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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.

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  • $\begingroup$ Is player 1 the one picking between X and Y or L and R? $\endgroup$
    – cmitch
    Jul 16, 2020 at 18:34
  • $\begingroup$ Player 1 chooses between X and Y and player 2 between L and R $\endgroup$
    – Moritz123
    Jul 16, 2020 at 18:40

1 Answer 1

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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

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  • $\begingroup$ 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? $\endgroup$
    – Moritz123
    Jul 16, 2020 at 19:56
  • $\begingroup$ 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. $\endgroup$
    – Moritz123
    Jul 16, 2020 at 20:13

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