Strategic Form Games
It is used to model situations in which players choose strategies without
knowing the strategy choices of the other players
Also known as normal form games
Microeconomics II
Definition (strategic form game)
Strategic Form Games
A strategic form game is composed of
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Koç University
1
Set of players: N = {1, 2, . . . , n}
2
A set of strategies: Si for each player i
3
A payoff function: ui : S → R for each player i
It is denoted by G = (N, (Si )i∈N , (ui )i∈N )
An outcome (or a strategy profile) s = (s1 , . . . , sn ) is a collection of
strategies, one for each player
Outcome space
S = ×i∈N Si = {(s1 , . . . , sn ) : si ∈ Si , i = 1, . . . , n}
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Nash Equilibrium
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Remarks
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Nash Equilibrium
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Prisoners’ Dilemma
N = {1, 2}
We will sometimes write G = (N, (Si ), (ui )), or simply G
Si = {C, N}, i = 1, 2
Payoff functions represent preferences over the set of outcomes and are
ordinal (for now)
Player 2
If Si is finite for all i ∈ N , then the game is called a finite game
The game is common knowledge
Player 1
Finite strategic form games with two players can be represented by a
bimatrix
C
−5, −5
−6, 0
C
N
N
0, −6
−1, −1
The following also represents the same game whenever a < b < c < d.
Player 2
Player 1
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U
D
L
u1 (U, L), u2 (U, L)
u1 (D, L), u2 (D, L)
Nash Equilibrium
R
u1 (U, R), u2 (U, R)
u1 (D, R), u2 (D, R)
Player 2
Player 1
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C
N
C
b, b
a, d
Nash Equilibrium
N
d, a
c, c
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Other Examples
Other Examples
Hawk-Dove
Battle of the Sexes
Player 2
Player 1
D
H
D
3, 3
5, 1
Player 2
H
1, 5
0, 0
Player 1
Stag Hunt
B
2, 1
0, 0
B
S
S
0, 0
1, 2
Matching Pennies
Player 2
Player 2
Player 1
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S
H
S
2, 2
1, 0
H
0, 1
1, 1
Nash Equilibrium
Player 1
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Guessing Game
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H
1, −1
−1, 1
T
−1, 1
1, −1
Nash Equilibrium
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Cournot Duopoly Model
There are n players: N = {1, 2, . . . , n}
There are only two firms
Each player picks a number between 1 and 99:
Inverse (market) demand function: p(q1 + q2 )
◮ p : R+ → R+
◮ p′ < 0
Si = {1, 2, . . . , 99} for all i ∈ N
Cost function of firm i = 1, 2: ci (qi )
◮ c i : R+ → R+
◮ c′ > 0, c′′ ≥ 0
i
i
A players wins if her number is (among the) closest to 2/3 of the average
( 1, si − 23 s ≤ sj − 23 s for all j = 1, . . . , n
ui (s1 , . . . , sn ) =
0, otherwise
where
The strategic form is given by
n
s=
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H
T
1
∑ si
n i=1
Nash Equilibrium
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1
N = {1, 2}
2
Si = R+
3
ui (q1 , q2 ) = p(q1 + q2 )qi − ci (qi ) for each (q1 , q2 ) ∈ S
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Nash Equilibrium
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Nash Equilibrium
Prisoners’ Dilemma (PD)
A solution concept for a strategic for game G = (N, (Si )i∈N , (ui )i∈N ) is a
strategy profile s ∈ S
A Nash equilibrium is a strategy profile such that given the strategies of all
the other players each player’s strategy maximizes her payoff
Let
s−i
(s′i , s−i )
Player 1
= (s1 , s2 , . . . , si−1 , s′i , si+1 , . . . , sn )
S−i = ×j∈N\{i} Sj
A strategy profile
each player i ∈ N
C
N
C
−5, −5
−6, 0
N
0, −6
−1, −1
Is (N, N) a Nash equilibrium?
Definition (Nash Equilibrium)
s∗
Player 2
= (s1 , s2 , . . . , si−1 , si+1 , . . . , sn )
How about (C, C)?
∈ S is a Nash equilibrium of G = (N, (Si )i∈N , (ui )i∈N ) if for
ui (s∗i , s∗−i ) ≥ ui (si , s∗−i )
N(PD) = {(C, C)}
for all si ∈ Si
The set of Nash equilibria is denoted N(G)
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Nash Equilibrium
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More Examples
Battle of the Sexes
Player 2
D
H
D
3, 3
5, 1
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One way to find Nash equilibria is to first find the best response
correspondence for each player
Player 2
H
1, 5
0, 0
Player 1
N(HD) = {(H, D), (D, H)}
Stag Hunt
B
S
B
2, 1
0, 0
◮
S
0, 0
1, 2
S
2, 2
1, 0
Definition (best response correspondence)
N(BS) = {(B, B), (S, S)}
The best response correspondence of player i ∈ N is given by Bi : S−i ⇒ Si
such that
Bi (s−i ) = {si ∈ Si : ui (s) ≥ ui (s′i , s−i ) for all s′i ∈ Si }
Player 2
H
0, 1
1, 1
H
Player 1
T
T
−1, 1
1, −1
Proposition
Let G = (N, (Si )i∈N , (ui )i∈N ) be a strategic form game. A strategy profile s∗ ∈ S
is a Nash equilibrium of G iff for each player i ∈ N
N(MP) = 0/
N(SH) = {(S, S), (H, H)}
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H
1, −1
−1, 1
Best response correspondence gives the set of payoff maximizing
strategies for each strategy profile of the other players
... and then find where they “intersect”
Matching Pennies
Player 2
S
Player 1
H
Nash Equilibrium
Best Response Correspondences
Hawk-Dove
Player 1
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Nash Equilibrium
s∗i ∈ Bi (s∗−i )
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Nash Equilibrium
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Examples
Nash Equilibria of Cournot Duopoly
Let
Hawk-Dove
Battle of the Sexes
D
H
D
3, 3
5, 1
H
1, 5
0, 0
B
S
B
2, 1
0, 0
p(q1 + q2 ) =
S
0, 0
1, 2
B1 (D) = {H}, B1 (H) = {D}
B1 (B) = {B}, B1 (S) = {S}
B2 (D) = {H}, B2 (H) = {D}
B2 (B) = {B}, B2 (S) = {S}
N(HD) = {(H, D), (D, H)}
N(BS) = {(B, B), (S, S)}
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Nash Equilibrium
(
a − b(q1 + q2 ), q1 + q2 ≤ a/b
0,
q1 + q2 > a/b
and for each i = 1, 2
ci (qi ) = cqi
where a > c ≥ 0 and b > 0
Therefore, payoff function of firm i = 1, 2 is given by
ui (q1 , q2 ) =
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(
(a − c − b(q1 + q2 ))qi , q1 + q2 ≤ a/b
−cqi ,
q1 + q2 > a/b
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Nash Equilibrium
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Claim
Best response correspondence of firm i 6= j is given by
( a−c−bq
j
Bi (qj ) =
2b
0,
, qj <
qj ≥
Claim
a−c
b
a−c
b
The set of Nash equilibria of the Cournot duopoly game is given by
N(G) =
Proof.
If q2 ≥ a−c
b , then u1 (q1 , q2 ) < 0 for any q1 > 0. Therefore, q1 = 0 is the
unique payoff maximizer.
a−c a−c
,
3b
3b
Proof.
If q2 < a−c
b , then the best response cannot be q1 = 0 (why?).
a
Furthermore, it must be the case that q1 + q2 ≤ a−c
b ≤ b , for otherwise
u1 (q1 , q2 ) < 0. So, the following first order condition must hold
Suppose (q∗1 , q∗2 ) is a Nash equilibrium and q∗i = 0. Then,
q∗j = (a − c)/2b < (a − c)/b. But, then q∗i ∈
/ Bi (q∗j ), a contradiction. Therefore,
we must have 0 < q∗i < (a − c)/b, for i = 1, 2. The rest follows from the best
response correspondences.
∂u1 (q1 , q2 )
= a − c − 2bq1 − bq2 = 0
∂q1
Similarly for firm 2.
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Nash Equilibrium
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Nash Equilibrium
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Cournot Nash Equilibrium
Existence of Nash Equilibrium
q2
Definition (correspondence)
A correspondence Γ : X ⇒ Y is any mapping that associates with each x ∈ A a
subset Γ(x) of Y .
a−c
b
BR1 (q2 )
Definition
Γ is nonempty and convex-valued if Γ(x) is nonempty and convex for each
x ∈ X.
a−c
2b
a−c
3b
Nash Equilibrium
Definition
Γ : X ⇒ Y has a closed graph if, for all x ∈ X ,
xm → x, ym ∈ Γ(xm ), ym → y ⇒ y ∈ Γ(x)
BR2 (q1 )
q1
a−c
3b
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a−c
2b
a−c
b
Nash Equilibrium
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Nash Equilibrium
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Theorem (Nash’s Existence Theorem)
Let G = (N, (Si )i∈N , (ui )i∈N ) be a strategic form game such that
Si is a nonempty, convex and compact subset of Rmi
Theorem (Kakutani’s Fixed Point Theorem)
ui is continuous on S and quasi-concave on Si , i = 1, . . . , n
Let S ⊂ Rn be a compact and convex set. If Γ : S ⇒ S is a nonempty and
convex-valued correspondence with a closed graph, then there exists an s ∈ S
such that s ∈ Γ(s).
Then N(G) 6= 0/ .
Proof.
Define the correspondence B : S ⇒ S by
Proof.
See Claude Berge, Topological Spaces (1963, p. 74-76).
B(s) ≡ {x ∈ S : x1 ∈ B1 (s−1 ), x2 ∈ B2 (s−2 ), . . . , xn ∈ Bn (s−n )}.
We will show that Kakutani’s theorem applies to B.
S is compact and convex (since each Si is)
B is nonempty-valued (Weierstrass’ theorem)
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Nash Equilibrium
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Nash Equilibrium
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Proof (continued)
B is convex-valued: Take any x, y ∈ B(s). Then, for all i ∈ N ,
ui (xi , s−i ) = ui (yi , s−i ) ≥ ui (si , s−i ), for all si ∈ Si . Quasi-concavity of ui on
Si implies
ui (λxi + (1 − λ)yi , s−i ) ≥ ui (xi , s−i ) ≥ ui (si , s−i ), ∀si ∈ Si , λ ∈ [0, 1].
Nash Equilibrium
sm → s, bm → b, bm ∈ B(sm )
Need to show: b ∈ B(s).
Intuitively:
m
m
1
ui (bm
i , s−i ) ≥ ui (xi , s−i ) for all xi ∈ Si
m
m
2
(bm
i , s−i ) → (bi , s−i ) and (xi , s−i ) → (xi , s−i ) imply
3
ui (bi , s−i ) ≥ ui (xi , s−i ) for all xi ∈ Si , by continuity of ui
Therefore, λx + (1 − λ)y ∈ B(s).
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Proof (continued)
B has closed graph: Take any
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Nash Equilibrium
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Parametric Games
Proof (continued)
Assume that ui : S × Θ → R
More formally... Suppose not:
Then the Nash equilibrium set would depend on θ
∃i ∈ N, xi ∈ Si : α ≡ ui (xi , s−i ) − ui (bi , s−i ) > 0
What happens to the equilibrium set as a result of small changes in θ?
ui continuous: ∀ε > 0, ∃Mε such that
ui (bm , sm ) − ui (bi , s−i ) < ε and ui (xi , sm ) − ui (xi , s−i ) < ε, ∀m ≥ Mε
i
−i
−i
Proposition
Let Θ 6= 0/ be a compact set in Rk and let Si 6= 0/ be a convex and compact
subset of Rmi , i = 1, . . . , n. Assume that ui : S × Θ → R is a continuous
function which is quasi-concave on Si and consider the game
G(θ) ≡ (N, (Si ), (ui (., θ))), where θ ∈ Θ. Define the correspondence
Γ : Θ ⇒ S by Γ(θ) ≡ N(G(θ)). Then, Γ is a nonempty-valued correspondence
with a closed graph.
So, for any ε > 0 and any m ≥ Mε
m m
ui (xi , sm
−i ) > ui (xi , s−i ) − ε = ui (bi , s−i ) + α − ε > ui (bi , s−i ) + α − 2ε
m m
ε = α/2 ⇒ ui (xi , sm
−i ) > ui (bi , s−i ), ∀m ≥ Mα/2
m
This contradicts bm
i ∈ Bi (s−i ) for all m.
Proof.
Exercise
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Nash Equilibrium
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Parametric Games: Example
Symmetric Games
Let A = [0, 1], Θ = [−1, 1], u(x, θ) = 1 + θx
Nash equilibrium correspondence:
Γ[θ] =
Definition
A strategic form game G = (N, (Si ), (ui )) is called symmetric if Si = Sj and
ui (s) = uj (s′ ) for all i, j = 1, . . . , n and all s, s′ ∈ S such that s′ is obtained from s
by exchanging si and sj .


θ<0
0,
[0, 1], θ = 0


1,
θ>0
As θ → 0, Nash equilibrium of G(θ) converges to an equilibrium of G(0)
Player 2
Player 2
Γ(θ)
1
Battle of the Sexes
Prisoners’ Dilemma
There may exist other equilibria of G(0)
Player 1
b
b
b
b
−1
0
1
C
N
C
−5, −5
−6, 0
N
0, −6
−1, −1
Player 1
Symmetric?
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B
S
B
2, 1
0, 0
S
0, 0
1, 2
Symmetric?
θ
Nash Equilibrium
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Symmetric Games
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Nash Equilibrium
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Strict Nash Equilibrium
Definition
A strategy profile s∗ ∈ S is a strict Nash equilibrium of G = (N, (Si )i∈N , (ui )i∈N )
if for each player i ∈ N
Proposition
ui (s∗i , s∗−i ) > ui (si , s∗−i )
Let G = (N, (Si ), (ui )) be a symmetric strategic form game such that
Si is a nonempty, convex and compact subset of Rmi
for all si ∈ Si with si 6= s∗i
Strict Nash equilibria are robust
ui is continuous on S and quasi-concave on Si , i = 1, . . . , n
◮
Then, there exists a strategy profile (s∗ , . . . , s∗ ) ∈ S such that
(s∗ , . . . , s∗ ) ∈ N(G).
they remain strict equilibria when the payoff functions are slightly perturbed
Not every game has a strict Nash equilibrium
Not every Nash equilibrium is strict
Proof.
U
D
Exercise
L
1, 1
0, 0
R
0, 0
0, 2
(U, L) is a strict Nash equilibrium
(D, R) is not
What happens if u1 (U, R) = ε, for any ε > 0?
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Nash Equilibrium
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Nash Equilibrium
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Existence Again
Mixed Strategies
For any finite set X with m elements, let ∆(X) denote the set of all probability
distributions over X
Does the Matching Pennies game have a Nash equilibrium?
Player 2
Player 1
H
T
H
1, −1
−1, 1
∆(X) = {σ ∈ Rm
+:
T
−1, 1
1, −1
Definition (Mixed Strategies)
How would you play?
Let G = (N, (Si ), (ui )) be a strategic form game. A mixed strategy σi for player
i ∈ N is a probability distribution over Si , i.e., σi ∈ ∆(Si ).
You should try to be unpredictable
Choose randomly
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We assume that mixed strategies are independent across players.
Nash Equilibrium
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Mixed Strategies
◮
Nash Equilibrium
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Assume that players’ preferences are defined over lotteries on S
and that ui is a von Neumann-Morgenstern utility function for each i ∈ N
This is the set of strategies that are played with positive probability
A pure strategy is a degenerate mixed strategy
◮
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Mixed Strategies
For any σi ∈ ∆(Si ) and si ∈ Si , let σi (si ) denote the probability assigned to
strategy si
support of a mixed strategy is supp(σi ) = {si ∈ Si : σi (si ) > 0}
◮
m
∑ σk = 1}
k=1
so that they can be represented by the expected payoff function
It assigns probability one to only one member of Si , say s′i
We sometimes refer to it as s′i
Ui (σ) = ∑ p(s|σ)ui (s)
A mixed strategy profile: σ = (σ1 , . . . , σn ) ∈ ×i∈N ∆(Si )
A mixed strategy profile induces a probability distribution p over the set of
outcomes S
s∈S
= ∑ (∏ σj (sj ))ui (s)
s∈S j∈N
p(s|σ) = ∏ σi (si )
i∈N
Definition
Let
Let G = (N, (Si ), (ui )) be a strategic form game. The mixed extension of G is
given by Γ(G) = (N, (Σi ), (Ui )).
Σi = ∆(Si )
Σ = ×i∈N ∆(Si )
Σ−i = ×j∈N\{i} ∆(Sj )
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Nash Equilibrium
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Nash Equilibrium
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Mixed Strategies
Mixed Strategy Equilibrium
Definition
Claim
Ui is multilinear in each component of σ: For any j ∈ N , σ′j , σ′′j ∈ Σj , σ−j ∈ Σ−j ,
and λ ∈ [0, 1], the following is true
The set of mixed strategy equilibria of a finite strategic form game G is the set
of Nash equilibria of the mixed extension of G. In other words, σ∗ ∈ Σ is a
mixed strategy equilibrium of G if, and only if, for every i ∈ N
Ui (λσ′j + (1 − λ)σ′′j , σ−j ) = λUi (σ′j , σ−j ) + (1 − λ)Ui(σ′′j , σ−j )
Ui (σ∗ ) ≥ Ui (σi , σ∗−i ), for all σi ∈ ∆(Si )
Claim
Let si be a pure strategy and σ a mixed strategy profile. Then
Ui (σ) =
Definition (best response correspondence)
The best response correspondence of player i ∈ N is given by Bi : Σ−i ⇒ Σi
such that
∑ σi (si )Ui (si , σ−i)
si ∈Si
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Nash Equilibrium
Bi (σ−i ) = {σi ∈ Σi : Ui (σ) ≥ Ui (σ′i , σ−i ) for all σ′i ∈ Σi }
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Nash Equilibrium
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Proposition
Let G be a finite strategic form game. A mixed strategy profile σ∗ ∈ Σ is a
mixed strategy equilibrium of G if, and only if, for each player i ∈ N and for any
Theorem
Every finite strategic form game has a mixed strategy equilibrium.
si ∈ supp(σ∗i )
Proof.
Exercise
From now on we use Nash equilibrium and mixed strategy equilibrium
interchangeably
If every player’s strategy in a Nash equilibrium is a pure strategy we call it
a pure strategy equilibrium
Matching pennies game does not have any pure strategy equilibrium
but by the above theorem it must have a (mixed strategy) Nash equilibrium
The following proposition is very useful in calculating mixed strategy
equilibria
si ∈ Bi (σ∗−i )
Proof.
(⇐)
Let σ∗ be a mixed strategy equilibrium. Suppose that for some player i and
some si ∈ supp(σ∗i ) we have si ∈
/ Bi (σ∗−i ), i.e., there exists s′i ∈ Si such that
Ui (s′i , σ∗−i ) > Ui (si , σ∗−i ). But then, player i can increase her payoff by shifting
some probability from si to s′i , a contradiction.
(⇒)
Suppose now that for each player i ∈ N and for any si ∈ supp(σ∗i ) we have
si ∈ Bi (σ∗−i ), but σ∗ is not a mixed strategy equilibrium. Then, there is an i ∈ N
and σ′i ∈ Σi such that Ui (σ′i , σ∗−i ) > Ui (σ∗i , σ∗−i ). But then, there must be a
s′i ∈ supp(σ′i ) and si ∈ supp(σ∗i ) such that Ui (s′i , σ∗−i ) > Ui (si , σ∗−i ), a
contradiction.
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Nash Equilibrium
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Nash Equilibrium
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Matching Pennies
Mixed and Pure Strategy Equilibria
H
1, −1
−1, 1
H
T
T
−1, 1
1, −1
How do you find the set of all (pure and mixed) Nash equilibria?
In 2 × 2 games we can plot the best response correspondences and find
where they intersect
Suppose σ∗ is a Nash equilibrium of this game.
It must be that supp(σ∗i ) = {H, T}, i = 1, 2. Why?
Consider the Battle of the Sexes game
By the above proposition U1 (H, σ∗2 ) = U1 (T, σ∗2 ) or
σ∗2 (H) × 1 + (1 − σ∗2(H)) × (−1) = σ∗2 (H) × (−1) + (1 − σ∗2 (H)) × 1
B
S
which implies σ∗2 (H) = 1/2
Similarly, we have σ∗1 (H) = 1/2. So,
N(MP) =
Levent Koçkesen (Koç University)
B
S
B
2, 1
0, 0
S
0, 0
1, 2
S
0, 0
1, 2
Simplify notation: p = σ1 (B) and q = σ2 (B)
1 1
1 1
,
,
,
2 2
2 2
Nash Equilibrium
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What is Player 1’s best response?
Expected payoff to
◮ B is 2q
◮ S is 1 − q
B
S
B
2, 1
0, 0
S
0, 0
1, 2
Nash Equilibrium
Expected payoff to
◮ B is p
◮ S is 2(1 − p)
If p > 2(1 − p) or p > 2/3
◮ best response is B (or equivalently q = 1)
If 2q < 1 − q or q < 1/3
◮ best response is S (or equivalently p = 0)
If p < 2(1 − p) or p < 2/3
◮ best response is S (or equivalently q = 0)
◮
◮
If p = 2(1 − p) or p = 2/3
he is indifferent
best response is any p ∈ [0, 1]
◮
◮
Player 1’s best response correspondence:
B1 (q) =


{1} , if q > 1/3
[0, 1], if q = 1/3


{0} , if q < 1/3
Nash Equilibrium
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What is Player 2’s best response?
If 2q > 1 − q or q > 1/3
◮ best response is B (or equivalently p = 1)
If 2q = 1 − q or q = 1/3
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B
2, 1
0, 0
she is indifferent
best response is any q ∈ [0, 1]
Player 2’s best response correspondence:
B2 (p) =
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Levent Koçkesen (Koç University)


{1} , if p > 2/3
[0, 1], if p = 2/3


{0} , if p < 2/3
Nash Equilibrium
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

{1} ,
[0, 1],


{0} ,


{1} ,
B2 (p) = [0, 1],


{0} ,
B1 (q) =
if q > 1/3
q
if q = 1/3
if q < 1/3
b
B1 (q)
if p > 2/3
if p = 2/3
if p < 2/3
1/3
Set of Nash equilibria
b
b
0
{(0, 0), (1, 1), (2/3, 1/3)}
Levent Koçkesen (Koç University)
B2 (p)
1
Nash Equilibrium
2/3
1
p
41 / 40