This question is about Pareto optimality (PO) in cake-cutting.
The basic definition is: an allocation is PO if there does not exist another allocation in which all players are weakly better off and at least one player is strictly better off. For example, consider the following cake (where "C" means Chocolate and "V" means Vanilla):
Suppose there are two players: Alice wants only chocolate and Bob wants only vanilla. Then, there is only one PO allocation, and that is to give slices 1 and 3 to Alice and pieces 2 and 4 to Bob.
Now, suppose we restrict the cake-cutting such that the pieces must be connected. With this restriction, there are two possible definitions of PO:
- If we remain with the previous definition, then there exits no PO allocations with connected pieces, since the only PO allocation requires disconnected pieces.
- On the other hand, we can change the definition and say that an allocation is PO if there does not exist another allocation with connected pieces in which all players are weakly better off and at least one player is strictly better off. Under this definition, of course there exist PO allocations, such as, giving slice 1 to Alice and slices 2-4 to Bob.
Both definitions are reasonable. My question is:
which definition is more common in the economics literature?