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Extensive-Form Games with Perfect Information

Yiling Chen

September 22, 2008

Logistics

In this unit, we cover 5.1 of the SLB book.

Problem Set 1, due Wednesday September 24 in class.

Example 1: The Sharing Game

Alice and Bob try to split two indivisible and identical gifts. First, Alice suggests a split: which can be “Alice keeps both”, “they each keep one”, and “Bob keeps both”. Then, Bob chooses whether to Accept or Reject the split. If Bob accepts the split, they each get what the split specifies. If Bob rejects, they each get nothing.

Alice

Bob Bob Bob

A R A R A R

Loosely Speaking...

Extensive Form

I A detailed description of the sequential structure of the

decision problems encountered by the players in a game.

I Often represented as a game tree

Perfect Information

I All players know the game structure (including the payoff

functions at every outcome).

I Each player, when making any decision, is perfectly informed

of all the events that have previously occurred.

Def. of Perfect-Information Extensive-Form Games

A perfect-information extensive-form game, G = (N, H, P, u)

I (^) H is a set of sequences (finite or infinite) F (^) Φ ∈ H F (^) h = (ak^ )k=1,...,K ∈ H is a history F (^) If (ak^ )k=1,...,K ∈ H and L < K , then (ak^ )k=1,...,L ∈ H F (^) (ak^ )∞ k=1 ∈ H if (ak^ )k=1,...,L ∈ H for all positive L F (^) Z is the set of terminal histories. H = {Φ, 2 − 0 , 1 − 1 , 0 − 2 , (2 − 0 , A), (2 − 0 , R), (1 − 1 , A), (1 − 1 , R), (0 − 2 , A), (0 − 2 , R)}

Alice

Bob Bob Bob

A R A R A R

Def. of Perfect-Information Extensive-Form Games

A perfect-information extensive-form game, G = (N, H, P, u)

I (^) P is the player function, P : H\Z → N. P(Φ)=Alice P(2 − 0)=Bob P(1 − 1) = Bob P(0 − 2) = Bob

Alice

Bob Bob Bob

A R A R A R

Pure Strategies in

Perfect-Information Extensive-Form Games

A pure strategy of player i ∈ N in an extensive-form game with

perfect information, G = (N, H, P, u), is a function that assigns

an action in A(h) to each non-terminal history h ∈ H\Z for

which P(h) = i.

I (^) A(h) = {a : (h, a) ∈ H}

I (^) A pure strategy is a contingent plan that specifies the action for player i at every decision node of i.

Pure Strategies for Example 1

Alice

Bob Bob Bob

A R A R A R

S = {S 1 , S 2 }

E.g. s 1 = (2 − 0 if h = Φ), s 2 = (A if h = 2 − 0; R if h = 1-1; R if h = 0 − 2).

Normal-Form Representation: Example 1

A perfect-information extensive-form game ⇒ A normal-form game

Alice

Bob Bob Bob

2-0 1-1 0-

A R A R A R

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

2-

1-

0-

(A,A,A) (A,A,R) (A,R,A) (A,R,R) (R,A,A) (R,A,R) (R,R,A) (R,R,R)

2,0 2,0 2,0 2,0 0,0 0,0 0,0 0,

1,1 1,1 0,0 0,0 1,1 1,1 0,0 0,

0,2 0,0 0,2 0,0 0,2 0,0 0,2 0,

A normal-form game ; A perfect-information extensive-form game

Normal-Form Representation: Example 2

A perfect-information extensive-form game ⇒ A normal-form game

Alice

Bob Bob

Alice

A B

C D E F

J K

(3,8) (8,3) (5,5)

(2,10) (1,0)

(A, J)

(A, K)

(B, J)

(B, K)

(C, E) (C, F) (D, E) (D, F)

3, 8

3, 8

5, 5

5, 5

3, 8

3, 8

2,

1, 0

8, 3

8, 3

5, 5

5, 5

8, 3

8, 3

1, 10

1, 0

A normal-form game ; A perfect-information extensive-form game

Pure Strategy Nash Equilibrium in

Perfect-Information Extensive-Form Games

A pure strategy profile s is a weak Nash Equilibrium if, for all agents i and for all strategies s′ i 6 = si , ui (si , s−i ) ≥ ui (s i′ , s−i ). (Same as in normal-form games)

Alice

Bob Bob

Alice

A B

C D E F

J K

(3,8) (8,3) (5,5)

(2,10) (1,0)

(A, J)

(A, K)

(B, J)

(B, K)

(C, E) (C, F) (D, E) (D, F)

3, 8

3, 8

5, 5

5, 5

3, 8

3, 8

2,

1, 0

8, 3

8, 3

5, 5

5, 5

8, 3

8, 3

1, 10

1, 0

Pure Strategy Nash Equilibrium in

Perfect-Information Extensive-Form Games

A pure strategy profile s is a weak Nash Equilibrium if, for all agents i and for all strategies s′ i 6 = si , ui (si , s−i ) ≥ ui (s i′ , s−i ). (Same as in normal-form games)

Alice

Bob Bob

Alice

A B

C D E F

J K

(3,8) (8,3) (5,5)

(2,10) (1,0)

(A, J)

(A, K)

(B, J)

(B, K)

(C, E) (C, F) (D, E) (D, F)

3, 8

3, 8

5, 5

5, 5

3, 8

3, 8

2,

1, 0

8, 3

8, 3

5, 5

5, 5

8, 3

8, 3

1, 10

1, 0

Pure Strategy Nash Equilibrium in

Perfect-Information Extensive-Form Games

A pure strategy profile s is a weak Nash Equilibrium if, for all agents i and for all strategies s′ i 6 = si , ui (si , s−i ) ≥ ui (s i′ , s−i ). (Same as in normal-form games)

Alice

Bob Bob

Alice

A B

C D E F

J K

(3,8) (8,3) (5,5)

(2,10) (1,0)

(A, J)

(A, K)

(B, J)

(B, K)

(C, E) (C, F) (D, E) (D, F)

3, 8

3, 8

5, 5

5, 5

3, 8

3, 8

2,

1, 0

8, 3

8, 3

5, 5

5, 5

8, 3

8, 3

1, 10

1, 0

Nash Equilibrium and Non-Credible Threat

Nash Equilibrium is not a very satisfactory solution concept for perfect-information extensive-form games.

Alice

Bob Bob

Alice

A B

C D E F

J K

(3,8) (8,3) (5,5)

(2,10) (1,0)

(A, J)

(A, K)

(B, J)

(B, K)

(C, E) (C, F) (D, E) (D, F)

3, 8

3, 8

5, 5

5, 5

3, 8

3, 8

2,

1, 0

8, 3

8, 3

5, 5

5, 5

8, 3

8, 3

1, 10

1, 0