Game Theoretic Approach in Computer Science CS3150, Fall 2002 Mixed Strategies

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Game Theoretic Approach in Computer
ScienceCS3150, Fall 2002Mixed Strategies

• Patchrawat Uthaisombut
• University of Pittsburgh

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Randomness in Payoff Functions

• 2002 US open Final match.
• Serena is about to return the ball.
• She can either hit the ball down the line (DL) or
  crosscourt (CC)
• Venus must prepare to cover one side or the other

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Mixes Strategies

• What is a mixed strategy?
• Suppose Ak is the set of pure strategies for
  player k.
• A mixed strategy for player k is a probability
  distribution over Ak.
• An actual move is chosen randomly according to
  the probability distribution.
• Example
• Ak stand, walk, run
• Stand 50, walk 25, run 25 is a mixed
  strategy for k.

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Need for Mixed Strategies

• Multiple pure-strategy Nash equilibria
• No pure-strategy Nash equilibria
• Games where players prefer opposite outcomes
• Matching Pennies
• Chicken
• Sports
• Attack and Defense
• Each player does very badly if her action is
  revealed to the other, because the other can
  respond accordingly.
• Want to keep the other guessing.
• Mixed strategy Nash equilibrium always exists.

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Expectation

• Suppose X is a random variable.
• Suppose X 5 with probability 0.5
• Suppose X 6 with probability 0.3
• Suppose X 0 with probability 0.2
• Then EX 50.5 60.3 00.2
• 2.5 1.8 0 4.3
• In general, if X vi with probability pi
• Then EX Sum vi pi

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Mixes Strategies in the Chicken Games

• Mixing 2 pure strategies
• Swerve with probability p and Straight with
  probability (1-p)
• A continuous range of mixed strategies.
• Shown as a function of p instead of one row for
  each value of p.

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Mixes Strategies in the Chicken Games
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Payoff Curves and Best-Response Curves
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Finding Mixed Strategy Nash Equilibrium

• Compute Rows payoffs as a function of q.
• Find q that make Rows payoffs indifferent no
  matter what pure strategy she chooses.
• Plot Rows best-response curve.
• Do steps 1-3 for the Column player and p.
• Plot Rows and Columns best-response curves
  together.
• Points where the 2 curves meet are Nash
  equilibria.

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Why it is an equilibrium?

• It is a Nash equilibrium because
• Apichai cant change his strategy to do better
  and
• Buncha cant change her strategy to do better
• Why cant Apichai do better?
• Buncha chooses a mix such that it doesnt matter
  what Apichai does.
• Why cant Buncha do better?
• Apichai chooses a mix such that it doesnt matter
  what Buncha does.

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Exercise

• Find mixed strategy Nash equilibrium in the
  following games.
• Battle of the Two Cultures
• Tennis match

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Mixing when a player has 2 strategies

• Suppose Serena has a third option, to lob.
• Need 2 variables for Serena
• p1 probability of choosing DL
• p2 probability of choosing CC
• (1-p1-p2) probability of choosing Lob.

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Tennis game with 3 Pure Strategies
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Reduction to 2-Pure-Strategy Game

• Serena will never use CC.
• Back to a game in which each player has only 2
  pure strategies.

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Coincidental Case

• 3 lines meet at a single point

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Coincidental Case

• Too few equalities to solve all variables.
• Serenas p-mix is indeterminate
• Serena has some flexibility to choose her mix and
  still get the best payoff.