There are three closed doors. One has a new car, one has the keys to the car and one has a goat. These prizes are randomly assigned.

There are two players: the first player has to find the car, the second player has to find the keys to the car. If both players succeed they win the car.

The first player enters the room and may open any two of the three doors, one after the other. If successful, the doors are closed again and the second player enters the room. The second player may also open two of the three doors, but cannot communicate with the first player.

Surprisingly there is a strategy where the probability of winning is better than (2/3)^{2}. Find it.

I had some ideas, but they depend on the detailed rules:

All of the below are assumptions about the rules. Are any wrong? Call the two players P1 and P2.

1) Can P1 quit after unsuccessfully opening just one door?

2) Can P1, when unsuccessful finding the car, after opening one or two doors, quit and leave one door (or two doors) open for P2 to see?

3) If he opened two, can he close one? (Open doors can act as signals...)

4) The fact that P1 and P2 cannot communicate means that they each must offer their best guess as to the door of the car and keys respectively and independently, right? That is what the success probability depends upon, yes?

*Edited on ***September 30, 2018, 6:46 pm**