Reverse Monty Hall

Suppose you're hosting a game show with the following rules:

  1. There are three doors. Behind one of the doors is a prize; you get to choose which door it is. The contestant picks one of these doors.
  2. Once you pick a door, the you must choose one door and shows you that there is nothing behind it. You must never choose the door that has a prize behind it, nor choose the door that the contestant chose.
  3. The contestant is then given the option to use the door you originally picked or to switch to the door they know nothing about.
  4. If the door the contestant chose in step 3 contains a prize, they win. If it does not, they lose.

What strategy (if any) will give the contestant the highest chance of receiving the prize, and what is their chance of winning the prize with this strategy? (Hint: it's not 1/2.)

The computer will use the strategy: