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Unformatted text preview: 6.207/14.15: Networks Lecture 10: Introduction to Game Theory2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Networks: Lecture 10 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies Existence of Mixed Strategy Nash Equilibrium in Finite Games Characterizing Mixed Strategy Equilibria Applications Reading: Osborne, Chapters 35. 2 Networks: Lecture 10 Nash Equilibrium Pure Strategy Nash Equilibrium Definition (Nash equilibrium) A (pure strategy) Nash Equilibrium of a strategic game I , ( S i ) i I , ( u i ) i I is a strategy profile s S such that for all i I u i ( s i , s i ) for all s i S i . i ) u i ( s i , s Why is this a reasonable notion? No player can profitably deviate given the strategies of the other players. Thus in Nash equilibrium, best response correspondences intersect. Put differently, the conjectures of the players are consistent : each player i chooses s i expecting all other players to choose s , and each i players conjecture is verified in a Nash equilibrium. 3 Networks: Lecture 10 Examples Examples: Bertrand Competition An alternative to the Cournot model is the Bertrand model of oligopoly competition. In the Cournot model, firms choose quantities. In practice, choosing prices may be more reasonable. What happens if two producers of a homogeneous good charge different prices? Reasonable answer: everybody will purchase from the lower price firm. In this light, suppose that the demand function of the industry is given by Q ( p ) (so that at price p , consumers will purchase a total of Q ( p ) units). Suppose that two firms compete in this industry and they both have marginal cost equal to c > (and can produce as many units as they wish at that marginal costs). 4 Networks: Lecture 10 Examples Bertrand Competition (continued) Then the profit function of firm i can be written as Q ( p i ) ( p i c ) if p i > p i i ( p i , p i ) = 1 2 Q ( p i ) ( p i c ) if p i = p i if p i < p i Actually, the middle row is arbitrary, given by some ad hoc tiebreaking rule. Imposing such tiebreaking rules is often not kosher as the homework will show. Proposition In the twoplayer Bertrand game there exists a unique Nash equilibrium given by p 1 = p 2 = c. 5 Networks: Lecture 10 Examples Bertrand Competition (continued) Proof: Method of finding a profitable deviation. Can p 1 c > p 2 be a Nash equilibrium? No because firm 2 is losing money and can increase profits by raising its price. Can p 1 = p 2 > c be a Nash equilibrium? No because either firm would have a profitable deviation, which would be to reduce their price by some small amount (from p 1 to p 1 )....
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This note was uploaded on 06/12/2010 for the course EECS 6.207J taught by Professor Acemoglu during the Fall '09 term at MIT.
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