Perfect Bayesian Equilibrium and Sequential Equilibrium

Maria Saez Marti

Extensive form games revisited. Example 1
N

0.5 out
1

0.5 out 1 1

in

in

2

L 2 0

R 0 1

L 2 2

R 0 0

In this example 2 can not have arbitrary beliefs. Since the game is common knowledge, he has to believe that he is in each of the nodes with the same probability: 1/2. (Out,R) is not a credible equilibrium. Let us substitute now Nature by player 3.

Example 2: (b,out,R) is Nash, is subgame perfect.

3

a
1 1 1

b
1

out

out

1 1 1

in

in

2

L
2 0 1

R
0 1 0

L
2 2 1

R
0 0 0

R is optimal if 2 believes that he is at the LHS node with probability larger than 2/3. Is there any problem with 2’s beliefs?

If that equilibrium is believed, 2 must form a theory of how he came to move. Clearly 1 did not play out (as suggested by his eq. strategy. One possibility is that 1 deviated. This would be the theory with minimum number of deviations.(other possibilities?). But then 2 must believe that he is at the RHS node and wants to pay L. At information sets that are unreached according to the proposed strategies and that are unexpectedly reached, the player must have a belief which is consistent not only with nature (if nature moves as in example 1) but also with a minimal theory of deviations from the proposed equilibrium.

Example 3. Look at (L,L’)

1 L
[p]

R M
[1-p]

1,3

2 L‘ 2,1 R‘ 0,0 L‘ 0,2 R‘ 0,1

Example 4. Look at (D,L,R’) (SPE) and (A,L,L’)

1 D 2 L
[p]

A 2,0,0

R
[1-p]

3 L‘ 1,2,1 R‘ L‘ R‘ 0,1,1

3,3,3 0,1,2

Example 5: (A,A’..)
1 D 2 L
[p]

A

A‘ R
[1-p]

3 L‘ R‘ L‘ R‘

The equilibrium concept which are used for these cases have two components: A strategy profile which prescribes what action to take at every information set, or probability distributions over actions A set of beliefs which assign probability distributions to nodes in every information set. The strategy must