# An Introduction to Fair Division

— **Lectures by Rohit Vaish**

### Questions based on the first lecture: Cake Cutting - I

- In the setting of Cake Cutting, which of the following is true?
- An envy-free allocation is also proportional.
- A proportional allocation is also envy-free.
- An allocation can be simultaneously envy-free and proportional.
- All of the above.
- Consider the Robertson Webb Query Model, for 2 agents, let the cut and evaluate queries be , (which returns ) and (which returns ). State true/false:
- , () and gives a proportional cake division.
- , () and gives both the agents a piece of cake that they like strictly more than .
- What is the query complexity of Dubins-Spanier protocol for proportional cake division?
- Consider a fair division problem involving agents. An agent receives a fair share implies:
- The agent receives a share that, in every agent’s opinion, has a value that is equal to 5% or more of the total.
- The agent receives a share that, in that agent’s own opinion, has a value that is exactly equal to 5% of the total.
- The agent receives a share that, in that agent’s own opinion, has a value that is at least 5% of the total.
- The agent receives at least of the total.
- Suppose there are agents fighting over a cake. Suppose cuts the cake into pieces say , such that she values them all equally, that is . Which of the following are true?
- If and , then, the under the allocation where gets respectively, is envious.
- If and , then, the allocation where gets respectively, is envy-free.
- If , and , then the allocation where gets can never be envy-free.
- Based on Q5c, suppose , and trims the piece into such that , then in order to achieve an envy-free allocation for the pieces , the correct picking order of the agents is:
- First , second , then
- First , second , then
- First , second , then
- None of the above orders will guarantee an envy-free allocation of the pieces
- Based on Q6, suppose get and are envy-free as yet and remains to be allocated.
- An envy-free division of among the agents combined with the envy free division where get may not be necessarily envy-free.
**will not be envious even if the entire goes to**- will not be envious even if the entire goes to
- If cuts into equal pieces (according to her) and , and pick one piece in that order, then all the agents are envy-free.
- If cuts into equal pieces (according to her) and , and pick one piece in that order, then all the agents are envy-free.