Static (or SimultaneousMove) Games of Complete
Information
Mixed Strategy Nash Equilibrium
F. Valognes - Game Theory - Chp 8
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Outline of Static Games of Complete
Information
  Introduction to games
  Normal-form (or strategic-form)
representation
  Iterated elimination of strictly dominated
strategies
  Nash equilibrium
  Review of concave functions, optimization
  Applications of Nash equilibrium
  Mixed strategy Nash equilibrium
F. Valognes - Game Theory - Chp 8
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Today’s Agenda
  Review of previous class
  Mixed strategy Nash equilibrium in Battle of
sexes
  Use indifference to find mixed strategy Nash
equilibria
F. Valognes - Game Theory - Chp 8
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Mixed strategy equilibrium
  Mixed Strategy:
  A
mixed strategy of a player is a probability
distribution over the player’s strategies.
  Mixed strategy equilibrium
  A probability distribution for each player
  The distributions are mutual best responses to
one another in the sense of expected payoffs
F. Valognes - Game Theory - Chp 8
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Battle of sexes
Pat
Opera (q)
Chris
Opera ( r )
Prize Fight (1-r)
2 ,
0
Prize Fight (1-q)
1
0 ,
0
, 0
1 ,
2
  Chris’ expected payoff of playing Opera: 2q
  Chris’ expected payoff of playing Prize Fight: 1-q
  Chris’ best response B1(q):
 
 
 
Prize Fight (r=0) if q<1/3
Opera (r=1) if q>1/3
Any mixed strategy (0≤r≤1) if q=1/3
F. Valognes - Game Theory - Chp 8
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Battle of sexes
Pat
Opera (q)
Chris
Opera ( r )
Prize Fight (1-r)
2 ,
0
Prize Fight (1-q)
1
0 ,
0
, 0
1 ,
2
  Pat’s expected payoff of playing Opera: r
  Pat’s expected payoff of playing Prize Fight: 2(1-r)
  Pat’s best response B2(r):
 
 
 
Prize Fight (q=0) if r<2/3
Opera (q=1) if r>2/3
Any mixed strategy (0≤q≤1) if r=2/3,
F. Valognes - Game Theory - Chp 8
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Battle of sexes
  Chris’ best response B1(q):
  Prize Fight (r=0) if q<1/3
 
 
Opera (r=1) if q>1/3
Any mixed strategy (0≤r≤1) if
q=1/3
  Pat’s best response B2(r):
 
 
 
Prize Fight (q=0) if r<2/3
Opera (q=1) if r>2/3
Any mixed strategy (0≤q≤1) if
r=2/3
Three Nash equilibria:
((1, 0), (1, 0))
((0, 1), (0, 1))
((2/3, 1/3), (1/3, 2/3))
1
r
2/3
1/3
F. Valognes - Game Theory - Chp 8
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q
7
Expected payoffs: 2 players each with
two pure strategies
Player 2
Player
1
s21 ( q )
s22 ( 1- q )
s11 ( r )
u1(s11, s21), u2(s11, s21)
u1(s11, s22), u2(s11, s22)
s12 (1- r )
u1(s12, s21), u2(s12, s21)
u1(s12, s22), u2(s12, s22)
  Player 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed
strategy ( q, 1- q ).
  Player 1’s expected payoff of playing s11:
EU1(s11, (q, 1-q))=q×u1(s11, s21)+(1-q)×u1(s11, s22)
  Player 1’s expected payoff of playing s12:
EU1(s12, (q, 1-q))= q×u1(s12, s21)+(1-q)×u1(s12, s22)
  Player 1’s expected payoff from her mixed strategy:
v1((r, 1-r), (q, 1-q))=r×EU1(s11, (q, 1-q))+(1-r)×EU1(s12, (q, 1-q))
F. Valognes - Game Theory - Chp 8
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Expected payoffs: 2 players each with
two pure strategies
Player 2
Player
1
s21 ( q )
s22 ( 1- q )
s11 ( r )
u1(s11, s21), u2(s11, s21)
u1(s11, s22), u2(s11, s22)
s12 (1- r )
u1(s12, s21), u2(s12, s21)
u1(s12, s22), u2(s12, s22)
  Player 1 plays a mixed strategy (r, 1- r ). Player 2 plays
a mixed strategy ( q, 1- q ).
 
 
Player 2’s expected payoff of playing s21:
EU2(s21, (r, 1-r))=r×u2(s11, s21)+(1-r)×u2(s12, s21)
Player 2’s expected payoff of playing s22:
EU2(s22, (r, 1-r))= r×u2(s11, s22)+(1-r)×u2(s12, s22)
  Player 2’s expected payoff from her mixed strategy:
v2((r, 1-r),(q, 1-q))=q×EU2(s21, (r, 1-r))+(1-q)×EU2(s22, (r, 1-r))
F. Valognes - Game Theory - Chp 8
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Mixed strategy equilibrium: 2-player
each with two pure strategies
Player 2
Player
1
s21 ( q )
s22 ( 1- q )
s11 ( r )
u1(s11, s21), u2(s11, s21)
u1(s11, s22), u2(s11, s22)
s12 (1- r )
u1(s12, s21), u2(s12, s21)
u1(s12, s22), u2(s12, s22)
  Mixed strategy Nash equilibrium:
  A pair of mixed strategies
((r*, 1-r*), (q*, 1-q*))
is a Nash equilibrium if (r*,1-r*) is a best response to
(q*, 1-q*), and (q*, 1-q*) is a best response to (r*,1-r*).
That is,
v1((r*, 1-r*), (q*, 1-q*)) ≥ v1((r, 1-r), (q*, 1-q*)), for all 0≤ r ≤1
v2((r*, 1-r*), (q*, 1-q*)) ≥ v2((r*, 1-r*), (q, 1-q)), for all 0≤ q ≤1
F. Valognes - Game Theory - Chp 8
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2-player each with two strategies
Player 2
Player
1
s21 ( q )
s22 ( 1- q )
s11 ( r )
u1(s11, s21), u2(s11, s21)
u1(s11, s22), u2(s11, s22)
s12 (1- r )
u1(s12, s21), u2(s12, s21)
u1(s12, s22), u2(s12, s22)
  Theorem 1 (property of mixed Nash equilibrium)
  A pair of mixed strategies ((r*, 1-r*), (q*, 1-q*)) is a
Nash equilibrium if and only if
v1((r*, 1-r*), (q*, 1-q*))
v1((r*, 1-r*), (q*, 1-q*))
v2((r*, 1-r*), (q*, 1-q*))
v2((r*, 1-r*), (q*, 1-q*))
≥
≥
≥
≥
EU1(s11, (q*, 1-q*))
EU1(s12, (q*, 1-q*))
EU2(s21, (r*, 1-r*))
EU2(s22, (r*, 1-r*))
F. Valognes - Game Theory - Chp 8
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Theorem 1: illustration
Matching pennies
Player 2
H (0.5)
Player 1
H (0.5)
T (0.5)
-1 ,
T (0.5)
1
1 , -1
1 , -1
-1 ,
1
  Player 1:
EU1(H, (0.5, 0.5)) = 0.5×(-1) + 0.5×1=0
  EU1(T, (0.5, 0.5)) = 0.5×1 + 0.5×(-1)=0
  v1((0.5, 0.5), (0.5, 0.5))=0.5×0+0.5×0=0
  Player 2:
  EU2(H, (0.5, 0.5)) = 0.5×1+0.5×(-1) =0
  EU2(T, (0.5, 0.5)) = 0.5×(-1)+0.5×1 = 0
  v2((0.5, 0.5), (0.5, 0.5))=0.5×0+0.5×0=0
 
F. Valognes - Game Theory - Chp 8
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Theorem 1: illustration
Matching pennies
Player 2
H (0.5)
Player 1
H (0.5)
T (0.5)
-1 ,
T (0.5)
1
1 , -1
1 , -1
-1 ,
1
  Player 1:
v1((0.5, 0.5), (0.5, 0.5)) ≥ EU1(H, (0.5, 0.5))
  v1((0.5, 0.5), (0.5, 0.5)) ≥ EU1(T, (0.5, 0.5))
  Player 2:
  v2((0.5, 0.5), (0.5, 0.5)) ≥ EU2(H, (0.5, 0.5))
  v2((0.5, 0.5), (0.5, 0.5)) ≥ EU2(T, (0.5, 0.5))
 
  Hence, ((0.5, 0.5), (0.5, 0.5)) is a mixed strategy Nash
equilibrium by Theorem 1.
F. Valognes - Game Theory - Chp 8
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Theorem 1: illustration
Employee Monitoring
Manager
Monitor (0.5)
Employee
Work (0.9)
50 ,
Shirk (0.1)
0
Not Monitor (0.5)
90
, -10
50 ,
100
100 , -100
  Employee’s expected payoff of playing “work”
EU1(Work, (0.5, 0.5)) = 0.5×50 + 0.5×50=50
  Employee’s expected payoff of playing “shirk”
  EU1(Shirk, (0.5, 0.5)) = 0.5×0 + 0.5×100=50
  Employee’s expected payoff of her mixed strategy
  v1((0.9, 0.1), (0.5, 0.5))=0.9×50+0.1×50=50
 
F. Valognes - Game Theory - Chp 8
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Theorem 1: illustration
Employee Monitoring
Manager
Monitor (0.5)
Employee
Work (0.9)
50 ,
Shirk (0.1)
0
Not Monitor (0.5)
90
, -10
50 ,
100
100 , -100
  Manager’s expected payoff of playing “Monitor”
EU2(Monitor, (0.9, 0.1)) = 0.9×90+0.1×(-10) =80
  Manager’s expected payoff of playing “Not”
  EU2(Not, (0.9, 0.1)) = 0.9×100+0.1×(-100) = 80
  Manager’s expected payoff of her mixed strategy
  v2((0.9, 0.1), (0.5, 0.5))=0.5×80+0.5×80=80
 
F. Valognes - Game Theory - Chp 8
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Theorem 1: illustration
Employee Monitoring
Manager
Monitor (0.5)
Employee
Work (0.9)
50 ,
Shirk (0.1)
0
No Monitor (0.5)
90
, -10
50 ,
100
100 , -100
  Employee
v1((0.9, 0.1), (0.5, 0.5)) ≥ EU1(Work, (0.5, 0.5))
  v1((0.9, 0.1), (0.5, 0.5)) ≥ EU1(Shirk, (0.5, 0.5))
  Manager
  v2((0.9, 0.1), (0.5, 0.5)) ≥ EU2(Monitor, (0.9, 0.1))
  v2((0.9, 0.1), (0.5, 0.5)) ≥ EU2(Not, (0.9, 0.1))
  Hence, ((0.9, 0.1), (0.5, 0.5)) is a mixed strategy Nash
equilibrium by Theorem 1.
 
F. Valognes - Game Theory - Chp 8
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Theorem 1: illustration
Battle of sexes
Pat
Opera (1/3)
Chris
Opera (2/3 )
Prize Fight (1/3)
2 ,
0
Prize Fight (2/3)
1
0 ,
0
, 0
1 ,
2
  Use Theorem 1 to check whether
((2/3, 1/3), (1/3, 2/3)) is a mixed strategy Nash
equilibrium.
F. Valognes - Game Theory - Chp 8
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Mixed strategy equilibrium: 2-player
each with two strategies
Player 2
Player
1
s21 ( q )
s22 ( 1- q )
s11 ( r )
u1(s11, s21), u2(s11, s21)
u1(s11, s22), u2(s11, s22)
s12 (1- r )
u1(s12, s21), u2(s12, s21)
u1(s12, s22), u2(s12, s22)
  Theorem 2
Let ((r*, 1-r*), (q*, 1-q*)) be a pair of mixed
strategies, where 0 <r*<1, 0<q*<1. Then ((r*, 1-r*), (q*, 1-q*)) is
a mixed strategy Nash equilibrium if and only if
EU1(s11, (q*, 1-q*)) = EU1(s12, (q*, 1-q*))
EU2(s21, (r*, 1-r*)) = EU2(s22, (r*, 1-r*))
  That is, each player is indifferent between her two strategies.
F. Valognes - Game Theory - Chp 8
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Use indifference to find mixed Nash
equilibrium (2-player each with 2 strategies)
  Use Theorem 2 to find mixed strategy Nash
equilibria
 
 
Solve EU1(s11, (q*, 1-q*)) = EU1(s12, (q*, 1-q*))
Solve EU2(s21, (r*, 1-r*)) = EU2(s22, (r*, 1-r*))
F. Valognes - Game Theory - Chp 8
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Use Theorem 2 to find mixed strategy
Nash equilibrium: illustration
Matching pennies
Player 2
H (q)
Player 1
H (r)
T ( 1–r )
-1 ,
T ( 1–q )
1
1 , -1
1 , -1
-1 ,
1
  Player 1 is indifferent between playing Head and Tail.
  EU1(H, (q, 1–q)) = q×(-1) + (1–q)×1=1–2q
  EU1(T, (q, 1–q)) = q×1 + ×(1–q) (-1)=2q–1
 
EU1(H, (q, 1–q)) = EU1(T, (q, 1–q))
1–2q = 2q–1
4q = 2
This give us q = 1/2
F. Valognes - Game Theory - Chp 8
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Use Theorem 2 to find mixed strategy
Nash equilibrium: illustration
Matching pennies
Player 2
H (q)
Player 1
H (r)
T ( 1–r )
-1 ,
T ( 1–q )
1
1 , -1
1 , -1
-1 ,
1
  Player 2 is indifferent between playing Head and Tail.
 
 
 
EU2(H, (r, 1–r)) = r ×1+(1–r)×(-1) =2r – 1
EU2(T, (r, 1–r)) = r×(-1)+(1–r)×1 = 1 – 2r
EU2(H, (r, 1–r)) = EU2(T, (r, 1–r))
2r – 1= 1 – 2r
4r = 2 This give us r = 1/2
  Hence, ((0.5, 0.5), (0.5, 0.5)) is a mixed strategy Nash equilibrium
by Theorem 2.
F. Valognes - Game Theory - Chp 8
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Use Theorem 2 to find mixed strategy
Nash equilibrium: illustration
Employee Monitoring
Manager
Monitor ( q )
Employee
Work (r)
50 ,
Shirk (1–r)
0
Not Monitor (1–q )
90
, -10
50 ,
100
100 , -100
  Employee’s expected payoff of playing “work”
EU1(Work, (q, 1–q)) = q×50 + (1–q)×50=50
  Employee’s expected payoff of playing “shirk”
  EU1(Shirk, (q, 1–q)) = q×0 + (1–q)×100=100(1–q)
  Employee is indifferent between playing Work and Shirk.
  50=100(1–q)
  q=1/2
 
F. Valognes - Game Theory - Chp 8
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Use Theorem 2 to find mixed strategy
Nash equilibrium: illustration
Employee Monitoring
Manager
Monitor ( q )
Employee
Work (r)
50 ,
Shirk (1–r)
0
Not Monitor (1–q )
90
, -10
50 ,
100
100 , -100
  Manager’s expected payoff of playing “Monitor”
EU2(Monitor, (r, 1–r)) = r×90+(1–r)×(-10) =100r–10
  Manager’s expected payoff of playing “Not”
  EU2(Not, (r, 1–r)) = r×100+(1–r)×(-100) =200r–100
  Manager is indifferent between playing Monitor and Not
100r–10 =200r–100 implies that r=0.9.
  Hence, ((0.9, 0.1), (0.5, 0.5)) is a mixed strategy Nash
equilibrium by Theorem 2.
 
F. Valognes - Game Theory - Chp 8
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Use Theorem 2 to find mixed strategy
Nash equilibrium: illustration
Battle of sexes
Pat
Opera (q)
Chris
Opera ( r )
Prize Fight (1-r)
2 ,
0
Prize Fight (1-q)
1
0 ,
0
, 0
1 ,
2
  Use Theorem 2 to find a mixed Nash
equilibrium
F. Valognes - Game Theory - Chp 8
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Use Theorem 2 to find mixed strategy
Nash equilibrium: illustration
Example
Player 2
L (q)
Player 1
T (r)
6 ,
B (1-r)
3
R (1-q)
4
2 ,
6
, 3
6 ,
1
  Use Theorem 2 to find a mixed Nash
equilibrium
F. Valognes - Game Theory - Chp 8
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Summary
  Mixed strategies
  Mixed Nash equilibrium
  Find mixed Nash equilibrium
  Next time
  2-player game each with a finite number of
strategies
  Reading lists
  Chapter 1.3 of Gibbons and Cha 4.3 of
Osborne
F. Valognes - Game Theory - Chp 8
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