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• Mixed strategy: player chooses a probability distribution (p1,p2,..,pN) over her set of actions rather than a single action
• If there is no NE without mixing, we will find at least one MSNE (Nash - proof)
• review example • best response functions – graphs
• Game
• Elimination of strategies that are strictly dominated by mixed strategies
• illustration • example
• Game
• Elimination of strategies that are strictly dominated by mixed strategies
• illustration • example
/ 32
2
Review Summary
MSNE
Elimination By Mixing
That is why we only eliminate strictly dominated strategies
/ 32
3
Review Summary
MSNE
Elimination By Mixing
Review
Mixed strategy NE
• need for making oneself unpredictable leads to mixing strategies
{0}
if q < 1/3
{p: 0≤p≤1} if q = 1/3
{1}
if q > 1/3
{0}
if p < 2/3
{q: 0≤q≤1} if p = 2/3
{1}
if p > 2/3
MSNE: {(2/3,1/3); (1/3,2/3)}
/ 32
7
Review Summary
MSNE
Elimination By Mixing
2 1
T (p)
B (1-p)
L (q) 0,1 2,2
R (1-q) 0,2 0,1
two NE: (T,R) and (B,L) any MSNE?
• P1 must be indifferent between T and B (otherwise not mixing, playing pure strategy):
q*0+(1-q)*0 = q*2+(1-q)*0 => q=0
T
B
• P2 must be indifferent between L and R: p*1+(1-p)*2 = p*2+(1-p)*1 => p=1/2
/ 32
9
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
• player 1 (2) chooses B with probability p (q) and S with probability 1-p (1-q)
/ 32
8
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
Mixed Strategy NE
2 1
T (p)
B (1-p)
L (q) 0,1 2,2
R (1-q) 0,2 0,1
If q>0: B is better than T If q=0: B is as good as S
B1(q) =
{0}
if q > 0
{p: 0≤p≤1} if q = 0
• If NE without mixing exists, we may find additional MSNE
/ 32
4
Review Summary
MSNE
Elimination By Mixing
Preview
• Review
• Mixed strategy Nash equilibrium
/ 32
wenku.baidu.com
5
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
2 1
B (p)
S (1-p)
B (q) 2,1 0,0
S (1-q) 0,0 1,2
two NE: (B,B) and (S,S) any MSNE?
• P1 must be indifferent between B and S (otherwise not mixing, playing pure strategy):
q*2+(1-q)*0 = q*0+(1-q)*1 => q=1/3
B
S
• P2 must be indifferent between B and S: p*1+(1-p)*0 = p*0+(1-p)*2 => p=2/3
/ 32
6
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
2 1
B (p)
S (1-p)
B (q) 2,1 0,0
S (1-q) 0,0 1,2
If q<1/3: S is better than B If q>1/3: B is better than S If q=1/3: B is as good as S
B1(q) = B2(p) =
Review
Strict vs. weak dominance
Elimination of weakly dominated strategies leads to:
• strict Nash equilibria • but can eliminate nonstrict Nash equilibria
Introduction to Game Theory
Lecture 4
Review Summary
MSNE
Elimination By Mixing
Preview
• Review
• Mixed strategy Nash equilibrium
• review example • best response functions – graphs
• If there is no NE without mixing, we will find at least one MSNE (Nash - proof)
• review example • best response functions – graphs
• Game
• Elimination of strategies that are strictly dominated by mixed strategies
• illustration • example
• Game
• Elimination of strategies that are strictly dominated by mixed strategies
• illustration • example
/ 32
2
Review Summary
MSNE
Elimination By Mixing
That is why we only eliminate strictly dominated strategies
/ 32
3
Review Summary
MSNE
Elimination By Mixing
Review
Mixed strategy NE
• need for making oneself unpredictable leads to mixing strategies
{0}
if q < 1/3
{p: 0≤p≤1} if q = 1/3
{1}
if q > 1/3
{0}
if p < 2/3
{q: 0≤q≤1} if p = 2/3
{1}
if p > 2/3
MSNE: {(2/3,1/3); (1/3,2/3)}
/ 32
7
Review Summary
MSNE
Elimination By Mixing
2 1
T (p)
B (1-p)
L (q) 0,1 2,2
R (1-q) 0,2 0,1
two NE: (T,R) and (B,L) any MSNE?
• P1 must be indifferent between T and B (otherwise not mixing, playing pure strategy):
q*0+(1-q)*0 = q*2+(1-q)*0 => q=0
T
B
• P2 must be indifferent between L and R: p*1+(1-p)*2 = p*2+(1-p)*1 => p=1/2
/ 32
9
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
• player 1 (2) chooses B with probability p (q) and S with probability 1-p (1-q)
/ 32
8
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
Mixed Strategy NE
2 1
T (p)
B (1-p)
L (q) 0,1 2,2
R (1-q) 0,2 0,1
If q>0: B is better than T If q=0: B is as good as S
B1(q) =
{0}
if q > 0
{p: 0≤p≤1} if q = 0
• If NE without mixing exists, we may find additional MSNE
/ 32
4
Review Summary
MSNE
Elimination By Mixing
Preview
• Review
• Mixed strategy Nash equilibrium
/ 32
wenku.baidu.com
5
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
2 1
B (p)
S (1-p)
B (q) 2,1 0,0
S (1-q) 0,0 1,2
two NE: (B,B) and (S,S) any MSNE?
• P1 must be indifferent between B and S (otherwise not mixing, playing pure strategy):
q*2+(1-q)*0 = q*0+(1-q)*1 => q=1/3
B
S
• P2 must be indifferent between B and S: p*1+(1-p)*0 = p*0+(1-p)*2 => p=2/3
/ 32
6
Review Summary
MSNE
Elimination By Mixing
Mixed Strategy NE
2 1
B (p)
S (1-p)
B (q) 2,1 0,0
S (1-q) 0,0 1,2
If q<1/3: S is better than B If q>1/3: B is better than S If q=1/3: B is as good as S
B1(q) = B2(p) =
Review
Strict vs. weak dominance
Elimination of weakly dominated strategies leads to:
• strict Nash equilibria • but can eliminate nonstrict Nash equilibria
Introduction to Game Theory
Lecture 4
Review Summary
MSNE
Elimination By Mixing
Preview
• Review
• Mixed strategy Nash equilibrium
• review example • best response functions – graphs