(Discrete second-price auction. Consider the following auction setting: o there are only 2 bidders: let's call them bidder A and B o each bidder knows how much they value the item, but are uncertain about their rival's valuation each bidder may have a valuation for the item of either $4 or $5 valuations are independent across bidders o the probability that a bidder has a valuation of $4 is 20% and the probability that a bidder has a valuation of $5 is 80% bidders can only bid whole amounts, that is, their bids can only take one of the following values 0, 1, 2, 3, ... o the auction's format is second-price sealed-bid (that is, bidders submit their bids in a sealed envelope, the highest bidder wins the item and pays an amount equal to the second-highest bid) o if the bidders are tied, the seller flips a (fair) coin to determine the winner (a) Explain, in words, what is a player's type in the context of this auction. (b) Find the bidder's payoff function (that is, the payoff the bidder gets as a function of her own and her rival's bids, and her own valuation: u;(bi,b_i; 0;)). Assume that the bidder who loses the auction gets a payoff of 0. (c) Explain, in words, what is a (pure) strategy in this game and give an example of such strategy. 2 (d) Define what we mean by a symmetric Bayesian Nash Equilibrium (BNE) in this game. (e) Show that bidding truthfully (that is, submitting a bid equal to your value for the item) is a (symmetric) BNE. (Discrete second-price auction. Consider the following auction setting: o there are only 2 bidders: let's call them bidder A and B o each bidder knows how much they value the item, but are uncertain about their rival's valuation each bidder may have a valuation for the item of either $4 or $5 valuations are independent across bidders o the probability that a bidder has a valuation of $4 is 20% and the probability that a bidder has a valuation of $5 is 80% bidders can only bid whole amounts, that is, their bids can only take one of the following values 0, 1, 2, 3, ... o the auction's format is second-price sealed-bid (that is, bidders submit their bids in a sealed envelope, the highest bidder wins the item and pays an amount equal to the second-highest bid) o if the bidders are tied, the seller flips a (fair) coin to determine the winner (a) Explain, in words, what is a player's type in the context of this auction. (b) Find the bidder's payoff function (that is, the payoff the bidder gets as a function of her own and her rival's bids, and her own valuation: u;(bi,b_i; 0;)). Assume that the bidder who loses the auction gets a payoff of 0. (c) Explain, in words, what is a (pure) strategy in this game and give an example of such strategy. 2 (d) Define what we mean by a symmetric Bayesian Nash Equilibrium (BNE) in this game. (e) Show that bidding truthfully (that is, submitting a bid equal to your value for the item) is a (symmetric) BNE