$.$ Consider a second price sealed bid auction for single good. There are two bidders, A and B$,$ with types greater than or equal to $0.$ Bidder A observes her privately known type vA and bidder B observes her privately known type vB$.$ The bidders simultaneously and independently submit sealed bids. The bidder submitting the higher bid wins the good and pays the losing bid. The bidder submitting the lower bid receives $0\mathrm{.}2$a$.$ Find the symmetric $($Bayesian$)$ Nash equilibrium in this game.