# Help with Bayesian Game with asymmetric information

I need help with this bayesian game. I tried it multiple times and watched several youtube tutorials but I just dont get it for this specific case.

Consider a two-player game with the following pay-off matrix: where $$\theta$$ (-2,2) is privately known by Player 1, and Pr ($$\theta$$ = -2) = 0.8. (There is no other private information.

Question: Find a Bayesian Nash Equilibrium of this game and verify that the profile you identified is indeed a Bayesian Nash equilibrium.

Bayesian theorem states that

$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

but I cannot connect the theorem with this example.

• Is player 1 the one picking between X and Y or L and R? Jul 16, 2020 at 18:34
• Player 1 chooses between X and Y and player 2 between L and R Jul 16, 2020 at 18:40

So it is called a bayesian problem because that area of probability using things like $$P(A|B)$$ is called bayesian probability, not that that explicit form of that equation is used. However, in this case we don't actually need it, as player 2 doesn't need to know $$\theta$$.
Looking at the payoff matrix, if $$\theta = -2$$, player 1 wants X regardless of the choice of player 2. However, if $$\theta = 2$$, player 1 want Y regardless of the choice of player 2. Therefore, the Nash Equilibrium is dependent on $$\theta$$:
If $$\theta = -2$$, player 1 is going to pick X, so player 2 prefers R, giving us a Nash Equilibrium of XR.
If $$\theta = 2$$, player 1 is going to pick Y, so player 2 prefers R, giving us the Nash Equilibrium of YR.
Therefore, player 1 will pick X if $$\theta = -2$$ or Y if $$\theta = 2$$, while player 2 will always pick R