Question

3. Angus, Mary and Jack are trying to decide where to go to eat tonight. There are three options: the local pub (P), a new dumpling place (D) and an Italian restaurant (I). Their preferences between these three alternatives are as follows, where the first alternative listed is the most preferred and the last is the least preferred:

Angus preferences: 1.P 2.D 3.I Mary preferences: 1.I 2.D 3.P

Jack preferences: 1.D 2.I 3.P

(a) Consider the following system for determining a social ranking between these alternatives. For each possible pairwise comparison, if 2 out of the 3 people prefer one option over the other, then that option is deemed strictly preferred over the other by “society”. For example, if Angus and Mary prefer A over B, “society” is said to prefer A over B. Using this rule, what is the social ranking between P, D and I? Is this ranking transitive? Explain briefly.

(b) Now suppose Mary has changed her mind: she now prefers P over D. Applying the social ranking rule described in part (a), is the new social ranking transitive? Explain briefly.

Answer 1

Here in this question the preference order are given of 3 individuals namely Angus, Mary and jack between choosing a restaurant to eat. The social choice rule is such that if majority like one restaurant over other than that restaurant will be chosen as the socially preferred restaurant.

(a) The preference of 3 individuals are as follows :

Angus- 1P, 2D, 3I

Mary- 1I, 2D, 3P

Jack- 1D, 2I, 3P

Where D=New dumpling place

P= local pub

I= Italian restaurant

Let's see the pairwise social preference between different restaurant.

Let's first considered P&D

Angus prefers P over D

Mary prefers D over P

And Jack prefers D over P

So the socially preferred restaurant between P&D is D over P because Mary and jack prefers D over P which is majority among 3 individuals.

Similarly let's see the preference order between D&I

Angus prefers D over I

Mary prefers I over D

Jack prefers D over I

So the socially preferred restaurant between D&I is D because Angus and Jack prefers D over I which is majority among 3 individuals

Similarly let's see the preference order between P & I

Angus prefers P over I

Mary prefers I over P

Jack prefers I over P

So the socially preferred restaurant between P & I is I because Mary and Jack prefers I over P and they are majority among 3 individuals.

Now let's see whether this social preference order is transitive or not.

The property of transitivity of preference says, if A is socially preferred over B and B is socially preferred over C then A must be socially preferred over C also.

In our question, D is socially preferred over I and I is socially preferred over P then for property of transitivity of preferences to hold D must be socially preferred over P, which is the case In our question that D is socially preferred over P. Hence in part a property of transitivity holds.

(b) In part B everything remains the same just that now Mary prefers P over D. So the preference order for Mary is 1I, 2P, 3D. And for other two preference order remains the same.

Now let's check for socially preferred restaurant between P&D

Angus prefers P over D

Mary prefers P over D

And Jack prefers D over P.

So the socially preferred restaurant between P&D in this case is P. The reason is the same as above because we are using majority rule if 2 person prefers P over D then it's majority preferring P over D in case of 3 person. That's why P is socially preferred over D

Let's see the social preference between D&I

Angus prefers D over I

Mary prefers I over D

Jack prefers D over I.

So D is socially preferred over I. Because Angus and jack prefers D over I.

Now let's see the preference order between I & P

Angus prefers P over I

Mary prefers I over P

Jack prefers I over P.

So the socially preferred restaurant between P&I is I. Because Mary and jack prefers I over P.

Now finally let's see of this social preference order satisfy the property of transitivity or not.

D is socially preferred over I and also I is socially preferred over P according to the property of transitivity of preference D must be socially preferred over P but in this case P is socially preferred over D which is the violation of transitivity of preferences.