note8 - Microeconomic Theory Econ 101A Fall 2008 GSI: Eva...

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Unformatted text preview: Microeconomic Theory Econ 101A Fall 2008 GSI: Eva Vivalt Section Notes 8: Intro to Game Theory - Nash Equilibrium 1 Nash Equilibrium For simplicity, consider two firms. Firm 1s profits 1 ( a 1 ,a 2 ) depend upon its own actions a 1 and the actions of firm 2, a 2 ; firm 2s profits are similarly denoted 2 ( a 1 ,a 2 ). Examples of possible actions the firms could take include setting a price, setting a level of output, or determining whether to enter or exit a market. 1.1 Definition A set of actions ( a N 1 ,a N 2 ) consitutes a Nash equilibrium if and only if: 1 ( a N 1 ,a N 2 ) 1 ( a 1 ,a N 2 ) for all a 1 and 2 ( a N 1 ,a N 2 ) 1 ( a N 1 ,a 2 ) for all a 2 In other words, a set of actions is a Nash equilibrium if each firm cannot do better for itself playing its Nash equilibrium action, given that each other firm is playing their Nash equilibrium action. 1.2 Solving for Nash Equilibria To find a Nash equilibrium solution, we must have firm 1 solve: max a 1 1 ( a 1 ,a 2 ) while firm 2 simultaneously solves: max a 2 2 ( a 1 ,a 2 ) Often, finding a Nash involves checking all the possible combinations ( a 1 ,a 2 ) and asking yourself is this a Nash equilibrium? Sometimes it is possible to eliminate (strictly not weakly) dominated actions itera- tively to narrow the cases that need to be checked (see a book on game theory for this). However, assuming that profit functions are continuously differentiable, concave, and that a N 1 and a N 2 are both positive, we can use first order conditions to find the Nash equilibria. The first order necessary conditions for a maximum in our example would be: FOC 1 : 1 ( a N 1 ,a N 2 ) a 1 = 0 FOC 2 : 2 ( a N 1 ,a N 2 ) a 2 = 0 1 which yield a system of two equations and two unknowns. The second order conditions for ( a N 1 ,a N 2 ) to be a global max of 1...
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This note was uploaded on 04/01/2009 for the course ECON 101a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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note8 - Microeconomic Theory Econ 101A Fall 2008 GSI: Eva...

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