Alice is selling her 2000 Chevrolet cavalier to her friends, Bob and Charles. It is commonly known that Bob attaches a value of $6000 to Alice’s old car, while Charles’s value of the car is $6500. Alice designs the following auction to sell her car: First, she asks each of them to write his bid on a piece of paper. Then Bob and Charles give their bids to Alice. Notice that when Bob and Charles write down their bids, they don’t know each other’s bid (hence the name "sealed bid"). After Alice receives the sealed bids, the bids are shown to everyone and the car will be sold to the person who has a higher bid. When there is a tie (Bob and Charles bid the same amount), then Alice would flip a fair coin to decide who will get the car.
(a) Write down the normal form representation of this game.
(b) Find weakly dominated strategies for both Bob and Charles. Is “bidding 0” a weakly dominated strategy for either player? Explain.
(c) Find the set of all pure-strategy Nash equilibria for this game.