week13-class - Auctions ECOS3012 Strategic Behavior School...

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Unformatted text preview: Auctions ECOS3012 - Strategic Behavior School of Economics, FASS University of Sydney Lecture 13, June 1, 2011 ECOS3012 - Strategic Behavior (USYD) Auctions Lecture 13, June 1, 2011 1 / 11 Two basic auctions I First price auction (FPA) Consider a symmetric environment with two bidders, whose types are distributed according to a probability distribution function F on an interval [0 , 1] Assume that both players bid according to a bidding function β : [0 , 1]-→ R . Payoff from submitting bid b to a player of type v is U F ( b , β | v ) = ( v- b ) Prob ( b > β ( w )) = v Prob ( w ∈ [0 , 1] | b > β ( w ))- b Prob ( w ∈ [0 , 1] | b > β ( w )) All-pay auction (APA) Payoff from submitting bid b to a player of type v is U A ( b , β | v ) = v Prob ( w ∈ [0 , 1] | b > β ( w ))- b Calculating the equilibrium of an APA is relatively straightforward using a simple trick. ECOS3012 - Strategic Behavior (USYD) Auctions Lecture 13, June 1, 2011 2 / 11 Equilibrium in APA I Suppose β A is the equilibrium bidding function. Equilibrium requires that type v submit the bid β A ( v ) . Therefore, if type v were to mimic the behavior of type v and instead bid β A ( v ) , her should not be better off. That is we must have for all v , v , U A ( β A ( v ) , β A | v ) ≥ U A ( β A ( v ) , β A | v ) (1) Now, let us search for an equilibrium in which higher types bid higher amounts. That is assume that β A is strictly increasing. In fact, let us also assume that β A is differentiable. Then, β A ( v ) > β A ( w ) ⇔ v > w and hence Prob ( w ∈ [0 , 1] β A ( v ) > β A ( w ) ) = F ( v ) (2) and H ( v , v ) = U A ( β A ( v ) , β A | v ) = vF ( v )- β A ( v ) . (3) ECOS3012 - Strategic Behavior (USYD) Auctions Lecture 13, June 1, 2011 3 / 11 Equilibrium in APA II Because of (1), we must have...
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week13-class - Auctions ECOS3012 Strategic Behavior School...

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