Question 3. (25 pts.) Two brothers, the old brother and the young one inherit a house from their parents, and the house

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Question 3 25 Pts Two Brothers The Old Brother And The Young One Inherit A House From Their Parents And The House 1 (206.87 KiB) Viewed 55 times

Question 3. (25 pts.) Two brothers, the old brother and the young one inherit a house from their parents, and the house is to be auctioned off between them.² The value of the house to (alternatively, the maximum willingness to pay by) the old brother is 1000000 units of money, while the young brother values the house at 800000 units of money. The rules of the auction (the decision protocol) is as follows: The order in which brothers are going to choose is determined by an arbitrator, arabulucu in Turkish, before the game starts. Thus, the player choosing first does not have to be the old brother. • The player who chooses first (as determined by the arbitrator) makes a bid b₁ ≥ 0. This becomes common knowledge among the brothers. • Observing b₁, the other brother, the second player, chooses next and has the option to "pass" or to submit a bid b₂ with the property that b₂ ≥ b₁ (that is, this is an ascending price auction).³ Consequently, the game/auction ends. These brothers are not wealth constraint and hence can submit any non-negative bid if they wish to do so. The winner of the auction is the brother with the highest bid, and when there is a tie (i.e., if they bid the very same amount), the winner is declared to be the old brother. On the other hand, the price to be paid for the house by the winner to the loser is as follows: If the second player has chosen the "pass" option, then the winner is the first player (according to the order of the auction) and the price equals the bid that player 1 has made, i.e., b₁. However, if the second player has updated the bid from by to b₂ with b₂ ≥ b₁, then the price equals the average bid, i.e., bb. Then, the utility of the player who loses the auction is equal to the price of the house determined in the auction, while the utility of the player who wins the auction is equal to his value of the house minus the price of the house, i.e., the payment he makes to his brother. In this question, we concentrate on pure strategies and you can use backward induction without the need for justifications. a. (10 pts.) Suppose that the arbitrator decides on the order of bids so that the old brother bids first, and this is followed by the choice of the younger brother. Find the set of subgame perfect equilibrium outcome (bids). b. (10 pts.) Suppose that the arbitrator decides on the order of bids so that the young brother bids first, and this is followed by the choice of the older brother. Find the set of subgame perfect equilibrium outcome (bids). c. (5 pts.) What is the subgame perfect level of "bribe" that the old brother would be willing to make to the arbitrator before the start of the game so that the arbitrator chooses the order of bids as the old brother wishes?