Chapter 14 – Game Theory
14.1 Nash Equilibrium
14.2 Repeated Prisoners’
Dilemma
14.3 Sequential-Move
Games and Strategic
Moves
1
Game Theory and Life
You are on a first date with the love of
your dreams. You can propose 2
activities:
1) Safe activity (Coffee)
2) Exciting Activity (Waterpark)
Your date could either want a safe
activity or an exciting activity. There
are different results if your ideas
match up or clash:
2
First Date Game
What is the outcome of this game?
Payoff format is (Left, Top)
You
Mr/Miss
Right
Coffee
Waterpark
Coffee
10,10
0,-5
Waterpark
-5,0
20,20
3
Chapter Fourteen
Game Theory Components
Players: agents participating in the game (You and Your
Date
Strategies: Actions that each player may take under any
possible circumstance (Coffee, Waterpark)
Outcomes: The various possible results of the game (four,
each represented by one cell of the payoff matrix)
Payoffs: The benefit that each player gets from each
possible outcome of the game (the profits entered in each
cell of the payoff matrix)
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Chapter Fourteen
Best Responses
In all game theory games, players choose
strategies without knowing with certainty
what the opposing player will do.
Players construct BEST RESPONSES
-optimal actions given all possible
actions of other players
5
First Date Game Best Responses
If you know your date will pick coffee, you should
pick coffee, since 10 > -5
If you know your date will pick waterpark, you
should pick waterpark, since 20 > 0
You
Mr/Miss
Right
Coffee
Waterpark
Coffee
10,10
0,-5
Waterpark
-5,0
20,20
6
Chapter Fourteen
First Date Game Best Responses
If your date knows you will pick coffee, they
should pick coffee, since 10 > -5
If your date knows you will pick waterpark, they
should pick waterpark, since 20 > 0
You
Mr/Miss
Right
Coffee
Waterpark
Coffee
10,10
0,-5
Waterpark
-5,0
20,20
Note that this
game is
SYMMETRICAL
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Chapter Fourteen
Nash Equilibrium
Definition: A Nash Equilibrium occurs when each player chooses
a strategy that gives him/her the highest payoff, given the
strategy chosen by the other player(s) in the game. ("rational
self-interest")
Nash Equilibria occur when best responses line up
The Date Game:
Nash equilibria: Each proposes coffee or each proposes
waterpark.
8
Chapter Fourteen
Game Theory
•A special kind of Best Response:
DOMINANT STRATEGY
•Strategy that is best no matter what
the other player does.
9
Advertising
B’s
STRATEGY
Don’t
advertise
Advertise
A’s
STRATEGY
Don’t advertise
Advertise
A’s profit=
$50 000
B’s profit =
$50 000
A’s loss =
$25 000
B’s profit =
$75 000
A’s profit=
$75 000
B’s loss =
$25 000
A’s profit =
$10 000
B’s profit =
$10 000
10
Dominant Strategy
B’s
dominant
strategy is
advertise
Don’t
advertise
Advertise
A’s dominant strategy is advertise
Don’t advertise
Advertise
A’s profit=
$50 000
B’s profit =
$50 000
A’s loss =
$25 000
B’s profit =
$75 000
A’s profit=
$75 000
B’s loss =
$25 000
A’s profit =
$10 000
B’s profit =
$10 000
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Prisoner’s Dilemma
• This is an example of a prisoner’s dilemma
type of game.
– There is dominant strategy.
– The dominant strategy does not result in the best
outcome for either player.
– It is hard to cooperate even when it would be
beneficial for both players to do so
– Cooperation between players is difficult to maintain
because cooperation is individually irrational.
• eg., The dominant strategy: advertise
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Classic Prisoners’ Dilemma
Rocky’s strategies
Deny
Deny
Ginger’s
strategies
Confess
1 year
Prison
1 year
Prison
7 years
Prison
Go free
Confess
Go free
7 years
Prison
5 years
Prison
Dominant
strategy:
confess, even
though they
would both be
better off if
they both
kept their
mouths shut.
5 years
Prison
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Dominant Strategy Equilibrium
Definition: A Dominant Strategy Equilibrium
occurs when each player uses a dominant strategy.
Toyota
Honda
Build a new
plant
Do not Build
Build a new
plant
16,16
20,15
Do not Build
15,20
18,18
Dominated Strategy
Definition: A player has a dominated strategy when
the player has another strategy that gives it a higher
payoff no matter what the other player does.
Toyota
Build a
Don’t
New Plant Build
Honda
Build a
12,4
New Plant
20,3
Don’t
Build
18,5
5,6
15
Chapter Fourteen
Dominant or Dominated Strategy
Why look for dominant or dominated strategies?
A dominant strategy equilibrium is particularly
compelling as a "likely" outcome
Similarly, because dominated strategies are unlikely to
be played, these strategies can be eliminated from
consideration in more complex games. This can make
solving the game easier.
16
Chapter Fourteen
Dominated Strategy
Toyota
Build
Large
Build
Small
Do Not
Build
Build Large 0,0
12,8
18,9
Build Small
8,12
16,16
20,15
Do Not
Build
9,18
15,20
18,18
Honda
"Build Large" is dominated for each player
By eliminating the dominated strategies, we can reduce 17
the game matrix.
Finding Nash Equilibrium Cases
1) Nash Equilibrium where Dominant
Strategies overlap
2) Nash Equilibrium with one Dominant
Strategy
3) Nash Equilibrium by eliminating Dominated
Strategy
4) Nash Equilibrium through Best Responses
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Chapter Fourteen
Nash Equilibrium – Dominant Overlap
Professor
Easy
Exam
Hard
Exam
No Exam
100,10
80,80
50,0
Don’t Study 50,5
30,60
20,0
Drop Out
50,30
0,20
Study
Student
30,10
19
Nash Equilibrium – One Dominant
Professor
Short
Exam
Long
Exam
No Exam
100,10
80,80
50,0
Don’t Study 50,40
30,10
20,0
Drop Out
50,30
0,20
Study
Student
30,10
20
Nash Equilibrium – Eliminate Dominated
Professor
Short
Exam
Long
Exam
Test Bank
100,10
80,80
50,0
Don’t Study 50,40
30,10
20,0
Cheat
0,30
0,0
Study
Student
0,10
21
Nash Equilibrium – Best Responses
Professor
Open
Book
Closed
Book
No Exam
20,10
80,80
50,0
Don’t Study 50,80
30,10
70,70
Drop Out
50, 0
0,20
Study
Student
10,10
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Nash Equilibrium
• However it is found, a Nash Equilibrium
ALWAYS occurs where Best Responses line up
• If Multiple Nash Equilibria exist, we can’t
conclude WHICH outcome will occur, only the
possible outcomes that can occur
• Also, it is often APPEARS that no Nash
Equilibria exist:
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No Nash Equilibrium
Fred
Rock
Paper
Scissors
Rock
0,0
-1 , 1
1, -1
Paper
-1 , 1
0, 0
-1, 1
Scissors
-1, 1
1, -1
0, 0
Barney
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Mixed Strategies
Pure Strategy – A specific choice of a strategy from
the player’s possible strategies in a game. (ie: Rock)
Mixed Strategy – A choice among two or more pure
strategies according to pre-specified probabilities.
(ie: Rock, Paper or Scissors each 1/3rd of the time)
If Pure Strategies can’t produce a Nash Equilibrium,
Mixed Strategies can:
If both players randomize each choice 1/3rd of the
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time, nether have an incentive to deviate.