I am in possession of two coins. One is fair so that it lands heads (H) and tails (T) with equal probability while the other coin is weighted so that it always lands H. Both coins are magical: if either is flipped and lands H then a $1 bill appears in your wallet, but when it lands T nothing happens. You may only flip a coin once per period. The interest rate is i per period. You are risk-neutral and thus only concern yourself with expected values (and not variance). For simplicity, in the questions below assume you will live forever.

1. How much are you willing to pay for such a coin that you know is fair?

2. How much are you willing to pay for such a coin that you know is weighted?

3. I currently own the coins and know which is fair and which is weighted, but you cannot tell which is which. You may make an offer to purchase a coin of your choosing, which I am free to accept or reject. What is the most you are willing to offer? Explain how you arrived at this answer.

4. Suppose now that I also do not know which coin is fair and which is weighted. You pick one of the two coins at random.

(a) What is your willingness to pay for this coin?

(b) What is your willingness to pay for an option1 to purchase the coin, where the option works as follows: you may flip the coin once and observe the outcome. Then, if you wish, you may purchase the coin from me for the amount you determined in part 4(a).

(c) What is your willingness to pay for an n-option, which works as follows: you may flip the coin n-times and observe the outcome. Then, if you wish, you may purchase the coin from me for the amount you determined in part 4(a).

(d) (Difficult) Suppose now you are competing in an auction against another bidder. You select one of the coins at random. Then, each of you get to flip the coin once for free and observe the outcome. Neither of you see the result of the flip for the other. You and the other bidder will simultaneously submit sealed bids to me. I will give the coin to the highest bidder, and the highest bidder pays his bid. The low bidder pays and receives nothing. In the event of a tie a winner will be selected with 50% probability. Find the symmetric equilibrium bidding strategies of each player. [Hint: what is the expected value of the coin given you win the auction?]