Mechansm Desgn Algorthms and Data Structures Wnter 2016 1 / 39
Vckrey Aucton Vckrey-Clarke-Groves Mechansms Sngle-Mnded Combnatoral Auctons 2 / 39
Mechansm Desgn (wth Money) Set A of outcomes to choose from. n bdders wth quantfable preferences. Preference of bdder s gven by a valuaton functon v : A R from a commonly known set V R A A mechansm determnes an outcome a A and payments, charges each bdder some amount m of money Utlty of bdder s v (a) m, quas-lnear utltes. Common currency enables utlty transfer between bdders. 3 / 39
Example: Sealed Bd Aucton A sngle tem s sold to one customer. Customer 1 2 3 4 5 Value 9 1 20 11 14 Agents report ther values as sealed bd. Socal Choce: Wnner s agent wth hghest bd. Payments: fnd payments to ensure truthful bddng No payments: Bdders strve to bd unboundedly hgh values. Payments = Bds: Bdders try to guess the second hghest bd and bd a slghtly hgher value. 4 / 39
Vckrey Second Prce Aucton Payment of the wnner s the second largest bd. Value 9 1 20 11 14 Payment 0 0 14 0 0 Utlty 0 0 6 0 0 Proposton The Vckrey aucton s truthful. 5 / 39
Example Value?? 20?? Bd 5 11 x 2 14 Payment 14 Utlty 6 Case 1: wns wth true value x = 20, then for all x 14 utlty 6, for x < 14 utlty 0. Value?? 20?? Bd 5 11 x 2 24 Payment 0 Utlty 0 Case 2: loses wth true value x = 20, then for all x < 24 utlty 0, for x 24 utlty 4. 6 / 39
Vckrey Aucton Vckrey-Clarke-Groves Mechansms Sngle-Mnded Combnatoral Auctons 7 / 39
Defntons Drect Revelaton Mechansm Denote V = V 1... V n and v V Socal choce functon f : V A A vector of payment functons p 1,..., p n Each functon p : V R specfes the amount bdder pays. Truthfulness / Incentve Compatblty (IC) For every bdder, profle v V, alternatve v V, Denote outcomes by a = f (v, v ) and b = f (v, v ) Mechansm (f, p 1,..., p n) s truthful (or ncentve compatble) f the utlty. v (a) p (v, v ) v (b) p (v, v ) 8 / 39
Sealed-Bd Aucton Bdder 1 2 3 4 5 Value 9 1 20 11 14 Outcomes A = {1, 2, 3, 4, 5}, where means wns Outcome 1 2 3 4 5 v 1 9 0 0 0 0 v 2 0 1 0 0 0 etc. Socal Choce: f (v) = arg max {v ()} Payments: p (v) = 0 f f (v), otherwse p (v) = max j v j (j). 9 / 39
VCG Mechansm Defnton A Vckrey-Clarke-Groves (VCG) mechansm s gven by f (v) arg max a A v (a), and for some functons h 1,..., h n wth h : V R and all v V we have p (v) = h (v ) j v j (f (v)) Observatons: VCG mechansm pcks outcome a that maxmzes socal welfare j v j(a) h does not depend on the own bd v Utlty of bdder when f (v) = a: v (a) p (v) = j v j (a) h (v ) 10 / 39
VCG s IC Theorem Every VCG mechansm s truthful. Proof: Gven types v, for bdder a le v, and let a = f (v) and b = f (v, v ). Utlty of declarng v s v (a) + j v j(a) h (v ) Utlty of declarng v s v (b) + j v j(b) h (v ) Utlty s maxmzed when outcome maxmzes socal welfare j v j(x). VCG mechansm maxmzes socal welfare, j v j(a) j v j(b). By declarng v bdder, VCG pcks b whch s optmzed for hs le, but possbly suboptmal for the real utlty. VCG algns each bdder ncentve wth the socal ncentves. 11 / 39
Desrable Propertes of Payments Defnton A mechansm s (ex-post) ndvdually ratonal f bdders always get non-negatve utlty,.e. for all v V we have v (f (v)) p (v) 0. A mechansm has no postve transfers f no bdder s ever pad money,.e. for all v V and all we have p (v) 0. Defnton (Clarke Rule) The choce h (v ) = max b A j v j(b) s called Clarke pvot payment. Then the payment of bdder becomes p (v) = max b A v j (b) v j (f (v)) j j Payment s the total damage to the other bdders caused by the presence of. Each bdder nternalzes externaltes. 12 / 39
Clarke Rule Lemma A VCG mechansm wth Clarke pvot payments makes no postve transfers. If v (a) 0 for all v V and a A, then t s ndvdually ratonal. Proof: Let a = f (v) and b = arg max a A j v j(a ) No postve transfers (by defnton) v j (b) v j (a) 0 j j Indvdually ratonal v (a) + j v j (a) j v j (b) j v j (a) j v j (b) 0 13 / 39
Example: Blateral Trade trade no-trade Seller v s 0 Buyer v b 0 Trade occurs f v b > v s, no-trade f v s > v b Analyze VCG Mechansm, should not subsdze trade. 14 / 39
Example: Blateral Trade trade no-trade Seller v s 0 Buyer v b 0 VCG payments for no-trade: Seller payments: h s(v b ) 0, Buyer payments: h b (v s) 0 No addtonal payments by the mechansms, so h s(v b ) = h b (v s) = 0. VCG payments for trade: Seller payments: h s(v b ) v b, Buyer payments: h b (v s) + v s Seller receves v b, but buyer pays only v s < v b. Not budget-balanced: VCG mechansm subsdzes trade! 15 / 39
Example: Procurement or Reverse Aucton Auctoneer buys servce, partcpants offer servce for a cost Auctoneer pays partcpants Negatve utlty, negatve payments Vckrey aucton: Pck cheapest partcpant, pay second smallest offered cost Corollary The Vckrey reverse aucton s ncentve compatble. 16 / 39
Vckrey Reverse Aucton s IC Case 1: If bddng hs true value, bdder wns. Value?? -7?? Bd -9-11 x -17-14 Payment -9 Utlty 2 Case 2: If bddng hs true value, bdder loses. Value?? -12?? Bd -9-11 x -17-24 Payment 0 Utlty 0 17 / 39
Example: Buyng a Path n a Network Reverse aucton: Bdders are edges n a network. Mechansm needs to buy an s-t-path. 18 / 39
Example: Buyng a Path n a Network Outcomes are s-t-paths n graph G VCG pcks the shortest path P for reported costs c e Payments for e P are h e(c e) + e e c e (P ) Clarke pvot payment: h e(c e) = mn P G e e P ce Total payment c(p e) c(p e), where P e s shortest path n G when t would not contan e. An edge e P gets no payment. 19 / 39
Truth-tellng s a domnant strategy! 20 / 39
Frugalty Incentve compatblty mght be EXPENSIVE! 21 / 39
Vckrey Aucton Vckrey-Clarke-Groves Mechansms Sngle-Mnded Combnatoral Auctons 22 / 39
Ad Auctons 23 / 39
General Settng Combnatoral Aucton: Set M of m ndvsble tems (e.g., ad slots) auctoned smultaneously n bdders, valuatons for each subset of tems Who should get whch tems and pay how much? General Allocaton Problem of Interrelated Resources Valuaton v for bdder : v (S) R when gettng assgned set S M free dsposal: S T v(s) v(t ) normalzed: v( ) = 0. 24 / 39
Allocaton Allocaton of the tems: S 1,..., S n where S M and S S j = for j. Valuaton of a bdder ndependent of tems receved by other bdders (no externaltes) Socal Welfare: v (S ). An optmal allocaton S 1,..., S n Quas-lnear utltes: v (S ) p (v, v ) maxmzes socal welfare. VCG s truthful wth S, but computng S s NP-hard! Let us restrct attenton to a specal case. 25 / 39
Sngle-Mnded Valuatons Defnton A valuaton v s sngle-mnded f there exsts a threshold bundle S t and value v t R + such that v (S) = v t for all S S t, and v (S) = 0 otherwse. A sngle-mnded bd s (S t, v t ), and a bdder can le about both v t and S t. Defnton The allocaton problem among sngle-mnded bdders s gven by: INPUT: (S t, v t ) for each bdder = 1,..., n OUTPUT: Set of wnners W {1,..., n} wth maxmum socal welfare W v t and such that S t Sj t = for each, j W wth j 26 / 39
Reducton from Independent Set Theorem The allocaton problem among sngle-mnded bdders s NP-hard. Proof: INDEPENDENT SET: Has a graph an ndependent set of sze at least k? Vertces Bdders, Edges Items (S t, v t ) = (Set of ncdent edges, 1) 27 / 39
Approxmaton Algorthms c-approxmaton algorthm: Returns an allocaton T such that wth the optmal allocaton S we have v (T ) v (S ) c Smple n-approxmaton algorthm: Bdder wth the maxmum valuaton gets M. Trvally yelds an IC mechansm, essentally sngle-tem VCG aucton. Theorem For any ɛ > 0 t s NP-hard to approxmate INDEPENDENT SET to wthn a factor of n 1 ɛ. Corollary For any ɛ > 0 t s NP-hard to approxmate the allocaton problem among sngle-mnded bdders to wthn a factor of n 1 ɛ. 28 / 39
Approxmaton Algorthms The graph n the reducton has at most m < n 2 edges/tems, hence Proposton For any ɛ > 0 t s NP-hard to approxmate the allocaton problem to wthn a factor of m 1/2 ɛ. Note: m < n for sparse nstances. So far, our best algorthm yelds an n-approxmaton. Can we get a truthful mechansm that returns an allocaton that s a m-approxmaton? 29 / 39
Greedy Mechansm for Sngle-Mnded Bdders INPUT: (S t, v t ) for each bdder OUTPUT: A set of wnners W, payments p j for all 1 j n. Intalzaton: 1. Reorder bds: v t 1 S t 1... v t n S t n 2. W, p = 0 for all Iteraton: 3. For = 1... n do: If S t ( j W S t j Payments: 4. For each W do ) = then W W {} 5. fnd smallest ndex j such that S t S t j and for all k < j, k t holds S t k S t j = 6. f j exsts, set p = vj t S j t / St 30 / 39
Example Phone Headset Power Mary John Jack x 50 0 0 x 0 0 0 x 0 0 0 x x 50 60 0 x x 50 0 65 x x 0 0 0 x x x 50 60 65 31 / 39
Example Reorderng: S t 1. Mary Phone 50 50 v t v t / S t 2. Jack Phone, Power 65 45.96... 3. John Phone, Headset 60 42.42... Algorthm determnes W and p : 1. Mary: W =, so W = {1} 2. Jack: S1 t S2 t = {Phone} 3. John: S1 t S3 t = {Phone} Wnner s Mary Frst bdder blocked by Mary, whch could be n W, s Jack (2) Payments: p 1 = v2/ t S2 t / S 1 t = 65/ 2/1 = 45.96... 32 / 39
Incentve Compatblty Lemma A mechansm for sngle-mnded bdders wth p = 0 whenever W s IC f and only f for every bdder and fxed other bds (S t, v t ) the followng holds: Monotoncty: If bdder wns wth (S t, v t any v > v t and S S t. ), then he remans a wnner for Crtcal Payment: A wnnng bdder pays the mnmum value needed for wnnng the nfmum of all values v such that (S t, v ) stll wns. 33 / 39
Greedy s IC Monotoncty: Crtcal Payment: Wnner pays nfmum of all v (S t, v t ) wns, then t wns wth any v > v t and S such that (S t, v S t ) wns. Does Greedy satsfy t? Increasng v t or reducng S t ncreases v t / S t moves up n order and remans wnnng Payment s the swtchng pont between and j: x S t v t j S x v t t Sj t j = S tj v t j S t j / S t 34 / 39
Proof of Lemma (f-part) Intal Observatons Truthful bdder has always postve utlty Bdder has (S, v) and bds (S, v ) (S, v) If (S, v ) s a losng bd, reportng (S, v) can only help. If S S, reportng (S, v ) can only help. Assumpton: (S, v ) s wnnng bd and S S. Wnner s never worse off to bd (S, v ): Denote payment p for (S, v ) and p for (S, v ). If (S, x) wth x < p loses, then (monotone) (S, x) loses. Thus, for the crtcal payments p p. (S, v ) causes at most the payments of (S, v ). It can wn n cases, n whch (S, v ) loses. 35 / 39
Proof of Lemma (f-part) If bdders reveal ther true sets S, truthful bddng of v follows: Assume (S, v ) wns and (S, v) also Crtcal payment p for (S, v) For v > p same payments, for v < p losng IC Assume (S, v ) wns and (S, v) loses v smaller than crtcal payments, negatve utlty for (S, v ) 36 / 39
Approxmaton of Socal Welfare Lemma The greedy mechansm computes a m-approxmaton for the correspondng allocaton problem. Proof: Denote optmal wnner set W, output of greedy W. For each W consder W = {j W, j S t j S t Every j W appears n at least one W, so Clam: j W v t j v t m. }. j W vj t W v t. Then lemma follows wth ntutve accountng argument: Consder value that greedy loses compared to optmum because of addng to W. Ths s at most a factor of m larger than the value he secures by addng to W. 37 / 39
Provng the Clam For every j W we have j, and so by the order v t j Summng over all j W t v Sj t S t we get j W v t j v t S t The followng Cauchy-Schwartz nequalty 2 1 Sj t j W j W v t j j W 1 2 v t S t j S t S t j. j W yelds a bound on the last term: S tj W Sj t. j W j W. ) 2 ( S tj 38 / 39
Provng the Clam Combnng the last two bounds we have so far: vj t v t W S t Sj t. j W Every S t j ntersects S t for j W. j W W yelds allocaton, so Sj t Sk t = for j, k W Ths means W S t. W s allocaton, so j W S t j m Ths gves and fnshes the proof. j W v t j v t Sj t v t m j W 39 / 39