1. The following game is called the magic refrigerator. It is a two-player dynamic game. There is a magic refrigerator. The first time it is opened, it contains two bottles of beer. Each successive time it is opened, two additional bottles appear by magic, up to a maximum of eight bottles. However, the refrigerator has rules: once any bottles are removed, the magic ceases and no additional bottles will be produced. Furthermore, when a given player opens the refrigerator, that player has the option of removing the bottles, and the player who removes the bottles is entitled to two more bottles than the other player, which we will call the "+2" rule. So, for example, if a player removes the bottles when there are six of them total, she receives four bottles and the other player receives two. The two players take turns opening the refrigerator and deciding whether or not to remove the bottles. When any player removes the bottles, the bottles are divided according to the "+2" rule given above, and the game ends. When a player chooses instead to close the refrigerator without removing any bottles, then the other player opens the refrigerator, which now contains two additional bottles, and chooses whether or not to remove the bottles. Assume that the first time a player opens the refrigerator and there are eight bottles, the player always removes them. Player 1 goes first, and a player's payoff is assumed to be the number of bottles of beer she receives. (a) Denote the action "remove" by R, and the action "don't remove" by D. Write down all of the possible outcomes (terminal histories) of this game, and write the players' payoffs for each outcome. (b) Represent this game in a game tree. (c) How many proper subgames does this game have?

• I am confused on how to structure the tree, whether we should increment the payoffs when ending each action. For example, if plaayer 1 goes first, when he removes, the payoff will be (2,0) 2 for player 1 and 0 for player 2. for not removing it goes to player 2, and when 2 removes, should it be (0,2) or (2,2)? could please explain the procedure? Thank you!