Question

Alice, Bob and Charlie rent an apartment with 3 rooms. The rooms are different from one another (say, one is larger, one has attached bathroom, etc.). Alice’s preferences over rooms is 1 > 2 > 3 (she likes room 1 the most, than room 2, than room 3). Bob’s preference is 2 > 1 > 3, and Charlie’s is 2 > 3 > 1. Make a list of the six 1
possible ways to assign tenants to rooms (one tenant in each room), and determine which of the assignments are Pareto efficient. Hint: Recall that an assignment is efficient if there is no other way to assign the rooms which improves the situation of some people and doesn’t hurt anyone. So if 2 tenants can trade their rooms to the benefit of both then the current assignment is inefficient.

Answer 1

6 possible ways are

Alice 1 Bob 2 Charlie 3

Alice 1 Bob 3 Charlie 2

Alice 2 Bob 1 Charlie 3

Alice 2 Bob 3 Charlie 1

Alice 3 Bob 1 Charlie 2

Alice 3 Bob 2 Charlie 1

We can see that it is not possible for all to get their most preferred rooms because , Bob and Charlie both prefer room 2. So, we will try that 2 of them get their first prefernces and the third person gets the second preferred room.

So we can allot the rooms in the following way,

Alice 1, Bob 2, Charlie 3.

So Alice and Bob get their most preffered rooms and Charlie gets his second preferred room.

If we did Alice 1, Bob 3 and Charlie 2,  then Alice and Charlie would get their most preferred rooms but Bob would get the least preferred room.

So the correct answer would be Alice 1, Bob 2, Charlie 3.