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# slides6 - Lecture 6 Applications of Rationalizability&Nash...

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Lecture 6 Applications of Rationalizability &Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Summary 2. Cournot Competition 3. Quiz 4. Simplified price competition 5. Two common games 6. Partnership Games 7. Mixed-strategy Nash Equilibrium 1

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Rationalizability Yes No Eliminate all the strictly dominated strategies. Any dominated strategy In the new game? Rationalizable strategies Nash Equilibrium Definition: A strategy-profile s* =(s 1 *,…,s n *) is a Nash Equilibrium iff, for each player i, and for each strategy s i , we have * * * * ( * i , s u 1 , K , s i 1 , s s i + 1 , K , s n ) i * * , s u 1 , K , s i * 1 , s s i + 1 , K , s ), i ( * i n i.e., no player has any incentive to deviate if he knows what the others play. 2
Cournot Oligopoly N = {1,2,…,n} firms; Simultaneously, each firm i P produces q i units of a good at marginal cost c, 1 and sells the good at price P = max{0,1-Q} where Q = q 1 +…+q n . Game = (S 1 ,…,S n ; π 1 ,…, π n ) Q where S i = [0, ), ), 1 π i (q 1 ,…,q n ) = q i [1-(q 1 +…+q n )-c] if q 1 +…+q n < 1, -q i c otherwise.

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