Finding Bayesian Nash Equilibrium

Question

I'm recently new to Game Theory and I've recently started teaching myself about Bayesian Nash Equilibirum. I've stumbled across a problem set that I can't seem to wrap my head around concerning Bayesian Nash Equilibrium.

The problem set is shown below:

For question 3, I initially tried to solve the first problem using Mixed Bayesian Nash Equilibrium but that doesn't make sense since both Player 1 and Player 2 have weakly dominated strategies, so why would they mix? Also when I combine the matrices I find no Pure Strategy Bayesian Equilibrium.

Is there something I'm missing here? How would I go about solving this question? Some example's would be very helpful

For question 4, I have no idea where to even start solving this question. I haven't come across any questions or tutorial on how to solve for Bayesian Nash Equilibria when BOTH players have don't know what game they're playing. Any and all help on how I would solve this problem would be greatly appreciated.

Answer 1

The Key to BNE is that players that know something (about the state of the world or their type) can condition their strategies with their information. That is, for example, in question 3, type A could choose strategy U, while type B could choose strategy D. Therefore from the second player's perspective, there are four possible pure strategies of his opponent. Namely UU, UD, DU, and DD. Which represents the actions of each type of player 1.

What you want to do is construct another table where player 1 has these four strategies and player 2 has strategies L and R whose payoffs are the expected payoff. For instance, in the cell (UU, L) the payoff to player 2 is $0.5(4)+0.5(2)=3$, and for (UU, R) it has a value of $0$. This table is only useful to asses player 2's payoffs since player 1 knows his type, so no need to fill a number for player 1. After constructing the table you realize that player 2 has a weakly dominant strategy (L). You have to go back and forth between the two tables (for type A and B) and the third table for player 2 to find all the fixed points.

For this game, there are several equilibria; so maybe that is throwing you off. For example, suppose player 2 plays her weakly dominant strategy, L, in that case, both types are indifferent between U or D, so they both could choose U. Given the strategy UU, player 2's unique best response is L, so this is an equilibrium. I.e. (UU, L) is an equilibrium.

Try to convince yourself that (DD,R), (DD,L) are the other equilibria, and there are no more equilibria.

As for question 4, it is actually simpler. Since no player knows the state of the world, they cannot condition their actions on the state of the world. so the pure actions for player 1 are U or D and for player 2 are L and R.

You will have to construct another table where players average out the payoffs of each of the four possible combinations of actions. For example for the cell (U,L), the payoff for player 1 is 1 and for player 2 is zero. Similarly, the payoffs on cell (U,R) is $(0,0.5)$. After constructing the new table you can find the BNE by the usual algorithm to find standard simultaneous-move NE.

Try to convince yourself that there is a continuum of equilibria for this game. In all of them, player 2 chooses $R$ since it is a dominant strategy, and player 1 is indifferent between both actions so chooses any mixed or pure strategy (which technically speaking is a type of mixed strategy too).