Find all the pure-strategy Bayesian Nash equilibria in the following
static Bayesian game:

1. Nature determines whether the payoffs are as in Game 1 or as in
Game 2, each game being equally likely.

2. Player 1 learns whether nature has drawn Game 1 or Game 2, but
player 2 does not.

3. Player 1 chooses either T or B; player 2 simultaneously chooses
either L or R.

4. Payoffs are given by the game drawn by nature.

         Game 1                   Game 2
        L    R                   L     R
    T  1,1  0,0               T 0,0   0,0
    B  0,0  0,0               B 0,0   2,2

vitaminc,

Hi again.

The important rules for a strategy to be a PBE are as follows :
1) Within each subgame (I'll explain what one is later) - the subgame
strategy must be a PBE.
2) Each player assigns probabilities to being at each node. The
probability of being at each node must obey Bayes rule - P(A and B) =
P(A|B) * P(B)

Here is the game drawn out in extensive form :

                  Nature
                 /      \
    (.5) Game 1 /        \ Game 2 (.5)
               /          \
           Player1      Player1
           /   \          /   \
        T /     \ B    T /     \ B
         /       \      /       \
       (P2-A     P2-B  P2-C     P2-D)
       /  \     / \    / \     /  \
     L/    \R L/   \RL/   \R L/    \R

P1    1    0   0   0  0   0   0    2
P2    1    0   0   0  0   0   0    2

There is only one subgame because of the uncertainty for Player 2 in
the final stage. A subgame is a piece of the game which contains a
starting node that is fully identified (ie. there is no uncertainty as
to which node one is in).

The best way of determining the best strategy is to write the game in
"normal form" as follows :

        L     R
G1  T  1,1   0,0
G1  B  0,0   0,0
G2  T  0,0   0,0
G2  B  0,0   2,2

For player 1, given G1 and L, the best strategy is to play T.
              given G1 and R, the best strategy is to play T or B.
              given G2 and L, the best strategy is to play T or B.
              given G2 and R, the best strategy is to play B.

Suppose player 2 plays L. Then player 1 would play T in game 1 and T
or B in game 2.

The probabilities of nodes A through D must be :
P(A) = 0.5
P(B) = 0
P(C) = 0.5*(plays T)
P(D) = 0.5*(plays B)

The expected payoff is : 0.5*1 + 0*0 + 0.5*(plays T)*0 + 0.5*(plays
B)*0 = 0.5

Suppose player 2 plays R. Then player 1 would play T or B in game 1
and B in game 2.

The probabilities of nodes A through D must be :
P(A) = 0.5*(plays T)
P(B) = 0.5*(plays B)
P(C) = 0
P(D) = 0.5

The expected payoff is : 0.5*(plays T)*0 + 0.5*(plays B)*0 + 0*0 +
0.5*2 + = 1

So player 2 should play R (as 1 > 0.5)

Player 1's stragegy should be to play:
If Game 1 : T or B (Both are equilibria)
If Game 2 : B

This game is relatively easy as PBE games go, so I took a lot of
shortcuts to arrive at this answer that you wouldn't be able to do on
a more complex game. For a more complex game - checking that the
probabilities (both true and perceived) are correct is vital to
solving the problem. I assume in a few weeks you will be studying the
Cho-Kreps Intuitive Criterion. Be sure to have the concepts in PBE
down before reaching that point.

Regards

calebu2-ga

Thanks!