Economic and Game Theory
|"Inside every small problem is a large problem struggling to get out."|
Thread and Full Text View
aa: ATTACK if WEAK and ATTACK if STRONG
an: ATTACK if WEAK and NOT ATTACK if STRONG
na: NOT ATTACK if WEAK and ATTACK if STRONG
nn: NOT ATTACK if WEAK and NOT ATTACK if STRONG
Then, each player has 4 pure strategies: aa,an,na,nn. So the matrix of this game is 4X4. Given that each type is equally likely (probability 1/2) and independent of each other, you just have to calculate the expected payoffs for each player based on that information. For example, player 1's expected payoff from playing "aa", given that player 2 plays "aa" is:
You'll see that calculating these payoffs will be easy. Once you get the matrix filled with the payoffs, you're 90% done. This is the most important part, theoretically speaking. All that remains to be done is finding out the Bayesian equilibria.
What is really boring about this problem is that we only have letters in the matrix, not specific numbers. Then you have to consider a lot of possible cases, a lot of possible relations between these letters: M,s,w.
Suppose you "guess" that (aa,an) is a Bayesian equilibrium. Then you have to write down the conditions for it to be true. For player 1, aa should be the best-response to player 2 playing an. Similarly for player 2. Then you'll get some inequalities involving the parameters M,s,w. If these inequalities don't lead to contradictions, then you just write the inequalities you got and say that under these inequalities, (aa,an) is a pure strategy bayesian equilibrium. Doing it will be a good exercise of patience.
From a theoretical point of view, I don't think that it is useful to consider all the possible cases with their corresponding equilibria. As I told you, the most important thing is to find the pure strategies (which I already did above) and to find out the payoffs, which is really easy now. Finding the equilibria is simple. My suggestion is: plug in some numbers in the place of the parameters and see what happens.