# Lecture 2 (Extensive and Strategic form representations)

```Extensive and Strategic Form
Games
Econ 171
Reminder: Course requirements
• Class website Go to economics department
then click on Econ 171
• Textbook: Games, Strategies, and Decision
Making by Joseph E. Harrington, Jr.
• Clicker: Available at campus bookstore
www.i&gt;clicker.com/registration
Vocabulary for Extensive form games
•
•
•
•
•
Decision Tree
Decision Node-Specifies whose turn
Branches-Options
Terminal Node—End of play
Payoffs—For each person at each terminal
node.
• Strategy—What will you do at each decision
node where it is your turn
Rock, Paper, Scissors
Rock-Paper-Scissors: Full information
version, Alice moves first
Alice
Scissors
Rock
Paper
Bob
Rock
0
0
Bob
Bob
Rock
Scissors Paper
Paper
Rock
Scissors
Paper
0
1
1 1
0 0
0
0
0
1
0
1
1
0
Scissors
0
0
Clicker Question
How many strategies are possible for Bob
in the perfect information version of RockPaper-Scissors when Alice moves first
A) 3
B) 6
C) 9
D) 12
E) 27
What is a strategy in a game of perfect
information?
• “A strategy is not a sequence of actions, but
rather a catalog of contingency plans, what to
do in every situation. ‘’ Harrington, page 34.
(Read this section with extra care.)
• A strategy is a list stating what you would do
at each possible decision node where it is your
turn.
Strategies for B in perfect information
rock, paper, scissors game
• A strategy for B in perfect rock, paper, scissors
– 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
A tougher game
Kidnapping Game
Vivica’s strategies
• How many strategies are possible for Vivica?
A) 2
B) 3
C) 4
D) 8
E) 9
Guy’s Nodes
• At how many decision nodes is it Guy’s turn?
A) 2
B) 3
C) 4
D) 5
E) 6
Clicker Question
• How many strategies are possible for Guy?
A) 3
B) 4
C) 6
D) 8
E) 16
Come Again? Why is that
• He has 3 nodes. He can do 2 things at each of
them. That give us 2x2x2=8 strategies.
Kidnap, Don’t Kill if R , Kill if No R
Kidnap, Don’t Kill if R, Don’t Kill if No R
Kidnap, Kill if R, Kill if No R
Kidnap, Kill if R, Don’t Kill if No \$
But also,
Don’t Kidnap, Don’t Kill if R, Kill if No R,
etc.
Game theorist’s convention
• A strategy lists what you would do at EVERY
decision node where it is your turn, even if
some things you can choose at an early node
prevent you reaching a later node.
• This is partly for convenience, but also it turns
out to be useful to specify what you would do
if you accidentally wound up at any node.
Sequences of Action are not strategies
• How many sequences of action are possible
for Guy?
A) 3
B) 4
C) 6
D) 8
E) 16
What are the sequences of actions
• Kidnap, Kill
• Kidnap, Don’t Kill
• Don’t Kidnap
Don’t confuse strategies with sequences of
actions. Strategies tell us what you would do in
any sequence of events. Historical sequences
tell you only what did happen.
Strategies and imperfect Information
Imperfect information
• The examples so far had perfect information.
• When it is your turn, you know everything
that has happened so far.
• Sometimes, when it is your turn, you don’t
know everything that has happened.
• Example: Real rock-paper scissors game has
simultaneous moves.
Information sets
• We deal with this by introducing the concept
of information set.
• When a player doesn’t know which node of
the game she is at, we include all of all the
nodes where she might be in a single
information set.
Rock-Paper-Scissors when
players don’t know other’s move
Alice
Scissors
Rock
Paper
Bob
Rock
0
0
Rock
Scissors Paper
Paper
Rock
Scissors
Paper
0
1
1 1
0 0
0
0
0
1
0
1
1
0
Scissors
0
0
Problem 3 a
Player 1
b
a
Player 2
Player 2
x
Player 3
High
Low
c
y
Player 3
High Low
x
y
Player2
x
y
Problem 3 b
b
a
Player 2
Player 2
x
y
Player 3
High
Low
c
High Low
x
y
Player2
x
y
Problem 3 c
Player 1
b
a
Player 2
Player 2
x
c
y
x
Player2
y
x
Player 3
Player 3
High
Low
High
High
Low
Low High
Low
y
What is a strategy in a game of
imperfect information?
• Recall that In a game of perfect information, a
strategy is a list stating what you would do at
each possible decision node where it is your
turn.
• In a game of imperfect information, a strategy
is a list stating what you would do at each
possible information set where it is your turn.
How many strategies does Bob have in
simultaneous-move rock-paper-scissors?
A)
B)
C)
D)
E)
3
6
9
27
4
Games in Strategic Form
Details of strategic form game
• Set of Players
• For 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.
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.
Example: Simultaneous Move
Matching Pennies
• In this case each player has only two possible
• Payoff to Player 1 (row chooser) is written
first, then payoff to Player 2.
Matching Pennies
Strategic Form of Game
Player 2
Player 1
Tails
Tails
-1, 1
1,-1
1,-1
-1,1
Rock, Paper, Scissors—
Simultaneous Move
Rock
Rock
Paper
Scissors
0,0
Paper
-1,1
Scissors
1,-1
More complicated game
Player 1
C
D
Player 2
Player 1
G
1
2
E
F
3
1
H
2
0
0
0
4 Possible Strategies for Player 1 :
What are they?
2 Possible Strategies for Player 2:
What are they?
Strategic Form
Player 2
E
Player 1
F
C,G
1, 2
3, 1
C,H
0, 0
3, 1
2, 0
2, 0
2, 0
2, 0
D,G
D,H
The game of Chicken
James Dean story. Two macho morons drive
Swerve or don’t swerve.
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.
Chicken a la Extensive Form
Player 1
Don’t Swerve
Swerve
Player 2
Swerve
0
0
Swerve
Don’t Swerve
0
1
1
0
Don’t Swerve
-10
-10
Strategic form of Chicken Game
Swerve
Swerve
Don’t Swerve
0, 0
Don’t Swerve
0 , 1
1, 0 -10, -10
Player 1
Problem 9
a
d
c
b
Player 2
Player 2
x
4
2
Player 1
x
0
6
3
1
1
4
5
2
Player 2
x/x
x/y
y/x
y/y
a
4,2
4,2
1,3
1,3
b
2,2
2,2
0,6
0,6
c
3,1
4,2
3,1
4,2
d
?
?
?
?
y
x
1
3
2
2
x
y
y
y
0
0
So long…at least for now.
```