105 20

Review Myerson's mechanism

Simplicity versus optimality

Vickreyauction with monopoly reservessimplifyingdependence on distributions

Non anonymouspricingcompletely removecompetition

Bulow Klemperer resultextra competition removedependenceon

Vickrey with anonymous reservedist

Prior independentmechanism

Hw1 due next Monday Oct12Info about projects coming soon

qesignN buyers Uno n Fi knownFeasibility constraintGoal Btc mechanism that maximizes revenue

Nigeria Btc mechs x is non decPaymentidentity

Theorem Exp revenue of any BK mechanismis equal to the expected virtual surplus

Rev a Eff seeVirtualvalues n Cvp Ui CIficvidf.eu

Theoremeo When all buyers have regularvaluedistributions pointwise maximizationofvirtualsurplusgives a BK mechanismf Optimal mech is the virtualsurplusmaximizer

Regularity dit is a non doc function

In the non regular case weneed to iron

8 Icq

i iiilow valve

highvalve to PitaRevenueofanyBICmech ironed virtualsurplusironed v.v are non decreasing

Rev ofany BK mech.that has a flattonstantallocationfunction over ironed regions

ironed virtual surplus

Revenue optimal mechanisms is the ironedvirtual surplus maximizer

Emile.uaYgwsHtmyvznnzucotDtbuEaiEftF Lt t b U 1411

fact L Felt 4613 t414t e 2T I 243 16 TECH314J

Atty lb t 3314

ay b t Hfit 413 E Wea

213 4149J

i t au.iq2T I 34,1

I Gilt

t.TTi

sq 112

VickeryBuyer1 wins if withu Kc and minxeseither Vi3314 and u3Vz Up42doe Ii cc's

3

smane

u.meaeEu

Restofthis lecture Single item and regularityShowed lasttime Vickrey with monopolyreserves gets a 2 approx to expected revenue

Eanism eGiven Fi Fn Determine pukes pi pinBuyers will arrive one at a time1st buyer with Vi pi will getthe item atpico

piSequential_PM_ f RandomorderPM fodderobliviousINIb whiformlyspecific bestcaseordering random worst case ordering

impose an order arrival

Ex AntonAB

Supplyconstraint AllocatetoBB B atmost onebuyer at every

value vectorV

AAe Ex ante relaxation Allocate11 atmost one item in expectation

A oven value vectors

Zi qe Maximum revenue we can getfrom agent 9 byselling with ex autoprob at mostgo to the agent

Ex ion

Eff max 7 Tgifhoming'atoptmettreffffp

s t Zi qi ElTg

QFftp.qqRa Reemn.negtfmagent

at apriceofEkodis

Whatis the best mech that sells with ex anteprobgIntheregularcase this correspondsexatoly to posting a

f price of G D2 g Rlg

Inthe non regular case 2Cop IcqAt quantiles q where Rlg f RG the

anlooptimal mech random

ex autooptimal mech randomizesbetween two potted prices

Suppose git grit is the solution to EA OPT

Y Set go I got and pi Feta gi RYndofggoffer prices pi on arbitrary summaziyi

EA OPT E Zig t

tzzicaox.sif

tie Zilgio 3 4by concavity

9

Ego E ai't e zTotal prob of selling the item is 112

Forany agent prob that item is stillavailablewhenkeyarrive is 3112

Revof0PM 3 Pilitemageinstofferedtg tffE Zilge

3 114 Zilgi TEA OPT4

3 OPT

PROPHETINEQUALITY

ME The he xD

m boxes with a reward in each

Rg can openboxes are at a timeobserve the rewarddecide to accept or reject it

game ends

Prophet knows theinstantiationsofall rewardsSo can accept the highestoneAlg competes against EImaxe Xie oPT

Theorem Canachieve 112of prophetsexpected rewardby picking a singlethreshold andacceptingthe 1st reward that crosses it

Alg E mine xistsX inany

Alg 3 42OPT for an appropriatechoiceoft

oftConnection toOPM's 8 the X's are virtualvaluesOPT virtualvalue maximizerAlg 0PM that places a priceofGuilt

for buyer

Proofofprophotinguality

Puffin x et Hzf median Max XD

OPT E maxieX E t t E maxieXi Efet t Xi tf

at maxlad

ALG It t Rfdidnotacceptarqm.geCho t

t

Pe x Xu atPn miyagiXii at

pn may x et

42

Et t Iz x t5 OPT

Non anonymous posted pricing different buyers areoffered differentpubs

Xmdimqmonpdition.eempcrerni.ird buyers with regularvalues

Optimal mechanism Vickrey auction with anon reserve

The revenueof the Vickreyauction over ntDbuyerswith no reserve price is at least as largeas revenue of optimal auction over n buyers

Vickreyauction overLntdbuyers sells to the buyerwith highest v v even whenthis is re

Among mechanisms that always selltheitem overloadbuyers Vickrey is the optimal one

Alternatemech over Mtl buyers

peer Run optimal much over n buyersopt If item remains unsold giffeitutuffutfger

Vickreynu 3 OPTn

extraneous DRY 10Jone buyer UNF Optimal mechanism offer 0 6

Draw p NF Offer p to the buyer

Claim ftp.fvlpD 3 I OPT if F isregulars

opt q qReq

tnIrandomquartile

fee correspondingto

40

1

Eu D Rl9D fRlqDdg1

3 area ofpurpleDPurple 0 has area 112OPTand lies below Rlg R is concave

ItartlineRoughgardenlo

Consider n non iid regularbuyers Ui vnSuppose we double thebuyers v un

Revenueofthe Vickreyauction over vi Vnv Unis at least 1 2 of the opt rev over u un