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?
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&amp;3; B gets slices 2&amp;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.