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Two buyers compete for an antique silver bowl in a sealed-bid auction. Each buyer's value is drawn independently from the range of $0 to$1,000 with all values in the range equally likely. Buyer 1's value for the bowl is $700, and he submits a sealed bid of $280. Buyer 2 has the value$450, makes a sealed bid of $300, and wins the bowl.
a. As profit maximizers, did the buyers make the appropriate sealed bids? Did the "right" buyer get the bowl? Compute the sum of the players' payoffs (i.e., the seller's revenue plus the winning buyer's profit).
b. What would have been the outcome (and total players' payoffs) under an English auction? Does the comparison of English and sealed-bid prices contradict revenue equivalence? Which auction provides the greater total "pie"?
c. Suppose buyer 1 offers to buy the bowl from buyer 2 and they negotiate a sale price of $600. Again, calculate the total of the player payoffs and provide a brief assessment. In his role as a middle man, does buyer 2 deserve the profit he makes?
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