ECO2007S
COOPERATION & COMPETITION
MODULE 2.1: CHAPTER 4
Rinelle Chetty
Rinelle.Chetty@uct.ac.za
Consultation Hours:
Tuesdays 12:00 – 14:00
Economics Satellite Office, Leslie Social, 3rd Floor
G a m e ta b l e
Introduction
SMGs with
d i s c re te
st rate g i e s
CHAPTER 4:
SIMULTANEOUSMOVE GAMES
S t r i c t l y b e tte r
v s J o i nt l y b e st
B e st re s p o n s e
Nash
Eq u i l i b r i u m
S i m u l ta n e o u s
m o ve s
B e l i e fs
INTRODUCTION
What are simultaneous move games?
Simultaneous move games occur when
• Players move with no knowledge of what their opponent has chosen
to do
• Players choose their action at the exact same time
• Players make choices in isolation
∴ These games have imperfect information
INTRODUCTION
Examples of simultaneous move games?
SIMULTANEOUS MOVE GAMES
WITH DISCRETE STRATEGIES
GAME TABLE
• Simultaneous move games are depicted with a
“game table” or “game matrix” or “payoff table”
• Each cell represents the payoffs for each player, for each outcome, under
the strategies
Row player’s
payoffs always
first
Player 1 has
4 strategies
Player 2 has
3 strategies
Column player’s
payoffs always
second
Player 1
Player 2
A
B
C
W
2,1
4,9
7,1
X
6,6
0,0
7,4
Y
4,2
2,7
2,3
Z
8,3
4,6
1,3
2 dimensions
for 2 players
GAME TABLE: EXAMPLE
• How many players are in this game table?
• Who is player 1 and who is player 2?
• How many strategies does each player have?
Player 2
Harry
Player 1
Ron
Stun
Attack
Run
Stun
1,1
4,6
5,1
Attack
6,4
0,0
7,1
Run
1,5
1,7
2,2
GAME TABLE
• The game table is called the normal form or strategic form of the game
• It can include any number of strategies, as long as it is reflected in the
dimensions of the table
Player 2
Harry
Player 1
Ron
Stun
Attack
Run
Stun
1,1
4,6
5,1
Attack
6,4
0,0
7,1
Run
1,5
1,7
2,2
Hide
2,4
2,3
2,2
GAME TABLE
• The number of rows (columns) relates to the number of strategies
available to the row (column) player
• How many strategies does each player now have available to them?
Harry
Ron
Stun
Attack
Run
Stun
1,1
4,6
5,1
Attack
6,4
0,0
7,1
Run
1,5
1,7
2,2
Hide
2,4
2,3
2,2
GAME TABLE: PAYOFFS
• Each cell represents the payoff to a player if he/she had taken that
strategy
• For example, suppose Ron chose to Stun and Harry chose to Attack
• Then Ron would receive a payoff of 4
• And Harry would receive a payoff of 6
Harry
Ron
Stun
Attack
Run
Stun
1,1
4,6
5,1
Attack
6,4
0,0
7,1
Run
1,5
1,7
2,2
Hide
2,4
2,3
2,2
GAME TABLE: ZERO-SUM GAME
• Zero-sum game: payoffs of one player are the opposite of payoffs of the
other player
• So we need only show the payoffs of one player (usually the first player)
Draco
Hermoine
Expelliarmus
Stupify
Levicorpus
5 , -5
4 , -4
Reducto
3 , -3
1 , -1
Draco
Expelliarmus
Hermoine
Levicorpus
Reducto
Stupify
NASH EQUILIBRIUM
NASH EQUILIBRIUM
A list of strategies, one for each player,
such that no player can get a better payoff
by switching to some other strategy that is available
while the other players adhere to the strategies specified for
them
BEST RESPONSE
• Best response is a method to solve for the Nash equilibrium
• It’s a comprehensive way of locating all possible NE of a game
• It’s also one of the easiest methods to employ, especially with games of
more than two players
BEST RESPONSE – THE LOGIC
• Suppose Player 1 chooses the Low strategy. What should Player 2 choose?
• Middle is Player 2’s best response to Player 1 choosing Low
• Suppose Player 2 chooses Middle. What should Player 1 choose?
• Low is Player 1’s best response to Player 2 choosing Middle
• So choices Low and Middle are each the chooser’s best response to the other’s
action
Player 2
NE =
NEO =
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
4,5
9,7
BEST RESPONSE ANALYSIS
• Involves circling a player’s best response to the other player’s strategies
• Starting with P1:
• If P2 chooses Left, what would P1’s best response be?”
• If P2 chooses Middle, what would P1’s best response be?”
• If P2 chooses Right, what would P1’s best response be?”
Player 2
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
4,5
9,7
BEST RESPONSE ANALYSIS
• We then apply the same thinking to P2:
• If P1 chooses Top, what would P2’s best response be?”
• If P1 chooses High, what would P2’s best response be?”
• If P1 chooses Low, what would P2’s best response be?”
• If P1 chooses Bottom, what would P2’s best response be?”
Player 2
NE =
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
4,5
9,7
BEST RESPONSE ANALYSIS
• What if there is a tie in payoffs?
• For example, if P1 chooses Low, P2 is best off receiving a payoff of 3 – which
occurs when P2 chooses both Left AND Right
• And if P2 chooses Left, P1 would want to receive the highest payoff of 5 –
which occurs when P1 chooses both High AND Bottom
• And if P2 chooses Middle, P1 would choose the highest possible payoff of 6
– which occurs when P1 chooses Top, Low, AND Bottom
Player 2
Player 1
Left
Middle
Right
Top
3,4
6,1
10 , 2
High
5,2
3,0
6,4
Low
2,3
6,2
12 , 3
Bottom
5,6
6,5
9,7
PRACTICE QUESTION 1
Which of the following can be a Nash Equilibrium?
A. (High, Left)
B. (Bottom, Left)
C. (Low, Middle)
D. All of the above
Player 2
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
4,5
9,7
STRICTLY BETTER NE
• Nash Equilibrium does not require choices to be strictly better than
other choices
• It is still true that given Player 2’s choice of Middle, Player 1 cannot do
any better that they do when choosing Low
• It is therefore still a Nash Equilibrium
Player 2
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
5,5
9,7
PRACTICE QUESTION 2
Is (Bottom, Middle) a NE?
A. Yes
B. No
Player 2
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
5,5
9,7
NE NOT JOINTLY BEST
• NE does not have to be jointly best for the players
• For example, compared to payoffs (5 , 4) under NE, (Bottom, Right) gives
payoffs (9 , 7) that are better for both players
• But choice (Bottom, Right) cannot be sustained
• If Player 2 chooses Right, then Player 1 is better off choosing Low (with
payoffs 12) than Bottom (with payoffs 9)
Player 2
Player 1
Left
Middle
Right
Top
3,1
2,3
10 , 2
High
4,5
3,0
6,4
Low
2,2
5,4
12 , 3
Bottom
5,6
5,5
9,7
SIMULTANEOUS (BUT
UNOBSERVABLE) MOVES
• Recall: in NE, players choose their best response to each other’s choice
• But each choice is made simultaneously
• So how does one respond to decisions that are unknown and cannot be
observed?
o Guess
o Use experience and observation
o Logical reasoning (putting yourself in their shoes)
BELIEFS
• Each player has beliefs about the other player’s actual choices
• Which comes with some uncertainty: uncertainty about what action the
other player is taking.
• So players form subjective views or estimates about the other player’s
actions
• In NE, those beliefs must be correct
• So we can then define NE as:
A set of strategies, one for each player, such that:
(1) each player has correct beliefs about the strategies of the others, and
(2) the strategy of each is the best for him/herself, given his/her beliefs
about the strategies of the others
PRACTICE QUESTION 3
Solve for the Nash Equilibrium of the game below
A. (B , E)
B. (C , G)
C. (D , F)
D. (B , F)
Summer
Seth
E
F
G
A
2 , -2
5 , -5
13 , -13
B
6 , -6
5.5 , -5.5
10.5 , -10.5
C
6 , -6
4.5 , -4.5
1 , -1
D
10 , -10
3 , -3
-2 , 2