From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian ......From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences

From Bayesian Nash Equilibrium (BNE) toPerfect Bayesian Equilibrium (PBE)

Félix Muñoz-GarcíaSchool of Economic SciencesWashington State University

BNEs and Sequential rationality

So far we have learned how to nd BNEs in incompleteinformation games.

We are doing great!

In settings where players are uncertain about their opponentstypes. . .

this is a fantastic solution concept.since it species optimal strategies, given the information everyplayer has access to.

BNEs and Sequential rationality

What if player interact in a sequential-move game?

Can the BNE prescribe "insensible" behavior? Yes!But, what do we mean by "insensible" behavior?

Strategies that are not sequentially rational.We will, hence, need a solution concept that guaranteessequential rationality (as SPNE, but applied to contexts ofincomplete information).

Lets show this with an example.

Use now the separate handout:

"Why do we need Perfect Bayesian Equilibrium? Asking forsequential rationality in sequential-move games withincomplete information."

More examples about how to nd PBEs

After nding the PBEs in the Gift game...

Lets now practice with another example (Monetary Authoritygame):

Now we will consider a Strong or Weak monetary authority,who makes an ination announcement.And a labor union (uninformed about the monetary authoritystrue commitment with low ination policies, either Strong orWeak). . .decides whether to demand large or moderate salary increases.

Monetary authority game

Example: Let us consider the following sequential game withincomplete information:

A monetary authority (such as the Federal Reserve Bank)privately observes its real degree of commitment withmaintaining low ination levels.After knowing its type (either Strong or Weak), the monetaryauthority decides whether to announce that the expectation forination is either High or Low.A labor union, observing the message sent by the monetaryauthority, decides whether to ask for high or low salary raises(denoted as H or L, respectively)

Monetary authority game

The game tree that represents this incomplete informationgame is, hence, as follows:

PBEs-Monetary Authority

Before starting to nd all possible PBEs. . .

Let us briey set up our "road map"

That is, lets recall the 5-step procedure that we need tofollow in order to nd PBEs.

Procedure to nd PBEs

1. Specify a strategy prole for the privately informed player,either separating or pooling.

In our above example, there are only four possible strategyproles for the privately informed monetary authority: twoseparating strategy proles, HighSLowW and LowSHighW ,and two pooling strategy proles, HighSHighW andLowSLowW .

(For future reference, it might be helpful to shade thebranches corresponding to the strategy prole we test.)

2. Update the uninformed players beliefs using Bayesrule,whenever possible.

In our above example, we need to specify beliefs µ and γ,which arise after the labor union observes a high or a lowination announcement, respectively.

Procedure to nd PBEs - Contd

3. Given the uninformed players updated beliefs, nd his optimalresponse.

In our above example, we rst determine the optimal responseof the labor union (H or L) upon observing a high-inationannouncement (given its updated belief µ),we then determine its optimal response (H or L) after observinga low-ination announcement (given its updated belief γ).

(Also for future reference, it might be helpful to shadethe branches corresponding to the optimal responses wejust found.)


4. Given the optimal response of the uninformed player, nd theoptimal action (message) for the informed player.

In our previous example, we rst check if the Strong monetaryauthority prefers to make a high or low ination announcement(given the labor unions responses determined in step 3).We then operate similarly for the Weak type of monetaryauthority.


5. Then check if this strategy prole for the informed playercoincides with the prole suggested in step 1.

If it coincides, then this strategy prole, updated beliefs andoptimal responses can be supported as a PBE of theincomplete information game.Otherwise, we say that this strategy prole cannot besustained as a PBE of the game.


Let us next separately apply this procedure to test each of thefour candidate strategy proles:

two separating strategy proles:

HighSLowW , andLowSHighW .

And two pooling strategy proles:

HighSHighW , andLowSLowW .

Separating equilibrium with (Low,High)

Let us rst check separating strategy prole: LowSHighW .

Step #1: Specifying strategy prole LowSHighW that wewill test.

(See shaded branches in the gure.)


Step #2: Updating beliefs

(a) After high ination announcement (left-hand side)

µ =0.6αStrong

0.6αStrong + 0.4αWeak=

0.6� 00.6� 0+ 0.4� 1 = 0



This implies that after high ination...the labor union restricts its belief to the lower left-hand corner(see box), since µ = 0 and 1� µ = 1


Step #2: Updating beliefs(b) After low ination announcement (right-hand side)

γ =0.6�1� αStrong

�0.6�1� αStrong

�+ 0.4

�1� αWeak

� = 0.6� 10.6� 1+ 0.4� 0 = 1



This implies that, after low ination...the labor union restricts its belief to the upper right-handcorner (see box), since γ = 1 and 1� γ = 0.


Step #3: Optimal response(a) After high ination announcement, respond with H since

0 > �100

in the lower left-hand corner of the gure (see blue box).


Step #3: Optimal response(b) After low ination announcement, respond with L since

0 > �100

in the upper right-hand corner of the gure (see box).


We can hence summarize the optimal responses we just found, byshading them in the gure:

H after high ination, but L after low ination.

(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)LaborU

nion

H

H

L

L

1-γ

μ

1-μ

γ


Step #4: Optimal messages by the informed player

(a) When the monetary authority is Strong, if it chooses Low(as prescribed), its payo¤ is $300,while if it deviates, its payo¤ decreases to $0.(No incentives to deviate).


Step #4: Optimal messages(b) When the monetary authority is Weak, if it chooses High(as prescribed), its payo¤ is $100,while if it deviates, its payo¤ decreases to $50.(No incentives to deviate either).


(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ

Since no type of privately informed player (monetaryauthority) has incentives to deviate,

The separating strategy prole LowSHighW can be sustainedas a PBE.

Separating equilibrium with (High,Low)

Let us now check the opposite separating strategy prole:HighSLowW .

Step #1: Specifying strategy prole HighSLowW that wewill test.




(a) After high ination announcement

µ =0.6αStrong


0.6� 10.6� 1+ 0.4� 0 = 1



Hence, after high ination...the labor union restricts its beliefs to µ = 1 in the upperleft-hand corner (see box).



(b) After low ination announcement



�+ 0.4

�1� αWeak

� = 0.6� 00.6� 0+ 0.4� 1 = 0



Hence, after low ination...the labor union restricts its beliefs to γ = 0 (i.e., 1� γ = 1)in the lower right-hand corner (see box).


Step #3: Optimal response

(a) After high ination announcement, respond with L since

0 > �100

in the upper left-hand corner of the gure (see box).



(a) After low ination announcement, respond with H since

0 > �100

in the lower right-hand corner of the gure (see box).


Summarizing the optimal responses we just found:

L after high ination, but H after high ination.

(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ


Step #4: Optimal messages of the informed player(a) When the monetary authority is Strong, if it chooses High(as prescribed), its payo¤ is $200,while if it deviates, its payo¤ decreases to $100.(No incentives to deviate).


Step #4: Optimal messages(b) When the monetary authority is Weak, if it chooses Low(as prescribed), its payo¤ is $0,while if it deviates, its payo¤ increases to $150.(Incentives to deviate!!).


(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ

Since we found one type of privately informed player (theWeak monetary authority) who has incentives to deviate...

The separating strategy prole HighSLowW cannot besustained as a PBE.

Pooling equilibrium with (High,High)

Let us now test the pooling strategy prole HighSHighW .

Step #1: Specifying strategy prole HighSHighW that wewill test.




(a) After high ination announcement

µ =0.6αStrong


0.6� 10.6� 1+ 0.4� 1 = 0.6

so the high ination announcement is uninformative.


Step #2: Updating beliefs(b) After low ination announcement (o¤-the-equilibrium path)



�+ 0.4

�1� αWeak

� = 0.6� 00.6� 0+ 0.4� 0 =

00

hence, γ 2 [0, 1].


Step #3: Optimal response(a) After high ination announcement (along the equil. path),respond with L since

EULabor (H jHigh) = 0.6� (�100) + 0.4� 0 = �60EULabor (LjHigh) = 0.6� 0+ 0.4� (�100) = �40


Step #3: Optimal response(a) After low ination announcement (o¤-the-equil.),

EULabor (H jLow) = γ� (�100) + (1� γ)� 0 = �100γEULabor (LjLow) = γ� 0+ (1� γ)� (�100) = �100+ 100γ

i.e., respond with H if γ < 12 .


Summarizing the optimal responses we found...

Note that we need to divide our analysis into two cases:Case 1, where γ < 12 , implying that the labor union respondswith H after observing low ination (right-hand side).

(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)LaborU

nion

H

H

L

L

1-γ

μ

1-μ

γ


and...Case 2, where γ � 12 , implying that the labor union respondswith L after observing low ination (right-hand side).

(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ


Case 1, where γ < 12

Step #4: Optimal messages

(a) When the monetary authority is Strong, if it chooses High(as prescribed), its payo¤ is $200,while if it deviates, its payo¤ decreases to $100.(No incentives to deviate).




(b) When the monetary authority is Weak, if it chooses High(as prescribed), its payo¤ is $150,while if it deviates, its payo¤ drops to $0.(No incentives to deviate either).



(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ

No type of monetary authority has incentives to deviate.

Hence, the pooling strategy prole HighSHighW can be sustainedas a PBE when o¤-the-equilibrium beliefs satisfy γ < 12 .


Case 2, where γ � 12


(a) When the monetary authority is Strong, if it chooses High(as prescribed), its payo¤ is $200,while if it deviates, its payo¤ increases to $300.(Incentives to deviate!!).




(b) When the monetary authority is Weak, if it chooses High(as prescribed), its payo¤ is $150,while if it deviates, its payo¤ drops to $50.(No incentives to deviate).



(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ

Since we found one type of privately informed player (theStrong monetary authority) who has incentives to deviate...

The pooling strategy prole HighSHighW cannot be sustainedas a PBE when o¤-the-equilibrium beliefs satisfy γ � 12 .

Pooling equilibrium with (Low,Low)

Let us now examine the opposite pooling strategy prole.

Step #1: Specifying strategy prole LowSLowW that we willtest.



Step #2: Updating beliefs(a) After a low ination announcement



�+ 0.4

�1� αWeak

� = 0.6� 10.6� 1+ 0.4� 1 = 0.6

so posterior and prior beliefs coincide.



(b) After a high ination announcement (o¤-the-equil. path)

µ =0.6αStrong


0.6� 00.6� 0+ 0.4� 0 =

00

hence, µ 2 [0, 1].



(a) After a low ination announcement (along the equilibriumpath), respond with L since

EULabor (H jLow) = 0.6� (�100) + 0.4� 0 = �60EULabor (LjLow) = 0.6� 0+ 0.4� (�100) = �40


Step #3: Optimal response(a) After a high ination announcement (o¤-the-equil.),

EULabor (H jLow) = µ� (�100) + (1� µ)� 0 = �100µEULabor (LjLow) = µ� 0+ (1� µ)� (�100) = �100+ 100µ

i.e., respond with H if µ < 12 .


Summarizing the optimal responses we found...

Note that we need to divide our analysis into two cases:Case 1, where µ < 12 , implying that the labor union respondswith H after observing high ination (left-hand side).

(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUnio

n

(100, 0)

(150, -100)LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ


and...

Case 2, where µ � 12 , implying that the labor union respondswith L after observing high ination (left-hand side).

(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUnio

n

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ


Case 1, where µ < 12

Step #4: Optimal messages(a) When the monetary authority is Strong, if it chooses Low(as prescribed), its payo¤ is $300,while if it deviates, its payo¤ decreases to $200.(No incentives to deviate).




(b) When the monetary authority is Weak, if it chooses High(as prescribed), its payo¤ is $50,while if it deviates, its payo¤ increases to $100.(Incentives to deviate!!).



(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUni

on

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ


The pooling strategy prole LowSLowW cannot be sustainedas a PBE when o¤-the-equilibrium beliefs satisfy µ < 12 .


Case 2, where µ � 12


(a) When the monetary authority is Strong, if it chooses Low(as prescribed), its payo¤ is $300,while if it deviates, its payo¤ decreases to $200.(No incentives to deviate).




(b) When the monetary authority is Weak, if it chooses Low(as prescribed), its payo¤ is $50,while if it deviates, its payo¤ increases to $150.(Incentives to deviate!!).



(0, -100)

(200, 0)

Nature

Strong

Weak

HighInflation

LowInflation

HighInflation

LowInflation

0.6

0.4

(100, -100)

(300, 0)

(0, 0)

(50, -100)

H

L

L

H

MonetaryAuthority

MonetaryAuthority

Labo

rUnio

n

(100, 0)

(150, -100)

LaborUnion

H

H

L

L

1-γ

μ

1-μ

γ


The pooling strategy prole LowSLowW cannot be sustainedas a PBE when o¤-the-equilibrium beliefs satisfy µ � 12 .

Hence, the pooling strategy prole LowSLowW cannot besustained as a PBE for any o¤-the-equilibrium beliefs µ.

Documents

From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian ......From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences