Alice and Bob are bidding to take over a promising internet startup company owne

Question

Alice and Bob are bidding to take over a promising internet startup company owned by Claire. The company has the same value v to both Alice and Bob, but Alice doesn’t know v. Bob is more informed and has a very accurate estimate of v that we will assume to be exact. From Alice’s point of view, v is random and is uniformly distributed between $0 and $10 million. Claire will run a second-price auction with a reserve price of $4 million. Assume that Bob bids the true value v (so if v < 4m, his bid is thrown out.) We will be determining Alice’s best strategy given this information.

(a) If Alice submits a bid under $4m, her bid automatically loses. If she submits a bid b ($4m, $10m), how likely is she to win the auction? The answer will be a formula in terms of b. The formula takes into account the randomness of v, but does not have a v in it.

(b) What is Alice’s average surplus, in terms of b? You will need to take into account three possibilities: Alice wins the auction and pays the reserve price, Alice wins the auction and pays Bob’s bid, or Alice loses the auction. In each case, multiply the probability by the surplus; then add up the cases.

(c) What is Alice’s best strategy? Why?

Explanation / Answer

b ($4m, $10m)

V is same for both Alice and Bob.

In this question, we know that the bid by bob is equal to v.

So, no matter if Alice bids true value or more, she will not face a loss.

But, if Alice bids less than v, she will lose the auction and have a utility of 0.

a)

Let h(b,2) represent that second highest bid is b.

F(b) = b/10million

F’(b) = 1 – b/10million (probability that b is not the second highest bid, I.e it is the winning bid)

b)

The average profit of Alice is 0.

If she wins the bid, she must pay v (bid by Bob), therefore total profit = v-v = 0

If she loses her profit = 0.

c)

Alice’s best strategy is to bid $10 million. By doing so she will have maximum chance to win the bid.

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