Second-price sealed-bid auction. There are 20 bidders. Each bidder i values the object at vi > 0. The indices are chosen in a way so that v1 > v2 > ... > v20 > 0.

Each bidder i can submit a bid bi ≥ 0. The bidder whose bid is the highest wins the object. If there are multiple highest bids, then the winner is the bidder whose valuation is the highest (or whose index is the smallest) among the highest bidders (for ex, if bidder 1 and bidder 2 have the highest bid, then bidder 1 is the winner).
The winner, say bidder i, gets a payoff vi - p, where p is the second highest bid (it equals the highest bid if there are multiple highest bids). Losing players receive zero payoff.

a. Show that (b1, ..., b20) = (v1, ..., v20), i.e. each player bidding her valuation, is a Nash equilibrium.

b. Find a Nash equilibrium in which player 10 wins the object. describe your answer.

c. (more difficult) Find all Nash equilibria in which player 9 wins the object.