# Lecture 7 Zero Sum Games HarvardFall 2018(1)

```ECON E-1040: GAME THEORY &
STRATEGIC GAMES
Lecture #7 Zero-Sum Games
Marion Laboure
Fall 2018
Games
with incomplete (asymmetric)
information
with complete (symmetric)
information
one-shot
games
sequential-move games
repeated
games
simultaneous-move
games
simultaneous-move games
sequential-move
games
simultaneous & sequential-moves
Nash Equilibrium
2
use
Do all players have a dominant strategy?
NO
Nash Equilibrium
is the
Nash Equilibrium
YES
Dominant strategy Equilibrium
use
Do players have strictly dominated
strategies?
YES
NO
Nash Equilibrium
if you find an equilibrium
Iteratively eliminate SDS
it is the
Nash Equilibrium
use
Do payoffs of both players add to the
same constant in all outcomes?
NO
Nash Equilibrium
finds the
Nash Equilibrium
YES
Minimax Method
3
Outline
Constant-sum / zero-sum games
Minimax method
Midterm Exam & Revision
4
Von Neumann and Morgenstern
John Von Neumann (19031957)
Oskar Morgenstern (19021977)
Princeton University Press,
1944
5
Games of pure conflict (strictly competitive games)
Two-player constant-sum game: players’ payoff always add
up to a constant.
Examples?
Zero-sum game: a special case of constant-sum game in
which players’ payoff add to zero.
In other words, payoffs of one player is the negative of the
payoffs of the other player.
6
Maximin strategies (a.k.a. playing cautiously)
You(Player1)areplayingazero-sumgamewithPlayer2.
yourstrategy).
Whatisthebestthatyoucando?
Remember,thisisazero-sumgame,soPlayer2willtryto
minimize yourpayoff(i.e.,willtrytomaximize his).
7
Example: Penalty kicks
“The strongest shooters can
kick at speeds of up to 80 mph
[~129 kmph]. This means that
the ball reaches the goal line in
500 milliseconds.”
Goalkeeper
Left
Right
0.2 0.8
0.7 0.3
1
0.5 0.5
Kicker
popularmechanics.com
Left
Right
0
prob. of scoring a goal + prob. of not scoring a goal = 1
A constant-sum game (payoffs add to 1)
8
Example: Penalty kicks
Transforming a
constant-sum game to
a zero-sum game
Goalkeeper
Kicker
Left
Left
Right
Right
0.2
0.7
1
0.5
Payoffs in each cell add to zero.
9
Example: Penalty kicks
Goalkeeper
Kicker
Transforming a
constant-sum game to
a zero-sum game
Left
Right
Left
Right
0.2 -0.2
0.7 -0.7
1
0.5 -0.5
-1
Payoffs in each cell add to zero.
Convention: write only Player 1’s (row player) payoffs
10
Example: Penalty kicks
Goalkeeper
Left
Right
0.2 -0.2
0.7 -0.7
1
0.5 -0.5
Kicker
Kicker’s perspective
Left
Right
-1
play his best-response.
He will choose the strategy that maximizes his payoff (which minimizes yours).
Play cautiously, and get the “better of the worse.”
11
Example: Penalty kicks
Goalkeeper
Left
Right
0.2 -0.2
0.7 -0.7
min= 0.2
1
0.5 -0.5
min= 0.5
Kicker
Kicker’s perspective
Left
Right
-1
Maximin
What is the minimum you can get from each strategy?
Which of the minima is better?
To guarantee yourself at least 0.5, you should play Right
12
Example: Penalty kicks
Goalkeeper’s perspective
Goalkeeper
Kicker
What is the minimum you can
get from each strategy?
Left
Right
Which of the minima is better?
Left
Right
0.2 -0.2
0.7 -0.7
1
-1
0.5 -0.5
min= -1
min= -0.7
Maximin
To guarantee yourself at least -0.7, you should play Right
13
Example: Penalty kicks
Kicker
Goalkeeper
Left
Right
Left
Right
0.2 -0.2
0.7 -0.7
1
0.5 -0.5
-1
This method fails to find the equilibrium
(0.5 and -0.7 are in different cells);
14
Pure-strategy Nash equilibrium of this game
A.
B.
C.
D.
E.
F.
G.
(Left, Left)
(Right, Right)
(Left, Right)
(Right, Left)
(Left, Left) and (Right, Right)
(Left, Right) and (Right, Left)
No pure-strategy Nash Equilibrium
15
Pure-strategy Nash equilibrium of this game
A.
B.
C.
D.
E.
F.
G.
(Left, Left)
(Right, Right)
(Left, Right)
(Right, Left)
(Left, Left) and (Right, Right)
(Left, Right) and (Right, Left)
No pure-strategy Nash Equilibrium
16
Why “Minimax method”?
Kicker
Goalkeeper
Left
Right
Left
Right
0.2 -0.2
0.7 -0.7
1
0.5 -0.5
-1
17
Why “Minimax method”?
Goalkeeper
Kicker
Left
Minimax
Left
Right
max=
0.2
1
Right
Minimax
0.7
Maximin
0.5
Maximin
min=
min=
max=
Minimax of the kicker is the Maximin of the goalkeeper
18
“Zero-sum games cannot have a Nash
equilibrium in pure strategies.”
A. True
B. False
19
“Zero-sum games cannot have a Nash
equilibrium in pure strategies.”
A. True
B. False
20
Can zero-sum games have equilibrium in
pure strategies?
Kicker
A player with a very strong
natural side
(Payoffs: scoring prob. in %)
Goalkeeper
Left
Right
Left
38
65
Right
93
70
Maximin
Minimax
Right is dominant strategy for the kicker.
N.E. in pure strategies (Right, Right)
21
Summing up
Toolkit:
Minimax method to find equilibrium in two-person zero-sum
games.
Minimax method will lead to a Nash Equilibrium in pure
strategies (provided that it exists).
This is true only for two-person zero-sum games.
22
Review for Midterm exam
23
Games
withincomplete(asymmetric)
information
withcomplete(symmetric)
information
one-shot
games
sequential-movegames
repeated
games
simultaneous-move
games
simultaneous-movegames
sequential-movegames
simultaneous&sequential-moves
24
1. Private Incentives to Control Pollution [30 points]
Review for Midterm exam
Suppose that a certain society consists of only two people and that each of these people
drives a car. Suppose that each person could voluntarily choose to put a pollution control
device on his or her car. Installing such a device on one car costs \$100 and provides a
benefit (in the form of cleaner air) that is worth \$70 to each of the two people (both people
benefit since both people breathe the cleaner air).
In Each
the following
find
allpossible
strictlyactions
dominant
strictly
person in thisgames,
society has
two
– Installand
or Not
Install.dominated
Assuming thispure
strategies.
Then solve one-time
the game.
game is a simultaneous,
game, draw a payoff matrix showing the net (dollar)
benefit (or cost) to each of the two people for all four possible outcomes of the game. How
would
describe this game?
Find
theyou
equilibrium.
2. Dominant and Dominated Strategies [15 points]
In the following games, find all strictly dominant and strictly dominated pure strategies.
(a)
Player 2
Player 1
Up
Straight
Down
Left
0,1
5,9
7,5
Middle
9,0
7,3
10 , 10
Right
2,3
1,7
3,5
(b)
Player 2
North
Straight Up
West
2,3
3,0
Center
8,2
4,5
East
10 , 6
6,4
25
device on his or her car. Installing such a device on one car costs \$100 and provides a
benefit (in the form of cleaner air) that is worth \$70 to each of the two people (both people
benefit since both people breathe the cleaner air).
Review for Midterm exam
Each person in this society has two possible actions – Install or Not Install. Assuming this
game is a simultaneous, one-time game, draw a payoff matrix showing the net (dollar)
benefit (or cost) to each of the two people for all four possible outcomes of the game. How
you describe
this game?
In would
the following
games,
find all strictly dominant and strictly dominated pure
strategies.
2. Dominant and Dominated Strategies [15 points]
In the
following
games, find all strictly dominant and strictly dominated pure strategies.
Find
the
equilibrium.
(a)
Player 2
Player 1
Up
Straight
Down
Left
0,1
5,9
7,5
Middle
9,0
7,3
10 , 10
Right
2,3
1,7
3,5
(b)
Player 1 has a dominant strategy of “Down.” For
player
Player
2 1 the strategies
“Straight” and “Up” are dominated
by “Down.”Center
West
East
North
2,3
8,2
10 , 6
Player 2 has neitherStraight
a strictly
Up dominant
3 , 0 nor a strictly
4 , 5 dominated6 ,strategy.
4
Player 1
Down
5,4
6,1
2,5
South
4,5
2,3
5,2
26
benefit (or cost) to each of the two people for all four possible outcomes of the game. How
would you describe this game?
Review for Midterm exam
2. Dominant and Dominated Strategies [15 points]
In the following games, find all strictly dominant and strictly dominated pure strategies.
(a)
In the following games, find all strictly dominant and strictly dominated pure
Player 2
strategies.
Find the equilibrium.
Player 1
Left
0,1
5,9
7,5
Up
Straight
Down
Middle
9,0
7,3
10 , 10
Right
2,3
1,7
3,5
(b)
Player 2
Player 1
West
2,3
3,0
5,4
4,5
North
Straight Up
Down
South
Center
8,2
4,5
6,1
2,3
East
10 , 6
6,4
2,5
5,2
(c)
Player 2
A
A
5,5
B
0,6
27
would you describe this game?
Review for Midterm exam
2. Dominant and Dominated Strategies [15 points]
In the following games, find all strictly dominant and strictly dominated pure strategies.
(a)
In the following games, find all strictly dominant Player
and strictly
dominated pure
2
strategies.
Find
the1equilibrium.
Player
Left
0,1
5,9
7,5
Up
Straight
Down
Middle
9,0
7,3
10 , 10
Right
2,3
1,7
3,5
(b)
Player 2
West
2,3
3,0
5,4
4,5
North
Straight Up
Down
South
Player 1
Center
8,2
4,5
6,1
2,3
East
10 , 6
6,4
2,5
5,2
(c)
There are no strictly dominant strategies or strictly dominated strategies,
in pure strategies.
Player 2
Player 1
A
B
A
5,5
8,4
B
0,6
3,1
28
Player 1
Up
Straight
Down
0,1
5,9
7,5
9,0
7,3
10 , 10
Review for Midterm exam
2,3
1,7
3,5
(b)
Player 2
In the following games, find all strictly dominant and strictly dominated pure
West
Center
East
strategies.
North
2,3
8,2
10 , 6
Straight Up
3,0
4,5
6,4
Find
the1 equilibrium.
Player
Down
5,4
6,1
2,5
South
4,5
2,3
5,2
(c)
Player 2
Player 1
A
B
C
A
5,5
8,4
4,5
B
0,6
3,1
5,3
29
Down
(b)
7,5
10 , 10
3,5
Review for Midterm exam
Player 2
West
Center
East
North
2,3
8 , 2 dominated 10
,6
In the following games, find
all strictly dominant
and strictly
pure
Straight Up
3,0
4,5
6,4
strategies.
Player
1
Down
5,4
6,1
2,5
South
4,5
2,3
5,2
Find the equilibrium.
(c)
Player 2
Player 1
A
B
C
A
5,5
8,4
4,5
B
0,6
3,1
5,3
There are no strictly dominant strategies.
For Player 1 “A” is strictly dominated by “B.”
30
Review for Midterm exam
“If a player has a strictly dominant strategy in a
simultaneous-move game, then she is sure to get
her best possible outcome.” True, or false?
Explain, and give an example of a game that
31
Review for midterm exam
“In a sequential-move game, the player who
moves first is sure to win.” Is this statement true or
brief sentences, and give an example of a game
32
Review for midterm exam
33
Review for Midterm exam: Sequential-Moves
Game
34
Review for Midterm exam: Sequential-Moves
Game
35
Review for Midterm exam: Sequential-Moves
Game
36
Review for Midterm exam: Sequential-Moves
Game
37
Review for Midterm exam: Sequential-Moves
Game
38
Review for Midterm exam: Sequential-Moves
Game
39
Review for Midterm exam: Sequential-Moves
Game
40
Review for Midterm exam: Sequential-Moves
Game
41
Review for Midterm exam: Sequential-Moves
Game
42
Review for Midterm exam: Sequential-Moves
Game
43
Review for Midterm exam: Sequential-Moves
Game
44
Review for Midterm exam: Sequential-Moves
Game
45
Review for Midterm exam: Sequential-Moves
Game
46
Review for Midterm exam: Sequential-Moves
Game
47
Review for Midterm exam: Sequential-Moves
Game
48
```