Suppose there are two bidders for a single object. Each bidder has a value for the object, v1 and v2, which is randomly drawn uniformly from 0 to 80. (Note that this means the probability that a value is 25 or lower is 25/80, and more generally, the probability that a value is k or lower is k/80). Bidders see their own values but not the values of their rival bidders. Consider a first price sealed bid auction where the winner of the object is the bidder who submits the highest bid and pays the price that he bid. Suppose that you are bidder 1 and you believe that your rival always bids a constant fraction α of his value. (For example, if bidder 2’ value is 15, he will bid 15α.). 1. What is your expected winning probability of the auction from any given bid b1? (You win the auction when your bid amount b1 is greater than your opponent’s bid).
Suppose there are two bidders for a single object. Each bidder has a value for the object,
v1 and v2, which is randomly drawn uniformly from 0 to 80. (Note that this means the
probability that a value is 25 or lower is 25/80, and more generally, the probability that a
value is k or lower is k/80). Bidders see their own values but not the values of their rival
bidders. Consider a first price sealed bid auction where the winner of the object is the bidder
who submits the highest bid and pays the price that he bid.
Suppose that you are bidder 1 and you believe that your rival always bids a constant
fraction α of his value. (For example, if bidder 2’ value is 15, he will bid 15α.).
1. What is your expected winning probability of the auction from any given bid b1? (You win the auction when your bid amount b1 is greater than your opponent’s bid).
2. What is your expected payoff from any given bid, b1? (Write answer
as a function of v1, b1, and α).
3. Compute your best response bid given your value v1. (Find the
bidding strategy that maximizes your expected payoff.)
4. Assuming the bidder 2 is acting in the same way, what should be the value of α?
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