(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...

(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.