# Pareto optimality when the allocations are restricted

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):

CVCV

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?

• The definition of Pareto-optimality did not change. The set of feasible allocations changed. Suppose the only possible allocation is giving Alice a vanilla slice and Bob a chocolate slice, throwing away all else. This is Pareto-optimal if no other allocations are feasible. Jul 27 '15 at 8:10
• Indeed, Denesp has it right. With the connected pieces restriction, giving CVC to Alice and V to Bob is Pareto optimal, as is giving C to Alice and VCV to Bob. Pareto optimality depends on alternative feasible allocations. Jul 28 '15 at 16:33