position auctions with budgets: existence and uniqueness

Position Auctions with Budgets: Existence and Uniqueness

Ron Lavi

Industrial Engineering and Management

Technion – Israel Institute of Technology

Joint work with Itai Ashlagi, Mark Braverman, Avinatan Hassidim, and Moshe Tennenholtz

Overview• Starting point: The elegant “generalized English auction”, of Edelman,

Ostrovsky, and Schwarz, for position auctions– Private values, incomplete information– Truthful, envy-free, Pareto-efficient

• Drawback: Not suitable for players with budget constraints– Realistic assumption

• Our work:– “Extend” the auction to support budgets– New format exhibits all above desired properties– Outcome is equivalent to another “extension”, of the DGS auction

(by Aggarwal, Muthukrishnan, Pal and Pal)– Turns out: This is the unique possible outcome satisfying above

properties

The Model• Player i has: private value vi ; private budget bi

• Seller has K “positions” ; worth of position j to player i is j vi

1 > 2 > …. > K – Same model of EOS (2007), Varian (2007)

• A player has quasi-linear utility if pays less than budget cap; negative utility otherwise:

• Goal: auction that satisfies– Ex-post equilibrium: regardless of values, if others follow strategy,

so do player i (has “no-regret”) [call this “truthful”]– Pareto-efficiency: cannot weakly improve all utilities– Envy-free: players do not want to switch positions+payments

ui(slot j, payment p) =j vi - p if p < bi

negative O/W

The Model• Player i has: private value vi ; private budget bi

• Seller has K “positions” ; worth of position j to player i is j vi

1 > 2 > …. > K – Same model of EOS (2007), Varian (2007)

• A player has quasi-linear utility if pays less than budget cap; negative utility otherwise:

• Goal: auction that satisfies– Ex-post equilibrium: regardless of values, if others follow

strategy, so do player i (has “no-regret”) [call this “truthful”]

Proposition: envy-free Pareto-efficient

ui(slot j, payment p) =j vi - p if p < bi

negative O/W

Related Work• Extensions of DGS:

– Van der Laan and Yang (2008)– Kempe, Mu’alem and Salek (2009)

– Aggarwal, Muthukrishnan, Pal, and Pal (2009)

• Hatfield and Milgrom (2005) – a more general setting for non-quasi-linearity, seems to subsume the above. Also viewed as an extension of DGS (as the authors note).

show envy-freeness

addtruthfulness

on top

Related Work• Extensions of DGS:

– Van der Laan and Yang (2008)– Kempe, Mu’alem and Salek (2009)

– Aggarwal, Muthukrishnan, Pal, and Pal (2009)

• Hatfield and Milgrom (2005) – a more general setting for non-quasi-linearity, seems to subsume the above. Also viewed as an extension of DGS (as the authors note).

• Q: what if we try to extend the generalized English auction?

show envy-freeness

addtruthfulness

on top

Budgets and the Generalized English Auction• The generalized English auction:

– Price ascends; players drop (rename players in reverse drop order)

– The i’th dropper wins slot i, pays price point of i+1 drop

• Example (no budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7

p = 0all players compete

p = 7player 3 drops






p = 7player 3 drops

p = 8p solves:

1 v2 - p = 2 v2 – 7

p = (1 - 2) v2 + 7

player 2 drops






p = 7player 3 drops

p = 8player 2 drops

Result:player 1 wins slot 1 and pays 8player 2 wins slot 2 and pays 7




• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =

9


p = 7player 3 drops ??



– The i’th dropper wins slot i, pays price point of i+1 drop• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7

b1 = 7.5, b2 = 7.6, b3 = 9


p = 7.5player 1 drops


Possible alternative:player 3 wins slot 1 and pays 7.6player 2 wins slot 2 and pays 7.5





9







9



However if p. 3 does not drop she can also end up with negative utility.

Conclusion: no ex-post equilibrium

Solution: The Generalized Position Auction


9

p = 0

all players compete

p = 7

SLOT 2 SLOT 1



9

p = 0

all players compete

p = 7

SLOT 2 SLOT 1

Player 3 no longer wants slot 2

Number of players interested in slot 2 is equal to slot number

p = 7



9

p = 0

p = 7

SLOT 2 SLOT 1

p = 7

p = 7.5p = 7.6

player 1 dropsplayer 2 drops

player 3 wins slot 1, pays 7.6



9

p = 0

p = 7

SLOT 2 SLOT 1

p = 7

p = 7.5p = 7.6


player 3 wins slot 1, pays 7.6Auction for slot 2 resumes; players 1 & 2 participate



9

p = 0

p = 7

SLOT 2 SLOT 1

p = 7

p = 7.5p = 7.6





The Generalized Position Auction

SLOTℓ

• (The direct version: players report types, and outcome is computed by the following algorithm)

SLOTK

. . . . .

SLOT1

. . . . . . . . . . . . . . .

(*) price ascent in auction ℓ stopswhen there are ℓ active players

(*) player i remains in auction ℓ untilprice = min(bi, (ℓ - ℓ’) vi + pℓ’)

[ℓ’> ℓ : last slot in which player i was active when price stopped]

pℓ

The Generalized Position Auction

SLOTℓ

• (the direct version: players report types, and outcome is computed by the following algorithm)

SLOTK

. . . . .

SLOT1

. . . . . . . . . . . . . . .

(*) when slot 1 is sold, auction for slot K resumes, for K-1 slots, with one less player.

THM: this is truthful and envy-free

Uniqueness• Result turns out to be always identical to the extended DGS

auction. (but different mechanism: )– Different price path– Ours is slightly faster (nk2 messages instead of nk3)

THM: Any mechanism that is truthful, envy-free, individually rational, and has no positive transfers, must yield the same outcome.

• Holds even if values are public and only budgets are private.

Proof Sketch• Use two properties of the generalized position auction:

– If player i wins slot ℓ and declares smaller budget still > Pℓ then she still wins slot ℓ.

– Slot prices are minimal among all mechanisms.

• Let M denote our auction, and fix another mechanism M’ that satisfies all properties. Fix arbitrary tuple of types.

Lemma: Let B={ s | Ps = P’s }. Then w(B) = w’(B).Proof: By contradiction i such that:

(1) i = w(ℓ) = w’(ℓ’) (2) Pℓ = P’ℓ (3) Pℓ’ < P’ℓ’

ℓvi - P’ℓ = ℓvi - Pℓ > ℓ’vi - Pℓ’ > ℓ’vi – P’ℓ’ contradicting envy-freeness of M’.

Proof Sketch• Use two properties of the generalized position auction:

– If player i wins slot ℓ and declares smaller budget still > Pℓ then she still wins slot ℓ.

– Slot prices are minimal among all mechanisms.

• Let M denote our auction, and fix another mechanism M’ that satisfies all properties. Fix arbitrary tuple of types.

Inductive claim: for slot ℓ = K,…,1:

– Set of winners of slots 1,.., ℓ is the same for M,M’

– For slot s > ℓ: (a) Ps = P’s (b) w(s) = w’(s)

• We need only prove (a) + (b) for some slot ℓ given correctness of inductive claim for slot ℓ+1.

Proof SketchProof for (a) Pℓ = P’ℓ

Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption.

Claim: Pℓ’ = P’ℓ’

Note:This implies (a) since i in w’(B) implies i in w(B) implies Pℓ = P’ℓ




Proof: Otherwise Pℓ’ > P’ℓ’

ℓvi - Pℓ > ℓ’vi - Pℓ’ > ℓ’vi – P’ℓ’





ℓvi - Pℓ > ℓ’vi – P’ℓ’ ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’






When player i declares budget = Pℓ + she still wins slot ℓ in M, and thus wins some slot ℓ’’ < ℓ in M’. She pays P’’ < Pℓ + .






When player i declares budget = Pℓ + she still wins slot ℓ in M, and thus wins some slot ℓ’’ < ℓ in M’. She pays P’’ < Pℓ + .Her utility in this case increases:

ℓ’’vi – P’’ > ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’

which contradicts truthfulness of M’.

Summary• Study position auctions with private values and private budget

constraints.

• Extend the generalized English auction to handle budgets, maintaining all its desired properties.

• Prove that the result is the unique possible truthful mechanism that satisfies:– Envy-freeness– Individual Rationality– No Positive Transfers

position auctions with budgets: existence and uniqueness

Documents

v2 p

dgs auction

slot i

payment p

players competep

generalized english

propertiesthe modelplayer

positions worth of position