Consider a slight generaliza- tion of the basic model of auction: A seller has a single unit of good to be sold which worth 0 to her. There are two buyers, 1 and 2, each values the good at either y or with >y> 0. Each buyer's value is with probability p and is u with probability 1-p where p = [0, 1]. We keep the assumption that the higher type buyer wins the object when the bids are equal. Second-Price Auction: (a) Prove informally that truthful bidding continues to be a dominant strategy in the second- price auction. (b) Calculate the expected payoffs for each type of buyer and the seller. First-Price Auction: (c) Show that the following strategies constitute a Bayes-Nash equilibrium. They type buyer bids her valuation. The type buyer uses a continuous mixed strategy on the interval of bids y to pu+ (1-p)u. The probability that her bid is less than a is given by x (d) Calculate the expected payoffs for each type of buyer and the seller to confirm that revenue equivalence principle holds. Consider a slight generaliza- tion of the basic model of auction: A seller has a single unit of good to be sold which worth 0 to her. There are two buyers, 1 and 2, each values the good at either y or with >y> 0. Each buyer's value is with probability p and is u with probability 1-p where p = [0, 1]. We keep the assumption that the higher type buyer wins the object when the bids are equal. Second-Price Auction: (a) Prove informally that truthful bidding continues to be a dominant strategy in the second- price auction. (b) Calculate the expected payoffs for each type of buyer and the seller. First-Price Auction: (c) Show that the following strategies constitute a Bayes-Nash equilibrium. They type buyer bids her valuation. The type buyer uses a continuous mixed strategy on the interval of bids y to pu+ (1-p)u. The probability that her bid is less than a is given by x (d) Calculate the expected payoffs for each type of buyer and the seller to confirm that revenue equivalence principle holds. Consider a slight generaliza- tion of the basic model of auction: A seller has a single unit of good to be sold which worth 0 to her. There are two buyers, 1 and 2, each values the good at either y or with >y> 0. Each buyer's value is with probability p and is u with probability 1-p where p = [0, 1]. We keep the assumption that the higher type buyer wins the object when the bids are equal. Second-Price Auction: (a) Prove informally that truthful bidding continues to be a dominant strategy in the second- price auction. (b) Calculate the expected payoffs for each type of buyer and the seller. First-Price Auction: (c) Show that the following strategies constitute a Bayes-Nash equilibrium. They type buyer bids her valuation. The type buyer uses a continuous mixed strategy on the interval of bids y to pu+ (1-p)u. The probability that her bid is less than a is given by x (d) Calculate the expected payoffs for each type of buyer and the seller to confirm that revenue equivalence principle holds. Consider a slight generaliza- tion of the basic model of auction: A seller has a single unit of good to be sold which worth 0 to her. There are two buyers, 1 and 2, each values the good at either y or with >y> 0. Each buyer's value is with probability p and is u with probability 1-p where p = [0, 1]. We keep the assumption that the higher type buyer wins the object when the bids are equal. Second-Price Auction: (a) Prove informally that truthful bidding continues to be a dominant strategy in the second- price auction. (b) Calculate the expected payoffs for each type of buyer and the seller. First-Price Auction: (c) Show that the following strategies constitute a Bayes-Nash equilibrium. They type buyer bids her valuation. The type buyer uses a continuous mixed strategy on the interval of bids y to pu+ (1-p)u. The probability that her bid is less than a is given by x (d) Calculate the expected payoffs for each type of buyer and the seller to confirm that revenue equivalence principle holds.