04.slides

Mixed Strategies

ECON2112

Mixed StrategiesIntroduction

Example

heads tails

heads 1,−1 −1, 1

tails −1, 1 1,−1

◮ This game does not have a Nash equilibrium in pure strategies.

◮ Both players randomizing and playing 12heads + 1

2tails is a Nash

equilibrium.

◮ Before that, we need to assume that players have preferences

defined over the set ∆(S) that are representable by means of a

von-Newman Morgenstern utility function.

Example

heads tails

heads 1,−1 −1, 1

tails −1, 1 1,−1

2tails is a Nash

equilibrium.

Example

heads tails

heads 1,−1 −1, 1

tails −1, 1 1,−1

2tails is a Nash

equilibrium.

General Setting. We will have:

◮ A set of players N = {1, . . . ,n},

◮ For each i ∈ N a finite set of pure strategies Si = {s1i , . . . ,s

◮ The set of pure strategy profiles is S = S1 ×·· ·×Sn.

◮ Players have preferences defined ∆(S). (Not simply over S as in

the previous lecture.) We need that because players are now

allowed to randomize between pure strategies.

◮ We will assume that such preferences are representable by

means of a von-Newman Morgenstern utility function.

◮ That means that we can represent preferences over ∆(S)assigning numerical values to each pure strategy profile.

Example (Matching Pennies (1))

T 1,−1 −1,1

B −1,1 1,−1

◮ S1 = {T ,B}, S2 = {L,R}.

◮ S = S1 ×S2. i.e. S = {(T ,L),(T ,R),(B,L),(B,R)}.

◮ Preferences are defined over ∆(S) (not over S).

◮ We assume that such preferences are representable by means of

a von-Newman Morgenstern utility function.

◮ Therefore, for each player, we assign numerical values:

u1(T ,L) = 1,u1(T ,R) = −1,u1(B,L) = −1,u1(B,R) = 1 and

u2(T ,L) = −1,u2(T ,R) = 1,u2(B,L) = 1,u2(B,R) = −1.

T 1,−1 −1,1

B −1,1 1,−1

◮ S1 = {T ,B}, S2 = {L,R}.

◮ S = S1 ×S2. i.e. S = {(T ,L),(T ,R),(B,L),(B,R)}.

u2(T ,L) = −1,u2(T ,R) = 1,u2(B,L) = 1,u2(B,R) = −1.

T 1,−1 −1,1

B −1,1 1,−1

◮ S1 = {T ,B}, S2 = {L,R}.

◮ S = S1 ×S2. i.e. S = {(T ,L),(T ,R),(B,L),(B,R)}.

u2(T ,L) = −1,u2(T ,R) = 1,u2(B,L) = 1,u2(B,R) = −1.

T 1,−1 −1,1

B −1,1 1,−1

◮ S1 = {T ,B}, S2 = {L,R}.

◮ S = S1 ×S2. i.e. S = {(T ,L),(T ,R),(B,L),(B,R)}.

u2(T ,L) = −1,u2(T ,R) = 1,u2(B,L) = 1,u2(B,R) = −1.

T 1,−1 −1,1

B −1,1 1,−1

◮ S1 = {T ,B}, S2 = {L,R}.

◮ S = S1 ×S2. i.e. S = {(T ,L),(T ,R),(B,L),(B,R)}.

u2(T ,L) = −1,u2(T ,R) = 1,u2(B,L) = 1,u2(B,R) = −1.

T 1,−1 −1,1

B −1,1 1,−1

◮ Now, given a particular probability distribution over S, we can

calculate expected utilities.

◮ Suppose player 2 plays σ2 = 12L+ 1

◮ Then player 1’s expected utility from playing T is

◮ U1(T ,σ2) = 121+ 1

2(−1) = 0.

◮ And player 1’s expected utility from playing B is

◮ U1(B,σ2) = 12(−1)+ 1

21 = 0.

T 1,−1 −1,1

B −1,1 1,−1

◮ U1(T ,σ2) = 121+ 1

2(−1) = 0.

◮ U1(B,σ2) = 12(−1)+ 1

21 = 0.

T 1,−1 −1,1

B −1,1 1,−1

◮ U1(T ,σ2) = 121+ 1

2(−1) = 0.

◮ U1(B,σ2) = 12(−1)+ 1

21 = 0.

T 1,−1 −1,1

B −1,1 1,−1

◮ U1(T ,σ2) = 121+ 1

2(−1) = 0.

◮ U1(B,σ2) = 12(−1)+ 1

21 = 0.

T 1,−1 −1,1

B −1,1 1,−1

◮ Therefore, if player 2 plays σ2 = 12L+ 1

2R, player 1 is indifferent

between T , B.

◮ Player 1, will also be indifferent between T , B and αT +(1−α)Bfor any value of α ∈ [0,1].

◮ We can repeat the same argument changing the roles of player 1

and player 2.

◮ Consequently, (12T + 1

12L+ 1

2R) is a Nash equilibrium.

T 1,−1 −1,1

B −1,1 1,−1

between T , B.

and player 2.

12L+ 1

T 1,−1 −1,1

B −1,1 1,−1

between T , B.

and player 2.

12L+ 1

T 1,−1 −1,1

B −1,1 1,−1

between T , B.

and player 2.

12L+ 1

Example (The Battle of the Sexes (1))

T 3,1 0,0

B 0,0 1,3

◮ The set of pure strategy profiles is

S = {(T ,L),(T ,R),(B,L),(B,R)}.

◮ Preferences are defined over ∆(S).

◮ Player i has to choose a strategy from ∆(Si).

◮ The game has two Nash equilibria in pure strategies (T ,L) and

(B,R).

◮ What about Nash equilibria in mixed strategies?

T 3,1 0,0

B 0,0 1,3

S = {(T ,L),(T ,R),(B,L),(B,R)}.

(B,R).

T 3,1 0,0

B 0,0 1,3

S = {(T ,L),(T ,R),(B,L),(B,R)}.

(B,R).

T 3,1 0,0

B 0,0 1,3

S = {(T ,L),(T ,R),(B,L),(B,R)}.

(B,R).

T 3,1 0,0

B 0,0 1,3

S = {(T ,L),(T ,R),(B,L),(B,R)}.

(B,R).

T 3,1 0,0

B 0,0 1,3

◮ We can compute player 1’s expected utilities from T :

◮ U1(T ,σ2) = 143+ 3

40 = 3

◮ And player 1’s expected utility from B:

◮ U1(B,σ2) = 140+ 3

41 = 3

T 3,1 0,0

B 0,0 1,3

◮ U1(T ,σ2) = 143+ 3

40 = 3

◮ U1(B,σ2) = 140+ 3

41 = 3

T 3,1 0,0

B 0,0 1,3

◮ U1(T ,σ2) = 143+ 3

40 = 3

◮ U1(B,σ2) = 140+ 3

41 = 3

T 3,1 0,0

B 0,0 1,3

◮ Player 1 is indifferent between T and B. Hence, he is also

indifferent between T , B, and randomizing between the two.

◮ If player 1 plays σ1 = 34T + 1

4B then player 2 is indifferent

between L, R and randomizing between the two.

◮ Therefore, σ = (σ1,σ2) = (34T + 1

14L+ 3

4R) is a Nash

equilibrium.

T 3,1 0,0

B 0,0 1,3

◮ Under the strategy profile (34T + 1

14L+ 3

◮ (T ,L) occurs with probability 316

◮ (T ,R) occurs with probability 916

◮ (B,L) occurs with probability 116

◮ (B,R) occurs with probability 316

Mixed StrategiesNormal Form Games

Definition (Finite Normal Form Game)

A finite normal form game G = (N,{Si}i ,{ui}i) consists of

◮ a set of players N,

◮ for each i ∈ N a set of pure strategies Si , and

◮ for each i ∈ N a utility function ui : S → R.

(Where we assume that the u′i s are bernullian utility functions.)

Mixed StrategiesMixed Strategy Set.

Consider a normal form game G = (N,{Si}i ,{ui}i):

◮ Si is player i ’s set of pure strategies.

◮ Σi = ∆(Si) is player i ’s set of mixed strategies

Example

◮ Suppose that player 1’s set of pure strategies is S1 = {T ,B}.

◮ Then player 1’s set of mixed strategies is:Σ1 = ∆(S1) = {αT +(1−α)B : α ∈ [0,1]}.

◮ For instance:

◮12T + 1

◮34T + 1

◮15T + 4

◮ 0T +1B = B

are mixed strategies.

Mixed StrategiesNash Equilibrium

Consider a game G = (N,{Si}i ,{ui}i), and denote the set of mixed

strategy profiles as Σ = Σ1 ×·· ·×Σn, where Σi = ∆(Si).

Definition (Nash Equilibrium)

A strategy profile σ∗ = (σ∗1, . . . ,σ

∗n) ∈ Σ is a Nash equilibrium if for

each i ∈ N

Ui(σ∗−i ,σ

∗i ) ≥ Ui(σ

∗−i ,σi) for every σi ∈ Σi .

Mixed StrategiesCarrier or Support of a Mixed Strategy

Definition (Carrier or Support)

The carrier or support of a mixed strategy σi , which we denote as

C (σi), is the set of pure strategies that receive strictly positive

probability from σi . That is, C (σi) = {si ∈ Si : σsi

i > 0}.

Example

◮ Let S1 = {T ,M,B} be the set of pure strategies of player 1.

◮ Consider the mixed strategy σ1 = 13T + 2

◮ The Carrier of σ1 is C (σ1) = {T ,M}.

i > 0}.

Example

i > 0}.

Example

i > 0}.

Example

Mixed StrategiesMixed Strategies in a Nash Equilibrium

Let σ be a Nash Equilibrium.

◮ Notice that no player can unilaterally deviate and obtain a higher

payoff.

◮ Suppose that for player i , the carrier of σi contains more than one

pure strategy

◮ Then if players other than i play according to σ−i , player i is

indifferent between any two elements of C (σi)

In other words, if a player plays a mixed strategy in equilibrium, then

he is indifferent among all his pure strategies that receive positive

probability.

Mixed StrategiesMixed Strategies in a Nash Equilibrium

Example

T 3,1 0,0

B 0,0 1,3

Under the Nash equilibrium (34T + 1

14L+ 3

4R) player 1 is indifferent

between T and B and player 2 is indifferent between L and R.

This implies that:

◮ An equilibrium in dominant strategies is always in pure strategies

◮ A strict equilibrium is always in pure strategies.

Mixed StrategiesDominated Strategies

Definition (Strictly Dominated Strategy)

A strategy σi is strictly dominated by σ′i if for every σ−i ∈ Σ−i

Ui(σ−i ,σi) < Ui(σ−i ,σ′i).

However, we only need to check against pure strategy profiles of the

opponents. This implies:

Definition (Strictly Dominated Strategy)

A strategy σi is strictly dominated by σ′i if for every s−i ∈ S−i

Ui(s−i ,σi) < Ui(s−i ,σ′i).

Mixed StrategiesWeakly Dominated Strategy

Definition (Weakly Dominated Strategy)

A strategy σi is weakly dominated by σ′i if for every σ−i ∈ Σ−i

Ui(σ−i ,σi) ≤ Ui(σ−i ,σ′i)

and there exists at least one σ′−i such that

Ui(σ′−i ,σi) < Ui(σ

′−i ,σ

′i).

Mixed StrategiesWeakly Dominated Strategy

Again, we only need to check against pure strategy profiles of the

opponents.

Definition (Weakly Dominated Strategy)

A strategy σi is weakly dominated by σ′i if for every s−i ∈ S−i

Ui(s−i ,σi) ≤ Ui(s−i ,σ′i)

and there exists at least one s′−i such that

Ui(s′−i ,σi) < Ui(s

′−i ,σ

′i).

Mixed StrategiesDominated strategies. Properties.

◮ If a pure strategy si is dominated then every mixed strategy that

gives si a strictly positive probability is also dominated.

Example

T 1,1 0,0

B 0,0 0,0

◮ B is dominated.

◮ σ1 = 12T + 1

2B is also dominated.

◮ If player 2 plays L, σ1 gives a payoff equal to 12

and T gives a

payoff equal to 1.◮ If player 2 plays R, both σ1 and T give a payoff equal to 0.

◮ More generally, if a mixed strategy σi is dominated then every

mixed strategy whose carrier coincides with or contains C (σi) is

also dominated.

◮ A mixed strategy that does not give strictly positive probability to

any dominated pure strategy may be dominated.

Example

T 10,10 0,0

M 0,0 10,10

B 6,6 6,6

◮ T is not dominated.

◮ M is not dominated.

◮ But 12T + 1

2M is strictly dominated by B

◮ Therefore, the following strategies are dominated:

◮ αT +(1−α)M, for every α ∈ (0,1)◮ αT +βM +(1−α−β)B, for every α > 0 and β > 0 such that

α+β < 1.

is dominated.

◮ A strategy (pure of mixed) σi may only be dominated by a mixed

strategy.

Example

T 10,10 0,0

M 0,0 10,10

B 4,4 4,4

◮ B is not dominated by T .

◮ B is not dominated by M.

◮ But B is dominated by 12T + 1

Mixed StrategiesAdmissible Equilibrium

Definition (Undominated Nash Equilibrium)

The strategy profile σ∗ is an undominated equilibrium, or admissible

equilibrium, if σ∗ is a Nash equilibrium where no player uses a

dominated strategy.

Mixed StrategiesBest Reply Correspondence

Definition (Pure Best Reply)

We say that player i ’s pure strategy si is a pure best reply against σ−i

if for all s′iUi(σ−i ,si) ≥ Ui(σ−i ,s

′i ).

The set of player i ’s best replies against σ−i is denoted PBRi(σ−i), or

simply PBRi(σ).

Mixed StrategiesBest Reply Correspondence

Definition (Best Reply)

We say that player i ’s pure strategy σi is a best reply against σ−i if for

all σ′i

Ui(σ−i ,σi) ≥ Ui(σ−i ,σ′i).

◮ We denote the set of player i ’s best replies against σ−i as

BRi(σ−i), or simply BRi(σ)

◮ Notice that BRi(σ−i) = BRi(σ) = ∆(PBRi(σ)).

Mixed StrategiesBest Reply Correspondence. Nash Equilibrium

Definition (Nash Equilibrium)

The strategy profile σ∗ is a Nash equilibrium if every player is playing a

best reply against σ∗. That is, σ∗i ∈ BRi(σ

∗) for every i ∈ N.

Mixed StrategiesBest Reply Correspondence. Example

Example (The Battle of the Sexes. (1))

T 3,1 0,0

B 0,0 1,3

◮ Suppose that player 2 plays L with probability y and R with

probability 1− y .

◮ We have that:

◮ U1(T ,y) = 3y◮ U1(B,y) = 1− y .

T 3,1 0,0

B 0,0 1,3

◮ We have that:

◮ U1(T ,y) = 3y◮ U1(B,y) = 1− y .

T 3,1 0,0

B 0,0 1,3

◮ We have that:

◮ U1(T ,y) = 3y◮ U1(B,y) = 1− y .

Mixed StrategiesBest Reply Correspondence. Example.

1− y

T 3,1 0,0

B 0,0 1,3

U1(T ,y) =3y

U1(B,y) =1− y .

T %1 B if and only if y ≥ 14.

f x represents the probability that player 1 plays T , we have that:

BR1(y)

x = 1 if y >14

x ∈ [0,1] if y = 14

x = 0 if y <14.

BR1(y)

x T 3,1 0,0

(1− x) B 0,0 1,3

We can also calculate BR2:

U2(L,x) =x

U2(R,x) =3(1− x).

L %2 R if and only if x ≥ 34.

(Remember that y represents the probability that 2 plays L)

BR2(x)

y = 1 if x >34

y ∈ [0,1] if x = 34

y = 0 if x <34

BR1(y)

BR2(x)

The set of Nash equilibria corresponds to all the points where BR1(y)and BR2(x) intersect.

◮ (T ,L) corresponds to x = 1 and y = 1.

◮ (B,R) corresponds to x = 0 and y = 0.

◮ (34T + 1

14L+ 3

4R) corresponds to x = 3

4and y = 1

04.slides

Documents

set of players n

set of pure strategy

representable bymeans

previous lecture

sassigning numerical

nash equilibrium