Normal Form Games

Julio Davila

2009

Julio Davila Normal Form Games

Normal form games...

1 I : a set of players

2 for all i ∈ I ,Si : the sets of actions available to each player

Σi : the set of probability distributions over Si

3 for all i ∈ I , ui ∈ R×j∈I Σi linear in each σj ∈ Σj , for all j ∈ I ,i.e. the payoffs for each player of each profile ofrandomizations over actions.

A normal form game is a collection {Si , ui}i∈I as above

Julio Davila Normal Form Games

Normal form games...

1 I : a set of players

2 for all i ∈ I ,Si : the sets of actions available to each player

Σi : the set of probability distributions over Si

3 for all i ∈ I , ui ∈ R×j∈I Σi linear in each σj ∈ Σj , for all j ∈ I ,i.e. the payoffs for each player of each profile ofrandomizations over actions.

A normal form game is a collection {Si , ui}i∈I as above

Julio Davila Normal Form Games

Normal form games...

1 I : a set of players

2 for all i ∈ I ,Si : the sets of actions available to each player

Σi : the set of probability distributions over Si

3 for all i ∈ I , ui ∈ R×j∈I Σi linear in each σj ∈ Σj , for all j ∈ I ,i.e. the payoffs for each player of each profile ofrandomizations over actions.

A normal form game is a collection {Si , ui}i∈I as above

Julio Davila Normal Form Games

Normal form games...

1 I : a set of players

2 for all i ∈ I ,Si : the sets of actions available to each player

Σi : the set of probability distributions over Si

3 for all i ∈ I , ui ∈ R×j∈I Σi linear in each σj ∈ Σj , for all j ∈ I ,i.e. the payoffs for each player of each profile ofrandomizations over actions.

A normal form game is a collection {Si , ui}i∈I as above

Julio Davila Normal Form Games

Normal form games...

1 I : a set of players

2 for all i ∈ I ,Si : the sets of actions available to each player

Σi : the set of probability distributions over Si

3 for all i ∈ I , ui ∈ R×j∈I Σi linear in each σj ∈ Σj , for all j ∈ I ,i.e. the payoffs for each player of each profile ofrandomizations over actions.

A normal form game is a collection {Si , ui}i∈I as above

Julio Davila Normal Form Games

Normal form games...

• utilities are linear in each σi ∈ Σi ,

• for finite Si ’s

ui (σ) =∑

s∈×i′∈I Si′

(∏j∈I

σj(sj))vi (s)

for some vi ∈ R×i∈I Si

• σi (si ) is the probability of i playing si according to σi .

Julio Davila Normal Form Games

Normal form games...

• utilities are linear in each σi ∈ Σi ,

• for finite Si ’s

ui (σ) =∑

s∈×i′∈I Si′

(∏j∈I

σj(sj))vi (s)

for some vi ∈ R×i∈I Si

• σi (si ) is the probability of i playing si according to σi .

Julio Davila Normal Form Games

Normal form games...

• utilities are linear in each σi ∈ Σi ,

• for finite Si ’s

ui (σ) =∑

s∈×i′∈I Si′

(∏j∈I

σj(sj))vi (s)

for some vi ∈ R×i∈I Si

• σi (si ) is the probability of i playing si according to σi .

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

payoff from

σ1 =”U with probability 1/3 and D with probability 2/3”,σ2 =”L with probability 1/4, C with probability 1/2, and Rwith probability 1/4”

u1(σ1, σ2) =1

3

(1

42 +

1

24 +

1

48

)+

2

3

(1

40− 1

21 +

1

44

)=

11

6

u2(σ1, σ2) =1

3

(1

46 +

1

24 +

1

40

)+

2

3

(1

41 +

1

22 +

1

45

)=

17

4

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

payoff from

σ1 =”U with probability 1/3 and D with probability 2/3”,σ2 =”L with probability 1/4, C with probability 1/2, and Rwith probability 1/4”

u1(σ1, σ2) =1

3

(1

42 +

1

24 +

1

48

)+

2

3

(1

40− 1

21 +

1

44

)=

11

6

u2(σ1, σ2) =1

3

(1

46 +

1

24 +

1

40

)+

2

3

(1

41 +

1

22 +

1

45

)=

17

4

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

payoff from

σ1 =”U with probability 1/3 and D with probability 2/3”,σ2 =”L with probability 1/4, C with probability 1/2, and Rwith probability 1/4”

u1(σ1, σ2) =1

3

(1

42 +

1

24 +

1

48

)+

2

3

(1

40− 1

21 +

1

44

)=

11

6

u2(σ1, σ2) =1

3

(1

46 +

1

24 +

1

40

)+

2

3

(1

41 +

1

22 +

1

45

)=

17

4

Julio Davila Normal Form Games

rational players

Julio Davila Normal Form Games

rational players

each player chooses his action so as to (try to) maximize his output

Julio Davila Normal Form Games

strategies not played by rational agents

Julio Davila Normal Form Games

strategies not played by rational agents

strictly dominated strategies:

σi is strictly dominatediff

there exists σ′i 6= σi such that, for all σ−i ,

ui (σi , σ−i ) < ui (σ′i , σ−i )

Julio Davila Normal Form Games

strategies not played by rational agents

which strategies are strictly dominated?

σi is strictly dominated if, and only if, it is strictly dominatedagainst pure strategies,

i.e.there exists σ′i 6= σi such that, for all σ−i ,

ui (σi , σ−i ) < ui (σ′i , σ−i )

iffthere exists σ′i 6= σi such that, for all s−i ,

ui (σi , s−i ) < ui (σ′i , s−i ).

Julio Davila Normal Form Games

strategies not played by rational agents

which strategies are strictly dominated?

σi is strictly dominated if, and only if, it is strictly dominatedagainst pure strategies,

i.e.there exists σ′i 6= σi such that, for all σ−i ,

ui (σi , σ−i ) < ui (σ′i , σ−i )

iffthere exists σ′i 6= σi such that, for all s−i ,

ui (σi , s−i ) < ui (σ′i , s−i ).

Julio Davila Normal Form Games

strategies not played by rational agents

which strategies are strictly dominated?

σi is strictly dominated if, and only if, it is strictly dominatedagainst pure strategies,

i.e.there exists σ′i 6= σi such that, for all σ−i ,

ui (σi , σ−i ) < ui (σ′i , σ−i )

iffthere exists σ′i 6= σi such that, for all s−i ,

ui (σi , s−i ) < ui (σ′i , s−i ).

Julio Davila Normal Form Games

strategies not played by rational agents

which strategies are strictly dominated?

σi is strictly dominated if, and only if, it is strictly dominatedagainst pure strategies,

i.e.for all σ′i 6= σi , there exists σ−i such that,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

ifffor all σ′i 6= σi , there exists s−i such that,

ui (σi , s−i ) ≥ ui (σ′i , s−i ).

Julio Davila Normal Form Games

strictly dominated strategies

”If”:

I assume that, for all σ′i 6= σi , there exists s−i such that

ui (σi , s−i ) ≥ ui (σ′i , s−i )

I then, for all σ′i 6= σi , there exists σ−i = s−i such that

ui (σi , σ−i ) ≥ ui (σ′i , σ−i ).

Julio Davila Normal Form Games

strictly dominated strategies

”If”:

I assume that, for all σ′i 6= σi , there exists s−i such that

ui (σi , s−i ) ≥ ui (σ′i , s−i )

I then, for all σ′i 6= σi , there exists σ−i = s−i such that

ui (σi , σ−i ) ≥ ui (σ′i , σ−i ).

Julio Davila Normal Form Games

strictly dominated strategies

”If”:

I assume that, for all σ′i 6= σi , there exists s−i such that

ui (σi , s−i ) ≥ ui (σ′i , s−i )

I then, for all σ′i 6= σi , there exists σ−i = s−i such that

ui (σi , σ−i ) ≥ ui (σ′i , σ−i ).

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥ ui (σ′i , σ−i )− ui (σi , σ−i )

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥ ui (σ′i , σ−i )− ui (σi , σ−i )

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥ ui (σ′i , σ−i )−

∑s∈×i′∈I Si′

(∏j∈I

σj(sj))vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥ ui (σ′i , σ−i )−

∑s∈×i′∈I Si′

(σi (si )

∏j 6=i

σj(sj))vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥∑

s∈×i′∈I Si′

(σ′i (si )

∏j 6=i

σj(sj))vi (s)

−∑

s∈×i′∈I Si′

(σi (si )

∏j 6=i

σj(sj))vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥∑

s−i∈×i′ 6=iSi′

∑si∈Si

(σ′i (si )

∏j 6=i

σj(sj))vi (s)

−∑

s−i∈×i′ 6=iSi′

∑si∈Si

(σi (si )

∏j 6=i

σj(sj))vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥∑

s−i∈×i′ 6=iSi′

(∏j 6=i

σj(sj)) ∑

si∈Si

σ′i (si )vi (s)

−∑

s−i∈×i′ 6=iSi′

(∏j 6=i

σj(sj)) ∑

si∈Si

σi (si )vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥∑

s−i∈×i′ 6=iSi′

(∏j 6=i

σj(sj)) [ ∑

si∈Si

σ′i (si )vi (s)−∑si∈Si

σi (si )vi (s)]

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Assume, for all σ′i 6= σi , there exists σ−i such that

0 ≥∑

s−i∈×i′ 6=iSi′

(∏j 6=i

σj(sj)) [

ui (σ′i , s−i )− ui (σi , s−i )

]

Julio Davila Normal Form Games

strictly dominated strategies

”Only if”: (when all Si are finite)

I Then, for all σ′i 6= σi , there exists s−i such that

0 ≥ ui (σ′i , s−i )− ui (σi , s−i )

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

C is strictly dominated by

σ2 =”L with probability .7, C with probability 0, and R withprobability .3”

u2(σ1,C ) = σ1(U) (0 · 6 + 1 · 4 + 0 · 0)

+ σ1(D) (0 · 1 + 1 · 2 + 0 · 5) = 4σ1(U) + 2σ1(D)

u2(σ1, σ2) = σ1(U) (.7 · 6 + 0 · 4 + .3 · 0)

+ σ1(D) (.7 · 1 + 0 · 2 + .3 · 5) = 4.2σ1(U) + 2.2σ1(D)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

C is strictly dominated by

σ2 =”L with probability .7, C with probability 0, and R withprobability .3”

u2(σ1,C ) = σ1(U) (0 · 6 + 1 · 4 + 0 · 0)

+ σ1(D) (0 · 1 + 1 · 2 + 0 · 5) = 4σ1(U) + 2σ1(D)

u2(σ1, σ2) = σ1(U) (.7 · 6 + 0 · 4 + .3 · 0)

+ σ1(D) (.7 · 1 + 0 · 2 + .3 · 5) = 4.2σ1(U) + 2.2σ1(D)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

C is strictly dominated by

σ2 =”L with probability .7, C with probability 0, and R withprobability .3”

u2(σ1,C ) = σ1(U) (0 · 6 + 1 · 4 + 0 · 0)

+ σ1(D) (0 · 1 + 1 · 2 + 0 · 5) = 4σ1(U) + 2σ1(D)

u2(σ1, σ2) = σ1(U) (.7 · 6 + 0 · 4 + .3 · 0)

+ σ1(D) (.7 · 1 + 0 · 2 + .3 · 5) = 4.2σ1(U) + 2.2σ1(D)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

C is strictly dominated by

σ2 =”L with probability .7, C with probability 0, and R withprobability .3”

u2(σ1,C ) = σ1(U) (0 · 6 + 1 · 4 + 0 · 0)

+ σ1(D) (0 · 1 + 1 · 2 + 0 · 5) = 4σ1(U) + 2σ1(D)

u2(σ1, σ2) = σ1(U) (.7 · 6 + 0 · 4 + .3 · 0)

+ σ1(D) (.7 · 1 + 0 · 2 + .3 · 5) = 4.2σ1(U) + 2.2σ1(D)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

U strictly dominates D

U strictly dominates any other strategy

u1(U, σ2) =1 · (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ 0 · (0 · 0− 1 · 1 + 0 · 4)

=2σ2(L) + 4σ2(C ) + 8σ2(R)

u1(σ1, σ2) =σ1(U) (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ σ1(D) (σ2(L) · 0− σ2(C ) · 1 + σ2(R) · 4)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

U strictly dominates D

U strictly dominates any other strategy

u1(U, σ2) =1 · (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ 0 · (0 · 0− 1 · 1 + 0 · 4)

=2σ2(L) + 4σ2(C ) + 8σ2(R)

u1(σ1, σ2) =σ1(U) (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ σ1(D) (σ2(L) · 0− σ2(C ) · 1 + σ2(R) · 4)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

U strictly dominates D

U strictly dominates any other strategy

u1(U, σ2) =1 · (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ 0 · (0 · 0− 1 · 1 + 0 · 4)

=2σ2(L) + 4σ2(C ) + 8σ2(R)

u1(σ1, σ2) =σ1(U) (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ σ1(D) (σ2(L) · 0− σ2(C ) · 1 + σ2(R) · 4)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

U strictly dominates D

U strictly dominates any other strategy

u1(U, σ2) =1 · (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ 0 · (0 · 0− 1 · 1 + 0 · 4)

=2σ2(L) + 4σ2(C ) + 8σ2(R)

u1(σ1, σ2) =σ1(U) (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ σ1(D) (σ2(L) · 0− σ2(C ) · 1 + σ2(R) · 4)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

U strictly dominates D

U strictly dominates any other strategy

u1(U, σ2) =1 · (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ 0 · (0 · 0− 1 · 1 + 0 · 4)

=2σ2(L) + 4σ2(C ) + 8σ2(R)

u1(σ1, σ2) =σ1(U) (σ2(L) · 2 + σ2(C ) · 4 + σ2(R) · 8)

+ σ1(D) (σ2(L) · 0− σ2(C ) · 1 + σ2(R) · 4)

Julio Davila Normal Form Games

strictly dominated strategies

if σi is strictly dominated, then any ρi playing σi with probabilityp > 0 is strictly dominated as well.

Julio Davila Normal Form Games

strictly dominated strategies

let σi be strictly dominated,

i.e. there exists σ′i 6= σi such that, for all σ−i ,∑s−i∈×j 6=iSj

∑si∈Si

σi (si )∏j 6=i

σj(sj)vi (s) <∑

s−i∈×j 6=iSj

∑si∈Si

σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

let σi be strictly dominated,

i.e. there exists σ′i 6= σi such that, for all σ−i ,∑s−i∈×j 6=iSj

∑si∈Si

σi (si )∏j 6=i

σj(sj)vi (s) <∑

s−i∈×j 6=iSj

∑si∈Si

σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

Consider

I a strategy ρi consisting of playing σhi with probability ph, with

σ1i = σi and p1 = p > 0

I the strategy ρ′i for i consisting of playing σhi with probability

ph, with σ1i = σ′i and σh

i = σhi otherwise.

Julio Davila Normal Form Games

strictly dominated strategies

Consider

I a strategy ρi consisting of playing σhi with probability ph, with

σ1i = σi and p1 = p > 0

I the strategy ρ′i for i consisting of playing σhi with probability

ph, with σ1i = σ′i and σh

i = σhi otherwise.

Julio Davila Normal Form Games

strictly dominated strategies

Consider

I a strategy ρi consisting of playing σhi with probability ph, with

σ1i = σi and p1 = p > 0

I the strategy ρ′i for i consisting of playing σhi with probability

ph, with σ1i = σ′i and σh

i = σhi otherwise.

Julio Davila Normal Form Games

strictly dominated strategies

then, for all σ−i ,

ui (ρi , σ−i ) =∑s−i∈×j 6=iSj

∑si∈Si

∑h

ph · σhi (si )

∏j 6=i

σj(sj)vi (s) <

∑s−i∈×j 6=iSj

∑si∈Si

∑h

ph · σhi (si )

∏j 6=i

σj(sj)vi (s)

= ui (ρ′i , σ−i )

Julio Davila Normal Form Games

strictly dominated strategies

since, for all σ−i ,∑s−i∈×j 6=iSj

∑si∈Si

p · σi (si )∏j 6=i

σj(sj)vi (s) <

∑s−i∈×j 6=iSj

∑si∈Si

p · σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

strictly dominated strategies

since, for all σ−i ,∑s−i∈×j 6=iSj

∑si∈Si

σi (si )∏j 6=i

σj(sj)vi (s) <

∑s−i∈×j 6=iSj

∑si∈Si

σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

I D is strictly dominated

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 5, 2 4, 5

I D is not strictly dominated anymore

I C is strictly dominated

I D is strictly dominated, given that C will be played with 0probabilty

I R is strictly dominated, given that D will be played with 0probabilty

players will play U and L

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 5, 2 4, 5

I D is not strictly dominated anymore

I C is strictly dominated

I D is strictly dominated, given that C will be played with 0probabilty

I R is strictly dominated, given that D will be played with 0probabilty

players will play U and L

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 5, 2 4, 5

I D is not strictly dominated anymore

I C is strictly dominated

I D is strictly dominated, given that C will be played with 0probabilty

I R is strictly dominated, given that D will be played with 0probabilty

players will play U and L

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 5, 2 4, 5

I D is not strictly dominated anymore

I C is strictly dominated

I D is strictly dominated, given that C will be played with 0probabilty

I R is strictly dominated, given that D will be played with 0probabilty

players will play U and L

Julio Davila Normal Form Games

Normal form games... an example

L C R

U 2, 6 4, 4 8, 0D 0, 1 5, 2 4, 5

I D is not strictly dominated anymore

I C is strictly dominated

I D is strictly dominated, given that C will be played with 0probabilty

I R is strictly dominated, given that D will be played with 0probabilty

players will play U and L

Julio Davila Normal Form Games

Normal form games... an example

it does not always solves the game:

L C R

U 2, 1 1, 2 1, 0M 1, 2 2, 1 1, 1D 0,−1 0, 0 10,−2

I Σ1 = {pU , pM , pD} and Σ2 = {pL, pC , pR}I R is strictly dominated

I Σ11 = {pU , pM , pD} and Σ1

2 = {pL, pC , 0}I D is (iteratively) strictly dominated

I Σ21 = {pU , pM , 0} and Σ2

2 = {pL, pC , 0}

Julio Davila Normal Form Games

Normal form games... an example

it does not always solves the game:

L C R

U 2, 1 1, 2 1, 0M 1, 2 2, 1 1, 1D 0,−1 0, 0 10,−2

I Σ1 = {pU , pM , pD} and Σ2 = {pL, pC , pR}

I R is strictly dominated

I Σ11 = {pU , pM , pD} and Σ1

2 = {pL, pC , 0}I D is (iteratively) strictly dominated

I Σ21 = {pU , pM , 0} and Σ2

2 = {pL, pC , 0}

Julio Davila Normal Form Games

Normal form games... an example

it does not always solves the game:

L C R

U 2, 1 1, 2 1, 0M 1, 2 2, 1 1, 1D 0,−1 0, 0 10,−2

I Σ1 = {pU , pM , pD} and Σ2 = {pL, pC , pR}I R is strictly dominated

I Σ11 = {pU , pM , pD} and Σ1

2 = {pL, pC , 0}I D is (iteratively) strictly dominated

I Σ21 = {pU , pM , 0} and Σ2

2 = {pL, pC , 0}

Julio Davila Normal Form Games

Normal form games... an example

it does not always solves the game:

L C R

U 2, 1 1, 2 1, 0M 1, 2 2, 1 1, 1D 0,−1 0, 0 10,−2

I Σ1 = {pU , pM , pD} and Σ2 = {pL, pC , pR}I R is strictly dominated

I Σ11 = {pU , pM , pD} and Σ1

2 = {pL, pC , 0}

I D is (iteratively) strictly dominated

I Σ21 = {pU , pM , 0} and Σ2

2 = {pL, pC , 0}

Julio Davila Normal Form Games

Normal form games... an example

it does not always solves the game:

L C R

U 2, 1 1, 2 1, 0M 1, 2 2, 1 1, 1D 0,−1 0, 0 10,−2

I Σ1 = {pU , pM , pD} and Σ2 = {pL, pC , pR}I R is strictly dominated

I Σ11 = {pU , pM , pD} and Σ1

2 = {pL, pC , 0}I D is (iteratively) strictly dominated

I Σ21 = {pU , pM , 0} and Σ2

2 = {pL, pC , 0}

Julio Davila Normal Form Games

Normal form games... an example

it does not always solves the game:

L C R

U 2, 1 1, 2 1, 0M 1, 2 2, 1 1, 1D 0,−1 0, 0 10,−2

I Σ1 = {pU , pM , pD} and Σ2 = {pL, pC , pR}I R is strictly dominated

I Σ11 = {pU , pM , pD} and Σ1

2 = {pL, pC , 0}I D is (iteratively) strictly dominated

I Σ21 = {pU , pM , 0} and Σ2

2 = {pL, pC , 0}

Julio Davila Normal Form Games

common ”knowledge” of rationality

players

I are rational,

I believe that all other players are rational,

I believe that all other players believe that all other players arerational,

I believe that all other players believe that all other playersbelieve that all other players are rational,

I ...

Julio Davila Normal Form Games

common ”knowledge” of rationality

players

I are rational,

I believe that all other players are rational,

I believe that all other players believe that all other players arerational,

I believe that all other players believe that all other playersbelieve that all other players are rational,

I ...

Julio Davila Normal Form Games

common ”knowledge” of rationality

players

I are rational,

I believe that all other players are rational,

I believe that all other players believe that all other players arerational,

I believe that all other players believe that all other playersbelieve that all other players are rational,

I ...

Julio Davila Normal Form Games

common ”knowledge” of rationality

players

I are rational,

I believe that all other players are rational,

I believe that all other players believe that all other players arerational,

I believe that all other players believe that all other playersbelieve that all other players are rational,

I ...

Julio Davila Normal Form Games

common ”knowledge” of rationality

players

I are rational,

I believe that all other players are rational,

I believe that all other players believe that all other players arerational,

I believe that all other players believe that all other playersbelieve that all other players are rational,

I ...

Julio Davila Normal Form Games

iterative elimination of strictly dominated strategies

I strategies undominated after n rounds

Σni =

{σi ∈ Σn−1

i |∀σ′i ∈ Σn−1i , ∃σ−i ∈ ×j 6=iΣ

n−1j such that

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

}with Σ0

i = Σi , for all i .

I strategies never dominated

Σ∞i = ∩n∈NΣni

I if each Σ∞i is a singleton, then the game is solvable byiterated strict dominance

Julio Davila Normal Form Games

iterative elimination of strictly dominated strategies

I strategies undominated after n rounds

Σni =

{σi ∈ Σn−1

i |∀σ′i ∈ Σn−1i , ∃σ−i ∈ ×j 6=iΣ

n−1j such that

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

}with Σ0

i = Σi , for all i .

I strategies never dominated

Σ∞i = ∩n∈NΣni

I if each Σ∞i is a singleton, then the game is solvable byiterated strict dominance

Julio Davila Normal Form Games

iterative elimination of strictly dominated strategies

I strategies undominated after n rounds

Σni =

{σi ∈ Σn−1

i |∀σ′i ∈ Σn−1i , ∃σ−i ∈ ×j 6=iΣ

n−1j such that

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

}with Σ0

i = Σi , for all i .

I strategies never dominated

Σ∞i = ∩n∈NΣni

I if each Σ∞i is a singleton, then the game is solvable byiterated strict dominance

Julio Davila Normal Form Games

strategies not played by rational players

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 2 4, 5

I C is strictly dominated

Julio Davila Normal Form Games

strategies not played by rational players

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 3 4, 5

IIII C is not strictly dominated anymore

I L is optimal if U is played

I R is optimal if D is played

I C is never optimal

Julio Davila Normal Form Games

strategies not played by rational players

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 3 4, 5

I C is not strictly dominated anymore

I L is optimal if U is played

I R is optimal if D is played

I C is never optimal

Julio Davila Normal Form Games

strategies not played by rational players

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 3 4, 5

I C is not strictly dominated anymore

I L is optimal if U is played

I R is optimal if D is played

I C is never optimal

Julio Davila Normal Form Games

strategies not played by rational players

L C R

U 2, 6 4, 4 8, 0D 0, 1 −1, 3 4, 5

I C is not strictly dominated anymore

I L is optimal if U is played

I R is optimal if D is played

I C is never optimal

Julio Davila Normal Form Games

strategies not played by rational players

never optimal strategies:

σi is a never optimal strategyiff

for all σ−i , there exists σ′i 6= σi such that

ui (σi , σ−i ) < ui (σ′i , σ−i )

Julio Davila Normal Form Games

strategies not played by rational players

strictly dominated strategies:

σi is strictly dominatediff

there exists σ′i 6= σi such that, for all σ−i ,

ui (σi , σ−i ) < ui (σ′i , σ−i )

Julio Davila Normal Form Games

strategies not played by rational players

a strictly dominated strategy is never optimal

but a never optimal strategy needs not be strictly dominated

Julio Davila Normal Form Games

strategies not played by rational players

a strictly dominated strategy is never optimal

but a never optimal strategy needs not be strictly dominated

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

if σi is never optimal, then it is never optimal strategy againstpure strategies,

i.e.if, for all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )

thenfor all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

if σi is never optimal, then it is never optimal strategy againstpure strategies,

i.e.if, for all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )

thenfor all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

if σi is never optimal, then it is never optimal strategy againstpure strategies,

i.e.if, for all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )

thenfor all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

if σi is never optimal, then it is never optimal strategy againstpure strategies,

i.e.if, for all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )

thenfor all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

if σi is never optimal, then it is never optimal strategy againstpure strategies,

2 thenthere exists σ−i = s−i such that, for all σ′i 6= σi ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

1 in effectif there exists s−i such that, for all σ′i 6= σi ,

ui (σi , s−i ) ≥ ui (σ′i , s−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

if σi is never optimal, then it is never optimal strategy againstpure strategies,

2 thenthere exists σ−i = s−i such that, for all σ′i 6= σi ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

1 in effectif there exists s−i such that, for all σ′i 6= σi ,

ui (σi , s−i ) ≥ ui (σ′i , s−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

In effect, is it true thatif, for all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

thenfor all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )?

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

In effect, is it true thatif, for all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

thenfor all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )?

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

In effect, is it true thatif, for all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

thenfor all σ−i , there exists σ′i 6= σi such that,

ui (σi , σ−i ) < ui (σ′i , σ−i )?

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

In effect, is it true thatif, for all s−i , there exists σ′i 6= σi such that,

ui (σi , s−i ) < ui (σ′i , s−i )

1 In effect, is it true thatif, there exists σ−i such that, for all σ′i 6= σi ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

2 thenthere exists s−i such that, for all σ′i 6= σi ,

ui (σi , s−i ) ≥ ui (σ′i , s−i )?

1 In effect, is it true thatif, there exists σ−i such that, for all σ′i 6= σi ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

2 thenthere exists s−i such that, for all σ′i 6= σi ,

ui (σi , s−i ) ≥ ui (σ′i , s−i )?

1 In effect, is it true thatif, there exists σ−i such that, for all σ′i 6= σi ,

0 ≥ ui (σ′i , σ−i )− ui (σi , σ−i )

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

2 thenthere exists s−i such that, for all σ′i 6= σi ,

ui (σi , s−i ) ≥ ui (σ′i , s−i )?

1 In effect, is it true thatif, there exists σ−i such that, for all σ′i 6= σi ,

0 ≥∑

s−i∈×j 6=iSj

∏j 6=i

σj(sj)[ui (σ

′i , s−i )− ui (σi , s−i )

]

Julio Davila Normal Form Games

strategies not played by rational players

which strategies are never optimal?

but if σi is never optimal against pure strategies, then it needsnot be never optimal against all strategies,

2 thenfor all σ′i 6= σi , there exists s−i such that,

ui (σi , s−i ) ≥ ui (σ′i , s−i )!

1 In effect, is it true thatif, there exists σ−i such that, for all σ′i 6= σi ,

0 ≥∑

s−i∈×j 6=iSj

∏j 6=i

σj(sj)[ui (σ

′i , s−i )− ui (σi , s−i )

]

Julio Davila Normal Form Games

never optimal strategies

if σi is never optimal, then any ρi that plays σi with probabilityp > 0 is never optimal as well

Julio Davila Normal Form Games

never optimal strategies

let σi be never optimal,

i.e. for all σ−i , there exists σ′i 6= σi such that∑s−i∈×j 6=iSj

∑si∈Si

σi (si )∏j 6=i

σj(sj)vi (s) <∑

s−i∈×j 6=iSj

∑si∈Si

σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

never optimal strategies

let σi be never optimal,

i.e. for all σ−i , there exists σ′i 6= σi such that∑s−i∈×j 6=iSj

∑si∈Si

σi (si )∏j 6=i

σj(sj)vi (s) <∑

s−i∈×j 6=iSj

∑si∈Si

σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

never optimal strategies

Consider

I a strategy ρi consisting of playing σhi with probability ph, with

σ1i = σi and p1 = p > 0

I the strategy ρ′i for i consisting of playing σhi with probability

ph, with σ1i = σ′i and σh

i = σhi otherwise.

Julio Davila Normal Form Games

never optimal strategies

Consider

I a strategy ρi consisting of playing σhi with probability ph, with

σ1i = σi and p1 = p > 0

I the strategy ρ′i for i consisting of playing σhi with probability

ph, with σ1i = σ′i and σh

i = σhi otherwise.

Julio Davila Normal Form Games

never optimal strategies

Consider

I a strategy ρi consisting of playing σhi with probability ph, with

σ1i = σi and p1 = p > 0

I the strategy ρ′i for i consisting of playing σhi with probability

ph, with σ1i = σ′i and σh

i = σhi otherwise.

Julio Davila Normal Form Games

never optimal strategies

then, for all σ−i ,

ui (ρi , σ−i ) =∑s−i∈×j 6=iSj

∑si∈Si

∑h

ph · σhi (si )

∏j 6=i

σj(sj)vi (s) <

∑s−i∈×j 6=iSj

∑si∈Si

∑h

ph · σhi (si )

∏j 6=i

σj(sj)vi (s)

= ui (ρ′i , σ−i )

Julio Davila Normal Form Games

never optimal strategies

since, for all σ−i ,∑s−i∈×j 6=iSj

∑si∈Si

p · σi (si )∏j 6=i

σj(sj)vi (s) <

∑s−i∈×j 6=iSj

∑si∈Si

p · σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

never optimal strategies

since, for all σ−i ,∑s−i∈×j 6=iSj

∑si∈Si

σi (si )∏j 6=i

σj(sj)vi (s) <

∑s−i∈×j 6=iSj

∑si∈Si

σ′i (si )∏j 6=i

σj(sj)vi (s)

Julio Davila Normal Form Games

iterative elimination of never optimal strategies

I strategies not never optimal after n rounds

Σni =

{σi ∈ Σn−1

i |∃σ−i ∈ ×j 6=i Σn−1j such that, ∀σ′i ∈ Σn−1

i ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

}with Σ0

i = Σi , for all i

I strategies never never optimal, a.k.a. rationalizablestrategies

Σ∞i = ∩n∈NΣni

I if each Σ∞i is a singleton, then the game is solvable inrationalizable strategies

Julio Davila Normal Form Games

iterative elimination of never optimal strategies

I strategies not never optimal after n rounds

Σni =

{σi ∈ Σn−1

i |∃σ−i ∈ ×j 6=i Σn−1j such that, ∀σ′i ∈ Σn−1

i ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

}with Σ0

i = Σi , for all i

I strategies never never optimal, a.k.a. rationalizablestrategies

Σ∞i = ∩n∈NΣni

I if each Σ∞i is a singleton, then the game is solvable inrationalizable strategies

Julio Davila Normal Form Games

iterative elimination of never optimal strategies

I strategies not never optimal after n rounds

Σni =

{σi ∈ Σn−1

i |∃σ−i ∈ ×j 6=i Σn−1j such that, ∀σ′i ∈ Σn−1

i ,

ui (σi , σ−i ) ≥ ui (σ′i , σ−i )

}with Σ0

i = Σi , for all i

I strategies never never optimal, a.k.a. rationalizablestrategies

Σ∞i = ∩n∈NΣni

I if each Σ∞i is a singleton, then the game is solvable inrationalizable strategies

Julio Davila Normal Form Games

rationalizable strategies

σ is a profile of rationalizable strategiesiff

for all i ∈ I and all σ′i 6= σi ,there exists σ′−i such that

ui (σ′i , σ′−i ) ≤ ui (σi , σ

′−i )

Julio Davila Normal Form Games

rationalizable strategies

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

I C is never optimal

I M is never optimal

I thus Σ11 = {(pU , 0, pD)} and Σ1

2 = {(pL, 0, pR)}I can more strategies be eliminated?

Julio Davila Normal Form Games

rationalizable strategies

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

I C is never optimal

I M is never optimal

I thus Σ11 = {(pU , 0, pD)} and Σ1

2 = {(pL, 0, pR)}I can more strategies be eliminated?

Julio Davila Normal Form Games

rationalizable strategies

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

I C is never optimal

I M is never optimal

I thus Σ11 = {(pU , 0, pD)} and Σ1

2 = {(pL, 0, pR)}I can more strategies be eliminated?

Julio Davila Normal Form Games

rationalizable strategies

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

I C is never optimal

I M is never optimal

I thus Σ11 = {(pU , 0, pD)} and Σ1

2 = {(pL, 0, pR)}

I can more strategies be eliminated?

Julio Davila Normal Form Games

rationalizable strategies

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

I C is never optimal

I M is never optimal

I thus Σ11 = {(pU , 0, pD)} and Σ1

2 = {(pL, 0, pR)}I can more strategies be eliminated?

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I since u1(σ) = (3pL + 1pR)pU + (1pL + 3pU)pD if{pL, pR} = {.5, .5}, then any {pU , pD} is optimal

I since u2(σ) = (2pU + 4pD)pL + (4pU + 1pD)pR if{pU , pD} = {.6, .4}, then any {pL, pR} is optimal

I thus Σn1 = {(pU , 0, pD)} and Σn

2 = {(pL, 0, pR)}, for all n

I rationalizabilty does not solve this game

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I since u1(σ) = (3pL + 1pR)pU + (1pL + 3pU)pD if{pL, pR} = {.5, .5}, then any {pU , pD} is optimal

I since u2(σ) = (2pU + 4pD)pL + (4pU + 1pD)pR if{pU , pD} = {.6, .4}, then any {pL, pR} is optimal

I thus Σn1 = {(pU , 0, pD)} and Σn

2 = {(pL, 0, pR)}, for all n

I rationalizabilty does not solve this game

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I since u1(σ) = (3pL + 1pR)pU + (1pL + 3pU)pD if{pL, pR} = {.5, .5}, then any {pU , pD} is optimal

I since u2(σ) = (2pU + 4pD)pL + (4pU + 1pD)pR if{pU , pD} = {.6, .4}, then any {pL, pR} is optimal

I thus Σn1 = {(pU , 0, pD)} and Σn

2 = {(pL, 0, pR)}, for all n

I rationalizabilty does not solve this game

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I since u1(σ) = (3pL + 1pR)pU + (1pL + 3pU)pD if{pL, pR} = {.5, .5}, then any {pU , pD} is optimal

I since u2(σ) = (2pU + 4pD)pL + (4pU + 1pD)pR if{pU , pD} = {.6, .4}, then any {pL, pR} is optimal

I thus Σn1 = {(pU , 0, pD)} and Σn

2 = {(pL, 0, pR)}, for all n

I rationalizabilty does not solve this game

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I since u1(σ) = (3pL + 1pR)pU + (1pL + 3pU)pD if{pL, pR} = {.5, .5}, then any {pU , pD} is optimal

I since u2(σ) = (2pU + 4pD)pL + (4pU + 1pD)pR if{pU , pD} = {.6, .4}, then any {pL, pR} is optimal

I thus Σn1 = {(pU , 0, pD)} and Σn

2 = {(pL, 0, pR)}, for all n

I rationalizabilty does not solve this game

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I {pL, pR} = {.5, .5} is optimal only if {pU , pD} = {.6, .4} isexpected to be played

I {pU , pD} = {.6, .4} is optimal only if {pL, pR} = {.5, .5} isexpected to be played

I {pL, pR} = {.5, .5} and {pU , pD} = {.6, .4} is the only profileof rationalizable strategies no regreted ex post

or in which expectations are correct

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I {pL, pR} = {.5, .5} is optimal only if {pU , pD} = {.6, .4} isexpected to be played

I {pU , pD} = {.6, .4} is optimal only if {pL, pR} = {.5, .5} isexpected to be played

I {pL, pR} = {.5, .5} and {pU , pD} = {.6, .4} is the only profileof rationalizable strategies no regreted ex post

or in which expectations are correct

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I {pL, pR} = {.5, .5} is optimal only if {pU , pD} = {.6, .4} isexpected to be played

I {pU , pD} = {.6, .4} is optimal only if {pL, pR} = {.5, .5} isexpected to be played

I {pL, pR} = {.5, .5} and {pU , pD} = {.6, .4} is the only profileof rationalizable strategies no regreted ex post

or in which expectations are correct

Julio Davila Normal Form Games

rationalizable strategies

L R

U 3, 2 1, 4D 1, 4 3, 1

I {pL, pR} = {.5, .5} is optimal only if {pU , pD} = {.6, .4} isexpected to be played

I {pU , pD} = {.6, .4} is optimal only if {pL, pR} = {.5, .5} isexpected to be played

I {pL, pR} = {.5, .5} and {pU , pD} = {.6, .4} is the only profileof rationalizable strategies no regreted ex post

or in which expectations are correct

Julio Davila Normal Form Games

rationalizable strategies

σ is a profile of rationalizable strategiesiff

for all i ∈ I and all σ′i 6= σi ,there exists σ′−i such that

ui (σ′i , σ′−i ) ≤ ui (σi , σ

′−i )

Julio Davila Normal Form Games

Nash equilibrium

σ is a Nash equilibrium profile of strategiesiff

for all i ∈ I and all σ′i 6= σi ,there exists σ′−i such that

ui (σ′i , σ′−i ) ≤ ui (σi , σ

′−i )

and σ′−i = σ−i

Julio Davila Normal Form Games

Nash equilibrium

σ is a Nash equilibrium profile of strategiesiff

for all i ∈ I and all σ′i 6= σi ,

ui (σ′i , σ−i ) ≤ ui (σi , σ−i )

Julio Davila Normal Form Games

Nash equilibrium

which strategy profiles are Nash equilibria?

if σ is a Nash equilibrium,then,

for all i , all si such that σi (si ) > 0, and all s ′i ,

ui (s′i , σ−i ) ≤ ui (si , σ−i )

Julio Davila Normal Form Games

Nash equilibrium

which strategy profiles are Nash equilibria?

if σ is a Nash equilibrium,then,

for all i , all si such that σi (si ) > 0, and all s ′i ,

ui (s′i , σ−i ) ≤ ui (si , σ−i )

Julio Davila Normal Form Games

Nash equilibrium

I assume not: for some i , some si such that σi (si ) > 0, andsome s ′i ,

ui (s′i , σ−i ) > ui (si , σ−i ).

I playing σ player i gets

σi (si )ui (si , σ−i ) +∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I deviating to s ′i whenever i should play si player i gets

σi (si )ui (s′i , σ−i ) +

∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I σi is not a best reply to σ−i

I σ is not a Nash equilibrium

Julio Davila Normal Form Games

Nash equilibrium

I assume not: for some i , some si such that σi (si ) > 0, andsome s ′i ,

ui (s′i , σ−i ) > ui (si , σ−i ).

I playing σ player i gets

σi (si )ui (si , σ−i ) +∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I deviating to s ′i whenever i should play si player i gets

σi (si )ui (s′i , σ−i ) +

∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I σi is not a best reply to σ−i

I σ is not a Nash equilibrium

Julio Davila Normal Form Games

Nash equilibrium

I assume not: for some i , some si such that σi (si ) > 0, andsome s ′i ,

ui (s′i , σ−i ) > ui (si , σ−i ).

I playing σ player i gets

σi (si )ui (si , σ−i ) +∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I deviating to s ′i whenever i should play si player i gets

σi (si )ui (s′i , σ−i ) +

∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I σi is not a best reply to σ−i

I σ is not a Nash equilibrium

Julio Davila Normal Form Games

Nash equilibrium

I assume not: for some i , some si such that σi (si ) > 0, andsome s ′i ,

ui (s′i , σ−i ) > ui (si , σ−i ).

I playing σ player i gets

σi (si )ui (si , σ−i ) +∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I deviating to s ′i whenever i should play si player i gets

σi (si )ui (s′i , σ−i ) +

∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I σi is not a best reply to σ−i

I σ is not a Nash equilibrium

Julio Davila Normal Form Games

Nash equilibrium

I assume not: for some i , some si such that σi (si ) > 0, andsome s ′i ,

ui (s′i , σ−i ) > ui (si , σ−i ).

I playing σ player i gets

σi (si )ui (si , σ−i ) +∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I deviating to s ′i whenever i should play si player i gets

σi (si )ui (s′i , σ−i ) +

∑s′′i 6=si

σi (s′′i )ui (s

′′i , σ−i )

I σi is not a best reply to σ−i

I σ is not a Nash equilibrium

Julio Davila Normal Form Games

Nash equilibrium

which strategy profiles are Nash equilibria?

if σ is a Nash equilibrium,then,

for all i , all si , s′i such that σi (si ), σi (s

′i ) > 0,

ui (s′i , σ−i ) = ui (si , σ−i )

Julio Davila Normal Form Games

Nash equilibrium

which strategy profiles are Nash equilibria?

if σ is a Nash equilibrium,then,

for all i , all si , s′i such that σi (si ), σi (s

′i ) > 0,

ui (s′i , σ−i ) = ui (si , σ−i )

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 2pC + 1pR = 2pL + 0pC + 2pR = 1pL + 2pC + 3pR

pL + 2pC − pR = 0

pL − 2pC − pR = 0

pL + pC + pR = 1

{pU , pM , pD} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 2pC + 1pR = 2pL + 0pC + 2pR = 1pL + 2pC + 3pR

pL + 2pC − pR = 0

pL − 2pC − pR = 0

pL + pC + pR = 1

{pU , pM , pD} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 2pC + 1pR = 2pL + 0pC + 2pR = 1pL + 2pC + 3pR

pL + 2pC − pR = 0

pL − 2pC − pR = 0

pL + pC + pR = 1

{pU , pM , pD} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 2pC + 1pR = 2pL + 0pC + 2pR = 1pL + 2pC + 3pR

pL + 2pC − pR = 0

pL − 2pC − pR = 0

pL + pC + pR = 1

{pU , pM , pD} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 3pM + 4pD = 3pU + 0pM + 2pD = 4pU + 3pM + 1pD

pU − 3pM − 2pD = 0

pU + 3pM − pD = 0

pU + pM + pD = 1

{pL, pC , pR} = { 9

14,− 1

14,

6

14}!!

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 3pM + 4pD = 3pU + 0pM + 2pD = 4pU + 3pM + 1pD

pU − 3pM − 2pD = 0

pU + 3pM − pD = 0

pU + pM + pD = 1

{pL, pC , pR} = { 9

14,− 1

14,

6

14}!!

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 3pM + 4pD = 3pU + 0pM + 2pD = 4pU + 3pM + 1pD

pU − 3pM − 2pD = 0

pU + 3pM − pD = 0

pU + pM + pD = 1

{pL, pC , pR} = { 9

14,− 1

14,

6

14}!!

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 3pM + 4pD = 3pU + 0pM + 2pD = 4pU + 3pM + 1pD

pU − 3pM − 2pD = 0

pU + 3pM − pD = 0

pU + pM + pD = 1

{pL, pC , pR} = { 9

14,− 1

14,

6

14}!!

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 1pR = 1pL + 3pR

2pL − 2pR = 0

pL + pR = 1

{pL, pC , pR} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 1pR = 1pL + 3pR

2pL − 2pR = 0

pL + pR = 1

{pL, pC , pR} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 1pR = 1pL + 3pR

2pL − 2pR = 0

pL + pR = 1

{pL, pC , pR} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

3pL + 1pR = 1pL + 3pR

2pL − 2pR = 0

pL + pR = 1

{pL, pC , pR} = {.5, 0, .5}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 4pD = 4pU + 1pD

2pU − 3pD = 0

pU + pD = 1

{pU , pM , pD} = {.6, 0, .4}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 4pD = 4pU + 1pD

2pU − 3pD = 0

pU + pD = 1

{pU , pM , pD} = {.6, 0, .4}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 4pD = 4pU + 1pD

2pU − 3pD = 0

pU + pD = 1

{pU , pM , pD} = {.6, 0, .4}

Julio Davila Normal Form Games

Nash equilibrium

L C R

U 3, 2 2, 3 1, 4M 2, 3 0, 0 2, 3D 1, 4 2, 2 3, 1

2pU + 4pD = 4pU + 1pD

2pU − 3pD = 0

pU + pD = 1

{pU , pM , pD} = {.6, 0, .4}

Julio Davila Normal Form Games