I need some help with the following Bayesian Game. I tried it a few times and watched some tutorials but I just cant figure out how to solve it.
Consider a two player game with this payoff matrix
where $\theta \in \{0,3\}$ is a parameter known by Player 1. Player 2 believes that $\theta = 0$ with probability $\frac{1}{2}$ and $\theta = 3$ with probability $\frac{1}{2}$. Everything above is common knowledge. Player 1 chooses between T and B, Player 2 chooses between L and R.
Compute two Bayesian Nash equilibria of this game.
Player 1's strategy consists of two actions, one specifying what to do when $\theta=0$, the other when $\theta=3$. Let $XY$ denote the strategy "choose $X$ if $\theta=0$ and choose $Y$ if $\theta=3$. Hence, player 1 has four pure strategies: $\{TT,TB,BT,BB\}$.
To find a (pure strategy) BNE, we first conjecture that player 1 follows one of the four strategies, call it $s_1$. Then we derive player 2's best response. Lastly we check if $s_1$ is a best response to $s_2$; if it is, $(s_1,s_2)$ is a BNE, if not, we move on to player 1's next strategy and repeat the process.
Alternatively, we can create the Bayesian normal form:
\begin{array}{|c|c|c|c|c|} \hline &L&R\\\hline TT&2,2&0,\frac12(0+3)\\\hline TB&\frac12(2+3),\frac12(2+0)&\frac12(0+1),\frac12(0+1)\\\hline BT&\frac12(0+2),\frac12(0+2)&\frac12(1+0),\frac12(1+3)\\\hline BB&\frac12(0+3),0&1,1\\\hline \end{array}
From here it's easy to verify that the only two pure strategy BNEs are $(TB,L)$ and $(BB,R)$.