Mixed Strategy Equilibriums

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Mixed Strategy Equilibriums

• Anyone for tennis?
• Should you serve to the forehand or the backhand?

2
Tennis Payoffs
3
Zero Sum Game
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Solving for Servers Optimal Mix

• What would happen if the the server always served
  to the forehand?
• A rational receiver would always anticipate
  forehand and 90 of the serves would be
  successfully returned.

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Solving for Servers Optimal Mix

• What would happen if the the server aimed to the
  forehand 50 of the time and the backhand 50 of
  the time and the receiver always guessed
  forehand?
• (0.50.9) (0.50.2) 0.55 successful returns

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Solving for Servers Optimal Mix

• What is the best mix for each player?

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of Successful Returns Given Server and Receiver
Actions
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of Successful Returns Given Server and Receiver
Actions

• If 20 of the serves are aimed at the forehand
  and the receiver is anticipating forehand then
  the of successful returns is
• (0.2 0.9) (0.8 0.2) 0.34
• Therefore, 34 of the serves are returned
  successfully.

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of Successful Returns Given Server and Receiver
Actions

• More generally, when the receiver anticipates
  forehand the of successful returns is defined
  by
• X of serves aimed at forehand
• 1-X of serves aimed at backhand
• of Successful Returns 0.90X 0.20(1-X)

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Servers Point of View
Receiver Anticipates Forehand
Y of Successful Returns
90 30
60 20
Y 0.9X 0.2(1-X)
X of Serves Aimed at Forehand
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Servers Point of View
Receiver Anticipates Backhand
Y of Successful Returns
90 30
60 20
Y 0.3X 0.6(1-X)
X of Serves Aimed at Forehand
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Servers Point of View
Receiver Anticipates Backhand
Y of Successful Returns
90 30
60 20
Receiver Anticipates Forehand
X of Serves Aimed at Forehand
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Best Response

• If the server aims to the forehand 20 of time,
  what is the receivers best response?
• If the server aims to the forehand 80 of time,
  what is the receivers best response?

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Servers Point of View
Receiver Anticipates Backhand
Y of Successful Returns
90 30
60 20
Receiver Anticipates Forehand
X of Serves Aimed at Forehand
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Best Response

• Where can the server minimize the receivers
  maximum payoff?

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Solving for Mixed Strategy Equilibrium

• Set the linear equations equal to each other and
  solve
• 0.9X 0.2(1-X) 0.3X 0.6(1-X)
• X 0.40

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Servers Point of View
Y of Successful Returns
90 30
60 20
40
X of Serves Aimed at Forehand
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Solving for Mixed Strategy Equilibrium

• If the server mixes his serves 40 forehand / 60
  backhand, the receiver is indifferent between
  anticipating forehand and anticipating backhand
  because her payoff ( of successful returns) is
  the same.

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Receivers Point of View

• Now we have to do the same thing from the
  receivers point of view to determine how often
  the receiver should anticipate forehand/backhand.
• Luckily for us there is a shortcut.

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Solving for the Optimal Mix

• In equilibrium, if one player is optimally mixing
  then the other player is indifferent to the
  action he selects. If a player is not optimally
  mixing then he can be taken advantage of by his
  opponent. This fact allows us to easily solve
  for the optimal mix in zero sum, 2x2 games.

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Zero Sum Game
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Receivers Optimal Mix

• If the receiver is optimally mixing her
  anticipation of forehand (Y) and backhand (1-Y),
  then the server is indifferent between aiming
  forehand/backhand because his payoff is the same.

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Receivers Optimal Mix

• This means that if the receiver is optimally
  mixing then the servers payoff for aiming
  forehand is equal to his payoff for aiming
  backhand.

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Receivers Optimal Mix

• Algebraically
• Servers payoff function for aiming to the
  forehand
• 90Y 30(1-Y)
• Servers payoff function for aiming to the
  backhand
• 20Y 60(1-Y)

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Receivers Optimal Mix

• Solving for Y
• 90Y 30(1-Y) 20Y 60(1-Y)
• Y 30
• Thus the receiver should anticipate forehand 30
  of the time and backhand 70.

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J.D. Williams Solution for a Zero Sum Game
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J.D. Williams Solution for a Zero Sum Game

• In equilibrium, to solve for the optimal mix for
  the column player
• (XA) ((1-X) B) (XC) ((1-X) D)
• XA (1-X)B XC (1-X)D
• XA - XC (1-X)D - (1-X)B
• X(A-C) (1-X) (D-B)
• X/(1-X) (D-B)/(A-C)

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J.D. Williams Solution for a Zero Sum Game

• In equilibrium, to solve for the optimal mix for
  the row player
• (YA) ((1-Y) C) (YB) ((1-Y ) D)
• YA (1-Y)C YB (1-Y)D
• YA - YB (1-Y)D - (1-Y)C
• Y(A-B) (1-Y) (D-C)
• Y/(1-Y) (D-C)/(A-B)

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J.D. Williams Solution for a Zero Sum Game

• Column Player X/(1-X) (D-B)/(A-C)
• (60-20)/(90-30)
• 40/60
• The equilibrium ratio of X to (1-X) is 40 to 60.
• X 40 / (4060) 40

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J.D. Williams Solution for a Zero Sum Game

• Row Player Y/(1-Y) (D-C)/(A-B)
• (60-30)/(90-20)
• 30/70
• The equilibrium ratio of Y to (1-Y) is 30 to 70.
• Y 30 / (3070) 30
• 1-Y 70

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Overall of Successful Returns
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Overall of Successful Returns
48 successful returns
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Or you can just plug X or Y back into the payoff
equation...

• 90Y 30(1-Y) 20Y 60(1-Y)
• Y 30
• (0.90.3) ( 0.3(1-0.3)) 0.48

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Zero Sum Game
35
Zero Sum Game