A single unit of a good is to be sold via an auction. There are two bidders, A and B. They are assumed to be risk-neutral. The seller knows that there are five possible values of willingness to pay, $100, $200, $300, $400 and $500. Bids are also restricted to those values. Suppose A's willingness to pay is $400 whereas B's willingness to pay is $200.

Conisider that the unit is sold via all-pay auction in which ties are broken by a coin flip. In all-pay auction, each bidder has to pay her bid, regardless of winning or losing. Write down the payoff matrix, and find all the pure-strategy Nash equilibria (if any)