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

1 Answer

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

• Thank you for your answer! You solution seems to make sense but are you sure the probabilities are not relevant for the estimation of the Nash Equilibrium? Jul 16, 2020 at 19:56
• I've posted another question concerning Nash Equlibria in Bayesian Games. I think in this case the probabilites are definitely relevant. I would really appreciate it, if you could also have a look at this example. Jul 16, 2020 at 20:13