MIT14_126S10_lec10a - Supermodularity 14. 126 Game Theory...

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Supermodularity 14. 126 Game Theory Muhamet Yildiz Based on Lectures by Paul Milgrom 1 Road Map ± Definitions: lattices, set orders, supermodularity… ± Optimization problems ± Games with Strategic Complements ² Dominance and equilibrium ² Comparative statics 2 1
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Two Aspects of Complements ± Constraints ² Activities are complementary if doing one enables doing the other… ² …or at least doesn’t prevent doing the other. ± This condition is described by sets that are sublattices. ± Payoffs ² Activities are complementary if doing one makes it weakly more profitable to do the other… ± This is described by supermodular payoffs. ² …or at least doesn’t change the other from being profitable to being unprofitable ± This is described by payoffs satisfying a single crossing condition. 3 2 Example – Peter-Diamond search model BR( a -i ) ± A continuum of players ± Each i puts effort a i , costing a i 2 /2; ± Pr i finds a match = a i g ( a -i ), θ g ( a -i ) ² a -i is average effort of others ± The payoff from match is θ . U i ( a ) = θ a i g ( a -i ) – a i 2 /2 ± Strategic complementarity: BR( a -i ) = θ g ( a -i ) a -i θ ’> θ 4
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Definitions: “Lattice” ± Given a partially ordered set ( X , ), define ² " join " : x ∨= inf { z X | z , } The y x z y . ² " m e e t " : xy sup { z X | z , } . The = x zy ± ( X , ) is a “ lattice ” if it is closed under meet and join: ( X ) x y , ∨∈ x , y ∧∈ X x y X ± Example: X = R N , x y if x y i , = 1,. .., N i i ( x y ) i = min( x i , y i ); i = .., N ( x y ) i = max( x i , y i ); i = .., N 5 Definitions, 2 ± ( X , ) is a “ complete lattice ” if for every non-empty subset S , a greatest lower bound inf( S ) and a least upper bound sup( S ) exist in X . ± A function f : X Æ R is “ supermodular ”if ( , Xfx ) ( ) + ( fx y ) f ( x y ) fy ( ∧+ ± A function f is “ submodular” if – f is supermodular. ± (if X = R , then f is supermodular.) 6 3
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Complementarity ± Complementarity/supermodularity has equivalent characterizations: ² Higher marginal returns ( x y ) f ( ) ( f ( x y f xf y ) y x y differences ² Nonnegative mixed second [ f ( x y ) ( ] − ( f ( x y ) ] ≥ f x [ f 0 x ² For smooth objectives, non-negative mixed second derivatives: x y 2 f 0 for i j ∂∂ xx i j 7 4 Definitions, 3 ± Given two subsets S,T X, “S is as high as T,” written S T, means [x S & y T] [x y S & x y T] ± A function x* is “ isotone ”(or ±“ weakly increasing ”) if t t’ x*(t) x*(t’) ± A set S is a “ sublattice ”if±S S.
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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_lec10a - Supermodularity 14. 126 Game Theory...

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