Applications of Myerson s Lemma

Transcription

1 Applcatons of Myerson s Lemma Professor Greenwald We apply Myerson s lemma to solve the sngle-good aucton, and the generalzaton n whch there are k dentcal copes of the good. Our objectve s welfare maxmzaton. Welfare-Maxmzng Auctons Myerson s lemma gves us a recpe for desgnng an IC and IR welfare-maxmzng aucton. The frst step s to construct a allocaton functon that s monotonc n values, and the second step s to plug that functon nto the payment formula. When that monotonc allocaton functon also acheves economc effcency (.e., t optmzes, or approxmately optmzes, welfare), and s also computatonally effcent, we say that the aucton s solved (or approxmately solved). 2 Sngle-Good Aucton Our frst applcaton of Myerson s lemma s a smple santy check. We have already dscussed a DSIC aucton desgn for the sngleparameter settng wth one good: the second-prce aucton, n whch the hghest bdder wns and pays the second-hghest bd. Here, we comfrm that Myerson s lemma leads us to the same concluson. Welfare Maxmzaton Recall that welfare s the quantty x (v), where x {, } n and x. Ths quantty s maxmzed by awardng the good to a bdder wth the hghest value:.e., a bdder s.t. arg max, Monotoncty Fx a bdder and a profle v. The necessary and suffcent condton for to be allocated s that bd hgher than, the hghest bd among bdders other than :.e., max v j, j = Ths allocaton rule s plotted n Fgure. Proposton 2.. Ths allocaton rule s monotoncally non-decreasng. Proof. If b <, then x (b, v ) =, so ncreasng the bd cannot possbly lower the allocaton. Indeed, for all ɛ >, x (b + ɛ, v )

2 applcatons of myerson s lemma 2 x (, v ) Fgure : Bdder s allocaton rule, for a fxed v. x (b, v ). On the other hand, f b s a wnnng bd, so that x (b, v ) =, then for all ɛ >, x (b + ɛ, v ) stll equals. In partcular, x (b + ɛ, v ) x (b, v ). Payments By the payment formula, f x =, then p =. Therefore, only the wnner of the aucton wll make a payment to the auctoneer. Assumng bdder s a wnner, ther payment s as follows: v p (, v ) = x (, v ) x (z, v ) dz, [ v ] = dz + dz = ( ) =. We splt up the ntegral n ths way because the allocaton for bddng less than bd s, whle the allocaton for bddng more s. Ths payment s the shaded regon n Fgure 2. x (, v ) Fgure 2: Bdder s payment functon for a gven v. We conclude that the combnaton of an allocaton rule that allocates to a hghest bdder together wth chargng the wnner of the aucton the second-hghest bd s IC and IR. Snce ths allocaton rule s economcally and compuatonally effcent, ths aucton s solved.

3 applcatons of myerson s lemma 3 3 k-good Aucton In ths aucton, there are k dentcal copes of a good and n k bdders, each wth a prvate value for exactly one copy of the good (.e., ths s another sngle-parameter aucton). Welfare Maxmzaton Problem Generalzng the sngle-good case, welfare s the quantty x (v), where x {, } n and x k. Ths quantty s maxmzed by awardng the goods to the k hghest bdders:.e., by settng precsely those entres of x that correspond to the k largest bds to, and all others to. Monotoncty Fx a bdder and a profle v. The necessary and suffcent condton for to be allocated s that bd hgher than, the kth-hghest bd among bdders other than :.e., max v j, j = Snce the condton for beng allocated s the same as t was n the sngle-good case smply bddng hgher than some crtcal value the allocaton rule s the same as t was n the sngle-good case. Ths allocaton rule s plotted n Fgure 3. x (, v ) Fgure 3: Bdder s allocaton rule, for a fxed v. Proposton 3.. Ths allocaton rule s monotoncally non-decreasng. Proof. If b <, then x (b, v ) =, so ncreasng the bd cannot possbly lower the allocaton. Indeed, for all ɛ >, x (b + ɛ, v ) x (b, v ). On the other hand, f b s a wnnng bd, so that x (b, v ) =, then for all ɛ >, x (b + ɛ, v ) stll equals. In partcular, x (b + ɛ, v ) x (b, v ). Payments By the payment formula, f x =, then p =. Therefore, only the wnners of the aucton make a payment to the auctoneer. Assumng bdder s a wnner, ther payment s as follows: p (, v ) = x (, v ) v x (z, v ) dz,

4 applcatons of myerson s lemma 4 [ v ] = dz + dz = ( ) =. Snce the condton for beng allocated s the same as t was n the sngle-good case smply bddng hgher than ths payment calculaton s the same as t was n the sngle-good case. Ths payment s the shaded regon n Fgure 4. x (, v ) Fgure 4: Bdder s payment functon for a gven v. We conclude that to the combnaton of an allocaton rule that allocates to the k hghest bdders together wth chargng the wnners of the aucton the kth-hghest bd s IC and IR. Snce ths allocaton rule s economcally and compuatonally effcent, ths aucton s solved. Ths soluton s called the k-vckrey aucton. A two-good example. Imagne three bdders, b, b 2 and b 3, and two goods. The bdders values are unformly dstrbuted on closed ntervals, but wth dfferent bounds: each bdder s value s unformly dstrbuted on the closed nterval [, ], so f (v) =, and F (v) =, for all v [, ]. Let represent bdder s realzed value. Suppose v = 3/4, v 2 = 2, and v 3 = 3/2. What happens n ths example n the welfare-maxmzng aucton, IC, IR, and ex-post feasble aucton? To answer ths queston, we do the followng:. Sort the bdders values. 2. Fnd the wnners:.e., the bdders wth the two hghest values. 3. Determne the crtcal value, and hence the wnners payments. These steps are llustrated n Table. Bdders 2 and 3 are allocated the goods, because they have the two hghest values. They each pay the crtcal value, whch n ths example s the thrd-hghest value. Observe that ths aucton s pror-free. No knowledge of the value dstrbutons s needed to mplement the optmal aucton.

5 applcatons of myerson s lemma 5 Rank Wnner? Crtcal bd Payment 5/6 3 no n/a n/a 2 2 yes 5/6 5/6 3 7/4 2 yes 5/6 5/6 Table : Example Two-Good Aucton