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MIT14_126S10_lec10a

# 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 Dominance and equilibrium Comparative statics 2 1

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Two Aspects of Complements Constraints Activities are complementary if doing one enables doing the other… …or at least doesn’t prevent doing the other. This condition is described by sets that are sublattices. Payoffs Activities are complementary if doing one makes it weakly more profitable to do the other… This is described by supermodular payoffs. …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 ) 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
Definitions: “Lattice” Given a partially ordered set ( X , ), define " join " : x = inf { z X | z , } The y x z y . " meet " : x y sup { z X | z , } . The = x z y ( 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 = 1,..., N ( x y ) i = max( x i , y i ); i = 1,..., 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 ( , X f x ) ( ) + ( ) f x y ) f ( x y ) x y f y ( + 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: Higher marginal returns ( x y ) f ( ) ( ) f ( x y f x f y ) y x y differences Nonnegative mixed second [ f ( x y ) ( ) ] ( ) f ( x y ) ] f x [ f y 0 x For smooth objectives, non-negative mixed second derivatives: x y 2 f 0 for i j x x 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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MIT14_126S10_lec10a - Supermodularity 14 126 Game Theory...

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