Three contestant Monty Hall problem

The simplest children in the streets have long known that one should switch doors when faced with the Monty Hall problem. A variant is proposed as follows:

Monty Hall presents three doors to Xavier, you, and Zebediah. Two doors conceal a goat, and one door conceals a car (assume that all contestants prefer a car to a goat). Each contestant chooses a door and communicates this choice to Monty without revealing this choice to the other contestants; if more than one contestant chooses the same door, all contestants choosing that door are awarded an instance of the prize behind it.

After each contestant has chosen a door, Monty reveals that each door has been chosen by exactly one person. He then opens the door chosen by Xavier, and there is a goat behind it. Monty offers you and Zebediah the choice to keep your respective doors or switch (again, if you both wind up with the same door you each get an instance of the prize behind it).

Should you stick or switch? Should Zebediah stick or switch?