# Method of Majority Decision

# Method of Majority Decision

Midterm

PHIL 339

Due 4:50pm on T 11/2

Answer five of the following seven questions. You are welcome to use the book and

handouts, but no other resources. Collaboration is prohibited. Please upload to Blackboard

or leave in my mailbox in MHP 113. Late work will not be accepted.

1. Suppose that R is a quasi-ordering over the set X = {x, y, z}, with xPy, ¬xRz, ¬zRx,

¬yRz, ¬zRy. What is the maximal set? What is the choice set?

2. Supose that a choice function C(⋅) satisfies both contraction consistency (α) and expansion consistency (β). Define R as follows: xRy iff x ∈ C({x, y}). Prove that R is

transitive.

3. Daniel Kahneman and Amos Tversky find that most people prefer a certain payoff

of $3000 to a lottery that offers them an 80% chance of winning $4000 (otherwise

winning $0), but would also prefer a lottery that offers them a 20% chance of winning

$4000 (otherwise $0) to a lottery that offers them a 25% chance of winning $3000

(otherwise $0). Prove, using inequalities, that these preferences are incompatible with

expected utility theory, if utility is solely a function of money.

4. Prove that, when there are only two people, the Method of Majority Decision always

delivers a quasi-transitive social preference relation.

5. Assume there are three people (1, 2, 3) and three alternatives (x, y, z). Consider the

following social welfare function f: for any profile (R1,R2,R3), if person 1 and person 2 have exactly the same rankings of all three alternatives, then social preference

coincides with their ranking (i.e., R = f(R1,R2,R3) = R1 = R2); if their rankings are

different in any way, then social preference is determined by person 3’s ranking (i.e.,

R = f(R1,R2,R3) = R3). Which axiom of Arrow’s Impossibility Theorem does this rule

violate? Prove your answer.

1

6. “Absolute Majority Rule” says that xPy if and only if more than half the population

strictly prefers x to y; yPx iff more than half the population strictly prefers y to x; xIy

otherwise.

(a) Is Absolute Majority Rule a decisive collective choice rule? Explain.

(b) Is it a social decision function? Explain.

(c) Does it satisfy Positive Responsiveness? If yes, prove it; if no, state a preference

profile in which Absolute Majority Rule violates Positive Responsiveness.

7. Consider the following profile U = (u1, u2) of utility functions for persons 1 and 2:

u1 u2

x 1 1

y 4 0

(a) State the social preference relation between x and y assigned by the leximin social

welfare functional to profile U.

(b) State the social preference relation between x and y assigned by the utilitarian

social welfare functional to profile U.

(c) Give an example of another profile U′ = (u

′

1

, u

′

2

) to which the leximin social welfare functional, but not utilitarianism, assigns the same social preference relation

that it assigns to U.

(d) Give an example of another profile U′′ = (u

′′

1

, u

′′

2

) to which the utilitarian social

welfare functional, but not leximin, assigns the same social preference relation

that it assigns to U.

2

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