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MIT14_126S10_lec11

# MIT14_126S10_lec11 - Global Games 14.126 Game Theory...

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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

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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 θ,θ θ −1, 0 Not-Invest 0,θ−1 0,0 2
θ is common knowledge 0,0 0,θ−1 Not-Invest θ −1, 0 θ,θ Invest Not-Invest Invest θ < 0 θ is common knowledge 0,0 0,θ−1 NotInvest θ −1, 0 θ,θ Invest NotInvest Invest θ > 1 3

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θ is common knowledge 0,0 0,θ−1 Not-Invest θ −1, 0 θ,θ Invest Not-Invest Invest 0 < θ < 1 Multiple Equilibria!!! θ is common knowledge Invest Not-Invest Multiple Equilibria θ 4
θ is not common knowledge θ is uniformly distributed over a large interval Each player i gets a signal x i = θ + εη i ( η 1 , η 2 ) is bounded, Independent of θ , iid with continuous F (common knowledge), E[ η i ] = 0. Conditional Beliefs given x i θ = d x i εη i i.e. Pr( θ θ| x i ) = 1- F (( x i - θ )/ ε ); x j = d x i + ε(η j - η i ) Pr( x j x | x i ) = Pr( ε ( η j η i ) x – x i ); F ( θ , x j | x i ) is decreasing in x i E[ θ | x i ] = x i 5

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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) 0 0 Not-Inv x i -1 x i Invest Not-Inv Invest 0 0 Not-Inv θ -1 θ Invest Not-Inv Invest Monotone BNE Best reply: Invest iff x i Pr( s j = Not-Invest| x i ) Assume supp( θ ) = [ 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. θ is not common knowledge but the noise is very small It is very likely that Not-Invest Invest θ 1/2 7

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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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MIT14_126S10_lec11 - Global Games 14.126 Game Theory...

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