Bayesian Nash Equilibria: Strong and Weak Types

Question

I need a little help with the question. I understand that since both players have two types each, there will be 4 different payoff matrices to be considered, $(S,S)$,$(S,W)$,$(W,S)$,$(W,W)$, each type occuring with equal probability. I found that in each of these four payoff matrices, $(A,NA),(NA,A),(N,N)$ will have the respective payoffs: $(10,0),(0,10),(0,0)$. The $(A,A)$ action will have payoffs $(-6,-6),(4,-8),(-8,4),(-8,-8)$ respectively. Have I modelled the question correctly? Also, please help me with the BNE as well. I understand that we need to check for deviations, using prior beliefs; however, it seems too tedious to work out. How do I find all Pure Startegy BNE?

Answer 1

Modeling the game as four separate matrices does not capture the fact that each general knows his army strength but not the strength of the other army.

Given the small action space, it may help if you could first visualize the game in an extensive form, with Nature deciding army strengths at the beginning, and each army general observing their type before making the attack or not decisions. (Of course, this step is not necessary if you have a good mental image of what the game looks like.)

You can then try to convert the extensive form game into its Bayesian normal form. Hint: each general has four pure strategies: attack or not when their army is strong and attack or not when their army is weak. The BNE can then be solved easily in the Bayesian normal form.

Answer 2

Each army general has 4 actions to take (or pure strategies) $(A,A), (A,NA), (NA,A), (NA,NA)$. So, construct a $4 \times 4$ matrix and calculate the payoffs for each case.

Say, we have $\textbf{(A,NA)}$ as army 1's strategy and $\textbf{(A,A)}$ as army 2's strategy, it means, Army 1 attacks ($A$) when it is of type "Strong" and does not attack ($NA$) when it is of type "Weak". Similarly, Army 2 attacks ($A$) in both types in this case.

The payoffs, in this case, are calculated as below:

The payoff for Army 1 = \begin{equation} \frac{1}{4}* \{Strong Army1 (A) \textrm{vs} Strong Army 2 (A)\} + \frac{1}{4}\{Strong Army1 (A) \textrm{vs} Weak Army 2 (A)\}+\frac{1}{4}* \{Weak Army1 (NA) \textrm{vs} Strong Army 2 (A)\}+\frac{1}{4}* \{Weak Army1 (NA) \textrm{vs} Weak Army 2 (A)\} \end{equation} \begin{equation} =\frac{1}{4}*(-6)+\frac{1}{4}*(10-6)+\frac{1}{4}*(0)+\frac{1}{4}*(0) =\frac{-1}{2} \end{equation}

That is the payoff for Army 1 in one $(A,NA),(A,A)$ of the 16 cases. Calculate all the payoffs and then solve for BNE just like you solve for NE solution in a normal form game.