1. Consider a scaled-bid auction where the bid for the object is either a or b where 0 sa b. The higher bidder gets the object at the price of . If the two bids are identical, the winner is decided by a toss of an unbiased coin. Suppose there are two types of possible buyers, those who value the object at & units (Type H) and those who value it at a units (Type Z). Assume that a bidder makes the bid independently of the other bidder. a) By considering the gain each bidder will receive on making his bid, find all Nash equilibrium points in pure strategies for this auction. Which, if any, of these NEPs produce the fairest outcome to both players? Note: to obtain the strategic form for the game, the payoff (or gain) for each player can be computed as gain = valuation of object - expected payment or 0 in case the player does not win the object. (Hint: each player belongs to the type space {H. LJ. each player has strategy set (a, b). the strategic form should include all the possible types that each player can assume. For example, players' type is Type = (L. H). Le., Player I's type is L, and Player 2's type is H. iv. suppose Type = (L, H). the possible payoffs can be obtained by considering the following examples: . L bids a, H bids a: L gains - -= 3 H gains b - = = > 0; . L bids a, H bids b: L gains 0, H gains b - ath b) Assuming that Pr(/) = 1 - q and Pr(Z) = q, and that the bidders report their true valuations, what can the seller expect to receive under this auction? (Hint: the expected value can be obtained by considering the following scenarios: Both player are type H; Exactly one player is type H; iii. Both player are type L.) c) Compare the expected selling price of this auction with the first best outcome mechanism and the Vickrey auction for various q values where 0 5q s 1. (Hint: To compare the expected selling price of this auction with the first best outcome mechanism and Vickrey auction, you may assume that a = 3 and b = 4, Le., the lower bid value is 3 units and higher bid value is 4 units.)