Game Theory Course: Jackson, Leyton-Brown & Shoham. Vickrey-Clarke-Groves Mechanisms: Definitions

Transcription

1 Vckrey-Clarke-Groves Mechansms: Defntons Game Theory Course: Jackson, Leyton-Brown & Shoham

2 A postve result Recall that n the quaslnear utlty settng, a drect mechansm conssts of a choce rule and a payment rule A VCG mechansm: has truth as a domnant strategy (satsfes truthfulness, s strategy-proof) makes effcent choces (not ncludng payments)

3 A postve result Recall that n the quaslnear utlty settng, a drect mechansm conssts of a choce rule and a payment rule A VCG mechansm: has truth as a domnant strategy (satsfes truthfulness, s strategy-proof) makes effcent choces (not ncludng payments) And, under addtonal assumptons about the settng, can satsfy: weak budget balance nterm ndvdual ratonalty

4 Groves Mechansms Defnton (Groves mechansms) Drect mechansms, (, p), such that (ˆv) arg ma ˆv () p (ˆv) = h (ˆv ) j ˆv j ( (ˆv)) Some people refer to these as VCG mechansms, although that name has more recently started to be used to refer to a specfc mechansm wthn ths class

5 The Vckrey-Clarke-Groves Mechansm Defnton (A Vckrey-Clarke-Groves (VCG) mechansm, aka a Pvotal mechansm) A Vckrey-Clarke-Groves mechansm or a pvotal mechansm s a Groves mechansm (, p), such that (ˆv) arg ma ˆv () p (ˆv) = ma ˆv j () ˆv j ( (ˆv)) j j

6 VCG dscusson (ˆv) arg ma ˆv () p (ˆv) = ma ˆv j () j j ˆv j ( (ˆv)) You get pad everyone s utlty under the allocaton that s actually chosen ecept your own, but you get that drectly as utlty Then you get charged everyone s utlty n the world where you don t partcpate Thus you pay your socal cost

7 VCG dscusson Questons: who pays 0? (ˆv) = arg ma p (ˆv) = j ˆv () ˆv j ( (ˆv )) j ˆv j ( (ˆv))

8 VCG dscusson (ˆv) = arg ma p (ˆv) = j ˆv () ˆv j ( (ˆv )) j Questons: who pays 0? agents who don t affect the outcome ˆv j ( (ˆv))

9 VCG dscusson (ˆv) = arg ma p (ˆv) = j ˆv () ˆv j ( (ˆv )) j Questons: who pays 0? agents who don t affect the outcome who pays more than 0? ˆv j ( (ˆv))

10 VCG dscusson (ˆv) = arg ma p (ˆv) = j ˆv () ˆv j ( (ˆv )) j Questons: who pays 0? agents who don t affect the outcome ˆv j ( (ˆv)) who pays more than 0? (pvotal) agents who make thngs worse for others by estng

11 VCG dscusson (ˆv) = arg ma p (ˆv) = j ˆv () ˆv j ( (ˆv )) j Questons: who pays 0? agents who don t affect the outcome ˆv j ( (ˆv)) who pays more than 0? (pvotal) agents who make thngs worse for others by estng who gets pad?

12 VCG dscusson (ˆv) = arg ma p (ˆv) = j ˆv () ˆv j ( (ˆv )) j Questons: who pays 0? agents who don t affect the outcome ˆv j ( (ˆv)) who pays more than 0? (pvotal) agents who make thngs worse for others by estng who gets pad? (pvotal) agents who make thngs better for others by estng

13 VCG and Groves Mechansms: Truthfulness Theorem Truth tellng s a domnant strategy under any Groves mechansm ncludng the pvotal mechansm (a VCG mechansm) Consder agent s problem of choosng the best strategy ˆv A best strategy for s solves ( ma v ( (ˆv, ˆv )) p(ˆv, ˆv ) ) ˆv Substtutng n the payment functon for a Groves mechansm ths becomes: ma ˆv v ( (ˆv)) h (ˆv ) + j ˆv j ( (ˆv)) Snce h (ˆv ) does not depend on ˆv, t s suffcent to solve ma ˆv v ( (ˆv)) + j ˆv j ( (ˆv))

14 VCG Truthfulness So, would lke to pck a declaraton ˆv that wll lead the mechansm to pck an X whch solves ( ) ma v () + j ˆv j () (1) Under a Groves mechansm, (ˆv) arg ma ( ˆv () + j ˆv j () ) A Groves mechansm wll choose n a way that solves the mamzaton problem n Equaton (1) when ˆv = v Thus, truth-tellng s a domnant strategy for agent

15 Groves Unqueness Theorem (Green Laffont) Suppose that for all agents any v : X R s a feasble preference Then an effcent mechansm (, p) (such that (ˆv) arg ma ˆv ()) has truthful reportng as a domnant strategy for all agents and preferences only f t s Groves mechansm: p (v) = h(v ) j v j( (v)) A proof can be found at

16 Summary Groves mechansms, and VCG mechansms n partcular, have nce domnant strategy propertes Agents payments nclude the mpact of ther announcements on other agents Internalze the eternaltes and lead to effcent decsons ( s) But may burn payments to do so!