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
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 = 8p solves:
1 v2 - p = 2 v2 – 7
p = (1 - 2) v2 + 7
player 2 drops
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 = 8player 2 drops
Result:player 1 wins slot 1 and pays 8player 2 wins slot 2 and pays 7
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 (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
9
p = 0all players compete
p = 7player 3 drops ??
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 (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7
b1 = 7.5, b2 = 7.6, b3 = 9
p = 0all players compete
p = 7.5player 1 drops
p = 7.6player 2 drops
Possible alternative:player 3 wins slot 1 and pays 7.6player 2 wins slot 2 and pays 7.5
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 (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
9
p = 0all players compete
p = 7player 3 drops ??
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 (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
9
p = 0all players compete
p = 7player 3 drops ??
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
• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
9
p = 0
all players compete
p = 7
SLOT 2 SLOT 1
Solution: The Generalized Position Auction
• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
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
Solution: The Generalized Position Auction
• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
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
Solution: The Generalized Position Auction
• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
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 7.6Auction for slot 2 resumes; players 1 & 2 participate
Solution: The Generalized Position Auction
• Example (with budget): 1 = 1.1, 2 = 1 ; v1 = 20, v2 = 10, v3 = 7 b1 = 7.5, b2 = 7.6, b3 =
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
p = 7.5player 1 drops
player 2 wins slot 2, pays 7.5
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 SketchProof for (a) Pℓ = P’ℓ
Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption.
Claim: Pℓ’ = P’ℓ’
Proof: Otherwise Pℓ’ > P’ℓ’
ℓvi - Pℓ > ℓ’vi - Pℓ’ > ℓ’vi – P’ℓ’
Proof SketchProof for (a) Pℓ = P’ℓ
Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption.
Claim: Pℓ’ = P’ℓ’
Proof: Otherwise Pℓ’ > P’ℓ’
ℓvi - Pℓ > ℓ’vi – P’ℓ’ ℓvi – (Pℓ + ) > ℓ’vi – P’ℓ’
Proof SketchProof for (a) Pℓ = P’ℓ
Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption.
Claim: Pℓ’ = P’ℓ’
Proof: Otherwise Pℓ’ > 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 + .
Proof SketchProof for (a) Pℓ = P’ℓ
Denote i = w(ℓ) = w’(ℓ’). We have ℓ > ℓ’ by inductive assumption.
Claim: Pℓ’ = P’ℓ’
Proof: Otherwise Pℓ’ > 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 + .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
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