2 Bidding Strategy in Different Auctions (16 pts) Suppose that Alice's valuation for an item is...

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2 Bidding Strategy in Different Auctions (16 pts) Suppose that Alice's valuation for an item is 2.5. There is one other bidder, Bob, who Alice believes will bid 1, 2, 3, or 4, each with probability 1/4. Alice is deciding how much to bid, and can bid any real number. Assume that in a tie, Alice always wins the auction. Part (a) (4 points) If the item is auctioned using a second-price auction, what is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? Hint: The function you give in this part should be a piecewise function, and so your solution should be in a form similar to the below (note that the solution below is itself incorrect): f(x) = 5x+2 0 x < 3 -12x 3 < x <50 x > 50 (1) Part (b) (4 points) If the item is auctioned using a first-price auction, what is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? Part (c) (4 points) Now suppose the item is auctioned using a first-price auction with a reserve price of 1.5. This means that the highest bid wins as long as it is at least as big as the reserve price. For example, if Alice bids 4 and Bob bids 3, Alice wins and pays 4 for the item; if Alice bids 1.4 and Bob bids 1, nobody wins the auction since the highest bid did not exceed 1.5. What is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? Part (d) (4 points) An all-pay auction is one in which the highest bidder receives the item, but everyone pays their own bid. For example, if Alice bid 2 and Bob bid 1, Alice receives the item and pays 2, and Bob pays 1 but receives nothing. If the item is auctioned using an all-pay auction, what is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? 2 Bidding Strategy in Different Auctions (16 pts) Suppose that Alice's valuation for an item is 2.5. There is one other bidder, Bob, who Alice believes will bid 1, 2, 3, or 4, each with probability 1/4. Alice is deciding how much to bid, and can bid any real number. Assume that in a tie, Alice always wins the auction. Part (a) (4 points) If the item is auctioned using a second-price auction, what is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? Hint: The function you give in this part should be a piecewise function, and so your solution should be in a form similar to the below (note that the solution below is itself incorrect): f(x) = 5x+2 0 x < 3 -12x 3 < x <50 x > 50 (1) Part (b) (4 points) If the item is auctioned using a first-price auction, what is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? Part (c) (4 points) Now suppose the item is auctioned using a first-price auction with a reserve price of 1.5. This means that the highest bid wins as long as it is at least as big as the reserve price. For example, if Alice bids 4 and Bob bids 3, Alice wins and pays 4 for the item; if Alice bids 1.4 and Bob bids 1, nobody wins the auction since the highest bid did not exceed 1.5. What is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward? Part (d) (4 points) An all-pay auction is one in which the highest bidder receives the item, but everyone pays their own bid. For example, if Alice bid 2 and Bob bid 1, Alice receives the item and pays 2, and Bob pays 1 but receives nothing. If the item is auctioned using an all-pay auction, what is Alice's expected reward when she bids x? (Write your answer as a simplified function of x.) What is one value of x that maximizes her expected reward?