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.
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