordering for person C. Any preference ordering for person C in which p is ranked above r would make the resulting profile and corresponding social preference ordering (1) consistent with the claim that M makes the set consisting of person A quasi-decisive for p over r; for if C ranks p above r, then the profile and its corresponding social preference ordering are prevented from being a counter-example to that claim. And any preference ordering for person C would make the resulting profile and corresponding social preference ordering (2) not consistent with the claim that M makes the set consisting of person A decisive for p over r, because with r being ranked above p in the social preference ordering (and A ranking p above r), we know that A is not decisive for p over r. Therefore, any preference ordering for person C in which p is ranked above r will do.

  1. Is there a preference ordering for person C such that the resulting profile and corresponding social preference ordering proves that person A is quasi-decisive for p over q but not decisive for p over q? If so, what is one such preference ordering?

Answer:

No, there is no such preference ordering. A single profile and its corresponding preference ordering can’t prove a universal statement such as a claim of quasi-decisiveness.

  1. Which of the following conditions if such that if a social welfare function satisfies it, then there is some set of individuals that is quasi-decisive for some alternative over another?
    1. condition U
    2. condition I
    3. condition P
    4. condition ND

correction added November 19: The first ‘if’ should have been ‘is’.

Answer: c

  1. Which of the following claims is part of the proof of the claim that if a social welfare function satisfies conditions U, I, and P, then there is an individual who is quasi-decisive for some alternative over another?
    1. If the set of all individuals is quasi-decisive for some alternative over another, then there is a smaller set of one or more individuals that is quasi-decisive for some alternative over another.
    2. If the set of all individuals is quasi-decisive for some alternative over another, then there is a smaller set of one or more individuals that is quasi-decisive for that first alternative over the second.

Answer: a

  1. If we are assuming that the set of all individuals is quasi-decisive for a over b, how would we fill in the following profile so that we could use it in proving the claim that is the correct answer to question 56?
George everyone except George
   
   
   

Answer:

George everyone except George
x a
a b
b x

 or

George everyone except George
a x
b a
x b

problems due Wednesday, November 28:

special instructions: If you turn in your answers to my office by the end of Tuesday, November 27, or send them to me by e-mail by that time (in which case I will waive the usual penalty for submitting answers electronically), I will grade them and return them in class on Wednesday, November 28.

  1. Consider the following profile for a two-person society:
1 2
e d
c a
d b
b c
a e

Suppose conditions P and L are in force, and person 1 has control (in the condition-L sense) over a versus b, and person 2 has control (in the condition-L sense) over c versus d. Does this profile show that condition U cannot be satisfied? If so, exhibit a set of cyclical preferences that follow from applying conditions P and L, and indicate, for each preference in the set of cyclical preferences, which condition it follows from. If not, give a social preference ordering for this profile that is consistent with conditions P and L.

Answer:

No, it does not. The following social preference ordering for this profile is consistent with conditions P and L:

d
b
a
c
e

Any social preference ordering consistent with the following constraints will work:

d P a (Pareto)
d P b (Pareto)
b P a (L—person 1)
d P c (L—person 2)

  1. same instructions as for the previous problem, but with this profile:
1 2
c b
a d
e c
b e
d a

Answer:

Yes, it does. Conditions P and L imply the following preferences, which are cyclical and therefore do not admit of being represented by a social preference ordering:

a P b (L—person 1)
b P d (P)
d P c (L—person 2)
c P a (P)

  1. Suppose there are just three alternatives (a, b, and c) and just two people (person 1 and person 2). Suppose person 1 has control (in the condition-L sense) over a versus b, and person 2 has control (in the condition-L sense) over b versus c. Is there a profile that shows that conditions U, P, and L cannot all be satisfied? If so, write it (or one of them). If not, give some indication of why such a profile cannot be provided (or, perhaps more realistically, why we should not expect that such a profile could be provided).

Answer:

Yes, there is. Here it is:

1 2
c b
a c
b a