ECO222 Behavioral Economics Problem Set 1a | Solutions
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ECO222 Behavioral Economics
Problem Set 1a | Solutions
1. In order for the weak preference relation ≿ to be rational, two conditions must be satisfied. (a) What is the first condition called and how is it defined?
(b) What is the second condition called and how is it defined?
(c) We consider a universe formed by all students registered in ECO222. Provide an example of relations that are not rational, and an example of relations that are rational – explain your answers with regards to the two conditions mentioned in (a) and (b).
Solutions:
(a) - Completeness: For all x, y in the universe, either x ≿ y or y ≿ x (or both). (b) - Transitivity: For all x, y, z in the universe, if x ≿ y and y ≿ z, then x ≿ z. (c) - Irrational relation:
“weighs more than” - incomplete and transitive.
- Rational relation:
“is at least as tall as” - complete and transitive.
2. (a) What does it mean to say that the strict preference relation ≻ is irreflexive? (b) Prove that the strict preference relation ≻ is in fact irreflexive.
Solutions:
(a) Irreflexive: not x≻x (for all x)
(b)
1. 
≻ 
2. 
≿ 
& ¬
≿ 
3. ⊥ ∴ ¬
≻ 
by assumption, for proof by contradiction
from (1), by definition of ≻
from (2)
QE
3. Prove the following principle: If x≻y and y≿z, then x≻z (for all x, y, z). This proof will have two parts. Please justify each step of your proof.
(a) First, prove that if x≻y and y≿z, then x≿z.
Solutions:
1. 
≻ 
& 
≿ 
2. 
≿ 
& ¬
≿ 
by assumption
from (1), by definition of ≻
3. 
≿ 
from (1) and (2), by transitivity of ≿
(b) Second, prove that if x≻y and y≿z, then not z≿x.
Solutions:
4. z ≿ x by assumption, for proof by contradiction
5. y ≿ x from (1) and (4), by transitivity of ≿
6. ⊥ from (2) and (5)
7. ¬z ≿ x from (4)-(6), by contradiction
8. x ≻ z from (3) and (7), by definition of ≻
∴ x ≻ y & y ≿ z → x ≻ z QE
□
4. (Difficult) Condorcet’s Paradox or Voting Paradox
Our committee consists of three individuals and makes decisions using majority voting. When they compare two alternatives x and y they simply take a vote, and the winner is said to be “preferred” by the committee to the loser.
Suppose that the preferences of the individuals are as follows: Person 1 likes x best, y second best, and z third best. We write this in the following way: Person 1: x, y, z. Assume the preferences of the other two people are: Person 2: y, z, x; and Person 3: z, x, y.
Show that in this example the committee preferences produced by majority voting violate transitivity.
Solutions:
Using majority voting, it is easy to show that for the committee, x≻y because Person 1 and Person 3 prefer x to y. Similarly, we can obtain y≻z, and z≻x.
However, if transitivity holds, given that x≻y and y≻z, it is expected to have x≻z. This leads to a contradiction.
2022-05-23