Auction:

Two bidders are about to compete in an auction for an object. Each of them has a private value for the item that the other believes is distributed uniformly from zero to fifty. Thus, a bidder i with valuation vi believes his value is higher than his opponent −i with the following probability: Pr(vi > v−i) = vi /50

1. Suppose that the auction format is a first-price sealed-bid auction. Prove that both bidders playing a strategy of β(v) = v/2 is a Nash equilibrium of this game.

2. Suppose that the auction format is an all-pay auction in which all bidders must pay their bids, win or lose. Prove that both bidders playing a strategy of β(v) = v^2/100 is a Nash equilibrium of this game.

3. Suppose that the auction format is a second-price sealed-bid auction. What is the Nash equilibrium of this game? For full marks, show that this Nash equilibrium is also a dominant strategy equilibrium.

4. A bidder is said to be shading her bid if she bids a number that is less than her true value. Which type of auction doesn’t see shading in equilibrium? Which types do? Which type of auction includes the most shading in equilibrium? For full marks, explain the intuition behind your answers.