Extensive and Strategic Form Games - PowerPoint Presentation

Slide1

Extensive and Strategic Form Games

Econ 171Slide2

Reminder: Course requirements

Class website

Go to economics departmen

t home page. Under Links, find Class pages, then click on Econ 171

Textbook:

Games, Strategies, and Decision Making 

by Joseph E. Harrington, Jr.

Clicker:

Available at campus bookstore

i

>clicker Register your clicker at www.i>clicker.com/registration Slide3

Matching Pennies

On blackboard, draw the extensive form game tree for matching pennies with perfect information where A moves first.

Then do the same for simultaneous move version.Slide4

Clicker question

How many strategies are available to the player that moves second in the perfect information version of matching pennies?

2

4

6

8Slide5

Rock, Paper, ScissorsSlide6

Rock-Paper-Scissors

First let’s draw the game tree on the blackboard for the game played with perfect information, where Player A moves first.

Let Payoff be 1 if you win, -1 if you lose, 0 if you tie.

How would we draw the game tree for this game if the players move simultaneously?Slide7

Clicker Question

How many (pure) strategies are possible for Player B in the perfect information version of Rock-Paper-Scissors

3

6

9

12

27Slide8

Strategies for B in perfect information

rock, paper, scissors game

A strategy for B in perfect information rock, paper, scissors answers 3 questions:

what will I do if I see rock?,

what will I do if I see paper?

What will I do if I see scissors?

There are 3 possible answers to each question. Hence there are 3x3x3=27 possible strategies.

Examples:

Paper if rock, rock if paper, rock if scissors

Or Rock if rock, scissors if paper, paper if scissors

And so on… 27 possibilities

Slide9

What is a strategy?

“A strategy is not a sequence of actions, but rather a catalog of contingency plans, what to do in every situation. ‘’ Harrington, page 34.

For games with perfect information, this must specify a player’s action at each node at which it would be that player’s turn.

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Information sets in imperfect information games

Information set—A collection of nodes such that you the player whose turn it is does not know which of these nodes he is currently at.

See Blackboard Examples:

Simultaneous move Rock, paper, scissors

Simultaneous move Matching pennies Slide11

Strategy in game with incomplete information

A strategy must specify a player’s action at each

information set

at which it would be that player’s turn.Slide12

Clicker question

How many (pure) strategies are available to each player in the simultaneous move matching pennies game?

2

4

6

8Slide13

Clicker question

How many strategies are available to each player in the simultaneous move rock-paper-scissors game?

2

3

6

9

27Slide14

Details of strategic form game

Set of PlayersFor each player a strategy set—list of all the strategies that the player could choose. Remember that a strategy tells everything you would do on any occasion when its your turn.

Strategy profile: List of strategies chosen by every player.

Payoff to each player depends on the strategy profile that was chosen.Slide15

Two player game matrix in strategic form

Make a two-by-two table with one row for each strategy that player 1 could choose and one column for every strategy that player 2 could choose.

Enter payoffs to players 1 and 2 in appropriate spots.Slide16

Example: Simultaneous Move Matching Pennies

In this case each player has only two possible strategies. Choose Heads, Choose tails.

Payoff to Player 1 (row chooser) is written first, then payoff to Player 2.Slide17

Simultaneous move Matching PenniesStrategic Form of Game

-1, 1

1,-1

1,-1

-1,1

Heads

Tails

Heads

Tails

Player 2

Player 1Slide18

Rock, Paper, Scissors—Simultaneous Move

0,0

-1, 1

1,-1

1,-1

0,0

-1,1

Rock

Paper

Scissors

Rock

Paper

ScissorsSlide19

More complicated game

Player 1

Player 1

Player 2

2

0

3

1

0

0

1

2

D

C

F

E

G

H

4 Possible Strategies for Player 1 :

What are they?

2 Possible Strategies for Player 2:

What are they?Slide20

Strategic Form

1, 2

3, 1

0, 0

3, 1

2, 0

2, 0

2, 0

2, 0

Player 2

E

F

Player 1

C,G

C,H

D,G

D,HSlide21

Prisoners’ Dilemma Game

10, 10

0,

11

11, 0

1, 1

Cooperate

Defect

Cooperate

Defect

P

LAyER

1

Player 2Slide22

Two rounds of Prisoners’ Dilemma

Players 1 and 2 play two rounds of prisoners’ dilemma.Draw the game tree with information sets if they move simultaneously in each round, but can see results of round 1 before starting round 2.

Now draw the game tree with information sets if they play two rounds but only see other’s plays at the end of the game.Slide23

Clicker Question

Players A and B play two rounds of simultaneous move prisoners’ dilemma.

They don’t get to see how the other player played until both rounds are over.

How many strategies are possible for each player?

2

4

8

16

32Slide24

Clicker Question

Players A and B play two rounds of simultaneous move prisoners’ dilemma. Each gets to see the other’s move in round 1 before choosing an action for round 2. How many strategies are possible for each player?

2

3

4

16

32Slide25

The game of Chicken

James Dean story.Alternatively—Two animals both want a resource. Each has two possible strategies.

Fight or give up. A fight is very bad for both of them. How do we make an interesting game of this?Slide26

0 , 0

0 , 1

1, 0

-10, -10

Swerve

Swerve

Don’t Swerve

Don’t SwerveSlide27

So long…at least for now

.Slide28

Common Knowledge of a fact

Three ladies in a railway car. All have dirty faces. They can see each other’s faces, but not their own. Each would blush visibly if she knew her own face was dirty.

All are brilliant logicians and they all know this.

The conductor comes into the car and announces for all to hear.

“Someone in this car has a dirty face.”Slide29

Common Knowledge

Why should this news matter

?

All three can see two dirty faces.

In fact, all three know that the others can see at least one dirty face.

Lady 1 says, Suppose that my face is clean. Then Lady 2 will see exactly one dirty face—that of Lady 3. Lady 2 will reason, “if my own face is clean, then Lady 3 will see 2 clean faces.” If Lady 3 saw 2 clean faces, she would know her face was dirty and would blush.

If Lady 3 doesn’t blush, lady 2 would conclude that her own face is dirty and would blush. Therefore if Ladies 2 and 3 don’t blush, Lady 1 must conclude that her own face if dirty.

Slide30

Moral of the story

In this example, all of the ladies know that there is a dirty face in the car, but until the conductor comes, none of them know that the others all know that there is a dirty face.

This additional bit of common knowledge leads them to a conclusion otherwise unavailable.