(Optional) This question looks at the equilibrium bidding strategies of all-pay auctions, in which bidders have private values for the good, as opposed to the discussion in Section 4, where the all-pay auction is for a good with a publicly known value. For the all-pay auction with private values distributed uniformly between 0 and 1, the Nash equilibrium bid function is 

(a) Plot graphs of b(v) for the case n = 2 and for the case n = 3.

(b) Are the bids increasing in the number of bidders or decreasing in the number of bidders? Your answer might depend on n and v. That is, bids are sometimes increasing in n, and sometimes decreasing in n.

(c) Prove that the function given above is really the Nash-equilibrium bid function. Use a similar approach to that of Exercise S4. Remember that in an all-pay auction, you pay your bid even when you lose, so your payoff is v 2 b when you win, and 2b when you lose.