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Introduction145 - v i = 1 I Mean/Expected Value E V = ∑ n...

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Introduction Ichiro Obara UCLA January 5, 2009 Obara (UCLA) Introduction January 5, 2009 1 / 8
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What do we learn? We study the exchange of indivisible goods. In particular, we examine the performance of a variety of mechanisms in different contexts. Along the way, we learn some useful math. Obara (UCLA) Introduction January 5, 2009 2 / 8
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Topics Assignment of dormitory rooms to students. Organ transplants. Housing markets. Auctions. Matchmaking. College Admission. etc .... Obara (UCLA) Introduction January 5, 2009 3 / 8
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Required Math Combination Pick k balls out of n balls. How many ways to do so? n k = n ! k !( n - k )! = n ( n - 1)( n - 2) · · · ( n - k + 1) k ( k - 1) · · · 2 × 1 Obara (UCLA) Introduction January 5, 2009 4 / 8
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Required Math Calculus Differentiation: ( x n ) 0 = nx n - 1 , ( logx ) 0 = 1 / x , ( f - 1 ( y ) ) 0 = 1 f 0 ( f - 1 ( y )) . Integration: R b a f ( x ) dx = F ( x ) | b a = F ( b ) - F ( a ). Chain Rule: ( f ( g ( x ))) 0 = f 0 ( g ( x )) g 0 ( x ). Integration by Parts: R b a F ( x ) g ( x ) dx = F ( x ) G ( x ) | b a - R b a f ( x ) G ( x ) dx . Differential equation... Obara (UCLA) Introduction January 5, 2009 5 / 8
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Required Math Probability Discrete Case I Random Variable: V takes a value from { v 1 , ..., v n } . I Probability: p ( v i ) = Pr( V = v i ) [0 , 1], n i =1
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Unformatted text preview: v i ) = 1. I Mean/Expected Value: E [ V ] = ∑ n i =1 v i p ( v i ). I Conditional Probability. .. Obara (UCLA) Introduction January 5, 2009 6 / 8 Required Math Probability Continuous Case I Random Variable: V take one of the value from [ v , v ]. I Cumulative Probability: F ( v ) = Pr( V ≤ v ). I Density Function: f ( v ) = F ( v ). So F ( v ) = R v v f ( x ) dx . I Expected Value: E [ V ] = R v v xf ( x ) dx . I Conditional Expectation. .. Obara (UCLA) Introduction January 5, 2009 7 / 8 Required Math Vector and Matrix Vector and Matrix Operations a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 x 1 x 2 = a 1 , 1 x 1 + a 1 , 2 x 2 a 2 , 1 x 1 + a 2 , 2 x 2 . Inner Product: x · y = x 1 y 1 + x 2 y 2 = k x kk y k cos θ . Inverse: A-1 A = I . etc. .. Obara (UCLA) Introduction January 5, 2009 8 / 8...
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