the effect of false-name bids in combinatorial the effect of false-name bids in combinatorial auctions:auctions:

new fraud in internet new fraud in internet auctionsauctions

Makoto Yokoo Makoto Yokoo

Yuko Sakurai Yuko Sakurai

Shigeo MatsubaraShigeo Matsubara

presented by: presented by: Wenjie XiaoWenjie Xiao

outlineoutline

IntroductionIntroduction FormalizationFormalization Effect of false-name bids in Effect of false-name bids in

combinatorial auctioncombinatorial auction Sufficient condition for VCG to be Sufficient condition for VCG to be

false-name-prooffalse-name-proof DiscussionDiscussion Conclusions and future workConclusions and future work

introductionintroduction

Internet auction in electronic Internet auction in electronic commercecommerce

Conventional auctions VS Conventional auctions VS Combinatorial auctionsCombinatorial auctions

Excellent environment for executing Excellent environment for executing combinatorial auctioncombinatorial auction

New cheating type!!New cheating type!!

introductionintroduction

False-name bidsFalse-name bids

bids submitted by a single bidder bids submitted by a single bidder using multiple identifiers such as using multiple identifiers such as multiple e-mail addresses.multiple e-mail addresses.

False-name-proofFalse-name-proof

we call a protocol is false-name-proof we call a protocol is false-name-proof if truth-telling without using false-if truth-telling without using false-name bids is a dominant strategy for name bids is a dominant strategy for each bidder.each bidder.

introductionintroduction

Collusion VS false-name bidsCollusion VS false-name bids restricted subclass of collusion:restricted subclass of collusion:

- false-name bids- false-name bids

- group-strategy-proof- group-strategy-proof False-name proof != group-strategy-False-name proof != group-strategy-

proofproof

(talked about later)(talked about later)

FormalizationFormalization

Model of a combinatorial auction in Model of a combinatorial auction in which false-name bids are possiblewhich false-name bids are possible

Modified the presented model so Modified the presented model so that it can handle false-name bidsthat it can handle false-name bids

Assume a quasi-linear private value Assume a quasi-linear private value model which no allocative model which no allocative externalityexternality

Formalization Formalization

Set of bidders N={1,2,..n} Set of bidders N={1,2,..n} Agent 0 is an auctioneer, who is Agent 0 is an auctioneer, who is

willing to sell a set of goods of willing to sell a set of goods of A={aA={a1,a,a2 …a…al}}

Each bidder i has his own Each bidder i has his own preferences over subset of Apreferences over subset of A

Privately observes a type Privately observes a type θi for each for each bidder i, bidder i, θi is drawn from a set Θ

Formalization Formalization

Definition 1:Definition 1:

the utility of bidder i the utility of bidder i is represented as v(B, θi )+ ti

Where:

B is the subset of A which bidder i B is the subset of A which bidder i obtainedobtained

ti isti is a monetary transfer v(.) is evaluation value of given set of

goods and type.

Formalization Formalization

Assumption:Assumption:

1). V is normalized by 1). V is normalized by v(∅, θi ) = 0.

2). Free disposal v(B’, θi ) ≥ v(B, θi ) for all B’ ⊆ B

3). For auctioneer (agent 0) 3). For auctioneer (agent 0) v(B, θ0)=0

for any subset of goods B

FormalizationFormalization Definition 2. There exists a set of identifiers M = {id1, id2, . . . ,

idm}. Furthermore, there exists a mapping function φ, where φ :N →

2M\ { }. 2∅ M is a power set of M. φ(i) represents a set of identifiers a bidder i can

use. And is private information of bidder i Assume - for all i , |φ(i)| ≥ 1 and - Ui φ(i) =M hold - for all i ≠ j , φ(i)∩φ(j)= holds.∅ the set of signals are represented as T = Θ × (2M \

), where the signal of bidder i is (θi ,φ(i)) ∈ T∅

FormalizationFormalization

Define a combinatorial auction Define a combinatorial auction protocolprotocol

Restrict our attention to:Restrict our attention to:

- almost anonymous mechanisms- almost anonymous mechanisms

- auction protocol, in which the set - auction protocol, in which the set of messages for each identifier is of messages for each identifier is Θ ∪ {0},

Formalization Definition 3. A combinatorial auction protocol is defined by Γ =

(k(·), t (·)). We call k(·) allocation function and t (·) transfer

function. Let us represent a profile of types, each of which is declared under each identifier as θ = (θid1, θid2, . . . ,θidm), where θidi ∈ Θ ∪ {0}.

0 is a special type declaration used when a bidder is not willing to participate in the auction:

k(θ) = (k0(θ ), kid1(θ ), . . . , kidm(θ )), where kidi(θ ) ⊆ A, t (θ) = (t0(θ ), tid1(θ ), . . . , tidm(θ )), where t0(θ ), tidi(θ ) ∈

R. R denotes the set of real numbers. Here, t0(θ )

represents the revenue of the auctioneer and −tidi(θ ) represents the payment of idi

Formalization Formalization Assumption:Assumption: Allocation feasibility constraints: For all i ≠ j , kidi(θ ) ∩ kidj(θ )= , k∅ idi(θ ) ∩

k0(θ )= ,∅

kidj(θ ) ∩ k0(θ )= . ∅

Ui=1m kidi(θ ) ∪ k0(θ ) = A.

Budget constraint: t0(θ )=− ∑1≤i≤mtidi (θ ). Non-participation constraint: For all θ, if θidi= 0, then kidi(θ )= and t∅ idi(θ ) =

0.

FormalizationFormalization

Assumption for case of tiesAssumption for case of ties in an almost anonymous mechanism,

the following condition is satisfied: For a declared type profile θ = (θid1,

θid2, . . . ,θidm),

if θidi = θidj , then v(kidi(θ ), θidi )+tidi(θ ) = v(kidj(θ ), θidj)+ tidj(θ ) holds.

Formalization Formalization

Definition 4. For an allocation function k(.), if for all k

= (k0, kid1, . . . ,kidm), which satisfies the allocation feasibility constraints, and

∑1≤i≤mv(kidi(θ), θidi)≥ ∑1≤i≤mv(kidi, θidi) holds. then k(·) is Pareto efficient Let us denote a Pareto efficient

allocation function as k∗(·).

Formalization Formalization

Definition 5. A strategy s of bidder i is a function

s : T → (Θ∪{0})M such that s(θi ,φ(i)) ∈ (Θ ∪{0})|φ(i)| for every (θi

,φ(i)) ∈ T . That is, s(θi ,φ(i)) = (θi,1, . . . ,θi,mi ),

where θi,j ∈ Θ ∪ {0} and |φ(i)| = mi.

Formalization Formalization

Definition 6. For bidder i, a strategy s*(θi,φ(i)) =

(s*i,1, . . . , s*i,mi) is a dominant strategy

if for all type profiles θ∼i , (θi,1, . . . ,θi,mi ), where θ=((s*i,1, . . . , s*i,mi),θ∼i), θ’=((θi,1, . . . ,θi,mi ),θ∼i),

v(ski(θ ), θi)+ sti(θ ) ≥ v(ski (θ’), θi)+ sti (θ’) holds.

Formalization Formalization

Definition 7. We say a mechanisms is false-name-

proof when for all bidder i , s*(θi ,φ(i)) =(θi , 0, . . ., 0) is a dominant strategy.

Effect of false-name bids in Effect of false-name bids in combinatorial auctionscombinatorial auctions

First consider the effect in VCG Definition 8. In the VCG mechanism, k∗(·) is used for

determining the allocation, and the transfer function is determined as follows.

tidi(θ)=[∑j≠iv(k*(θ), θidj)]-[∑j≠iv(k*-idi(θ), θidj)].

Where k*-idi(θ) is an allocation k that

maximizes ∑j≠iv(kidj, θidj) i.e. in VCG, each bidder is required to pay

the decreased amount of the surplus.

Effect of false-name bids in Effect of false-name bids in combinatorial auctionscombinatorial auctions

Example:Example: Proposition 1: Proposition 1:

(generally in combinatorial auctions)(generally in combinatorial auctions)

in combinatorial auctions, there exists no in combinatorial auctions, there exists no false-name-proof auction protocol that false-name-proof auction protocol that satisfied pareto efficiency.satisfied pareto efficiency.

Proof by counter exampleProof by counter example

proof:proof:

assuming there exists a false-name-proof, assuming there exists a false-name-proof, pareto efficiency protocol.pareto efficiency protocol.

Effect of false-name bids in Effect of false-name bids in combinatorial auctionscombinatorial auctions

Proof (cont.)Proof (cont.) a1 a2 both a1 and a2a1 a2 both a1 and a2

bidder 1: bidder 1: (b, (b, 00, b), b);;

bidder 2: bidder 2: ((00, , 00, b , b + + c)c);;

bidder 3: bidder 3: ((00, b, b), b, b)..

- b>c, so by pareto efficiency, bidder 1 gets good a1 and bidder - b>c, so by pareto efficiency, bidder 1 gets good a1 and bidder 3 get good a23 get good a2

- let P- let Pbb denote the payment of bidder 1. denote the payment of bidder 1.

- if bidder 1 declares his evaluation value for good a1 as b’=c+e - if bidder 1 declares his evaluation value for good a1 as b’=c+e and the allocation does not change. And let payment to be Pand the allocation does not change. And let payment to be Pbb’’

- P- Pbb’ <= b’’ <= b’

- P- Pbb<=P<=Pbb’ (by dominant strategy) => P’ (by dominant strategy) => Pbb <= c+e (same for <= c+e (same for bidder 3)bidder 3)

Effect of false-name bids in Effect of false-name bids in combinatorial auctionscombinatorial auctions

Proof (cont.)Proof (cont.) a1 a2 both a1 and a2a1 a2 both a1 and a2 bidder 1: bidder 1: (b, b, (b, b, 22b)b);; bidder 2: bidder 2: ((00, , 00, b , b + + c)c);; bidder 3: bidder 3: ((00, , 00, , 00))..

- payment is P- payment is P2b2b for bidder 1 and P for bidder 1 and P2b2b <= 2*P <= 2*Pbb <= 2c+2e <= 2c+2e

a1 a2 both a1 and a2a1 a2 both a1 and a2 bidder 1: bidder 1: (d, d, (d, d, 2d2d));; bidder 2: bidder 2: ((00, , 00, b , b + + c)c);; bidder 3: bidder 3: ((00, , 00, , 00)).. - c+e<d<b and b+c >2*d, good got by bidder 2- c+e<d<b and b+c >2*d, good got by bidder 2 - bidder 1 declare his evaluation value as (b,b,2b) instead of - bidder 1 declare his evaluation value as (b,b,2b) instead of

(d,d,2d), payment is P(d,d,2d), payment is P2b2b<=2c+2e (bidder 1) <=2d<=2c+2e (bidder 1) <=2d

Effect of false-name bids in Effect of false-name bids in combinatorial auctionscombinatorial auctions

Bidder 1 can increase the utility by Bidder 1 can increase the utility by overstating his true evaluation valuesoverstating his true evaluation values

in combinatorial auctions, there exists no in combinatorial auctions, there exists no false-name-proof auction protocol that false-name-proof auction protocol that satisfied pareto efficiency.satisfied pareto efficiency.

Note:Note: - proposition 1 holds in more general setting.- proposition 1 holds in more general setting. - Relies on the model defined before, but not - Relies on the model defined before, but not

for free disposal.for free disposal. - Not rely on the fact that the mechanism is - Not rely on the fact that the mechanism is

almost anonymousalmost anonymous

Sufficient condition where Sufficient condition where VCG is false-name-proofVCG is false-name-proof

Definition 9. For a set of bidders and their types Y =

{(y1, θy1), (y2, θy2), . . .} and a set of goods B ⊆ A, we define surplus function U as follows. Let us denote KB,Y as a set of feasible allocations of B to Y :

U(B,Y) = max k∈KB,Y ∑ (yi ,θyi )∈Y v(kyi, θyi ).

Define U(A,Y) as UA(Y ).

Sufficient condition where Sufficient condition where VCG is false-name-proofVCG is false-name-proof

Definition 10. We say UA(·) is concave over bidders if for all

possible sets of bidders Y,Z, and W, where Y ⊆ Z, the following condition holds:

UA(Z ∪ W) − UA(Z) ≥UA(Y ∪ W) − UA(Y ).

Proposition 2. The VCG mechanism is false-name-proof if the

following conditions are satisfied: • Θ satisfies that UA(·) is concave for every

subset of bidders with types in Θ. • Each declared type is in Θ ∪ {0}.

Sufficient condition where Sufficient condition where VCG is false-name-proofVCG is false-name-proof

Definition 11. Given a price vector p = (pa1,…, pal), we denote

Di(p)={B⊂A: v(B, θi)−∑ aj∈B Paj≥ v(C, θi)−∑aj∈C paj ,∀C ⊂ A}.

Di(p) represents the collection of bundles that maximize the net utility of bidder i under price vector p.

- if for any two price vectors p and p’ such that p’ ≥ p, p’aj = paj , and aj ∈ B ∈ Di(p),

- then there exists B ∈ Di(p) such that aj ∈ B. - so gross substitutes condition is satisfied.

Sufficient condition where Sufficient condition where VCG is false-name-proofVCG is false-name-proof

Another sufficient condition – Another sufficient condition – submodularity.submodularity.

Definition 12:Definition 12: We say U is submodular for a set of bidders

X, if the following condition is satisfied for all sets B ⊆ A and C ⊆ A:

U(B,X) + U(C,X) ≥ U(B ∪ C,X) +U(B ∩ C,X).

Proposition 3: (sufficient condition)Proposition 3: (sufficient condition) If U is submodular for all set of bidders X ⊆

N, then UA is concave.

Sufficient condition where Sufficient condition where VCG is false-name-proofVCG is false-name-proof

Proposition 4. (necessary condition) If U is not submodular for a set of bidders

X and a set of goods B and C, i.e., U(B,X) + U(C,X) <U(B ∪ C,X) + U(B ∩ C,X),

then we can create a situation where for a set of bidders Y , although U is submodular for Y , UA is not concave for X∪Y .

where A = B ∪ C.

DiscussionDiscussion

Group-strategy-proof protocolsGroup-strategy-proof protocols Group-strategy-proof is not false-Group-strategy-proof is not false-

name proofname proof Also, false-name proof not necessary Also, false-name proof not necessary

to be group-strategy-proof.to be group-strategy-proof.

DiscussionDiscussion

Definition: (group-strategy-proof)Definition: (group-strategy-proof) An auction protocol is group-

strategy proof if there exists no group of bidders that satisfies the following condition.

• if telling the false information - Uf(i)≥ Ut(i) for all i in N

- at least one i in N s.t Uf(i) > Ut(i)

DiscussionDiscussion if a protocol is false-name-proof, but it is if a protocol is false-name-proof, but it is

not necessary to be a group-strategy-proof.not necessary to be a group-strategy-proof. Example: 2 bidders and 2 goods, Θ = {θ1, θ2, θ3,

θ4}, • θ1: (10, 9, 18); • θ2: (9, 10, 18); • θ3: (10, 0, 10); • θ4: (0, 10, 10). v({a1}, θi) + v({a2}, θi) ≥ v({a1, a2}, θi)

DiscussionDiscussion

A protocol is group-strategy-proof, A protocol is group-strategy-proof, but not false-name-proofbut not false-name-proof

Protocol:Protocol: The auctioneer sets a reservation

price p. The winner is chosen randomly from the bidders whose declared evaluation value is larger than p. The winner pays p.

conclusionconclusion

False-name bids & False-name proofFalse-name bids & False-name proof Defined modelDefined model Connected to VCGConnected to VCG There exists no false-name-proof combinatorialThere exists no false-name-proof combinatorial

auction protocol that satisfies pareto efficiency.auction protocol that satisfies pareto efficiency. Point the sufficient condition where the VCG Point the sufficient condition where the VCG

mechanism is false-name-proofmechanism is false-name-proof Group-strategy-poofGroup-strategy-poof

Future workFuture work

Consider the false-name bids in:Consider the false-name bids in: no theoretical bound on the

efficiency loss Multi-unit auctionMulti-unit auction Situation where multiple buyers and Situation where multiple buyers and

sellers bit to exchange a designated sellers bit to exchange a designated goodgood

Question?Question?