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.

  • 1
    $\begingroup$ Welcome to MSE. Please type your questions instead of posting images. Images can't be browsed and are not accessible to those using screen readers. If you need help formatting math on this site, here's a tutorial $\endgroup$
    – saulspatz
    Jul 16, 2020 at 20:12

1 Answer 1


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.

Alternatively, we can create the Bayesian normal form:

\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 $(BB,R)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.