Consider an $$all$-$pay auction$$ with two players $($the bidders$).$ Player $1$s valu$-$ ation v $1$ for the object being auctioned is uniformly distributed between $0$ and $1.$ That is$,$ for any x in $[0,1],$ 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$$ valuations, each player observes his$/$her own valuation but not that of the other player. Simultane$-$ ously and independently, the players submit bids. The player who bids higher wins the object, but both players must pay their bids. That is$,$ if player i bids b i $,$ then his$/$her payoff is $$ b i if he$/$she does not win the auction; his$/$her payoff is v i $$ b i if he$/$she wins the auction. Calculate the Bayesian Nash equilibrium strategies $($bidding functions$).($Hint: The equilibrium bidding function for player i is of the form b i $($v i $)=$ kv i $2$ for some number k$.$