Static or SimultaneousMove Games of Complete Information

1
Static (or Simultaneous-Move) Games of Complete
Information

• Mixed Strategy Nash Equilibrium

2
Battle of Sexes

• Chris expected payoff of playing Opera 2q
• Chris expected payoff of playing Prize Fight
  1-q
• Chris best response B1(q)
• Prize Fight (r0) if qlt1/3
• Opera (r1) if qgt1/3
• Any mixed strategy (0?r?1) if q1/3

3
Battle of sexes

• Pats expected payoff of playing Opera r
• Pats expected payoff of playing Prize Fight
  2(1-r)
• Pats best response B2(r)
• Prize Fight (q0) if rlt2/3
• Opera (q1) if rgt2/3
• Any mixed strategy (0?q?1) if r2/3,

4
Battle of sexes

• Chris best response B1(q)
• Prize Fight (r0) if qlt1/3
• Opera (r1) if qgt1/3
• Any mixed strategy (0?r?1) if q1/3
• Pats best response B2(r)
• Prize Fight (q0) if rlt2/3
• Opera (q1) if rgt2/3
• Any mixed strategy (0?q?1) if r2/3

Three Nash equilibria ((1, 0), (1, 0)) ((0, 1),
(0, 1)) ((2/3, 1/3), (1/3, 2/3))
2/3
1/3
5
2-player each with two strategies

• 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)) ? EU1(s11, (q,
  1-q))v1((r, 1-r), (q, 1-q)) ? EU1(s12,
  (q, 1-q)) v2((r, 1-r), (q, 1-q)) ?
  EU2(s21, (r, 1-r))v2((r, 1-r), (q, 1-q))
  ? EU2(s22, (r, 1-r))

6
Theorem 1 illustration

• Player 1
• EU1(H, (0.5, 0.5)) 0.5(-1) 0.510
• EU1(T, (0.5, 0.5)) 0.51 0.5(-1)0
• v1((0.5, 0.5), (0.5, 0.5))0.5?00.5?00
• Player 2
• EU2(H, (0.5, 0.5)) 0.510.5(-1) 0
• EU2(T, (0.5, 0.5)) 0.5(-1)0.51 0
• v2((0.5, 0.5), (0.5, 0.5))0.500.500

7
Theorem 1 illustration

• 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.

8
Theorem 1 illustration

• Use Theorem 1 to check whether ((2/3, 1/3),
  (1/3, 2/3)) is a mixed strategy Nash equilibrium.

9
Mixed strategy equilibrium 2-player each with
two strategies

• Theorem 2 Let ((r, 1-r), (q, 1-q)) be a
  pair of mixed strategies, where 0 ltrlt1, 0ltqlt1.
  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.

10
Use Theorem 2 to find mixed strategy Nash
equilibrium

• Player 1 is indifferent between playing Head and
  Tail.
• EU1(H, (q, 1q)) q(-1) (1q)112q
• EU1(T, (q, 1q)) q1 (1q) (-1)2q1
• EU1(H, (q, 1q)) EU1(T, (q, 1q)) 12q 2q1
  4q 2 This give us q 1/2

11
Use Theorem 2 to find mixed strategy Nash
equilibrium

• Player 2 is indifferent between playing Head and
  Tail.
• EU2(H, (r, 1r)) r 1(1r)(-1) 2r 1
• EU2(T, (r, 1r)) r(-1)(1r)1 1 2r
• EU2(H, (r, 1r)) EU2(T, (r, 1r)) 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.

12
Use Theorem 2 to find mixed strategy Nash
equilibrium

• Chris expected payoff of playing Opera
• EU1(O, (q, 1q)) q2 (1q)0 2q
• Chris expected payoff of playing Prize Fight
• EU1(F, (q, 1q)) q0 (1q)1 1q
• Chris is indifferent between playing Opera and
  Prize
• EU1(O, (q, 1q)) EU1(F, (q, 1q)) 2q 1q3q
  1 This give us q 1/3

13
Use Theorem 2 to find mixed strategy Nash
equilibrium

• Pats expected payoff of playing Opera
• EU2(O, (r, 1r)) r 1(1r)0 r
• Pats expected payoff of playing Prize Fight
• EU2(F, (r, 1r)) r0(1r)2 2 2r
• Pat is indifferent between playing Opera and
  Prize
• EU2(O, (r, 1r)) EU2(F, (r, 1r)) r 2 2r
  3r 2 This give us r 2/3

14
Use Theorem 2 to find mixed strategy Nash
equilibrium

• 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.

15
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
  dont 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 dont match (heads,
  tails or tails, heads) then Bruce decides.

16
Example 1

• Bruces expected payoff of playing Head
• EU1(H, (q, 1q)) q1 (1q)4 43q
• Bruces expected payoff of playing Tail
• EU1(T, (q, 1q)) q4 (1q)1 13q
• Bruce is indifferent between playing Head and
  Tail
• EU1(H, (q, 1q)) EU1(T, (q, 1q)) 43q 13q
  6q 3 This give us q 1/2

17
Example 1

• Sheilas expected payoff of playing Head
• EU2(H, (r, 1r)) r 4(1r)1 3r 1
• Sheilas expected payoff of playing Tail
• EU2(T, (r, 1r)) r1(1r)4 4 3r
• Sheila is indifferent between playing Head and
  Tail
• EU2(H, (r, 1r)) EU2(T, (r, 1r)) 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.

18
Example 2

• Player 1s expected payoff of playing T
• EU1(T, (q, 1q)) q6 (1q)0 6q
• Player 1s expected payoff of playing B
• EU1(B, (q, 1q)) q3 (1q)6 6-3q
• Player 1 is indifferent between playing T and B
• EU1(T, (q, 1q)) EU1(B, (q, 1q)) 6q 6-3q
  9q 6 This give us q 2/3

19
Example 2

• Player 2s expected payoff of playing L
• EU2(L, (r, 1r)) r 0(1r)2 2- 2r
• Player 2s expected payoff of playing R
• EU2(R, (r, 1r)) r6(1r)0 6r
• Player 2 is indifferent between playing L and R
• EU2(L, (r, 1r)) EU2(R, (r, 1r)) 2- 2r 6r
  8r 2 This gives us r ¼
• ( (1/4, 3/4), (2/3, 1/3) ) is a mixed strategy
  Nash equilibrium.

20
Example 3Market 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

21
Example 3Market entry game

• 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 r5/6 and q5/6

22
Example 4

• 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 r2/3 and q1/2

23
Example 5 Rock, paper and scissors

• Can you guess a mixed strategy Nash equilibrium?

24
Example 5 Rock, paper and scissors

• Check whether there is
• a mixed strategy Nash equilibrium in which
  p11gt0, p12gt0, p13gt0, p21gt0, p22gt0, p23gt0.

25
Example 5 Rock, paper and scissors

• If each player assigns positive probability to
  every of her/his pure strategy, then each player
  is indifferent among her three pure strategies.

26
Example 5 Rock, paper and scissors

• Player 1 is indifferent among her/his three pure
  strategies EU1(Rock, p2) 0?p21(-1)? p221?
  p23EU1(Paper, p2) 1? p210? p22(-1)?
  p23EU1(Scissors, p2) (-1)? p211? p220? p23
• EU1(Rock, p2) EU1(Paper, p2) EU1(Scissors, p2)
• Together with p21 p22 p231, we have three
  equations and three unknowns.

27
Example 5 Rock, paper and scissors

• 0?p21(-1)? p221? p23 1? p210? p22(-1)?
  p230?p21(-1)? p221? p23 (-1)? p211? p220?
  p23 p21 p22 p231
• The solution is p21 p22 p231/3

28
Example 5 Rock, paper and scissors

• Player 2 is indifferent among her/his three pure
  strategies EU2(Rock, p1)0?p11(-1)? p121?
  p13EU2(Paper, p1)1? p110? p12(-1)?
  p13EU2(Scissors, p1)(-1)? p111? p120? p13
• EU2(Rock, p1) EU2(Paper, p1) EU2(Scissors,
  p1)
• Together with p11 p12 p131, we have three
  equations and three unknowns.

29
Example 5 Rock, paper and scissors

• 0?p11(-1)? p121? p131? p110? p12(-1)?
  p130?p11(-1)? p121? p13(-1)? p111? p120?
  p13 p11 p12 p131
• The solution is p11 p12 p131/3

30
Example 5 Rock, paper and scissors

• Player 1 EU1(Rock, p2) 0?(1/3)(-1)?(1/3)1?(1/
  3)0 EU1(Paper, p2) 1?(1/3)0?(1/3)(-1)?(1/3)
  0 EU1(Scissors, p2) (-1)?(1/3)1?(1/3)0?(1/3)
  0
• Player 2 EU2(Rock, p1)0?(1/3)(-1)?(1/3)1?(1/3)
  0 EU2(Paper, p1)1?(1/3)0?(1/3)(-1)?(1/3)0
  EU2(Scissors, p1)(-1)?(1/3)1?(1/3)0?(1/3)0
• Therefore, (p1(1/3, 1/3, 1/3), p2(1/3, 1/3,
  1/3)) is a mixed strategy Nash equilibrium by
  Theorem 4.