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: j vi - p if p < bi ui(slot j, payment p) = negative O/W • 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
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: j vi - p if p < bi ui(slot j, payment p) = negative O/W • 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
Related Work • Extensions of DGS: – Van der Laan and Yang (2008) – Kempe, Mu’alem and Salek (2009) show envy-freeness – Aggarwal, Muthukrishnan, Pal, and Pal (2009) add truthfulness on top • Hatfield and Milgrom (2005) – a more general setting for nonquasi-linearity, seems to subsume the above. Also viewed as an extension of DGS (as the authors note).
Related Work • Extensions of DGS: – Van der Laan and Yang (2008) – Kempe, Mu’alem and Salek (2009) show envy-freeness – Aggarwal, Muthukrishnan, Pal, and Pal (2009) add truthfulness on top • Hatfield and Milgrom (2005) – a more general setting for nonquasi-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?
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 ; v 1 = 20, v 2 = 10, v 3 = 7 player 3 drops p=7 all players compete p=0
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 ; v 1 = 20, v 2 = 10, v 3 = 7 player 2 drops player 3 drops p=8 p=7 all players compete p=0 p solves: 1 v 2 - p = 2 v 2 – 7 p = ( 1 - 2) v 2 + 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 (no budget): 1 = 1. 1, 2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 Result: player 1 wins slot 1 and pays 8 player 2 wins slot 2 and pays 7 player 2 drops p=8 player 3 drops p=7 all players compete p=0
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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 player 3 drops ? ? p=7 all players compete p=0
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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 Possible alternative: player 3 wins slot 1 and pays 7. 6 player 2 wins slot 2 and pays 7. 5 player 2 drops p = 7. 6 player 1 drops p = 7. 5 all players compete p=0
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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 player 3 drops ? ? p=7 all players compete p=0
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 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 However if p. 3 does not drop she can also end up with negative utility. Conclusion: no ex-post equilibrium player 3 drops ? ? p=7 all players compete p=0
Solution: The Generalized Position Auction • Example (with budget): 1 = 1. 1, 2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 p=7 all players compete p=0 SLOT 2 SLOT 1
Solution: The Generalized Position Auction • Example (with budget): 1 = 1. 1, 2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 Player 3 no longer wants slot 2 Number of players interested in slot 2 is equal to slot number p=7 all players compete p=7 p=0 SLOT 2 SLOT 1
Solution: The Generalized Position Auction • Example (with budget): 1 = 1. 1, 2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 player 3 wins slot 1, pays 7. 6 player 2 drops player 1 drops p=7 p = 7. 6 p = 7. 5 p=7 p=0 SLOT 2 SLOT 1
Solution: The Generalized Position Auction • Example (with budget): 1 = 1. 1, 2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 Auction for slot 2 player 3 wins slot 1, pays 7. 6 resumes; players 1 & 2 participate p = 7. 6 player 2 drops p = 7. 5 player 1 drops p=7 p=0 SLOT 2 SLOT 1
Solution: The Generalized Position Auction • Example (with budget): 1 = 1. 1, 2 = 1 ; v 1 = 20, v 2 = 10, v 3 = 7 b 1 = 7. 5, b 2 = 7. 6, b 3 = 9 player 2 wins slot 2, pays 7. 5 player 1 drops p = 7. 5 p=7 player 3 wins slot 1, pays 7. 6 player 2 drops player 1 drops p = 7. 6 p = 7. 5 p=7 p=0 SLOT 2 SLOT 1
The Generalized Position Auction • (The direct version: players report types, and outcome is computed by the following algorithm) (*) price ascent in auction ℓ stops when there are ℓ active players (*) player i remains in auction ℓ until price = min(bi, ( ℓ - ℓ’) vi + pℓ’) pℓ . . . SLOT K [ℓ’> ℓ : last slot in which player i was active when price stopped] SLOT ℓ . . . . SLOT 1
The Generalized Position Auction • (the direct version: players report types, and outcome is computed by the following algorithm) (*) 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 . . . SLOT K SLOT ℓ . . . . SLOT 1
Uniqueness • Result turns out to be always identical to the extended DGS auction. (but different mechanism: ) – Different price path – Ours is slightly faster (nk 2 messages instead of nk 3) 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’ ℓ vi - P’ℓ = ℓ vi - 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 Sketch Proof 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 Sketch Proof 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’ ℓ vi - Pℓ > ℓ’ vi - Pℓ’ > ℓ’ vi – P’ℓ’
Proof Sketch Proof 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’ ℓ vi - Pℓ > ℓ’ vi – P’ℓ’ ℓ vi – (Pℓ + ) > ℓ’ vi – P’ℓ’
Proof Sketch Proof 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’ ℓ vi - Pℓ > ℓ’ vi – P’ℓ’ ℓ vi – (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 Sketch Proof 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’ ℓ vi - Pℓ > ℓ’ vi – P’ℓ’ ℓ vi – (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’’ > ℓ vi – (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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