All pay auctions with certain and uncertain prizes a comment


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1 CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No All py uctions with certin nd uncertin prizes comment Christin Riis
2 All py uctions with certin nd uncertin prizes comment Christin Riis Norwegin Business School April 24, 2015 Abstrct In the importnt contribution "All py uctions with certin nd uncertin prizes" published in Gmes nd Economic Behvior My 2014, Minchu nd nd Sel nlyze n ll py uction with multiple prizes. The specific feture of the model is tht ll vlutions re common except for the vlution of one of the prizes, for which contestnts hve privte vlutions. owever, the equilibrium chrcteriztion derived in the pper is incorrect. This note derives the correct equilibrium of the model. Key words: All py uctions, uncertin prizes JE codes: D 44, D82, J31, J41 Minchu nd Sel (2014) (herefter MS) consider n ll py uction with multiple prizes. The specific feture of their model is tht ll vlutions re common except for the vlution of one of the prizes. For this prticulr prize contestnts hve privte vlutions, independently drwn from common distribution. The uthors clim tht the equilibrium bid function is symmetric nd monotone in the vlution of the uncertin prize. owever this is only the cse if the uncertin prize hs the highest or lowest vlue. It is not if the uncertin prize hs n intermedite vlue, which is MS min cse. et me first provide intuition for why MS result fils. I follow MS nottion. There re n contestnts competing for m different prizes. The highest bidder wins the most vluble prize, with common vlue of v n. 1
3 The second highest bidder wins the second most vluble prize v n 1 nd so on. The uncertin prize is indexed n j +1, with privte vlue, denoted by, drwn independently from distribution F ( ) with support on [, +2 ]. Suppose now tht the equilibrium bid function β() is strictly incresing. The probbility tht plyer who bids ccording to vlution s will win the uncertin prize is then (n 1)! (j 1)! (n j)! F (s)n j (1 F (s)) j 1. Obviously this probbility is nonmonotonic function of s, strictly incresing in s for low bid levels, nd strictly decresing for high bid levels. The probbility of winning the uncertin prize reches mximum t vlution â implicitly determined by F (â) = n j n 1 It follows from stndrd single crossing conditions 1 tht strictly incresing bid function on bid segment is prt of seprting equilibrium only if the win probbility is incresing in vlution s the plyer with high vlution hs stronger incentive to bid ggressively thn bidder with lower vlution. In other words, strictly incresing bid function is incomptible with optiml bidding behvior if the probbility of winning the uncertin prize declines with the bid level in which cse bidder with prticulrly high vlution for the uncertin prize will lower her bid. Thus, seprting equilibrium cnnot be monotone in the vlution of the uncertin prize in our setting. I will now by construction derive the seprting equilibrium. Suppose the number of plyers strictly exceeds the number of prizes, ming generliztion of the result strightforwrd. A stndrd chrcteristic of seprting equilibrium in ll py uctions with ex nte identicl contestnts is the following: The plyer with the lowest possible vlution for the uncertin prize obtins zero pyoff. An impliction of this is tht the bid function must hve upper support equl to v n, the vlue of the highest prize. Otherwise the plyer with the 1 For t detiled exposition see Athey (2001) 2
4 lowest possible vlution obtins strictly positive rent by jumping to the upper support. As the lower support must be zero, seprting equilibrium will be mpping from vlution ɛ [, +2 ] to bids β on [0, v n ]. 2 Observe tht the equilibrium cn be nchored in the following observtion: For high bids, the probbility of winning the uncertin prize declines s the bid level increses. Therefore, single crossing indictes tht the bid function must hve declining segment t high vlutions. For low vlutions the equilibrium bid function must be incresing. A conjecture would be tht the equilibrium bid function hs the shpe illustrted in figure 1, consisting of two bid segments: declining segment, β (), for those bid levels t which the win probbility (for the uncertin prize) declines with the bid; nd n incresing segment β () for bid levels t which the win probbility increses. A contestnt with vlution then rndomizes between the two bid levels, nd chooses β () with probbility q(). With one exception: the contestnt with the highest possible vlution for the uncertin prize, = +2, bids certin mount, corresponding to the bid level tht mximizes the probbility of winning the uncertin prize. Note tht the plyer with the lowest possible vlution for the uncertin prize, =, rndomizes between bidding zero nd v n. Figure 1 I will now show tht the seprting equilibrium indeed stisfies this pt 2 It is lso stndrd tht the bid distribution must be tomless. 3
5 tern. Denote by i () the equilibrium probbility tht plyer with vlution wins the prize indexed i, given tht she bids ccording to bid segment β (), =,. The probbility tht this plyer will win the uncertin prize, indexed n j + 1, is then () = (1) ( ) n j ( (n 1)! ) j 1 q(z)f(z)dz 1 q(z)f(z)dz (i 1)! (n i)! if she bids β () 3, nd () = (2) ( ) n j ( (n 1)! ) j 1 1 (1 q(z)) f(z)dz (1 q(z)) f(z)dz (j 1)! (n j)! if she bids β (). et us chrcterize the equilibrium bid functions following MS procedure, which is lso the stndrd procedure. A plyer with vlution behves s plyer with vlution s in order to mximize ( =, ) mx s i i (s)v i + (s) β (s) The necessry conditions yields the following pir of differentil equtions β (s) = i d i(s) v i + ds d ds (s) 3 To see this, note tht bid s wins the prize n j + 1 if n exct number j 1 of the bidder s contestnts outbid her (note tht there is probbility 1 s q(z)f(z)dz tht single contestnt will choose bid bove s), nd the remining n j contestnts will bid below s. 4
6 The first order condition evluted t s = yields the cndidte bid functions 4 d β () = β ( ) + i (x) dx v d (x) i + x dx dx = β ( ) + i i i ()v i i i ( )v i + [ d If = we hve β ( ) = 0 nd thus i () = 0, s the plyer loses with certinty. If = we hve β ( ) = v 1 nd thus n() = 1, s the plyer in this cse is certin to win the most vluble prize. In both cses, the cndidte bid function is (where the second equlity follows from integrtion by prts) β () = = i i This yields ssocited utilities i ()v i + i ()v i + [ d dx () ] (x) xdx [ ] (x) dx dx ] (x) xdx U () = (x)dx Thus the bidder s rent is ssocited with vlution of the uncertin prize bove the lower support, exctly s described by MS. Note tht β (+2 ) = β (+2 ) since i (+2) = i (+2) for ll i = n m + 1,.., n 5, which confirms tht the two bid segments meet t = +2. It remins to determine q(). The equilibrium probbility function, q( ), mes ech contestnt indifferent between choosing β () nd β (). This 4 Note tht i () = i ( ) + d i (x) dx dx. 5 This cn esily be checed by inserting +2 in the integrl limits in (1) nd (2), nd generlizing to ny i. 5
7 mens tht we hve to find function q( ) such tht U () = U () for every type, thus (x)dx = (x)dx (3) must lwys hold. Obviously this is equivlent to the following condition: for ny ɛ[, +2 ] we hve () = () (4) ence, in seprting equilibrium, for ech type, the probbility of winning the uncertin price is independent of the choice between β () nd β (). Accordingly, the net cost of submitting high bid, β () β (), cncels out with the net gin from the higher probbility of winning one of the certin nd more vluble prizes. Substituting from (1) nd (2) condition (4) cn be written = ( ( q(z)f(z)dz 1 ) j 1 ( (1 q(z)) f(z)dz 1 q(z)f(z)dz ) j 1 ( ) n j (5) (1 q(z)) f(z)dz ) n j which must hold for ll ɛ[, +2 ]. q( ) is the solution to the functionl eqution given by (5). One cse is prticulrly simple. If the uncertin prize is the medin prize, j = (n + 1)/2, the equilibrium q is constnt, q() = 1/2 for ll. To see this, insert q() = 1/2 in (5) nd solve the integrls, which yields 6 ( ) (n 1)/2 ( 1 2 F () 1 1 ) (n 1)/2 2 F () = (1 12 ) (n 1)/2 ( ) (n 1)/2 12 F () F () 6 An intuitive rgument for existence more generlly goes s follows: note tht the equilibrium condition hs recursive structure in tht (5) determines q( ) up to ny rbitrry, independent of the shpe of q( ) bove. Furthermore since () = () is n identity, their derivtives re equl, d ()/d = d ()/d, nd since d ()/d is proportionl to q() nd d ()/d is proportionl to 1 q(), nd given the equilibrium function q( ) up to, q() is uniquely determined by the condition d ()/d = d ()/d. 6
8 Note tht the equilibrium is fully reveling, despite the fct tht plyers rndomize bid levels, s ech pir of possible bids is unique for the plyer s vlution. 1 References Athey, Susn. "Single crossing properties nd the existence of pure strtegy equilibri in gmes of incomplete informtion." Econometric 69.4 (2001): Minchu, Yizhq, nd Aner Sel. "Allpy uctions with certin nd uncertin prizes." Gmes nd Economic Behvior 88 (2014):
9 Center for Reserch in Economics nd Mngement (CREAM) ndelshøysolen BI / BI Norwegin Business School 0442 Oslo, Norwy The objective of CREAM is to provide reserch nd nlysis in the re of industril economics nd lbor economics with pplictions to mngement, nd provide reserchbsed nlysis for decision mers in public nd privte sector. ISSN X
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