Consider an "all-pay auction" with two players (the bidders). Player 1's valuation v1 for the object being auctioned is uniformly distributed between 0 and 1. That is, for any x ∈ [0, 1], player 1's valuation is below x with probability x. Player 2's valuation is also uniformly distributed between 0 and 1, so the game is symmetric.

After nature chooses the players' valuations, each player observes his/her own valuation but not that of the other player. Simultaneously and independently, the players submit bids. The player who bids higher wins the object, but both players must pay their bids.

That is, if player i bids bi , then his/her payoff is -bi if he/she does not win the auction; his/her payoff is i - bi if he/she wins the auction. Calculate the Bayesian Nash equilibrium strategies (bidding functions player