Lecture13 - 3/3/2009 Mixed Strategies Matching Pennies We...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
3/3/2009 1 Mixed Strategies – Matching Pennies ± We denote the strategies on the game bi-matrix Guildenstern q1 - q Heads Tails ± Rosencrantz’s payoff to H : (1)q + (-1)(1-q) = 2q-1 T : (-1)q + (1)(1-q) = 1-2q Rosencrant zp Heads 1,-1 -1 , 1 1-p Tails -1 , 1 1 , -1 Mixed Strategies – Matching Pennies ± Rosencrantz’s & Guildenstern’s best responses: q 1 The best response curves intersect at the Nash p 01 1/2 1/2 equilibrium p=1/2, q=1/2 Mixed Strategies – Matching Pennies ± Suppose R’s payoff to H increases Guildenstern - q Heads Tails ± Rosencrantz’s payoff to H : (2)q + (-1)(1-q) = 3q-1 T : (-1)q + (1)(1-q) = 1-2q ± Now, 3q-1 = 1-2q requires q=2/5 Rosencrant Heads 2,-1 -1 , 1 1-p Tails -1 , 1 1 , -1 Mixed Strategies – Matching Pennies ± Rosencrantz’s & Guildenstern’s best responses: q 1 The best response curves intersect at the Nash p 1/2 1/2 the Nash equilibrium p=1/2, q=2/5 2/5 Mixed Strategies – Matching Pennies ± Suppose R’s payoff to H increases Rt Guildenstern - q Heads Tails ± Rosencrantz’s payoff to H : (2)q + (-1)(1-q) = 3q-1 T : (-1)q + (1)(1-q) = 1-2q ± Now, 3q-1 = 1-2q requires q=2/5 Rosencrant Heads -1 , 1 1-p Tails -1 , 1 1 , -1 Mixed Strategies – Chicken ± Since we allow for randomization over pure strategies, so we can consider mixed-strategy Nash equilibria even in non-zero-sum games ± Recall the game of Chicken… ± We have two pure-strategy NE Daisy Duke Swerve Don’t Swerve Billy-Jo Bob Swerve 1 , 1 1 , 2 Don’t Swerve 2 , 1 0 , 0
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
3/3/2009 2 Daisy Duke q1 - q Swerve
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

Lecture13 - 3/3/2009 Mixed Strategies Matching Pennies We...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online