Microeconomics: Auctions

Transcription

1 Mcroeconomcs: Auctons Frédérc Robert-coud ovember 8, Abstract We rst characterze the PBE n a smple rst prce and second prce sealed bd aucton wth prvate values. The key result s that the expected revenue from both auctons s the same. We then look at a smple case of an aucton wth mperfectly known common value. The key result here s that a wnner s curse arses n some states of the world. Auctons wth prvate and ndependent values: The revenue equvalence theorem Auctons wth prvate and ndependent values can take several forms (see BD p67): Englsh aucton Dutch aucton Frst-prce sealed bd aucton Second-prce sealed bd aucton (aka Vckrey aucton) They all yeld the same expect revenue to the seller. We llustrate ths pont wth an exemple.. Exemple One seller. The set of buyers s ; :::; +. Buyers prvate valuaton for an object s and are d ~U[; ]. Ther acton s a bd, wth b ( ), all.

2 Buyer s expected payo s E ; fb j g + j Prfb b j g ( B ) ; j where B b n the rst prce sealed bd aucton and B j6 b j n the Vckrey aucton. See below for the seller s expected payo.. Frst-prce sealed bd aucton Let us guess the form of a symmetrc PBE equlbrum. Wth a unform dstrbuton, assume b j ( j ) j ; some > to be determned at equlbrum. Assumng that b ( ) ; we want to ensure that s a best reply for. Under the bdng strateges lad out above, Prfb b j g Y Prfb j < b g j j6 Pr j < : Thus yelds so that, for all, E ; fb j g + j ( [ ( ) ] + : ) Ths s the unque symmetrc BE wth strctly ncreasng and d erentable strategy functons; see Gbbons Appendx 3..B (p57).

3 + Y Usng G () Prf < g Prf < g +, the expected revenue for the seller s E ( seller ) Z Z : dg () ( + ) d.3 Second-prce sealed bd aucton (aka Vckrey aucton) Here t s a weakly domnant strategy to bd for all. The expected revenue for the seller s E ( seller ) Z Z Z " Z ( + ) b, dg (j < ) dg ( ) # d ( + ) d Z ; whch s the same as above, as was to be shown. + + d Auctons wth mperfectly known common values: The wnner s curse Wnnng the aucton brngs mxed news: the loosers may have got a very pessmstc sgnal. We llustrate ths pont wth a smple exemple. 3

4 . Exemple Players: buyers and one seller. The value of the object can take two values wth equal probablty, fh; Lg, wth H > L > : Each buyer gets a sgnal s () fs H ; s L g, wth Pr fhj s H g Pr flj s L g p > so that and H E [j s H ] ph + ( p) L L + p (H L) L E [j s L ] pl + ( p) H L + ( p) (H L) : Strateges are contngent on the sgnal so that we may wrte b b (s ()). In order to compute expected payo s, we may wrte so that E [j s L ; s L ] b (s L ) p L + ( p) H p + ( p) b (s L ) [ L b (s L )] b (s L ) < L ; p ( p) p (H L) ; + ( p) for otherwse the bdder who got a negatve sgnal would make a loss. Put d erently, f I wn the aucton havng receved a low sgnal then ths mples that the other buyer also got a low sgnal, so that the state of nature s of type L wth a probablty hgher than the ex-ante probablty p; hence the curse. Also, so that E [j s H ; s H ] E [j s H ; s H ] b (s H ) p H + ( p) L p + ( p) b (s H ) b (s H ) > f [ H b (s H )] + p b (s H ) H ; (p ) ( p) p + ( p) (H L) ; that s, f both players receve a postve sgnal (and knew about t) then they would revse ther belefs upwards ths s because the ex-post probablty of a hgh state of nature s more than p, a blessng. 4

5 Fnally, E [j s L ; s H ] b (s H ) (under the reasonable condton, ver ed at equlbrum, that b (s H ) > b (s L ), so that the player recevng a hgh sgnal wns the aucton) and E [j s H ; s L ] b (s H ) H + L b (s H ) [ H b (s H )] + p (H L) so that E [j s H ; s L ] sgnal would revse her belefs upwards. b (s H ) > f b (s H ) H ;that s, f the player recevng a postve Thus, n a Wckley aucton, the PBE equlbrum bds are and b (s L ) E [j s L ; s L ] b (s H ) E [j s H ; s H ] ; (p ) 3 whch mples E [j s H ; s L ] b (s H ) >, and p +( p) E ( seller ) p (H L) + p L + ( p) H p + ( p) : 5