Last semester, teaching some maths to engineers, I decided to play (via email) the Guess 2/3 of the Average game. Two players won (with guesses of 22) and so I needed a tie breaker.

I came up with a hybrid of Monty Hall, not too dissimilar to the game below (the prize was €5).

## Rules

In this game, the host presents four doors to the players Alice and Bob:

Behind three of the doors is an empty box, and behind one of the doors is €100.

The host flips a coin and asks the Alice would she like heads or tails. If she is correct, she gets to choose whether to go first or second.

The player that goes first picks a door, then the second player gets a turn, picking a different door.

Then the host opens a door revealing an empty box.

Now the first player has a choice to stay or switch.

The second player then has a choice to stay or switch (the second player can go to where the first player was if the first player switches).

Questions:

1. What is the best strategy for the player who goes second:
• if the first player switches?
• if the first player stays?
2. Should the person who wins the toss choose to go first or second? What assumptions did you make?
3. How much would you pay to play this game? What assumptions did you make?
4. If there is a bonus for playing second, how much should the bonus be such that the answer to question 1. is “it doesn’t matter”.