Position Auctions Paper by Hal Varian, Presented by Rahul Jain

The Assignment Game N agents: Types v n P positions: rates x 1 > x 2 >  > x P Agent utilities: u n (p)=v n x p Make bids b n b1b1 b2b2 bPbP bNbN x1x1 x2x2 xPxP

The VCG Mechanism Allocation: Position 1 to highest bidder for position 1, etc. Payment of 1 = The Mechanism is incentive-compatible, individual-rational, and results in efficient allocation  Complicated with general utility functions with high-dimensional message space

The Next-Price Mechanism Allocation: nth position to the nth highest bidder Payment of n = b n+1 (price of position n) Payoff to n = (v n -p n )x n

Nash Equilibrium Assume Let prices be A Nash Equilibrium is a set of prices s.t. There is a continuum of such equilibria Maximum and Minimum Revenue bounds

Equilibrium Refinements A symmetric N.E. is a set of prices s.t. Easy to calculate, nicely behaved. (What is the intuition?) Properties of SNE:  v n ¸ p n  v n-1 ¸ v n  p n-1 >p n and p n-1 x n-1 > p n x n  SNE ½ NE

Calculating SNE Fact: If a set of bids satisfies the SNE inequalities for m=n+1 and m=n-1, then it satisfies these inequalities for all m, and n. proof: v 1 (x 1 -x 2 )¸ p 1 x 1 -p 2 x 2 v 2 (x 2 -x 3 )¸ p 2 x 2 -p 3 x 3 ) (by v 1 ¸ v 2 ) v 1 (x 2 -x 3 )¸ p 2 x 2 -p 3 x 3 ) v 1 (x 1 -x 3 )¸ p 1 x 1 -p 3 x 3

Calculating SNE Recursively obtain upper/lower bounds: b n U x n-1 =  m ¸ n v m-1 (x m-1 -x m ) b n L x n-1 =  m ¸ n v m (x m-1 -x m ) Get b L P+1 x P =v P+1 (x P -x P+1 )=v P+1 x P, i.e., b L P+1 =v P+1

NE and SNE Revenue SNE Revenue: R L = v 2 (x 1 -x 2 )+2v 3 (x 2 -x 3 )+3v 4 x 3 R U = v 1 (x 1 -x 2 )+2v 2 (x 2 -x 3 )+3v 3 x 3 Fact: Max NE Revenue same as upper bound on SNE revenue. Idea: p n N x n · p N n+1 x n+1 +v n (x n -x n+1 ) p n U x n · p U n+1 x n+1 +v n (x n -x n+1 ) p P N · v P =p P U ) p n N · p n U