Introduction to game theory

An introduction to game
theory
Today: The fundamentals of
game theory, including Nash
equilibrium

Today
 Introduction to game theory
 We can look at market situations with two
players (typically firms)
 Although we will look at situations where
each player can make only one of two
decisions, theory easily extends to three or
more decisions

John Nash, the person portrayed
in “A Beautiful Mind”

John Nash
 One of the early
researchers in
game theory
 His work resulted
in a form of
equilibrium
named after him

Three elements in every game
 Players
 Two or more for most games that are
interesting
 Strategies available to each player
 Payoffs
 Based on your decision(s) and the
decision(s) of other(s)

Game theory: Payoff matrix
 A payoff
matrix
shows the
payout to
each
player,
given the
decision of
each
player
Action C Action D
Action A 10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

How do we interpret this box?
 The first number in
each box determines
the payout for
Person 1
 The second number
determines the
payout for Person 2
Action
C
Action
D
Action
A
10, 2 8, 3
Action
B
12, 4 10, 1
Person
1
Person 2

How do we interpret this box?
 Example
 If Person 1
chooses Action A
and Person 2
chooses Action D,
then Person 1
receives a payout
of 8 and Person 2
receives a payout
of 3
Action
C
Action
D
Action
A
10, 2 8, 3
Action
B
12, 4 10, 1
Person
1
Person 2

Back to a Core Principle:
Equilibrium
 The type of equilibrium we are looking
for here is called Nash equilibrium
 Nash equilibrium: “Any combination of
strategies in which each player’s strategy
is his or her best choice, given the other
players’ choices” (F/B p. 322)
 Exactly one person deviating from a NE
strategy would result in the same payout or
lower payout for that person

How do we find Nash
equilibrium (NE)?
 Step 1: Pretend you are one of the players
 Step 2: Assume that your “opponent” picks a
particular action
 Step 3: Determine your best strategy (strategies),
given your opponent’s action
 Underline any best choice in the payoff matrix
 Step 4: Repeat Steps 2 & 3 for any other opponent
strategies
 Step 5: Repeat Steps 1 through 4 for the other
player
 Step 6: Any entry with all numbers underlined is NE

Steps 1 and 2
 Assume that
you are
Person 1
 Given that
Person 2
chooses
Action C,
what is
Person 1’s
best
choice?
Action
C
Action D
Action
A
10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

Step 3:
 Underline
best payout,
given the
choice of the
other player
 Choose
Action B,
since
12 > 10 
underline 12
Action
C
Action D
Action
A
10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

Step 4
 Now
assume
that Person
2 chooses
Action D
 Here,
10 > 8 
Choose
and
underline
10
Action C Action
D
Action A 10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

Step 5
 Now,
assume you
are Person 2
 If Person 1
chooses A
 3 > 2 
underline 3
 If Person 1
chooses B
 4 > 1 
underline 4
Action C Action D
Action
A
10, 2 8, 3
Action
B
12, 4 10, 1
Person
1
Person 2

Step 6
 Which
box(es) have
underlines
under both
numbers?
 Person 1
chooses B
and Person
2 chooses
C
 This is the
only NE
Action C Action D
Action A 10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

Double check our NE
 What if
Person 1
deviates
from NE?
 Could
choose A
and get 10
 Person 1’s
payout is
lower by
deviating 
Action C Action D
Action A 10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

Double check our NE
 What if
Person 2
deviates
from NE?
 Could
choose D
and get 1
 Person 2’s
payout is
lower by
deviating 
Action C Action D
Action A 10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

Dominant strategy
 A strategy is
dominant if that
choice is
definitely made
no matter what
the other
person chooses
 Example:
Person 1 has a
dominant
strategy of
choosing B
Action C Action D
Action A 10, 2 8, 3
Action B 12, 4 10, 1
Person
1
Person 2

New example
 Suppose in
this example
that two
people are
simultaneousl
y going to
decide on this
game
Yes No
Yes 20, 20 5, 10
No 10, 5 10, 10
Person
1
Person 2

New example
 We will go
through the
same steps to
determine NE
Yes No
Yes 20, 20 5, 10
No 10, 5 10, 10
Person
1
Person 2

Two NE possible
 (Yes, Yes) and
(No, No) are
both NE
 Although (Yes,
Yes) is the
more efficient
outcome, we
have no way to
predict which
outcome will
actually occur
Yes No
Yes 20, 20 5, 10
No 10, 5 10, 10
Person
1
Person 2

Two NE possible
 When there are multiple NE that are
possible, economic theory tells us little
about which outcome occurs with
certainty

Two NE possible
 Additional information or actions may
help to determine outcome
 If people could act sequentially instead of
simultaneously, we could see that 20, 20
would occur in equilibrium

Sequential decisions
 Suppose that decisions can be made
sequentially
 We can work backwards to determine
how people will behave
 We will examine the last decision first and
then work toward the first decision
 To do this, we will use a decision tree

Decision tree in a sequential
game: Person 1 chooses first
A
B
C
Person
1
chooses
yes
Person
1
chooses
no
Person 2
chooses
yes
Person 2
chooses
yes
Person 2
chooses no
Person 2
chooses no
20, 20
5, 10
10, 5
10, 10

 Given point B,
Person 2 will
choose yes
(20 > 10)
 Given point C,
Person 2 will
choose no
(10 > 5)
A
B
C
Person
1
chooses
yes
Person
1
chooses
no
Person 2
chooses
yes
Person 2
chooses
yes
Person 2
chooses no
Person 2
chooses
no
20, 20
5, 10
10, 5
10, 10

 If Person 1 is
rational, she will
ignore potential
choices that
Person 2 will not
make
 Example: Person
2 will not choose
yes after Person 1
chooses no
A
B
C
Person
1
chooses
yes
Person
1
chooses
no
Person 2
chooses
yes
Person 2
chooses
yes
Person 2
chooses no
Person 2
chooses
no
20, 20
5, 10
10, 5
10, 10

 If Person 1 knows
that Person 2 is
rational, then she
will choose yes,
since 20 > 10
 Person 2 makes a
decision from point
B, and he will
choose yes also
 Payout: (20, 20)
A
B
C
Person
1
chooses
yes
Person
1
chooses
no
Person 2
chooses
yes
Person 2
chooses
yes
Person 2
chooses no
Person 2
chooses
no
20, 20
5, 10
10, 5
10, 10

Summary
 Game theory
 Simultaneous decisions  NE
 Sequential decisions  Some NE may not
occur if people are rational

Can you think of ways game theory
can be used in these games?

Introduction to game theory

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