# 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
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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.
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