Consider the first price sealed bid auction. In the FPSB auction, the higher of the two bids wins, and the winner pays her own bid. Is bidding your value a dominant strategy? If yes, prove it. If no, show an example where it is better to bid something other than your value. In this case, be sure to specify what the other bid is, what your value is, and explain why you should not bid your value.

Next, consider a kind of a mix of the first and second price sealed bid auctions: the average price sealed bid auction. In the APSB auction, there are two bidders, and the one who submits the higher bid wins the object. However, the winner pays the average of the two bids. Is bidding your value a dominant strategy in the APSB auction? If yes, prove it. If no, show an example where it is better to bid something other than your value. In this case, be sure to specify what the other bid is, what your value is, and explain why you should not bid your value.

Next, read this. Write 5-8 paragraphs on several points made in that chapter that you found most interesting or surprising.

Finally, if you are feeling adventurous, you may attempt this last part. However, this part is not required. Answer the following questions about an auction in which Ann and Bob participate. Suppose that they have independent and identically distributed valuations of the object being auctioned. Each has a valuation that is uniformly distributed in the interval [0,2].

1. If the auction is a second price sealed bid auction, what is Ann's optimal bid if her value is v?
2. If Ann's value is v, what is the probability that she wins? If she does win, how much can she expect to pay? Multiply her probability of winning by her expected payment if she wins to find her overall expected payment when her value is v. Bob's overall expected payment when his value is v will be the same, as their situations are symmetric.
3. Now suppose the auction is a first price sealed bid auction. Suppose that Bob always bids half his actual value. (This means that Ann should never bid more than 1, so assume henceforth that her bid is always at most 1.) If Ann's value is v , and she bids b, how much does she gain if she wins? And what is the probability that she wins with this bid b? Multiply her gain if she wins by the probability that she wins to find her expected gain. What bid b will maximize her expected gain?
4. Calculate Ann's expected payment in the first price auction if her value is v.
5. Compare your answers to 2 and 4 above, and comment. What does this say about the revenue generated by the two different auction types?