LectureFeb24_MATH4321_12S

# LectureFeb24_MATH4321_12S - Equilibrium Principle: BR to...

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Unformatted text preview: Equilibrium Principle: BR to each other Maximin Principle: Safety First For Player I: Find p so that Min q p T Aq is largest. p is called the Safety Strategy or Optimal Strategy. Min q p T Aq is the lower value. For Player II: Find q so that Max p p T Aq is smallest. q is called the Safety Strategy or Optimal Strategy. Max p p T Aq is the upper value. Minimax Theorem: Maximin=Minimax Solution of All 2 by 2 Matrix Games. Consider the general 2 × 2 game matrix We will use the Maximin Principle to find the safety strategies and the value. Given a mixed strategy of Player I, , we will find the minimum of the payoff. Recall that the minimum is achieved by a pure strategy of Player II. a b A d c        The minimum is then Min(pa+(1-p)d),pb+(1-p)c), for 0  p  1. The graph of the function p  Min(pa+(1-p)d),pb+(1-p)c) is the lower envelope of the graphs of the two linear functions p  pa+(1-p)d, for 0  p  1 and p  pb+(1-p)c, for 0  p  1 1 p a b p d c  The maximum of Min(pa+(1-p)d),pb+(1-p)c) is...
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## This note was uploaded on 03/13/2012 for the course MATH 4321 taught by Professor Cheng during the Spring '12 term at HKUST.

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LectureFeb24_MATH4321_12S - Equilibrium Principle: BR to...

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