ec319 economic theory and its applications, part ii: lecture...

EC319 Economic Theory and Its Applications,Part II: Lecture 6

Leonardo Felli

NAB.2.14

20 February 2014

Dynamic Games of Imperfect Information

1. Set of players N = {N , 1, ..., n}

I {1, ..., n} are the players in the game

I Nodes assigned to N are nodes controlled by Nature

I Nodes assigned to i 6= N are decision nodes controlled by i

2. Each alternative at an N node has a label specifying itsprobability:

I Nature’s probabilities are nonnegative and add to 1

Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 20 February 2014 2 / 40

Dynamic Games of Imperfect Information (cont’d)

3. Each node controlled by player i is associated to aninformation set:

I Thus two nodes associated to the same information set arecontrolled by the same player

I Two nodes belong to the same information set if and only if icannot distinguish between them

4. Each alternative at a decision node is associated with anaction by the player whose label is associated with theinformation set:

I If two nodes x , y belong to the same information set, for anyalternative at x there must be exactly one alternative at y withthe same action label


Dynamic Games of Imperfect Information (cont’d)

5. Each terminal node z is associated to a vector of n payoffs{ui (z)}i∈{1,...,n} such that:

I The number ui (z) specifies the payoff to i if the game ends atnode z

6. All players have perfect recall of the moves they chose


Perfect Recall

The following is a game of perfect recall:

b.......................................

.....................LLLLLLr

rr

rr

r(0, 0)

1

L

2

A B

R

U1 D1

1

(6, 6) (1, 0)(1, 1)

D2��

EEEEE

��r r

rU1

%%%%%r

eeeee

D1

%%%%%

eeeee

��

U2

��

ccccccc

.....................

r r r r

L R

U2 D2

(6, 6)(2, 0) (3, 0) (3, 10)

LLLLLL

1LLLLLL

��


Imperfect Recall

The following is a game of imperfect recall:

b.......................................

........................................LLLLLLr

rr

rr

r(0, 0)

1

L

2

A B

R

U D

(6, 6) (1, 0)(1, 1)

D��

EEEEE

��r r

rU

%%%%%r

eeeee

D

%%%%%

eeeee

��

U

��

ccccccc

.....................

r r r r

L R

U D

(6, 6)(2, 0) (3, 0) (3, 10)

LLLLLL

1LLLLLL

��


Behavioral Strategies

Denote:

I Pi the set of the information sets associated with player i ∈ N

I Api be the action set of player i at info state p ∈ Pi

A behavioral strategy for player i maps information sets intoprobability distributions over actions


Behavioral Strategies (cont’d)

In particular σpi (ai ) is the probability that player i at informationset p chooses action ai ∈ Ap

i

Then

I a behavioral strategy of player i is σi ={σpi}p∈Pi

I a profile of behavioral strategy is σ = {σi}i∈N

I the Nature’s strategy/probabilities is π ={πpN}p∈PN


Expected Payoffs

I For any terminal node z and any behavioral strategy profile σ,

p(z |σ)

denotes the probability that the game ends at node z

I Let Z be the set of terminal notes, the expected payoff ofplayer i is then:

Ui (σ) = Ui (σi , σ−i ) =∑z∈Z

p(z |σ)ui (z)


Bayesian Nash Equilibrium

Definition (Bayesian Nash Equilibrium)

A Nash Equilibrium of a dynamic game of imperfect information isa profile of behavioral strategies σ such that:

Ui (σ) ≥ Ui (σ′i , σ−i ) for any σ′i ∈ ×p∈Pi

∆(Api )


Bayesian Nash Equilibrium: Example I

Consider the Bayesian game (of imperfect information).

b,,,,,,

,,,,,

@@@@

@@@@@

R

(2, 1)

rr

rr

(0, 1)

1

2

U M

L L

(0, 2)(0, 0)

R

r r...............................JJJJ

r (1, 3)D


Bayesian Nash Equilibrium: Example I (cont’d)

I The normal form associated to this game is:

L R

U 2, 1 0, 0

M 0, 2 0, 1

D 1, 3 1, 3

I Clearly there exist two pure strategy (Bayesian) Nashequilibria of this game:

(U, L), (D,R)

I The first Bayesian Nash equilibrium (U, L) is a reasonableprediction.


Non-Credible Threats

I Consider however the second Bayesian Nash equilibrium(D,R).

I Player 1’s choice of D is sustained by the threat by player 2that if player 1 chooses either U or M player 2 will choose R.

I This choice of action by player 2 is a best reply to player 1’schoice of D.

I However it is also a non-credible threat.


Subgame Perfection

Recall however that the role of Subgame Perfect equilibrium was torule out equilibria sustained by non-credible threats.

Definition

A Nash equilibrium s∗i of a game Γ is a Subgame Perfectequilibrium if and only if it is a Nash equilibrium for every propersubgame.

Where a proper subgame of an extensive form game is a subgamethat starts at a decision node that is a singleton information set(contains only one decision node).


Perfect Bayesian Equilibrium

I In our example there exist only one proper subgame, henceboth Bayesian Nash equilibria of the game are SubgamePerfect.

I We therefore need to strengthen the equilibrium concept to beable to rule out non-credible threats in a Bayesian game.

I One way to proceed is to restrict attention to PerfectBayesian equilibria.

I A perfect Bayesian equilibrium strengthen the equilibriumconcept so as to rule out the Bayesian Nash equilibrium(D,R) by imposing the following restrictions.


Players’ Beliefs

I At each information set the player that is required to choosean action must have beliefs about the node in the informationset that has been reached by the play in the game.

I Notice that whenever the information set includes more thanone node beliefs are a probability distribution over the nodesin the information set.

I Trivially, if the information set contains only one node theonly feasible belief is to put probability one on that node.


Players’ Beliefs: Example I

In the example above let p and (1− p) denote player 2’s beliefsthat the play has reached the left, respectively right, node in theinformation set:

b,,,,,,

,,,,,

@@@@@

@@@@@

R

(2, 1)

rr

rr

(0, 1)

1

2

U M

L L

(0, 2)(0, 0)

R

r r...............................JJJJJ

r (1, 3)D

[p] [1− p]


Sequential Rationality

Given their beliefs the players’ strategies must be sequentiallyrational meaning that at each information set the action taken bythe player must be:

I optimal given the player’s beliefs at the information set;

I optimal given the other players’ strategies in the rest of thegame.

Moreover each player’s strategy must be optimal given his/her(correct) beliefs on the other players’ beliefs at every informationset.


Sequential Rationality: Example I

I In the example given player 2’s beliefs the expected payoff toplayer 2 from playing R is:

p 0 + (1− p) 1 = 1− p

I while the expected payoff to player 2 from choosing L is:

p 1 + (1− p) 2 = 2− p

I Since clearly 2− p > 1− p for every p ∈ [0, 1] sequentialrationality implies that player 2 will not choose R at theinformation set in question.


Sequential Rationality: Example I (cont’d)

I Therefore we can rule out (D,R) since for this to be areasonable outcome we need player 1 to have incorrectexpectations on player 2’s behaviour.

I In our case the condition we just analyzed are enough to ruleout the undesired outcome (D,R).

I For player 2 the strategy R is strictly dominated by thestrategy L at the information set in which she is required tomove.


Example II: Weakly Dominated Strategies

Notice that the outcome (D,R) cannot be ruled out in thefollowing modification of the previous game.

b,,,,,,

,,,,,

@@@@@

@@@@@

R ′

(2, 0)

rr

rr

(0, 1)

1

2

U ′ M ′

L′ L′

(0, 2)(0, 0)

R ′

r r...............................JJJJJ

r (1, 3)D ′

[p′] [1− p′]


Example II: Weakly Dominated Strategies (cont’d)

I The normal form associated with this new Bayesian game is:

L′ R ′

U ′ 2, 0 0, 0

M ′ 0, 2 0, 1

D ′ 1, 3 1, 3

I The set of Bayesian Nash equilibria of this game is then:

(U ′, L′), (D ′,R ′)

I Let p′ and (1− p′) be player 2’s beliefs at the information setwhere she is required to move.



I The expected payoff to player 2 from playing R ′ is now:

p′ 0 + (1− p′) 1 = 1− p′

I while the expected payoff to player 2 from choosing L is:

p′ 0 + (1− p′) 2 = 2 (1− p′)

I Notice that now 2 (1− p′) ≥ 1− p′.

I Therefore, if p′ = 1 it is a best reply for player 2 to choosestrategy R ′.



I In other words the outcome (D ′,R ′) cannot be ruled out sinceit is sustained by player 2’s beliefs p′ = 1.

I In general, weakly dominated strategies cannot be ruled outby imposing sequential rationality by itself.

I This shows that having beliefs at each information node is notenough we also need these beliefs to be reasonable.

I To define reasonable beliefs we first define what we meanwhen an information set is on the equilibrium path.


Bayes Rule

I An information set is on the equilibrium path if it will bereached with strictly positive probability if the game is playedaccording to the equilibrium strategies.

I Alternatively an information set is off the equilibrium path if itis certain that it will not be reached if the game is playedaccording to the equilibrium strategies.

I We impose the following additional restriction on each player’sbeliefs at each information set.

I At the information sets that are on the equilibrium path,beliefs are determined by Bayes’ rule given the players’equilibrium strategies.


Perfect Bayesian Equilibrium:

A Perfect Bayesian equilibrium consists of both a set of strategiesfor each player and a set of beliefs for each player at eachinformation set.

Definition (Perfect Bayesian Equilibrium)

A Perfect Bayesian equilibrium is a set of strategies and a set ofbeliefs such that:

I each player’s strategy is a best reply to other players’strategies, given the player’s beliefs,

I beliefs are derived at all information sets on the equilibriumpath using Bayes rule given the players’ equilibrium strategies.


Example III

Consider now the Bayesian game:

b

,,,,,,

,,,,,

@@@@

@@@@@

R

(2, 1, 1)

rr

rr

(1, 1, 0)

3

2

U D

L L

(1, 2, 0)(3, 3, 3)

R

r r...............................JJJJ

r (0, 0, 2)A

r1

B

[p] [1− p]


Example III (cont’d)

I The normal form associated with this game is:

L R

U 0, 0, 2 0, 0, 2

D 0, 0, 2 0, 0, 2

L R

U 2, 1, 1 3, 3, 3

D 1, 2, 0 1, 1, 0

A B

I The pure strategy Bayesian Nash equilibria are:

(U, L,A), (D, L,A), (D,R,A), (U,R,B)



I Notice that[(U,R,B), p = 1]

is a Perfect Bayesian equilibrium of the game.

I But also[(U, L,A), p = 0]

is a Perfect Bayesian equilibrium of the game.

I Player 2’s expected payoff from choosing L is p + 2(1− p)while from choosing R is 3p + (1− p) clearly player 2 choosesL if and only if

p + 2(1− p) ≥ 3p + (1− p) or p ≤ 1

3

I But player 2’s information set is off the equilibrium path.



Consider the subgame that starts at the (singleton) node whereplayer 1 chooses his action:

1

,,,,,,

,,,,,

@@@@@

@@@@@

R

(2, 1, 1)

rr

rr

(1, 1, 0)

2

U D

L L

(1, 2, 0)(3, 3, 3)

R

r r...............................JJJJJ

r[p] [1− p]



I Had this been the game, given player 1’s strategy U the beliefsp = 0 are inconsistent with player 1’s equilibrium strategy.

I In this subgame, given player 1’ strategy choice U the onlybeliefs consistent with Bayes’ rule are p = 1.

I Therefore had player 2’s information set been on theequilibrium path the beliefs p = 0 would be inconsistent.

I In other words, the threat player 1 chooses U and player 2chooses L even with beliefs p = 0 is “non-credible”.


Non-Credible Threats

I However, the definition of Perfect Bayesian equilibrium is notenough to rule out these non-credible threats.

I The next step is therefore to strengthen further the notion ofPerfect Bayesian equilibrium by imposing restriction on beliefsat information sets that are off the equilibrium path.

I The restriction consists in requiring that at information sets offthe equilibrium path players’ beliefs are determined by Bayes’rule, and the players’ equilibrium strategies where possible.


Strong Perfect Bayesian or Sequential Equilibrium

We also require that each player’s strategy (given his beliefs) is abest response to other players strategies and beliefs in every propersubgame.

Definition (Strong Perfect Bayesian equilibrium)

A set of strategies and beliefs such that:

I they are a Perfect Bayesian equilibrium of the game,

I they are a Perfect Bayesian equilibrium of every propersubgame.

In every proper subgame, whether the information set is on or offthe equilibrium path we can use Bayes rule.



Consider now the restriction of the equilibrium strategies (U, L,A)on the subgame:

1

,,,,,,

,,,,,

@@@@@

@@@@@

R

(2, 1, 1)

rr

rr

(1, 1, 0)

2

U D

L L

(1, 2, 0)(3, 3, 3)

R

r r...............................JJJJJ

r[p] [1− p]



I Clearly Bayes’ rule implies that in this subgame player 2 needto have belief: p = 1.

I Given these beliefs it is not a best reply for player 2 to choosethe strategy L.

I This implies that the Bayesian Nash equilibrium (U, L,A) is aPerfect Bayesian equilibrium of the game for any belief p ≤ 1

3but it is not a Strong Perfect Bayesian Equilibrium.

I Consider now the Bayesian Nash equilibrium (D, L,A).



In the same proper subgame, consider the restriction of theequilibrium strategies (D, L) on this subgame:

1

,,,,,,

,,,,,

@@@@@

@@@@@

R

(2, 1, 1)

rr

rr

(1, 1, 0)

2

U D

L L

(1, 2, 0)(3, 3, 3)

R

r r...............................JJJJJ

r[p] [1− p]



I Player 2’s beliefs p = 0 is consistent with Bayes rule in thissubgame.

I Notice however that given that player 2’s best reply is to playL in this subgame, then it is not a best reply for player 1 tochooses D in this subgame.

I Implying that [(D, L,A), p = 0] is a Perfect Bayesianequilibrium, but it is not a Strong Perfect Bayesianequilibrium of the game.

I Consider the Bayesian Nash equilibrium (D,R,A)



I Clearly the only player 2’s belief consistent with Bayes rule inthis subgame given player 1’s equilibrium strategy is p = 0.

I As seem before this implies that choosing R is not a bestreply for player 2.

I In other words [(D,R,A), p = 0] is a Perfect Bayesianequilibrium of the game but it is not a Strong PerfectBayesian Equilibrium.


Strong Perfect Bayesian Equilibrium (cont’d)

I The easy way to compute the Strong Perfect Bayesianequilibria of the game is therefore to compute the PerfectBayesian equilibrium of every proper subgame proceedingbackward.

I Start from the last proper subgame. This has the followingnormal form associated with it.

L R

U 2, 1, 1 3, 3, 3

D 1, 2, 0 1, 1, 0

I The unique Bayesian Nash equilibrium of this subgame isthen: (U,R).


Strong Perfect Bayesian Equilibrium (cont’d)

I This implies that all Strong Perfect Bayesian equilibria of thegame have to specify the strategies for players 1 and 2:(U,R).

I Moreover given the strategies (U,R) in the subgame inquestion player 2’s information set is on the equilibrium path.

I Then Bayes’ rule implies that the only beliefs for player 2consistent with strategies (U,R) are p = 1.

I This implies that the unique Strong Perfect Bayesianequilibrium of this game is:

[(U,R,B), p = 1]


ec319 economic theory and its applications, part ii: lecture...

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