Philosophy of Economic Preference and Rational Choice

Please turn in your homework at the start of class. Please staple or use a paperclip if you use multiple sheets. You may collaborate with (but not copy from!) others. But (a) please try them by yourself first, and
(b) acknowledge anyone you work with.

1. Suppose there are only two people: person 1 prefers x to z and z to y (xP1zP1y), and
person 2 prefers y to x and x to z (yP2xP2z). Suppose that our CCR ranks y above
x (yPx) for this profile of individual preferences. Suppose also that our CCR always
delivers a transitive weak social preference relation R, and satisfies the Strong Pareto
principle (defined by Sen on p. 101) and Independence of Irrelevant Alternatives.
Show that our CCR must always rank y above z whenever person 2 prefers y above z,
no matter what person 1 prefers.
2. Our proof of Arrow’s theorem in class (based on Sen’s pp. 286–289) appealed to two
lemmas: the Spread of Decisiveness and the Contraction of Decisive Sets. Suppose
we relax the assumption of completeness for social preference. Would the Spread of
Decisiveness still hold? Would Contraction of Decisive Sets still hold? Explain why
or why not.
3. Would the Method of Majority Decision qualify as a SWF if restricted to cases where
there are two individuals and three alternatives? Prove your answer.
4. Suppose that individuals are allowed to have incomplete preferences. Say that x is
pseudo-Pareto-superior to y if and only if someone prefers x to y (for some i, xPiy)
and no one prefers y to x (for all i, ¬yPix). Show that, when individuals can have
incomplete preferences, pseudo-Pareto-superiority is not a transitive relation.
5. Give an example in which the Pareto-extension rule violates Positive Responsiveness.