# Stanford MS&E 275 - Principles of Game Theory (14 pages)

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## Principles of Game Theory

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## Principles of Game Theory

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Pages:
14
School:
Stanford University
Course:
Ms&E 275 - Foundations for Large-Scale Entrepreneurship
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STANFORD UNIVERSITY Winter 2001 DEPT OF MANAGEMENT SCIENCE AND ENGINEERING MS E 241 ECONOMIC ANALYSIS PRINCIPLES OF GAME THEORY LECTURE 3 MIXED STRATEGIES The strategies si Si are referred to as Player i s pure strategies i e a particular s i is played with probability one A mixed strategy for Player i is a probability distribution over some or all of the strategies in Si See e g Luenberger p 268 Varian p 264 3 1 ExampleIn the Matching Pennies game for instance Si consists of the two pure strategies Heads and Tails A mixed strategy for Player i is the probability distribution q 1 q where q is the probability of playing Heads 1 q the probability of playing Tails and 0 q 1 The mixed strategy 0 1 is simply the pure strategy Tails likewise the mixed strategy 1 0 is the pure strategy Heads There is no Nash Equilibrium in this game if we restrict players to pure strategies 3 2 ExampleIn the following game payoff diagram on next page Player 2 has pure strategies L M and R Here a mixed strategy for Player 2 is the probability distribution q r 1 q r where q is the probability of playing L r the probability of playing M and 1 q r the probability of playing R We require 0 q 1 0 r 1 and 0 q r 1 The mixed strategy 1 3 1 3 1 3 for instance puts equal probability on L M and R The pure strategies are simply the limiting cases of the players mixed strategies Player 2 s pure strategy L for instance is the mixed strategy 1 0 0 Player i may in reality follow a pure strategy but it may be modelled as a mixed strategy to reflect uncertainty in the mind of Player j about which element of S i Player i will actually play Player i s strategy may not really be random but appear to be random to Player j Even explicitly random actions are not uncommon The IRS randomly selects which tax returns to audit and telephone companies randomly monitor their operators conversations to make sure they are polite to customers Player 2 Up Left Middle 1 0 1 2 Right 0 1 Player 1 Down 0 3 0 1 2 0 3 3 Definition

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