You are one of five risk-neutral bidders participating in an independent private values auction. Each bidder perceives that all other bidders' valuations for the item are evenly distributed between $20,000 and $50,000. For each of the following auction types, determine your optimal bidding strategy if you value the item at $35,000.

a. First-price, sealed-bid auction.

• Bid $35,000.
• Bid $20,000.
• Bid $32,000.
• Bid $50,000.

b. Dutch auction.

Let the auctioneer continue to lower the price until it reaches $35,000, and then yell "Mine!".

c. Second-price, sealed-bid auction.

Bid $50,000.

d. English auction.

Remain active until the price exceeds $32,000, and then drop out.