Static (or SimultaneousMove) Games of Complete
Information
Mixed Strategy Nash Equilibrium
F. Valognes - Game Theory - Chp 9
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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 9
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Today’s Agenda
  Review of previous class
  Examples
F. Valognes - Game Theory - Chp 9
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Mixed strategy Nash equilibrium
  Mixed Strategy:
  A
mixed strategy of a player is a probability
distribution over the player’s strategies.
  Mixed strategy Nash 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 9
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Mixed strategy Nash equilibrium: 2player 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 9
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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 9
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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 pure
strategies.
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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*))
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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
  Chris’ expected payoff of playing Opera
  EU1(O, (q, 1–q)) = q×2 + (1–q)×0 = 2q
  Chris’ expected payoff of playing Prize Fight
  EU1(F, (q, 1–q)) = q×0 + (1–q)×1 = 1–q
  Chris is indifferent between playing Opera and Prize
  EU1(O, (q, 1–q)) = EU1(F, (q, 1–q))
2q = 1–q
3q = 1
This give us q = 1/3
F. Valognes - Game Theory - Chp 9
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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
  Pat’s expected payoff of playing Opera
  EU2(O, (r, 1–r)) = r ×1+(1–r)×0 = r
  Pat’s expected payoff of playing Prize Fight
  EU2(F, (r, 1–r)) = r×0+(1–r)×2 = 2 – 2r
  Pat is indifferent between playing Opera and Prize
  EU2(O, (r, 1–r)) = EU2(F, (r, 1–r))
r = 2 – 2r
3r = 2
This give us r = 2/3
F. Valognes - Game Theory - Chp 9
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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
  Hence, ( (2/3, 1/3), (1/3, 2/3) ) is a mixed
strategy Nash equilibrium. That is,
  Chris chooses Opera with probability 2/3 and
Prize Fight with probability 1/3.
  Pat chooses Opera with probability 1/3 and
Prize Fight with probability 2/3.
F. Valognes - Game Theory - Chp 9
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Example 1
  Bruce and Sheila determine whether to go to the
opera or to a pro wrestling show.
  Sheila gets utility of 4 from going to the opera and 1
from pro wrestling.
  Bruce gets utility of 1 from going to the opera and 4
from pro wrestling.
  They agree to decide what to do in the following way:
 
Bruce and Sheila each puts a penny below an issue of
the TV guide on the coffee table (assume they don’t
cheat by looking at the other). They count to 3 and
simultaneously reveal which side of their penny is up. If
the pennies match (both heads, or both tails), Sheila
decides what to watch, while if the pennies don’t match
(heads, tails or tails, heads) then Bruce decides.
F. Valognes - Game Theory - Chp 9
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Example 1
Sheila
H (q)
Bruce
T ( 1–q )
H (r)
1 ,
4
4 ,
1
T ( 1–r )
4 ,
1
1 ,
4
  Bruce’s expected payoff of playing Head
  EU1(H, (q, 1–q)) = q×1 + (1–q)×4 = 4–3q
  Bruce’s expected payoff of playing Tail
  EU1(T, (q, 1–q)) = q×4 + (1–q)×1 = 1+3q
  Bruce is indifferent between playing Head and Tail
  EU1(H, (q, 1–q)) = EU1(T, (q, 1–q))
4–3q = 1+3q
6q = 3
This give us q = 1/2
F. Valognes - Game Theory - Chp 9
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Example 1
Sheila
H (q)
Bruce
T ( 1–q )
H (r)
1 ,
4
4 ,
1
T ( 1–r )
4 ,
1
1 ,
4
  Sheila’s expected payoff of playing Head
EU2(H, (r, 1–r)) = r ×4+(1–r)×1 = 3r + 1
  Sheila’s expected payoff of playing Tail
  EU2(T, (r, 1–r)) = r×1+(1–r)×4 = 4 – 3r
  Sheila is indifferent between playing Head and Tail
  EU2(H, (r, 1–r)) = EU2(T, (r, 1–r))
3r + 1 = 4 – 3r
6r = 3 This give us r = ½
  ( (1/2, 1/2), (1/2, 1/2) ) is a mixed strategy Nash equilibrium.
 
F. Valognes - Game Theory - Chp 9
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Example 2
Player 2
L (q)
Player 1
R ( 1–q )
T (r)
6 ,
0
0 ,
6
B ( 1–r )
3 ,
2
6 ,
0
  Player 1’s expected payoff of playing T
  EU1(T, (q, 1–q)) = q×6 + (1–q)×0 = 6q
  Player 1’s expected payoff of playing B
  EU1(B, (q, 1–q)) = q×3 + (1–q)×6 = 6-3q
  Player 1 is indifferent between playing T and B
  EU1(T, (q, 1–q)) = EU1(B, (q, 1–q))
6q = 6-3q
9q = 6 This give us q = 2/3
F. Valognes - Game Theory - Chp 9
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Example 2
Player 2
L (q)
Player 1
R ( 1–q )
T (r)
6 ,
0
0 ,
6
B ( 1–r )
3 ,
2
6 ,
0
  Player 2’s expected payoff of playing L
EU2(L, (r, 1–r)) = r ×0+(1–r)×2 =2- 2r
  Player 2’s expected payoff of playing R
  EU2(R, (r, 1–r)) = r×6+(1–r)×0 = 6r
  Player 2 is indifferent between playing L and R
  EU2(L, (r, 1–r)) = EU2(R, (r, 1–r))
2- 2r = 6r
8r = 2
This gives us r = ¼
  ( (1/4, 3/4), (2/3, 1/3) ) is a mixed strategy Nash equilibrium.
 
F. Valognes - Game Theory - Chp 9
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Example 3:Market entry game
  Two firms, Firm 1 and Firm 2, must decide
whether to put one of their restaurants in a
shopping mall simultaneously.
  Each has two strategies: Enter, Not Enter
  If either firm plays “Not Enter”, it earns 0 profit
  If one plays “Enter” and the other plays “Not
Enter” then the firm plays “Enter” earns
$500K
  If both plays “Enter” then both lose $100K
because the demand is limited
F. Valognes - Game Theory - Chp 9
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Example 3:Market entry game
Firm 2
Enter ( q )
Firm 1
Enter ( r )
Not Enter ( 1–r )
-100 ,
0
,
Not Enter ( 1–q )
-100
500
500 ,
0
0 ,
0
  How many Nash equilibria can you find?
  Two pure strategy Nash equilibrium
(Not Enter, Enter) and (Enter, Not Enter)
  One mixed strategy Nash equilibrium
((5/6, 1/6), (5/6, 1/6))
That is r=5/6 and q=5/6
F. Valognes - Game Theory - Chp 9
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Example 4
Player 2
L (q)
Player 1
R ( 1–q )
T (r)
1 ,
1
1 ,
2
B ( 1–r )
2 ,
3
0 ,
1
  How many Nash equilibria can you find?
  Two pure strategy Nash equilibrium
(B, L) and (T, R)
  One mixed strategy Nash equilibrium
((2/3, 1/3), (1/2, 1/2))
That is r=2/3 and q=1/2
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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
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