Research Methods in Political Science Formal Political Theory · Research Methods in Political Science Formal Political Theory Graduate School of Political Science at Waseda University

Research Methods in Political ScienceFormal Political Theory

Graduate School of Political Science at Waseda UniversitySpring Semester, 2013

Week 14:“Dynamic Games of Incomplete Information:

Perfect Bayesian Equilibrium in Signaling Games”

Shuhei Kurizaki

Perfect Bayesian Equilibrium: Definition

Perfect Bayesian Equilibrium

A perfect Bayesian equilibrium (PBE) consists of strategies andbeliefs satisfying the following requirements:

1 The player with the move must hold a belief

2 Players’ strategies must be sequentially rational.

3 Beliefs must be consistent with strategies.

Perfect Bayesian Equilibrium: A Potential Separating Equilibrium

Candidate

Incompetent(1 - p)

NotEndorse

Partyleadership

Competent(p)

NotEndorse

Not RunNot Run Run

Endorse ( C)

Run

-1, 0 -1, 01, -1 1, 1

0, 0 0, 0

Partyleadership

(q) (1 - q)

Endorse ( I)

Candidate’s posterior is: q = 0

Since q < 1/2, the candidate will Not Run.

If so, the incompetent leader has a deviation incentive.

Perfect Bayesian Equilibrium: A Potential Separating Equilibrium

Candidate

Incompetent(1 - p)

NotEndorse

Partyleadership

Competent(p)

NotEndorse

Not RunNot Run Run

Endorse ( C)

Run

-1, 0 -1, 01, -1 1, 1

0, 0 0, 0

Partyleadership

(q) (1 - q)

Endorse ( I)

Candidate’s posterior is: q = 1

Since q > 1/2, the candidate will Run.

Leader’s separating strategy is not incentive compatible forthe incompetent type, so this is not a separating equilibrium.

Perfect Bayesian Equilibrium: A Potential Pooling Equilibrium

Candidate

Incompetent(1 - p)

NotEndorse

Partyleadership

Competent(p)

NotEndorse

Not RunNot Run Run

Endorse ( C)

Run

-1, 0 -1, 01, -1 1, 1

0, 0 0, 0

Partyleadership

(q) (1 - q)

Endorse ( I)

Candidate’s posterior is: q = p

Candidate will Run if q > 1/2, Not Run if q < 1/2.

Leader’s pooling strategy is not incentive compatible if NotRun, so a pooling equilibrium exists only if q > 1/2.

Perfect Bayesian Equilibrium: A Potential Pooling Equilibrium

Candidate

Incompetent(1 - p)

NotEndorse

Partyleadership

Competent(p)

NotEndorse

Not RunNot Run Run

Endorse ( C)

Run

-1, 0 -1, 01, -1 1, 1

0, 0 0, 0

Partyleadership

(q) (1 - q)

Endorse ( I)

Candidate can have any posterior belief as long as thestrategy is a best response to it.Leader’s pooling strategy is not incentive compatible if thecandidate chooses Run but is compatible if Not Run.This pooling equilibrium exists only if q < 1/2.

Perfect Bayesian Equilibrium: A Potential Partial Equilibrium

Candidate

Incompetent(1 - p)

NotEndorse

Partyleadership

Competent(p)

NotEndorse

Not RunNot Run Run

Endorse ( C)

Run

-1, 0 -1, 01, -1 1, 1

0, 0 0, 0

Partyleadership

(q) (1 - q)

Endorse ( I)

Candidate’s posterior is:

q =p1

p1 + (1 − p)αI


Candidate will also mix so as to make the incompetent leaderindifferent about whether to Endorse, or EUI (E ) = EUI (NE ).

This indifference condition implies r(1) + (1− r)(−1) = 0 andsolving for r we get

r∗ =1

2Since the candidate mixes in this equilibrium, her posteriorbelief must also be q = 1/2.

Thus, it must be that pp+(1−p)αI

= 12 . Solving for αI , we get

α∗I =

p

1 − p

Since p1−p < 1 by definition, the equilibrium condition is

p <1

2.


Candidate

Incompetent(1 - p)

NotEndorse

Partyleadership

Competent(p)

NotEndorse

Not RunNot Run Run

Endorse ( C)

Run

-1, 0 -1, 01, -1 1, 1

0, 0 0, 0

Partyleadership

(q) (1 - q)

Endorse ( I)

Candidate’s posterior is:

q =pαC

pαC + (1 − p)1


Candidate will also mix so as to make the competent leaderindifferent about whether to Endorse, or EUI (E ) = EUI (NE ).This indifference condition implies r(1) + (1− r)(−1) = 0 andsolving for r we get

r∗ =1

2Since the candidate mixes, her posterior belief must also beq = 1/2.Thus, it must be that pαC

pαC+(1−p) = 12 . Solving for αC , we get

α∗C =

1 − p

p

Since 1−pp < 1 by definition, the equilibrium condition is

p >1

2.

Perfect Bayesian Equilibria to the Endorsement Game

Pooling Equilibrium if p > 1/2

Leader of both types Endorses.

Candidate Runs.

Candidate’s posterior is: q = p

Pooling Equilibrium

Leader of both types chooses Not Endorse.

Candidate chooses Not Run.

Candidate’s posterior must be q < 1/2

Perfect Bayesian Equilibria to the Endorsement Game

Semi-Separating Equilibrium if p < 1/2

Competent leader Endorses, but incompetent mixes withprobability α∗

I = p1−p .

Candidate Runs with probability r∗ = 12 .

Candidate’s posterior is: q = pp+(1−p)αI

.

Semi-Separating Equilibrium if p > 1/2

Incompetent leader Endorses, but competent mixes withprobability α∗

C = 1−pp .

Candidate Runs with probability r∗ = 12 .

Candidate’s posterior is: q = pαCpαC+(1−p) .

The Beer-Quiche Game

Cho and Kreps (1987)

Players: Player 1 and Player 2.

Player 1 is either a Real man (R) or Wimp (W), where

Pr(t = R) = 0.1.

Player 2’s action space: {Fight (F), Not Fight (N)}.Player 1 wats to fight a wimp, not a real man.

Player 1 chooses either to drink beer (B) or eat quiche (Q).

A real men prefers beer and wimps prefer quiche.


Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


A Potential Separating Equilibrium in the Beer-Quiche Game

Real man plays B and the wimp plays Q.

Player 2’s posterior belief: q(B) = 1 and q(Q) = 0.

Player 2’s sequentially rational play is N after B and F afterQ.


Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


Yet, the wimp would deviate from Q.

⇒ Thus, this is not an equilibrium.



Real man plays Q and the wimp plays B .

Player 2’s posterior belief: q(B) = 0 and q(Q) = 1.

Player 2’s sequentially rational play is F after B and N afterQ.


Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


Yet, the wimp would deviate from B .

⇒ Thus, this is not an equilibrium.


A Pooling Equilibrium in the Beer-Quiche Game

Both the real man and the wimp prefer B .

Player 2’s posterior belief: q(B) = 0.9

Sequential rationality suggests that Player 2 plays F after Bbecause EU2(N|B) > EU2(F |B).

Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


q(Q) is off the equilibrium path in this pooling equilibrium

Player 1 has no incentive to deviate if Player 2 plays F uponseeing Q.

What belief off the path would make F after Q a bestresponse? q(Q) < 1/2.

Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


Interpretation of this pooling equilibrium

The wimp acts like a real man (pooling capturesmisrepresentation of a true type).

Upon observing an action of a real man, Player 2 chickens out.

If Player 2 ever observes Quiche, he concludes that Player 1 ismore likely a wimp and fights.

⇒ A typical argument of strategic misrepresentation.


A Pooling Equilibrium in the Beer-Quiche Game

Both the real man and the wimp prefer Q.

Player 2’s posterior belief: q(Q) = 0.1.

Sequential rationality suggests that Player 2 plays N after Qbecause EU2(N|Q) > EU2(F |Q).

Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


q(B) is off the equilibrium path in this pooling equilibrium

Player 1 has no incentive to deviate if Player 2 plays F uponseeing B .

What belief off the path should Player 2 have to make F afterB a best response? q(B) < 1/2.

Nature(0.9)

(0.1)

R

W

Q

Q

B

B

F

F

F

F

N

N

N

N

1

1

2 2

0, 0

2, 1 3, 1

1, 0

0, 1

2, 0

1, 1

3, 0

1 - q(Q) 1 - q(B)

q(Q) q(B)


Interpretation of this pooling equilibrium

The real man acts like a wimp (pooling capturesmisrepresentation of a true type).

Upon observing an action of a wimp, Player 2 chickens out.

If Player 2 ever observes Beer, he concludes that Player 1 ismore likely a wimp and fights.

⇒ Not intuitively appealing.

⇒ Further refinement: Intuitive Criterion

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Research Methods in Political Science Formal Political Theory · Research Methods in Political Science Formal Political Theory Graduate School of Political Science at Waseda University