Microeconomics II

Lecture 2: Game Theory

Mohammad Vesal
Graduate School of Management and Economics
Sharif University of Technology

44706
Spring 2025

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Motivation

• In Micro I, we considered behavior of agents in competitive

settings.
Many agents but no inter-related decisions,
except through prices at the market level.
• Game theory: a more general way of modeling multi-person
interaction.
Agents want to find the strategies that give highest payoffs.
• Strategic interdependence
Agent A’s payoffs not only depends on her decision
But on agent B’s decision

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Outline

Introduction

Basic elements

Simultaneous-move games of complete information

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Outline

Introduction

Basic elements
Extensive form representation
Normal (strategic) form representation

Simultaneous-move games of complete information

Reference: MWG Ch 7, Jehle-Reny Ch 7.

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Four elements of a game

Game: representation of a situation of strategic interaction for a

number of agents.
• Players: who is involved?
• Rules: Timing, information sets, possible alternatives
• Outcomes: What happens given a set of actions?
• Payoffs: Specify utilities over outcomes.
If random/uncertain use expected utility form.

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Example: matching pennies

• Two players: A and B

• Simultaneously put down pennies: heads up or tails up.
• Outcomes:
both heads/tails
unmatched coins
• Payoffs:
if matched B pays A one dollar.
if unmatched A pays B one dollar.

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Example: Cournot duopoly

Consider two car manufacturers: I and S each with an identical

car.
• Simultaneously pick what scale to produce at.
• More production by one firm lowers the market price for
both.
decreasing demand
• If production is qI and qS then market clearing price would
be p(qI + qS )
• With zero cost, profits would be
πI (qI , qs ) = p(qI + qS ) × qI and πS (qI , qs ) = p(qI + qS ) × qS

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Example: Entry

• Two players: incumbent I and entrant E

• I is the current producer in an industry
• E is a potential producer thinking about entry
• E decides whether to enter
• I decides whether to fight or accommodate
• Outcomes:
No entry
Entry and fight OR Entry and accommodation
• Payoffs:
Price war will result in zero profits for both.
Accommodation results in sharing of profits.

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What do players know about each other?

• We assume players are rational and know other players are

rational too.
They also know that other players know they are rational. . .
Summarize this as “Rationality is common knowledge”.
• Structure of the game is also common knowledge

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Outline

Introduction

Basic elements
Extensive form representation
Normal (strategic) form representation

Simultaneous-move games of complete information

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Example: sequential matching pennies

• Consider matching pennies as before except that player B

moves after observing A’s action.

• Rather boring game, could we represent the more

interesting version in a game tree?

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Information set
• Information set: sub-set of a player’s decision nodes.
When reached one node in a given information set, the
player does not know which point she is actually at!
represent all decision situations.
• We can now represent the original matching pennies in
extensive form.

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Restrictions on information sets

• Set of possible actions are the same at each node within a

given information set.
• Perfect recall: players don’t forget what they once knew.
• Do you think these restrictions are intuitive?
• Could you draw game trees that violate either of these
restrictions?
• A game is a perfect information game if each information
set contains a single decision node. Otherwise, it is an
imperfect information game.

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Extensive form representation

ΓE = {X , A, I, p(.), α(.), H, H(.), ι(.), ρ(.), u}

• A finite set of nodes (X ), players (I), and actions (A)

• p : X → {X ∪ ∅} specifying a single immediate predecessor
of node x ∈ X
initial node x0 , terminal nodes T , decision nodes D = X \T .
• α : X \{x0 } → A action that leads to node x
• All information sets: H
H : X → H assign each x ∈ D to an information set
Hi collection of information sets for player i
• Assign information sets to players: ι : H → {0, 1, . . . , I}
• How nature plays: ρ : H0 × A → [0, 1]
• ui (z): payoff for player i at terminal node z ∈ T

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Example: Entry

ΓE = {X , A, I, p(.), α(.), H, H(.), ι(.), u}

• X = {all the nodes}, A =
Entrant {In,Out,Fight,Accommodate},
I = {Entrant,Incumbent}
• p: reflects the succession of
Out In
nodes in the tree, α: actions
Incumbent that gives the succession
• H: each node is a separate
0
5 Fight Accommodate information set
• ι: says which player is deciding
−1 2 at which information set (here
−1 2 nodes)
• u: The vector of payoffs

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Example: Matching pennies

ΓE = {X , A, I, p(.), α(.), H, H(.), ι(.), u}

P1
0
• X = {0, 1, 2, 3, 4, 5, 6},
A = {Head, Tail}, I = {P1, P2}
Head Tail
• p: reflects the succession of
1 P2 2 nodes in the tree, α: actions
that gives the succession
Head Tail Head Tail • H = {{0}, {1, 2}}
3 4 
5  6 
• ι ({1, 2}) = P 2, ι ({0}) = P 1
−1 −1 • u: The vector of payoffs
+1 +1
−1 +1 +1 −1

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Outline

Introduction

Basic elements
Extensive form representation
Normal (strategic) form representation

Simultaneous-move games of complete information

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Strategy

• Strategy: a complete contingent plan that says what a

player will do in every possible distinguishable circumstance
that she might be asked to move.
Information sets specified all such circumstances.
Strategy is a function si : Hi → A
▶ Hi collection of all information sets for player i.
▶ A collection of all available actions.
▶ for any H ∈ Hi , si (H) ∈ C(H) ⊂ A, where C(H) is the set
of available actions at H.
Strategy profile: a collection of all players’ strategies
(s1 , . . . , sI )

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Example: Entry

Entrant
• Each decision node is an
information set
Out In
• Entrant’s strategies
0 Incumbent s1E : In
s2E : Out
5 Fight Accommodate
• Incumbent’s strategies
−1 2 s1I : Fight
s2I : Accommodate
−1 2

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Example: Sequential Matching pennies
• P1’s strategies: s1P 1 : Head,
s2P 1 : Tail
only 1 information set
0 P1 • P2’s strategies: 2 information
sets, a strategy must pick an
Head Tail
action at each
( set
1 P2 2 T if P1 T
s1P 2 =
T if P1 H
(
Head Tail Head Tail H if P1 T
s2P 2 =
H if P1 H
3 −14 
5  6  (
+1 −1 +1 H if P1 T
−1 +1 +1 −1 s3P 2 =
T if P1 H
(
T if P1 T
s4P 2 =
H if P1 H

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The normal form representation

• Any strategy profile s = (s1 , . . . , sI ) induces an outcome of

the game
sequence of the moves and probability distribution over the
terminal nodes
write down the payoff of each player for each strategy
profile.
• For a game with I players, the normal form representation
ΓN specifies for each player i a set of strategies Si and a
payoff function ui (s1 , · · · , sI ) giving the payoff arising from
the strategy profile (s1 , · · · , sI ).

ΓN = [I, {Si } , {ui (.)}]

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Example: Entry

Figure: Extensive form Table: Normal form

Entrant Inc.
Fight Acc.
In (-1,-1) (2,2)
Out In Ent.
Out (0,5) (0,5)
0 Incumbent

5 Fight Accommodate

−1 2
−1 2

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Example: Matching pennies

• Normal form representation of matching pennies

Player B
Head Tail
Head (1,-1) (-1,1)
Player A
Tail (-1,1) (1,-1)

• What is the payoff associated with the strategy profile

(H, T )?

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Example: Sequential Matching pennies

0 P1
Table: Normal form
Head Tail

1 P2 2 P2
s1P 2 s2P 2 s3P 2 s4P 2
Head Tail Head Tail H (-1,1) (1,-1) (-1,1) (1,-1)
P1
3 4 
5  6  T (1,-1) (-1,1) (-1,1) (1,-1)
+1 −1 −1 +1
−1 +1 +1 −1

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Example: Entry version B

Figure: Extensive form Table: Normal form

Entrant Inc.
Fight Acc.
In, Fight (-1,-1) (-2,1)
Out In
In, Acc. (1,-2) (2,2)
Ent.
0 Incumbent Out, Fight (0,5) (0,5)

5 Out, Acc. (0,5) (0,5)

Fight Acc.

Entrant

Fight Acc. Fight Acc.

−1  1  −2 2

−1 −2 1 2

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Relation between normal and extensive form

• For any extensive form representation there is a unique

normal form representation.
The converse is not true: more than one extensive
representation might exist for one normal form.
• Example?
Normal form of the sequential matching pennies could be
the representation of a game where the two players
simultaneously act; P2 has four actions, while P1 has only
two actions.

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Mixed strategies

• Given player i’s pure strategy set Si , a mixed strategy

assigns to each pure strategy si ∈ Si a probability
Pi : Si → [0, 1] that it will be played, where
σ
si ∈Si σi (si ) = 1.
• Suppose player i has M pure strategies Si = {s1i , . . . , sM i },
then her possible set of mixed strategies are
M
( )
X
∆(Si ) = (σ1i , . . . , σM i ) | σmi ≥ 0 and σmi = 1
m=1

(σ1i , . . . , σM i ) ∈ ∆(Si ) is a mixed strategy for player i.

This is a simplex: mixed extension of Si .
Note: this set includes pure strategies too!
• Mixing results in uncertain outcomes→expected utility

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Example of mixed strategies

Player B
Head Tail
Head (1,-1) (-1,1)
Player A
Tail (-1,1) (1,-1)

• Player A mix strategy: choose H w.pr. 0.5 and T w.pr. 0.5

• Player B mix strategy: choose H w.pr. 0.9 and T w.pr. 0.1
• What are the expected payoffs of players if they choose
these strategies?

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Behavior strategies

• In the extensive form the player could randomize between

available actions in each information set.
This is called a Behavior strategy.
• In games with perfect recall the two concepts are
equivalent.
You can find a behavior strategy that gives the same
probability distribution over outcomes as a given mixed
strategy.
We will therefore focus on mixed strategies.

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Behavior vs. mixed strategies

Entrant:U1 Inc.
Fight Acc.

Out In In, Fight (-1,-1) (-2,1)

In, Acc. (1,-2) (2,2)
Ent.
0 Incumbent Out, Fight (0,5) (0,5)
5 Out, Acc. (0,5) (0,5)
Fight Acc.

Entrant:U2 • Entrant’s mix strategy:

(In,Fight) w.pr. 0.5 and
Fight Acc. Fight Acc. (In,Acc.) w.pr. 0.5
−1  1  −2 2
• Entrant’s behavior
−1 −2 1 2
strategy: In w.pr. 0.5 at U1
and Acc. w.pr. 0.5 at U2

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Outline

Introduction

Basic elements

Simultaneous-move games of complete information

Dominance
Nash equilibrium

Reference: MWG Ch 8.A-8.D.

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Simultaneous-move games

• Definition: players play a one-shot game (one round) and

move simultaneously.
Use the normal form representation ΓN = [I, {Si } , {ui (.)}]
Or ΓN = [I, {∆(Si )} , {ui (.)}] if we consider mixed
strategies as well.
• Simplest model that allow for strategic interaction.
• Use this to define the notion of equilibrium.
• Paves the way for static games of incomplete information
and dynamic games.

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Questions

• Assuming rationality and structure of the game are

common knowledge
Could we predict the set of strategies that will be never
played?
Is there a strategy that yields max payoff regardless of
other’s actions?
Could we predict the outcome of the game?

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Outline

Introduction

Basic elements

Simultaneous-move games of complete information

Dominance
Nash equilibrium

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Strictly dominant strategy

• Consider a game ΓN = [I, {Si } , {ui (.)}],

si ∈ Si is a strictly dominant strategy for player i if for all
s′i ̸= si , we have

ui (si , s−i ) > ui (s′i , s−i )

for all s−i ∈ S−i .

• A dominant strategy is the unique payoff maximizer
regardless of what others do.
Notation: we often drop the term “strict”.
• We expect rational players to play dominant strategies.

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Example: Prisoner’s Dilemma

Prisoner 2
Confess Don’t Confess
Confess (−4, −4) (−1, −10)
Prisoner 1
Don’t Confess (−10, −1) (−2, −2)

• Is there a dominant strategy for prisoner 1?

• How about prisoner 2?
• Could players improve their payoff by NOT playing their
dominant strategies?
• Would they do that?

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Strictly dominated strategy

• Consider a game ΓN = [I, {Si } , {ui (.)}],

si ∈ Si is a strictly dominated strategy for player i if there
exists s′i ̸= si such that

ui (si , s−i ) < ui (s′i , s−i )

for all s−i ∈ S−i .

s′i strictly dominates si .
• A dominated strategy gives lower payoffs than another
strategy regardless of what others do!
• Redefine strictly dominant strategy: a strategy that strictly
dominates all other strategies.

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Example: dominated strategies

Player 2
L R
U (1,-1) (-1,1)
Player 1 M (-1,1) (1,-1)
D (-2,5) (-3,2)

• Is there a dominant strategy for players 1 and 2?

• Are there dominated strategies?

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Weak dominance

• si ∈ Si is a weakly dominated strategy for player i if there

exists s′i ̸= si such that

∀s−i ∈ S−i ui (si , s−i ) ≤ ui (s′i , s−i )

∃s−i ∈ S−i ui (si , s−i ) < ui (s′i , s−i )

s′i weakly dominates si .

• A strategy is a weakly dominant strategy if it weakly
dominates every other strategy.

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Example: weakly but not strictly dominated

Player 2
L R
U (5,1) (4,0)
Player 1 M (6,0) (3,1)
D (6,4) (4,4)

• Is there a (weakly) dominant strategy for players 1 and 2?

• Are there (weakly) dominated strategies?

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Elimination of strictly dominated strategies

• Rationality is common knowledge→anticipate the strategies

that won’t be played by others.
• Eliminate strictly dominated strategies for each player.
Reduced (simplified) game with fewer strategies.
▶ In the reduced game eliminate strictly dominated strategies

Reduced game with fewer strategies.

. . . (repeat until no elimination is possible)
• Each round of deletion requires a deeper level of knowledge
of rationality.
• Rationality does not justify deletion of weakly dominated
strategies.

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Example: iterated deletion of dominated strategies

Table: Original game Table: Delete C for player 2

Player 2
Player 2 L R
L C R

Player 1
T (−1, −2) (0, −2)
Player 1

T (−1, −2) (1, −3) (0, −2)

M (−2, 1) (0, 0) (10, 0) M (−2, 1) (10, 0)
B (2, 2) (−4, 1) (3, −1) B (2, 2) (3, −1)

• P 1: no dominated strategy. • P 1: T is dominated by B.

• P 2: C dominated by L. • P 2: no dominated strategy.

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Example: Continue deletion

Table: Delete T for player 1 Table: Delete R for player 2

Player 2 Player 2
L R L
Player 1

Player 1
M (−2, 1) (10, 0) M (−2, 1)
B (2, 2) (3, −1) B (2, 2)

• P 1: no dominated strategy. • P 1: M is dominated by B.

• P 2: R dominated by L. • Seems in the reduced game
(B,L) will be played!

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Dominance in mixed strategies

• σi ∈ ∆(Si ) is a strictly dominated strategy for player i if

∃σi′ ∈ ∆(Si ), such that
Y
∀σ−i ∈ ∆(Sj ) then ui (σi , σ−i ) < ui (σi′ , σ−i )
j̸=i

σi′ strictly dominates σi .

• A strategy is strictly dominant if it strictly dominates all
other strategies.

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Example: dominance of a mixed strategy

Player 2
L R

Player 1
U (10, 1) (0, 4)
M (4, 2) (4, 3)
B (0, 5) (10, 2)

• Looking at pure strategies for P1: no dominated strategy.

• Allowing for mixing for P1: ( 12 U, 12 B) dominates M.

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Checking for dominance in mixed strategies

• σi is strictly dominated by σi′ if

ui (σi′ , σ−i ) > ui (σi , σ−i ) for all σ−i

which is equivalent to checking

ui (σi′ , s−i ) > ui (σi , s−i ) for all s−i

i.e. use only pure strategies for other players!

• Proof:
 
X Y
ui (σi′ , σ−i )−ui (σi , σ−i ) σk (sk ) ui (σi′ , s−i ) − ui (σi , s−i )


= 
s−i ∈S−i k̸=i

LHS is positive for all σ−i iff ui (σi′ , s−i ) − ui (σi , s−i ) is
positive for all s−i .

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Implications

• Player i’s pure strategy si is strictly dominated iff

∃σi′ ∈ ∆(Si ) such that

ui (σi′ , s−i ) > ui (si , s−i )

for all s−i ∈ S−i .

• If pure strategy si is strictly dominated so is any mixed
strategy that assigns a positive probability to si .
• Can apply iterated deletion to mixed dominated strategies.

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Outline

Introduction

Basic elements

Simultaneous-move games of complete information

Dominance
Nash equilibrium

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Best response and Nash equilibrium
• Iterated deletion of dominated strategies may not yield
definitive outcomes.
• Could we restrict the set of reasonable strategies further?
• What is the best strategy for player i if other players
choose s−i ∈ S−i

bi (s−i ) = si ∈ Si | ui (si , s−i ) ≥ ui (s′i , s−i ) for ∀s′i ∈ Si



in general this is a correspondence.

• s∗ = (s∗1 , . . . , s∗I ) is a Nash equilibrium iff

s∗i = bi (s∗−i )

for i = 1, . . . , I
• All players’ strategies are best responses to other players’
strategies.
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Example: best response and NE

Player 2
b1 b2 b3 b4
a1 (0, 7) (2,5) (7,0) (0, 1)
a2 (5,2) (3,3) (5,2) (0, 1)
Player 1
a3 (7,0) (2,5) (0,7) (0, 1)
a4 (0, 0) (0,-2) (0,0) (10, −1)

• What is the best response correspondence for P2?

• What is the best response correspondence for P1?
• What is the Nash equilibrium of this game?

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Nash equilibrium

• Definition: A strategy profile s = (s1 , . . . , sI ) is a Nash

equilibrium of game ΓN = {I, Si , ui (.)} if for every
i = 1, . . . , I

ui (si , s−i ) ≥ ui (s′i , s−i ) forall s′i ∈ Si

• Given what others are doing, the chosen strategy gives

highest payoff to player i.
• No incentive for any of the players to deviate from the
equilibrium, given what everyone else is doing.
• Existence: Vector of all best responses b = (b1 , . . . , bI )
QI QI
b : i=1 Si → i=1 Si
Nash equilibrium is a fixed point of b:
(s∗1 , . . . , s∗I ) = b ((s∗1 , . . . , s∗I ))

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Example: Stag hunt

• Two hunters, if help each other catch a stag, if goes on its

own can catch a hare:
Player 2
Stag Hare
Stag (2, 2) (0, 1)
Player 1
Hare (1, 0) (1, 1)
• What is the Nash equilibrium of this game?

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Discussion of Nash equilibrium

• Why is it reasonable to expect players’ conjectures about

each other’s play to be correct?
• Various perspectives on the concept of NE
A result of rational inference
A necessary condition if gives a unique prediction
A focal point
A self-enforcing agreement
A stable social convention

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Extension to mixed strategies

• Definition: A mixed strategy profile σ = (s1 , . . . , sI ) is a

Nash equilibrium of game ΓN = {I, ∆(Si ), ui (.)} if for every
i = 1, . . . , I

ui (σi , σ−i ) ≥ ui (σi′ , σ−i ) for all σi′ ∈ ∆(Si )

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Example: matching pennies

• Does matching pennies have a pure strategy Nash

equilibria?
Player B
Head Tail
Head (1,-1) (-1,1)
Player A
Tail (-1,1) (1,-1)
• Say (H,H) is a NE,
then given A is playing H, the best strategy for B is to play
T!
• Say (H,T) is a NE,
then given B is playing T, the best response for A is to play
T!

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Finding mixed strategy NE
• Say B plays H and T with probabilities σB = (q, 1 − q)
where q ∈ [0, 1]
What is A’s best response to the indicated strategy of B?
▶ If plays H: uA (H, σB ) = q × 1 + (1 − q) × (−1) = 2q − 1
▶ If plays T: uA (T, σB ) = q × (−1) + (1 − q) × 1 = 1 − 2q
▶ Choose H iff uA (H, σB ) > uA (T, σB )
A’s best response

(1, 0)
 if q > 1/2
(p, 1 − p) = (0, 1) if q < 1/2

{(p, 1 − p) | p ∈ [0, 1]} if q = 1/2


• If B chooses q ̸= 1/2, A’s best response is pure strategy,

but B’s best response to A’s pure strategy is either H or T.
• To get a mixed strategy equilibrium we must have q = 1/2.

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Finding mixed strategy NE

• Let Si+ ⊂ Si be the pure strategies played with positive

probability in a mixed strategy profile σ = (σ1 , . . . , σI ).
• σ is a NE iff for every i = 1, . . . , I
1. ui (si , σ−i ) = ui (s′i , σ−i ) for all si , s′i ∈ Si+
2. ui (si , σ−i ) ≥ ui (s′i , σ−i ) for all si ∈ Si+ and for all s′i ∈
/ Si+
• Proof?

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Example: Meeting in New York

Player 2
Grand Central Empire State
Grand Central (5,5) (0,0)
Player 1
Empire State (0,0) (1,1)

• What are the NE of this game?

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Usefulness of mixed strategy NE

• In mixed strategy NE each player is really indifferent

between probabilities assigned to different strategies he/she
plays.
probabilities are chosen so the other player is indifferent
between his/her strategies.
• Criticisms
Why randomize if there is a pure strategy with the same
payoff?
Players must randomize with correct probabilities but they
have no incentives to do so!

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Example: Employer-employee

• Cost of monitoring to employer: ϕ > 0

• Employee’s effort choice: e ∈ {0, ē}
(
1 if e = ē
• Output: y =
0 if e = 0
• Worker and employer simultaneously decide to put effort
and whether to monitor.
• Payment to worker
( w
no monitoring:
w if y = 1
monitoring:
0 if y = 0
• What are the strategies for the two players?
• What are the Nash Equilibria of this game?

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Example: Employer-employee

Employer
Monitor No monitor
Effort (w − ē, 1 − w − ϕ) (w − ē, 1 − w)
Worker
No effort (0, −ϕ) (w, −w)

• Assume w − ē > 0 and w + ϕ < 1.

• Could (effort,monitor) be a NE?
• Could (no effort,no monitor) be a NE?
To rule out: assume ϕ < w!
• Are there any pure/mixed strategy NE?

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Summary

• In this lecture, we discussed

the concept of a game
normal and extensive form representation of a game
best response and Nash equilibrium in the context of
simultaneous-move games.

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