Bayesian Nash Equilibria in Battle of Sexes Consider the static Bayesian game as described below. $\ t_1$ and  $\ _2$ are the types of the row and column player respectively, which are both uniformly distributed on the interval $[0,1]$. The first part of the question is asking us to find a Bayesian Nash Equilibrium. Trivially, don't the the top left and bottom right corners correspond to equilibrium outcomes? Unless I've misunderstood the definition of a BNE.
Yes, you are correct. All types $t_{1}$ choose O (B) and all types $t_{2}$
choose O (B) are both Bayesian equilibria.
Note that there are other Bayesian equilibrium in this game, if you are interested this is explained in detail here (p. 10, see reference below) for this particular battle of the sexes with two-sided incomplete information. The basic idea is to note that in this game, each player has a continuum of types, and so the set of types is infinite. You can look for a Bayesian equilibrium in which player 1 goes to the $Opera$ if $t_{1}$ exceeds some critical value $x_{1}$ and chooses $Fight$ otherwise, and player 2 chooses to $Fight$ if $t_{2}$ exceeds some critical value $x_{2}$ and goes to the $Opera$ otherwise. To find the values $x_{1}$, $x_{2}$ that make these strategies a Bayesian equilibrium you can calculate each player's expected payoffs given the other player's strategy and find the optimal values based on this.
Game Theory: Static and Dynamic Games of Incomplete Information
Branislav L. Slantchev Department of Political Science, University of California – San Diego