Rational Preference Relations and Transitivity

This document discusses the concept of rational preference relations and transitivity in decision making. It explains the importance of completeness and transitivity in preference relationships. The document also explores the Prospect Theory and its implications in decision making under risk.

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.

(a) What does standard theory predict about the average amounts written down in each of the
groups (those given the mug and those not given the mug).
Due to the endowment factor it would predicted that the mug owners priced the mugs at a
relatively high price than the non-mug owners were willing to offer. Both sides were trying to
avert losses and maximize gains.
b) What would you predict about the amounts written down by people in each group?
Which effect is at work?
Both the loss aversion effect and the endowment factor. Those who received the mugs were not
willing to sell at low prices because they valued the item in their possession at a greater value
and selling it at a lower price would mean losses. Those who did not possess the mugs priced
them at a lower value hence paying large amounts for a mug would mean losses.
(c) Which element of Prospect Theory is consistent with the observations typically found in
experiments of this type?
The prospect theory assumes that individuals aim at averting losses , hence given a choice that is
presented in terms of potential gains and another presented in relation to potential gains they will
choose the latter. Same case with the mug experiment, the mug owners were more attached to the
potential gains they would get from selling at high prices
2. Match each of the following descriptions to phenomena (for example heuristics) that you have
seen during the course, provide a brief description in each case:
(a) Hugo was offered a cable TV subscription service for e40 a month; initially he was reluctant
as he considered that the price was higher than his valuation of the service.
The cable company said he could enjoy the service for a month for free at which point he could
call and cancel the service; Hugo accepted expecting to make the call. At the end of the month he
considered that he would prefer to give up AUD 40 than the cable TV.
Prospect theory-; Endowment factor Hugo valued the service of the cable TV that he was
not willing to part with it.
(b) Sarah lost all her luggage the last time she checked it in on a flight. She is never going to
check her luggage in again, even if it means having unpleasant arguments with flight attendants.
Preference relation: Sarah would rather have arguments with the attendants than check in her
own luggage again
(c) Franz is contemplating the future success of the German men’s football team, he considers
that it is quite unlikely that Germany will win the World Cup in 2018, but he thinks there is

strong chance that Germany will win the European Championships in 2020. When an interviewer
asks him about the possibility that Germany could win both the European Championship and the
World Cup, he says he thinks it is more likely than Germany winning the World Cup.
Probability; Franz sees the probability of Germany winning the euro 2020 as an independent
event to their winning of the world cup 2018.The two events are independent.
(d) Steve and James are entering a contest; they have to guess the weight in kilograms of actor
Matt Damon, the nearest guess wins a prize. Prior to providing their answers
Steve and James were asked for their own weights, they weighed 63kilograms and 105kilograms
respectively. Steve guessed 75kg and James guessed 90kg
Base rate neglect; In such a case both James are assuming that since their weight is at some given
point so should all other actors weight. They are neglecting all other information and dependent
characteristics
3. Consider the following problem:
The probability of breast cancer is 1% for a woman at age forty who participates
in a routine screening. If a woman has breast cancer, the probability
is 80% that she will get a positive mammography. If a woman does not have
breast cancer, the probability is 9.6% that she will also get a positive mammography.
A woman in this age group had a positive mammography in a routine
screening. What is the probability that she actually has breast cancer?
(a) Using Bayes’ rule give the correct answer to this question.
Bayes rule provides a way to update the existing beliefs in the chance new ones come to light.
For example, if the probability that a person has cancer is trying to be provided, initially it would
be said to be whatever percent in the population has got cancer but due to the recent acquired
knowledge like if a person a smoker, then the probability can be updated since its more likely for
a smoker to get cancer than a non-smoker. (Sven Ove Hansson, 2017)
Hence:
Out of the 1000 women 10 have breast cancer ;
8 out of the 10 have positive mammography;
From the 1000, 990 do not have breast cancer and out of those 990, 95 will get a
positive mammography.
Total number of women having a positive mammography will hence be 95+8=103
Of the 103 with positive mammography 8 will have cancer
Hence expressed as a proportion :- 8∕103 = 0.0777 or 78%.
(b) When faced with this question respondents frequently report answers between 70%
and 80%. What type of error are respondents committing?

The omission error, they forget the initial percentage of women with breast cancer and also
forgetting to subtract those with false positive mammography.
Consider the following alternative formulation of the problem.
10 out of every 1,000 women at age forty who participate in a routine screening
have breast cancer.
8 of every 10 women with breast cancer will get a positive mammography.
4
95 out of every 990 women without breast cancer will also get a positive mammography.
There is a new representative sample of women at age forty who got a positive
mammography in a routine screening. What proportion of the sample do you
expect to have breast cancer?
95+8=103
Original %= 8
8÷103= 0.0777 representing 78%
(c) When faced with this alternative formulation respondents are typically much closer to the
correct answer. Provide an explanation for the difference in responses for the two questions.
The alternative formulation outlines clearly parts by parts of the different proportions unlike
the first where the proportions are all in a continuous statement.
4. A coroner in the UK investigated four cases of suicides among teenage boys in the UK within
a short space of time. During the fourth inquest he commented that all the boys were owners of
the well-known computer game Call of Duty and recommended that use the game by under 18s
should be restricted. In response to the coroner’s recommendation it was suggested that the
coroner was guilty of Base Rate Neglect in reaching his conclusion.
(a) Provide a suggestion of which base rate the coroner may be ignoring.
The base rate or class which is the whole community of subscribers to the online game, 4
kids represent a very small percentage of the total users. He ignored the general information and
concentrated on the specific information of the four boys and used it to judge the whole
phenomena.
(b) Provide a brief description of base rate neglect.
It’s a kind of a formal fallacy whereby if an individual is given data related to some
base rate information that is general and also given specific information on some parts of the
general information he or she tends to concentrate on the latter to make hypothesis.

(c) Suppose that in subsequent inquests into suicides of teenage boys the same coroner finds that
some owned Call of Duty while others had never played the game. Assuming that the coroner is
susceptible to Confirmation Bias, how would you expect his belief in the hypothesis that Call of
Duty is a causal factor in teenage suicides to be affected by the subsequent evidence?
He would be cautious before making conclusions from a small part of the whole
phenomena.
REFERENCES
Harold Kincaid, D.R., 2009. The Oxford handbook of philosophy of economics. Oxford
Handbooks.
Sven Ove Hansson, T.G.-Y., 2017. Stanford Encyclopedia of Philosophy. Substansive revision,
Tuesday 14 November. pp.10-30.