Dad baked a chocolate cake for his twins Jack and Mary on their birthday, and he wants Jack and Mary to divide it without any screaming and mess. Mom devises the following procedure: First, Jack (who was born first) proposes a division of the cake that Mary can accept or reject. If the first offer is accepted by Mary, then the game is over and Mary and Jack get the split of cake offered by Jack. If the offer is rejected by Mary, then Mom gets to eat half of the cake, and next Mary offers a division of the remaining cake to Jack that he can either accept or reject. If the second offer is accepted by Jack, then the game is over and Mary and Jack get the split of the remaining cake offered by Mary. If the kids do not reach an agreement in the second round (Jack rejects Mary's offer). Mom gets to eat the rest of the cake by herself. The cake has two equal parts: one with cherries and one with coconut. If Jack gets fraction x in [0, 1] of the cherry part and y in [0, 1] of the coconut part, his utility is x + 2y. Mary prefers cherries, so if she gets fraction x in [0, 1] of the cherry part and y in [0, 1] of the coconut part, her utility is 3x + y. You can assume that if the kids do not reach an agreement after the first round, Mom eats half of each part of the cake. a) Find the subgame perfect equilibrium strategies and payoffs in the subgame following Mary rejecting Jack's offer. b) Find the subgame perfect equilibrium strategies and payoffs of the two - round game. Does Mom get to eat any cake? C) Does this game have a Nash Equilibrium in which Mom gets some cake? If there is such an equilibrium, provide an example, and if there is not, show that there is not.
Dad baked a chocolate cake for his twins Jack and Mary on their birthday, and he wants Jack and Mary to divide it without any screaming and mess. Mom devises the following procedure: First, Jack (who was born first) proposes a division of the cake that Mary can accept or reject. If the first offer is accepted by Mary, then the game is over and Mary and Jack get the split of cake offered by Jack. If the offer is rejected by Mary, then Mom gets to eat half of the cake, and next Mary offers a division of the remaining cake to Jack that he can either accept or reject. If the second offer is accepted by Jack, then the game is over and Mary and Jack get the split of the remaining cake offered by Mary. If the kids do not reach an agreement in the second round (Jack rejects Mary's offer). Mom gets to eat the rest of the cake by herself. The cake has two equal parts: one with cherries and one with coconut. If Jack gets fraction x in [0, 1] of the cherry part and y in [0, 1] of the coconut part, his utility is x + 2y. Mary prefers cherries, so if she gets fraction x in [0, 1] of the cherry part and y in [0, 1] of the coconut part, her utility is 3x + y. You can assume that if the kids do not reach an agreement after the first round, Mom eats half of each part of the cake.
a) Find the subgame perfect equilibrium strategies and payoffs in the subgame following Mary rejecting Jack's offer.
b) Find the subgame perfect equilibrium strategies and payoffs of the two - round game. Does Mom get to eat any cake?
C) Does this game have a Nash Equilibrium in which Mom gets some cake? If there is such an equilibrium, provide an example, and if there is not, show that there is not.
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