$(See$ screenshot $)$ Consider the following bidder knows his $or$ her own valuation, and all the valuations are all distributed uniformly

$(thatis$ evenly$)$ between $$100$ million and $$400$ million. For example, the probability that a typical bidder's valuation lies between $$300$ million and $$400$ million $is$ one third.

$(PartA)$ Suppose you are one $of5$ bidders, your valuation $is$ $$235$ million, and you are bidding $in$ a first price sealed bid auction. What should you bid? What are your expected profits from bidding $in$ this auction $(thatis$ after you know your valuation, the number $of$ bidders $in$ the auction, but before you know the outcome $of$ the auction$)?$ Explain $or$ justify your answers.

$(PartB)$ Suppose you are one $of5$ bidders, your valuation $is$ $$235$ million, and you are bidding $inan$ ascending auction. How high should you $be$ prepared $to$ bid? What are your expected profits from bidding $in$ this auction $(thatis$ after you know your valuation,

the number $of$ bidders $in$ the auction, but before the bidding starts$)?$ Explain $or$ justify your answers.

$(PartC)$ Now suppose you are one $of2$ bidders, your valuation $is$ $$235$ million, and you are bidding $in$ a first price sealed bid auction. What should you bid? What are your expected profits from bidding $in$ this auction $(thatis$ after you know your valuation,

the number $of$ bidders $in$ the auction, but before you know the outcome $of$ the auction$)?$ Intuitively explain why your answer differs from your answer $in$ the first part $of$ this question.

$(PartD)$ Now suppose there are $5$ bidders, and you are bidding $in$ a first price sealed bid auction. However $in$ this new setup every bidder has the same valuation but $no$ one knows what that valuation $is.$ They all know that this common valuation $is$ distributed uniformly between $$100$ million and $$400$ million. Each bidder gets a noisy signal about what the valuation $is,$ that gives them some clue about the true valuation, but not $an$ exact reading. Suppose the reading you get $is$ $$235$ million. Should you bid more $or$ less than what you bid the first part $of$ this question?

$(ParE)$ What $if$ there were $10$ bidders? $In$ words, how would your answer change then, and why?

$(PartF)Asin$ the previous question every bidder has the same valuation. There are only two bidders. You know exactly what the valuation $is.$ The other bidder does not. The other bidder believes the valuation $is$ distributed between valuation $is$ distributed

uniformly between $$100$ million and $$400$ million. $He$ decides $to$ bid cautiously, bidding $$150$ million, thinking that the expected value $is$ $$250$ million, and $soifhe$ wins, $he$ would profit $on$ average $$100$ million. You know the actual value. $(Itis$ some number between $$100$ million and $$400$ million.$)$ Given his bidding strategy, what should you bid? Assuming the valuation $is$ distributed between valuation $is$ indeed initially distributed uniformly between $$100$ million and $$400$ million, what

are your expected pro$ts,$ what are his expected losses, and what $is$ the expected revenue $of$ the auctioneer?