Supplemental problems:1) Suppose that 3 risk-neutral bidders participate in an auction. The winner will be appointed as the interim senator by the Illinois governor. The bidders A, B and C value the senate seat at $8 million, $6 million and $6 million respectively. Each bidder knows everyone’s valuations (including his own and everyone else’s).The bidders will compete in auctions as described in parts (a) through (d); in each case, bids can be made in $1 thousand increments (1 thousand, not 1 million) at any value from $1 million to $10 million. Last, the winner must kiss the governor’s cheek when the transaction concludes, although he would rather not, all other things being equal.a) Which bidder wins in a first-price sealed-bid auction? How much will the governor receive?b) Which bidder wins in a Dutch auction? How much will the governor receive?c) In a second-price sealed-bid auction, explain why bidding $8 million is (weakly) dominated by bidding $6 million for bidder B.d) In a second-price sealed-bid auction, can bidder A improve his equilibrium payoff by keeping his valuation secret? Explain why.2-A corrupt governor from Illinois decides to conduct a “bribe†auction to determine which of two politicians to fill a vacant US senate seat. Everyone knows that each politician has a value for the seat equally likely to be from $0 to $1 million. The politician himself knows his own value, but doesn’t know his rival’s value. In a bribe auction, each pays cash (simultaneously) to the governor in the form of campaign contribution. The governor awards the seat to whoever pays him the most cash and the governor keeps the cash.a) Suppose that you conjecture that your rival will be bidding according to the strategy b(v) = av2 where a < 1.="" set="" up="" the="" profit="" maximization="" problem.="" (hint:="" the="" probability="" a="" bid="" of="" b=""> b(v) in this setting is √B / √a.b) Compute your best response when you have a valuation of V. (Helpful hint: the derivative of √B is 1/(2√B). It’s also useful to know that (√B)2 = B.)c) Figure out the equilibrium value of a. (Hint: If a depends on V, then you’ve done something wrong. The value of a should be a number, like 0.2.)d) Compute the expected revenues and show that revenue equivalence holds. (Hint: the expected revenue is 2∫01b(v)dv.)
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