# 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. – Giskard 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. – Shane Jul 28 '15 at 16:33

## 1 Answer

Let me elaborate on the comments quickly - maybe only for the sake of this question not being "unanswered". Pareto optimality only makes sense when defined with respect to feasible allocations. That is, your basic definition above lacks an important word:

an allocation is PO if there does not exist another feasible allocation in which all players are weakly better off and at least one player is strictly better off.

Without this word, you can easily find an allocation that dominates "A gets slices 1&3; B gets slices 2&4". For instance, you could give Alice eight additional full cakes of chocolate. However, there are no other cakes and, hence, this allocation is not feasible and therefore does not Pareto dominate what you suggest.

As a result of your feasibility restriction, any allocation in which one agent gets one slice of the preferred kind and the other one gets the other three is PO. If A gets the entire cake, she can make B better off by giving him a slice of V which she does not care about; analogously if B has the whole cake. If both agents get two connected slices, a similar improvement exists.