167hw8 - Game Theory Steven Heilman Please provide complete...

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Game Theory Steven Heilman Please provide complete and well-written solutions to the following exercises. Due March 8th, in the discussion section. Homework 8 Exercise 1. Prove the following Lemma from the notes: The set of functions { W S } S ⊆{ 1 ,...,n } is an orthonormal basis for the space of functions from {- 1 , 1 } n R , with respect to the inner product defined in the notes. (When we write S ⊆ { 1 , . . . , n } , we include the empty set as a subset of { 1 , . . . , n } .) (Also, for any x ∈ {- 1 , 1 } n , W S ( x ) = Q i S x i .) Exercise 2. Let f : {- 1 , 1 } 2 → {- 1 , 1 } such that f ( x ) = 1 for all x ∈ {- 1 , 1 } 2 . Compute b f ( S ) for all S ⊆ { 1 , 2 } . Let f : {- 1 , 1 } 3 → {- 1 , 1 } such that f ( x 1 , x 2 , x 3 ) = sign( x 1 + x 2 + x 3 ) for all ( x 1 , x 2 , x 3 ) {- 1 , 1 } 3 . Compute b f ( S ) for all S ⊆ { 1 , 2 , 3 } . The function f is called a majority function . Exercise 3. Let f : {- 1 , 1 } 3 → {- 1 , 1 } such that f ( x 1 , x 2 , x 3 ) = sign( x 1 + x 2 + x 3 ) for all ( x 1 , x 2 , x 3 ) ∈ {- 1 , 1 } 3 . In the previous homework, we computed b f ( S ) for all S ⊆ { 1 , 2 , 3 } .
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