MIT14_126S10_lec11 - Global Games 14.126 Game Theory...

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Unformatted text preview: Global Games 14.126 Game Theory Muhamet Yildiz Road map 1. Theory 1. 2 x 2 Games (Carlsson and van Damme) 2. Continuum of players (Morris and Shin) 3. General supermodular games (Frankel, Morris, and Pauzner) 2. Applications 1. Currency attacks 2. Bank runs 1 Motivation Outcomes may differ in similar environments. This is explained by multiple equilibria (w/strategic complementarity) Investment/Development Search Bank Runs Currency attacks Electoral competition But with introduction of incomplete information, such games tend to be dominance-solvable A simple partnership game Invest Not-Invest Invest , 1, 0 Not-Invest 0,1 0,0 2 is common knowledge 0,0 0,1 Not-Invest 1, 0 , Invest Not-Invest Invest < 0 is common knowledge 0,0 0,1 NotInvest 1, 0 , Invest NotInvest Invest > 1 3 is common knowledge 0,0 0,1 Not-Invest 1, 0 , Invest Not-Invest Invest 0 < < 1 Multiple Equilibria!!! is common knowledge Invest Not-Invest Multiple Equilibria 4 is not common knowledge is uniformly distributed over a large interval Each player i gets a signal x i = + i ( 1 , 2 ) is bounded, Independent of , iid with continuous F (common knowledge), E[ i ] = 0. Conditional Beliefs given x i = d x i i i.e. Pr( | x i ) = 1- F (( x i- )/ ); x j = d x i + ( j - i ) Pr( x j x | x i ) = Pr( ( j i ) x x i ); F ( , x j | x i ) is decreasing in x i E[ | x i ] = x i 5 Payoffs given x i Invest > Not-Invest U i (a i ,a j ,x i ) is supermodular. Monotone supermodular! There exist greatest and smallest rationalizable strategies, which are Bayesian Nash Equilibria Monotone (isotone) Not-Inv x i-1 x i Invest Not-Inv Invest Not-Inv -1 Invest Not-Inv Invest Monotone BNE Best reply: Invest iff x i Pr( s j = Not-Invest| x i ) Assume supp( ) = [ a , b ] where a < 0 < 1 < b . x i < 0 s i ( x i ) = Not Invest x i > 1 s i ( x i ) = Invest A cutoff x i * s.t. x i < x i * s i ( x i ) = Not Invest; x i > x i * s i ( x i ) = Invest; Symmetry: x 1 * = x 2 * = x * x * = Pr( s j = Not-Invest|x*) = Pr( x j < x *| x i = x *) = 1/2 Unique BNE, i.e., dominance-solvable 6 Questions What is the smallest BNE? What is the largest BNE? Which strategies are rationalizable? Compute directly. is not common knowledge but the noise is very small It is very likely that Not-Invest Invest 1/2 7 Risk-dominance In a 2 x 2 symmetric game, a strategy is said to be risk dominant iff it is a best reply when the other player plays each strategy with equal probabilities....
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This note was uploaded on 11/28/2011 for the course ECONOMICS - taught by Professor Muhammadyildiz during the Spring '05 term at University of Massachusetts Boston.

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MIT14_126S10_lec11 - Global Games 14.126 Game Theory...

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