Simultaneous Games - Mixed Strategies II.pdf - Simultaneous Games Mixed Strategies II Mixed-Strategy Nash Equilibrium ZERO-SUM GAME KICKER Left(pL

Simultaneous Games - Mixed Strategies II.pdf - Simultaneous...

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Simultaneous Games - Mixed Strategies II Mixed-Strategy Nash Equilibrium GOALIE Left ( q L ) Center Right ( q R ) KICKER Left ( p L ) 45 90 90 Center 85 0 85 Right ( p R ) 95 95 60 Figure 7.10 Penalty Kick Does the game have a Nash equilibrium in pure strategies? How does a mixed-strategy Nash equilibrium look like? Can the MSNE involve only two pure strategies? ... when you have eliminated the impossible, whatever remains, however improbable, must be truth. Sir Arthur Conan Doyle (Sherlock Holmes) 1. W/o Kicking Right (i.e. p R = 0) Goalie: Right’s expected probability = 90 p L +85(1 - p L ) > Left’s or Center’s expected probability Goalie will stop guessing Right But Kicker will kick Right ⇒⇐ 2. W/o Kicking Center (i.e. 1 - p L - p R = 0) Goalie: Center’s expected probability = 90 p L + 95 p R > Left’s or Right’s expected probability Goalie will stop guessing Center Kicker: Mixing Left and Right 45 q L + 90 q R = 95 q L + 60 q R (= 73 . 125, using q L + q R = 1) But Kicker will kick Center ⇒⇐ 3. W/o Kicking Left (i.e. p L = 0) Goalie: Left’s expected probability = 85(1 - p R ) + 95 p R > Cen- ter’s or Right’s expected probability Goalie will stop guessing Left 1 probability to goal (45,90,90) ZERO-SUM GAME
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