this is the game theory problem

Question 2. [40 total points] Auctions with complete information. Ann, Bar- bara and Carol are all interested in bidding on a house. All three know that Ann values the house at $1 million, Barbara values it at $800,000 and Carol values it at $750,000. The bank which has foreclosed on the mortgage held on this house conducts an auction in which Ann, Barbara and Carol are the only bidders. The auction works in the following way. Each bidder simultaneously puts her bid into a sealed envelope that has her name on it. The auctioneer then opens the three envelopes, reads out the bids and sells the house to the person who submitted the highest bid. The winner, however, does. not pay her bid, but rather pays an amount equal to the highest non-winning bid. [In the event that there are two or three highest bids, then the winner is determined randomly from the set of bidders who submitted the highest bid and the winner in this case ends up paying an amount equal to her bid.] Assume that players may bid zero or any positive multiple of $1000. If bi, the bid submitted by bidder z", is strictly greater than any other bid then her payoff is U,- 3-1: where v,- is her valuation and R,- is the highest bid submitted by a player other than 3'. If I),- is one of two highest bids then her (expected) payoff is (U,- 1),) f2. If I),- is equal to both other bids then her (expected) payoff is (1;,- 13,-) /3. If I),- is not one of the highest bids then 's payoff is zero. (a) [10 points] Show that a player submitting a bid b,- o, is also weakly dominated by a bid equal to 1,1,. [Hintz Show that by bidding more than her valuation she stands to win the auction in some cases in which she will incur a loss by witming the auction. And in all other cases she gets the same whether she bids more than her valuation or equal to her valuation] (c) [5 points] Who wins the auction and what does she pay if no one plays a weakly dominated strategy? (d) [8 points] Show that Ann submitting a bid of $900,000, Barbara submitting a bid of $1000 and Carol submitting a bid of $2000 is also a Nash equilibrium of the auction. (e) [7 points] Show that Ann submitting a bit of $800,000, Barbara submitting a bid of $1,000,000 and Carol subitting a bit of $0 is also a Nash equilibrium of the auction