$$$$$$$$ (7)
Behind the first door, there are three dollar bills, behind the second door, there are five dollar bills, and behind the third door there are seven dollar bills. Players know about this.
Player 1 moves first. He can open any door and remove as many dollar bills as he likes from behind that door. Player 2 moves next. He can select any door, open it and remove as many dollar bills as he likes from behind that door.
The game continues, until there is only one dollar bill left. The player who picks up the last dollar bill gets zero, the other player gets all the money.
Remember, when it is your turn to play, you can open any door, but can not open two doors! After you open any door, you would have to remove at least one dollar bill from behind that door! And when it is your turn to play, you can not pass, which means you would have to open one of the doors! Of course, if there are no dollar bills left behind a door, you can not open that door!
Let’s play!
Hint: Try to solve this game for (1,2,3) instead of (3,5,7), and you will see that the player who moves first will lose!
To learn about subgame perfect equilibrium, try to explicitly write down the equilibrium strategies in the (1,23) game.
I I have taught game theory and related material for 25+ years and used this game as an example of extensive-form games and subgame perfect equilibrium.