Rational Preference Relations and Transitivity

Ralph-Christopher Bayer
Assignment 1
Deadline: Beginning of week 6 (Sunday April 7th, midnight)
Multiple Choice Questions
1. A preference relation is rational if it is both transitive and reflexive.
⇤ False
Briefly explain:
A rational preference relation must have all the axioms and
majorly the two axioms of completeness and of transitivity. In regards
to transitivity, it means that an individual should have a preference
relationship such that a bundle X is preferred to a bundle Y and the
bundle Y preferred to a bundle Z then automatically bundle X is
preferred to bundle Z .On completeness it is assumed that an
household must have a preference relationship between any two
categories or models of goods, that is it must be clearly seen that
either they prefer Y to X, or X to Y or they are indifferent. With those
two a preference relationship can be termed rational
2. For a given set of alternatives X, the preference relation % is transitive if it satisfies x ≥ y,
y ≥ z, and x ≥ z for all x, y, z € X.
⇤ True
Briefly explain:
It shows the preference over a certain set of alternatives X ,by transitivity we refer to the
property that if x is preferred to y and y is preferred to z , then it means x is preferred to z.
3. For a given set of alternatives, the preference relation % is transitive when the following
Holds: if x ≥ y, and y ≥ z, then x ≥z for all x, y, z € X.
⇤ True
Briefly explain:
It shows the preference over a certain set of alternatives X, by transitivity we refer to the
property that if x is preferred to y and y is preferred to z, then it means x is preferred to z.

4. Let X denote the set of all car models which are currently available on the Australian market.
The relation “has at least as much horse power as” is complete and transitive.
⇤ True
Briefly explain:
It is complete in the sense that it explains what or which model is preferred more between two
car models and it is also transitive in the sense that in regards to the horse power it is preferred
than the other one which is better than the rest of the models. Say a consumer wants to buy a car,
he finds out that car A and B have better horse power strengths than the rest, but he prefers B to
A. This means A is better than all the rest.
5. Whenever the preferences of an individual can be represented by a utility function, the
individual must have rational preferences.
⇤ True
Briefly explain:
Every household as sets of wants that it wishes to satisfy. Utility is the level of satisfaction
got from consumption of a service or a good. Utility function therefore is a concept which
measures the preferences in a set of services and goods. Therefore a household must have a
utility function of the various goods it needs so as to satisfy its wants and thus to manage this
utility function sit must have a rational preference. (Harold Kincaid, 2009)
6. According to the Prospect Theory, for a given reference point; the value function is steeper in
the gain domain than in the loss domain.
⇤ True ⇤
Briefly explain:
Prospect theory is one for decision making under conditions of risk. It is also known as the
loss aversion theory and it assumes that if an individual is given a set of two choices, one
detailing on potentials losses and another detailing the potential gains he will opt for the latter.
That is individuals make decisions in respect to perceived gains rather than to perceived losses.
For questions 7 and 8, consider the situation of rolling a fair dice once such that the set of
possible outcomes is ⌦ = {1, 2, 3, 4, 5, 6}, and each realization occurs with probability of 1/6
7. The event that the outcome is even is given by the set A = {2, 4, 6}, and hence Pr(A) =1∕2

⇤ True ⇤
Briefly explain:
Each possible outcome of say a 1 occurring per throw is 1/6 hence the probability of either
getting 2, 4 or 6 occurring will be given by 1/6 × 3= ½.
8. The intersection of the two events A1 = {1, 2, 3} and A2 = {4, 5} is empty. Hence, these
two events are disjoint and independent.
⇤ True ⇤
Briefly explain:
They are independent because the event of a 1 or 2 or 3 occurring will not affect the event of 4
or 5 appearing since they are two separate events.
That is P (A∩B) =P (A) P (B).
9. Suppose you roll the dice three times. Then the probability of rolling at least one even
number is strictly bigger than the probability of rolling an even number which is strictly
larger than 2.
⇤ True ⇤
Briefly explain:
Because the probability of getting at least one even number will 3/18 because of the chances
of a 2,3 or 4 occurring, yet probability of getting an even number greater than 2 will be 2/18
because of the likelihood of a 4 or 6 occurring.
10. The Linda Problem is an example of conjunction fallacy.
⇤ True ⇤
Briefly explain:
Yes because most scholars and students were biased on who Linda as person was because they
failed to count for the low base rates violating the law of probability. (Tversky, 1983)
Problems
1. Consider the following situation: 100 people are gathered in a room and 50 of them are
selected at random and given a mug, those that were given the mug are asked for the minimum
amount of money they would need to give up the mug and those without the mug are asked for
the maximum amount they would be prepared to pay for the mug.