# Bayesian Game with Asymmetric Information and 2 Players

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.

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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.
$$\begin{array}{|c|c|c|c|c|} \hline &L&R\\\hline TT&2,2&0,\frac12(0+3)\\\hline TB&\frac12(2+3),\frac12(2+0)&\frac12(0+1),\frac12(0+1)\\\hline BT&\frac12(0+2),\frac12(0+2)&\frac12(1+0),\frac12(1+3)\\\hline BB&\frac12(0+3),0&1,1\\\hline \end{array}$$
From here it's easy to verify that the only two pure strategy BNEs are $$(TB,L)$$ and $$(BB,R)$$.