Extensive Game with Imperfect Information

Part I: Strategy and Nash equilibrium

Adding new features to extensive games: A player does not know actions

taken earlier non-observable actions taken by

other players The player has imperfect recall--e.g.

absent minded driver The “type” of a player is unknown

to others (nature’s choice is non-observable to other players)

Player 1’s actions are non-observable to Player 2

LM

R

2,2

3,1 0,0 0,2 1,1

22

1

L R L R

Imperfect recall: Absent minded driver

Nature’s choice is unknown to third party

0,-1 0,-3-2,-1

c

high low1 1

MBA MBA

No MBA No MBA2

2

2 2

hirehire

hirehire

No hire No hire

No hireNo hire

0,00,0-2,2

1,1

1,2

1/2 1/2

Extensive game with imperfect information and chances Definition: An extensive game <N,H,P,fc,(Ti),(ui)>

consists of a set of players N a set of sequences H a function (the player function P) that assigns

either a player or "chance" to every non-terminal history

A function fc that associates with every history h for which P(h)=c a probability distribution fc(.|h) on A(h), where each such probability distribution is independent of every other such distribution.

For each player i, Ti is an information partition and Ii (an element of Ti) is an information set of player i.

For each i, a utility function ui.

Strategies DEFINITION: A (pure) strategy of

player i in an extensive game is a function that assigns to each of i's information sets Ii an action in A(Ii) (the set of actions available to player i at the information set Ii).

DEFINITION: A mixed strategy of player i in an extensive game is a probability distribution over the set of player i’s pure strategies.

Behavioral strategy DEFINITION. A behavioral strategy of

player i in an extensive game is a function that assigns to each of i's information sets Ii a probability

distribution over the actions in A(Ii), with the property that each probability distribution is independent of every other distribution.

Mixed strategy and Behavioral strategy: an example

(L,l) ½

(L,r) ½

(R,l) 0

(R,r) 0

β1(φ)(L)=1; β1(φ)(R)=0; β1({(L,A),(L,B)})(l)=1/2; β1({(L,A),(L,B)})(r)=1/2

non-equivalence between behavioral and mixed strategy amid imperfect recall

Mixed strategy choosing LL with probability ½ and RR with ½.

The outcome is the probability distribution (1/2,0,0,1/2) over the terminal histories. This outcome cannot be achieved by any behavioral strategy.

Equivalence between behavioral and mixed strategy amid perfect recall

Proposition. For any mixed strategy of a player in a finite extensive game with perfect recall there is an outcome-equivalent behavioral strategy.

Nash equilibrium

DEFINITION: The Nash equilibrium in mixed strategies is a profile σ* of mixed strategies so that for each player i,

ui(O(σ*-i, σ*i))≥ ui(O(σ*-i, σi))

for every σi of player i. A Nash equilibrium in behavioral

strategies is defined analogously.

Part II: Belief and Sequential Equilibrium

A motivating example

L R

L 2,2 2,2

M 3,1 0,2

R 0,2 1,1

LM

R

2,2

3,1 0,2 0,2 1,1

22

1

L R L R

Strategic game

The importance of off-equilibrium path beliefs (L,R) is a Nash equilibrium According to the profile,

2’s information set being reached is a zero probability event. Hence, no restriction to 2’s belief about which history he is in.

2’s choosing R is optimal if he assigns probability of at least ½ to M; L is optimal if he assigns probability of at least ½ to L.

Bayes’ rule does not help to determine the belief

LM

R

2,2

3,1 0,2 0,2 1,1

22

1

L R L R

belief From now on, we will restrict our attention to

games with perfect recall. Thus a sensible equilibrium concept should consist

of two components: strategy profile and belief system.

For extensive games with imperfect information, when a player has the turn to move in a non-singleton information set, his optimal action depends on the belief he has about which history he is actually in.

DEFINITION. A belief system μ in an extensive game is a function that assigns to each information set a probability distribution over the histories in that information set.

DEFINITION. An assessment in an extensive game is a pair (β,μ) consisting of a profile of behavioral strategies and a belief system.

Sequential rationality and consistency It is reasonable to require that

Sequential rationality. Each player's strategy is optimal whenever she has to move, given her belief and the other players' strategies.

Consistency of beliefs with strategies (CBS). Each player's belief is consistent with the strategy profile, i.e., Bayes’ rule should be used as long as it is applicable.

Perfect Bayesian equilibrium Definition: An assessment (β,μ) is a perfect

Bayesian equilibrium (PBE) (a.k.a. weak sequential equilibrium) if it satisfies both sequential rationality and CBS. Hence, no restrictions at all on beliefs at zero-

probability information set In EGPI, the strategy profile in any PBE is a SPE The strategy profile in any PBE is a Nash

equilibrium

Sequential equilibrium

Definition. An assessment (β,μ) is consistent if there is a sequence ((βn,μn))n=1,… of assessments that converge to (β,μ) and has the properties that each βn is completely mixed and each μn is derived from using Bayes’ rule.

Remark: Consistency implies CBS studied earlier

Definition. An assessment is a sequential equilibrium of an extensive game if it is sequentially rational and consistent. Sequential equilibrium implies PBE Less easier to use than PBE (need to

consider the sequence ((βn,μn))n=1,… )

Back to the motivating example The assessment (β,μ) in

which β1=L, β2=R and μ({M,R})(M)= for any (0,1) is consistent

Assessment (βε,με) with the following properties

βε1 = (1-ε, ε,(1-)ε)

βε2 = (ε,1- ε)

με ({M,R})(M)= for all ε As ε→0, (βε,με)→ (β,μ) For ≥1/2, this

assessment is also sequentially rational.

LM

R

2,2

3,1 0,2 0,2 1,1

22

1

L R L R

Two similar games

LM

R

3,3

0,1 0,0 1,0 5,1

22

1

L R L R

Game 1 Game 2

Game 1 has a sequential equilibrium in which both 1 and 2 play L

Game 2 does not support such an equilibrium

L

M R

3,3

0,1 0,0 1,0 5,1

22

1

L R L R

C

Structural consistency Definition. The belief system in an extensive

game is structurally consistent if for each information set I there is a strategic profile with the properties that I is reached with positive probability under and is derived from using Bayes’ rule.

Remark: Note that different strategy profiles may be needed to justify the beliefs at different information sets.

Remark: There is no straightforward relationship between consistency and structural consistency. (β,μ) being consistent is neither sufficient nor necessary for μ to be structurally consistent.

Signaling games A signaling game is an extensive game with

the following simple form. Two players, a “sender’ and a “receiver.” The sender knows the value of an uncertain

parameter and then chooses an action m (message)

The receiver observes the message (but not the value of ) and takes an action a.

Each player’s payoff depends upon the value of , the message m, and the action a taken by the receiver.

Signaling games

Two types Signals are (directly) costly Signals are directly not costly

– cheap talk game

Spence’s education game Players: worker (1) and firm (2) 1 has two types: high ability H with

probability p H and low ability L with probability p L .

The two types of worker choose education level e H and e L (messages).

The firm also choose a wage w equal to the expectation of the ability

The worker’s payoff is w – e/

Pooling equilibrium e H = e L = e* L pH (H - L) w* = pHH + pLL

Belief: he who chooses a different e is thought with probability one as a low type

Then no type will find it beneficial to deviate.

Hence, a continuum of perfect Bayesian equilibria

Separating equilibrium e L = 0 H (H - L) ≥ e H ≥ L (H - L) w H = H and w L = L

Belief: he who chooses a different e is thought with probability one as a low type

Again, a continuum of perfect Bayesian equilibria

Remark: all these (pooling and separating) perfect Bayesian equilibria are sequential equilibria as well.

When does signaling work? The signal is

costly Single crossing

condition holds (i.e., signal is more costly for the low-type than for the high-type)

Refinement of sequential equilibrium

There are too many sequential equilibria in the education game. Are some more appealing than others?

Cho-Kreps intuitive criterion A refinement of sequential

equilibrium—not every sequential equilibrium satisfies this criterion

An example where a sequential equilibrium is unreasonable

Two sequential equilibria with outcomes: (R,R) and (L,L), respectively

(L,L) is supported by belief that, in case 2’s information set is reached, with high probability 1 chose M.

If 2’s information set is reached, 2 may think “since M is strictly dominated by L, it is not rational for 1 to choose M and hence 1 must have chosen R.”

LM

R

2,2

1,3 0,0 0,0 5,1

22

1

L R L R

Beer or Quiche

1,0 0,13,0

c

strong weak1 1

Q Q

B B2

2N N

NN

F F

FF

0,01,11,0

1,1

3,1

0.9 0.1

Why the second equilibrium is not reasonable?

If player 1 is weak she should realize that the choice for B is worse for her than following the equilibrium, whatever the response of player 2.

If player 1 is strong and if player 2 correctly concludes from player 1 choosing B that she is strong and hence chooses N, then player 1 is indeed better than she is in the equilibrium.

Hence player 2’s belief is unreasonable and the equilibrium is not appealing under scrutiny.

1,0 0,13,0

cstrong weak

1 1

Q Q

B B2

2N N

NN

F F

FF

0,01,11,0

1,1

3,1

0.9 0.1

Spence’s education game All the pooling equilibria are eliminated by the Cho-

Kreps intuitive criterion. Let e satisfy w* – e*/ L > H – e/ L and w* – e*/ H > H –

e/ L (such a value of e clearly exists.) If a high type work deviates and chooses e and is

correctly viewed as a good type, then she is better off than under the pooling equilibrium

If a low type work deviates and successfully convinces the firm that she is a high type, still she is worse off than under the pooling equilibrium.

Hence, according to the intuitive criterion, the firm’s belief upon such a deviation should construe that the deviator is a high type rather than a low type.

The pooling equilibrium break down!

Spence’s education game Only one separating equilibrium survives the

Cho-Kreps Intuitive criterion, namely: e L = 0 and e H = L (H - L)

Why a separating equilibrium is killed where e L = 0 and e H > L (H - L)?

A high type worker after choosing an e slightly smaller will benefit from it if she is correctly construed as a high type.

A low type worker cannot benefit from it however.

Hence, this separating equilibrium does not survive Cho-Kreps intuitive criterion.