Only Do Qs 3 (Part a and b). It continues with the model from Qs 2 but the questions are different.

1. (20 points) Consider the game with two brothers with inherited antique in the previous assignment. Now, suppose the older brother proposes a price rst and the younger brother, after observing the price proposed by the older brother, proposes another price. The buyer's actions facing the prices are the same as in the previous assignment. Illustrate the extensive form of the game (you may omit part of the game tree, as long as your illustration gives sufcient information about the game). Find all the subgame perfect Nash equilibria of this modied game. 2. (20 points) Consider the twobidder auction example in class, where bidder i's valuation, V,, is equal to 1 with probability q 6 (0,1) and 0 with probability 1 q, for 1' = 1, 2. Assume V1 and V2 are mutually independent. The object is sold by a rst-price auction. The bidders are allowed to bid any nonnegative price. Whenever there is a tie, each bidder gets the good with probability 1/2. (a) Argue that, when 1/} = D, it is weakly dominant for player i to bid 3,-(0) = 0 for i = 1,2. (b) Show that there does not exist an equilibrium in which 31 (1) = 132(1) 2 3/4. (c) Find the mixed strategy Nash equilibrium when q = 1/4. (d) Argue that the expected revenue of the seller in a rstprice auc tion is lower than that in postedprice selling. 3. (20 points) Modify the twobidder example above and allow only three possible bids, 0, 1/2, or 1. (a) In a rstprice auction, what are the symmetric Nash equilibria? (b) In a secondprice auction, is bidding one't true valuation still a weakly dominant strategy? Conclude whether or not the strategy prole in which each bidder bids one's true valuation is a Nash equilibrium. Are there other symmetric Nash equilibria