Consider a simple simultaneous$-$bid poker game. First, nature selects num bers x$1$ and x$2.$ Assume that these numbers are independently and uniformly distributed between $0$ and $1.$ Player $1$ observes x$1$ and player $2$ observes x$2,$ but neither player observes the number given to the other player. Simultane ously and independently, the players choose either to fold or to bid. If both players fold, then they both get the payoff $1.$ If only one player folds, then he obtains $1$ while the other player gets $1.$ If both players elected to bid, then each player receives $2$ if his number is at least as large as the other player$$s number; otherwise, he gets $2.$ Compute the Bayesian Nash equilibrium of this game. $($Hint: Look for a symmetric equilibrium in which a player bids if and only if his number is greater than some constant a$.$ Your analysis will reveal the equilibrium value of a$.)$