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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

Perfect Bayesian Equilibrium: A Potential Partial Equilibrium

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.

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 =pαC

pαC + (1 − p)1

Perfect Bayesian Equilibrium: A Potential Partial Equilibrium

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.

The Beer-Quiche Game

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)

The Beer-Quiche Game

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.

The Beer-Quiche Game

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

Yet, the wimp would deviate from Q.

⇒ Thus, this is not an equilibrium.

The Beer-Quiche Game

A Potential Separating Equilibrium in the Beer-Quiche Game

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.

The Beer-Quiche Game

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

Yet, the wimp would deviate from B .

⇒ Thus, this is not an equilibrium.

The Beer-Quiche Game

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)

The Beer-Quiche Game

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)

The Beer-Quiche Game

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.

The Beer-Quiche Game

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)

The Beer-Quiche Game

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)

The Beer-Quiche Game

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