Bayesiens

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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 acredible 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 mustform 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 unexpectedlyreached, 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‘ 0,1,13,3,3 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 mustbe rational given the beliefs and the beliefs have to be derived, when possible, from the strategy profile. What does this mean? It is easy at information sets which are reached with positive probability under the prescribe strategy profile. The main differences between different equilibrium concepts are found in what to believe in information set that are reached with 0 probability.

The keyfeature is that beliefs are as important as strategies. The agents not only are requiered to choose credible strategies, but also to hold reasonable beliefs. In applicatons the most widely used equilibrium concept is (Weak) Perfect Bayesian equilibrium. We need that: 1. At every information set the agent with the move has some beliefs about about which node has been reached. This beliefs are describedby a probability distribution over the nodes. 2. Given the beliefs, the strategies are sequentially rational, namely the actions taken at any information set must be optimal given the beliefs and all other players’ subsequent strategies. 3. Beliefs are determined by Bayes’ Rule when possible.

The main diference of the different equilibrium concepts rests on the beliefs out of the equilibriumpath, namely at those information set reached with zero probability under the prescribed equilibrium. We are going to define (Weak) Perfect Bayesian Equilibrium and the Sequential Equilibrium. By means of an example I will show that the second has tougher requirements, in fact all sequential equilibria are WPBE but the opposite is not true. Sequential equilibrium is slightly stronger than PerfectBayesian equilibrium. It assumes that the beliefs in equilibrium are the limit of a system of ”consistent beliefs” given a sequence of completely mixed strategies (so that Bayes rule applies everywhere).

The theory

Γ = {P, {K1, K2, .., KI }, {H1, H2, .., HI },

{A(x)}x∈K\Z , {(π 1(z), ..., π I (z))}z∈Z }

Let Ψi be the set of behavioural strategies for player i and Ψ = Ψ1 × Ψ2.. × ΨI An...
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